Research ArticleNumerical Investigation on Vortex Shedding froma Hydrofoil with a Beveled Trailing Edge

Seung-Jae Lee1 Jun-Hyeok Lee2 and Jung-Chun Suh12

1Research Institute of Marine Systems Engineering Seoul National University Seoul 08826 Republic of Korea2Department of Naval Architecture and Ocean Engineering Seoul National University Seoul 08826 Republic of Korea

Correspondence should be addressed to Seung-Jae Lee sj38leegmailcom

Received 16 May 2015 Revised 10 August 2015 Accepted 17 August 2015

Academic Editor Jean-Michel Bergheau

Copyright copy 2015 Seung-Jae Lee et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To better understand the vortex shedding mechanism and to assess the capability of our numerical methodology we conductednumerical investigations of vortex shedding from truncated and oblique trailing edges of a modified NACA 0009 hydrofoil Thehybrid particle-mesh method and the vorticity-based subgrid scale model were employed to simulate these turbulent wake flowsThe hybrid particle-mesh method combines the vortex-in-cell and the penalization methods We have implemented numericalschemes to more efficiently use available computational resources In this study we numerically investigated vortex shedding fromvarious beveled trailing edges at a Reynolds number of 106 We then compared the numerical results with the experimental datawhich show good agreementWe also conducted numerical simulations of wakes behind the hydrofoil at rest in periodically varyingflows Results reveal that vortex shedding is affected by the periodicity of a free-stream flow as well as the trailing-edge shape

1 Introduction

The vortex shedding phenomenon is encountered in manypractical engineering applications and physical sciences andit is an important characteristic of flows past a bluff body Avortex sheet shed from a solid body consists of alternatingvortices of strength Γ that produce their own velocity fieldsuperimposed on the free-stream velocity These sheddingvortices cause an oscillating force component perpendicularto the direction of flow This force can induce a vibrationin the body Vortex shedding behind circular and squarecylinders is a benchmark problem that has been well investi-gated and is addressed in a vast amount of literature Severalscholarly reviews [1ndash4] are entirely devoted to the state ofthe art of this problem In contrast vortex shedding froma streamlined body such as a hydrofoil has been studied toa much lesser extent despite its being of direct relevance topractical engineering problems in hydraulic turbines pumpsand marine propellers Propeller-singing is well known as acritical vibration phenomenon generated by the interactionbetween a Karman vortex-shedding mechanism from thetrailing-edge of a blade and its natural frequencies [5]

Some laboratory experiments have been conducted onvortex shedding from the trailing edge of a hydrofoilBourgoyne et al [6 7] experimentally investigated thedominant features of flows over a two-dimensional (2D)hydrofoil with an antisinging trailing edge at Reynoldsnumbers ranging from 10

6 to 5 times 107 They concludedthat turbulent fluctuations in the near wake are Reynolds-number dependent because of the varying strengths of thestructured near-wake vortex shedding Mulvany et al [8]numerically simulated turbulent flows around a modifiedNACA 16 hydrofoil using four turbulent models available inthe commercial computational fluid dynamics (CFD) codeFLUENT They compared their numerical results with thosefrom the experimental data of Bourgoyne et al [6] Theauthors mainly discussed the capabilities of the four differentmodels and did not report any vortex shedding phenomenonAusoni et al [9] performed laboratory-scale experiments toinvestigate the von Karman vortex street generated in thewake of a 2D hydrofoil with a truncated trailing edge Theirexperiments were performed at a zero angle of attack for aReynolds number ranging from 50 times 105 to 20 times 106 basedon the hydrofoil chord length [9 10] Their experimental

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2015 Article ID 565417 13 pageshttpdxdoiorg1011552015565417

2 Modelling and Simulation in Engineering

results showed that a lock-in phenomenon occurs wherethe vortex shedding frequency is locked onto the naturalfrequency of a hydrofoil Interestingly it has been reportedthat cavitation has a minor effect on the wake dynamicRecently Zobeiri et al [11] investigated two NACA 0009hydrofoils with blunt and oblique trailing edges respectivelyfor Reynolds numbers ranging from 50 times 105 to 29 times 106and conducted high-speed visualization and flow-inducedvibrationmeasurementsThey confirmed experimentally thatflow-induced vibration was significantly reduced with anoblique trailing edge compared with a truncated edge Theyalso concluded that the collision between upper and lowervortices and the resulting vorticity redistribution were themain reasons for the obtained vibration reduction with theoblique trailing edge

This study numerically investigated vortex shedding fromtruncated and oblique trailing edges of a hydrofoil to betterunderstand the vortex shedding mechanism and to assess thecapability of our numerical methodology We employed thehybrid particle-meshmethod and the vorticity-based subgridscale model to undertake a numerical flow simulation Thehybrid particle-mesh method is a combination of the vortex-in-cell (VIC) method (see [12ndash15] and references therein)and the penalization technique [16] which has been devel-oped in recent years (see [17ndash24] for a review) The hybridparticle-mesh method enables the use of fast and efficienttechniques for computing differential operators therebyenabling large scale simulations However techniques tosave computational memory and CPU time consumptionare still required Thus we have introduced the implementa-tions of numerical schemes to more efficiently use availablecomputational resources [25 26] With these numericalschemes we numerically investigate vortex shedding fromvarious beveled trailing edges of a NACA 0009 hydrofoil ata Reynolds number of 106 We then compare our numericalresults with experimental data [10 11] to assess the capabilityof our numerical methodology In addition we investigatethe influence of periodically varying free-stream flows onvortex shedding to better understand the vortex sheddingmechanism

The organization of the remainder of this paper is asfollows we present a brief description of the numerical meth-ods used for the vortex shedding simulation in Section 2We describe our implementation of numerical schemes toefficiently use available computational resources in Section 3and the computational procedure followed in Section 4 Wepresent out vortex shedding results from a variety of beveledtrailing edges and discuss the influence of periodic free-streamflows in Section 5 In Section 6 we provide a summaryof this study and some perspectives for future work

2 Governing Equations andNumerical Methods

The vorticity-velocity formulation of the Navier-Stokes equa-tions allows a purely kinematical problem to be decoupledfrom the pressure term which is eliminated by applying thecurl operator Pressure which can be evaluated in an explicit

manner using the identified vorticity and velocity fields [27]is not part of the solution algorithm For a 2D flow parallel tothe 119909119910-plane the vorticity transport equation in terms of thevorticity can be expressed as

120597120596119911

120597119905+ u sdot nabla120596

119911= ]nabla2120596

119911 (1)

where120596119911is the scalar plane component of the vorticity vector

that is 120596 equiv 120596119911in two dimensions The evolution of a flow

is considered in discrete time steps The viscous splittingalgorithm can be expressed in a Lagrangian framework as

119863x119863119905= u (2a)

119863120596119911

119863119905= ]nabla2120596

119911 (2b)

where 119863119863119905 = 120597120597119905 + u sdot nabla is the material derivative 119873119901

discrete Lagrangian fluid particles with a finite core size 120598 arelinearly superposed to approximate the vorticity field as

120596119911 (x 119905) =

119873119901

sum

119901=1

Γ119901120577120598(x minus x

119901) (3)

where 120577120598is a mollification of the Dirac-delta function [12]

Finite core size vortices are used instead of point vorticesEach particle is characterized by its position x

119901and its

strength Γ119901 For a given vortex particle the circulation Γ

119901

is identical to the product of the vorticity and the area ofthe vortex particle 1205982 which also represents the contributionof the vortex particle to the vorticity field Γ

119901= int120596

119911119889119878 asymp

1205961199111205982 The vorticity-carrying fluid particle is advanced with

the velocity u and is gradually diffused because of viscouseffects

21 Vortex-in-Cell (VIC) Method for Convection Since thecontinuity equation and the definition of vorticity lead tonablasdotu = 0 andnablatimesu = 120596 the velocity u of (2a) can be expressedas the gradients of the stream-function120595 The velocity vectoris expressed as

u = Uinfin+ u120596= Uinfin+ nabla times 120595 (4)

where Uinfinis the free-stream velocity and u

120596is the rotational

velocity The Poisson equation for the stream-function isnabla2120595 = minus120596 In two dimensions the stream function is scalar

120595119911 and only the vorticity component perpendicular to the

plane is nonzero 120596 = 120596119911119890119896 The vector Poisson equation

reduces to a scalar Poisson equation

nabla2120595119911= minus120596119911 (5)

Rotational velocity is defined by 119906119909= 120597120595

119911120597119910 and 119906

119910=

minus120597120595119911120597119909 In the VIC method stream functions to evaluate

velocities are computed using a uniform Cartesian meshThefast Fourier transform- (FFT-) based Poisson solver reducesthe computational cost of obtaining the velocity field toO(119873log119873) where 119873 is the number of grid points [12] To

Modelling and Simulation in Engineering 3

compute the stream functions using the FFT-based Poissonsolver the vorticity 120596

119911is interpolated onto equally spaced

Cartesian grid using

120596119911(x119892) =

119873119901

sum

119901=1

120596119911(x119901)119882(

x119901minus x119892

ℎ) (6)

where ℎ is the mesh spacing using the following third orderinterpolation kernel [28 29]

119882(119909) = 1198721015840

4(119909)

=

0 for |119909| gt 21

2(2 minus |119909|)

2(1 minus |119909|) for 1 lt |119909| le 2

1 minus5

2|119909|2+3

2|119909|3 for |119909| le 1

(7)

in each coordinate direction This kernel conserves the 0th1st and 2nd order moments The subscripts

119892and119901denote

the grid and particle quantities respectively A vortex particlecontributes to the nearest 16 nodes through the1198721015840

4scheme

and the total vorticity at each node is obtained by summingthe vorticity contributions of all the vortex particles

Boundary conditions for the stream function are requiredto solve the Poisson equation If the computational domainboundaries are far enough from the particles homogeneousDirichlet boundary conditions (120595 = 0) may be usedHowever using a larger domain is inefficient since it requiresa regular grid with too many points To compactly place asquare FFT-domain Ω we used nonhomogeneous Dirichletboundary conditions in this study Unknown stream func-tions at the domain boundaries (120597Ω) can be directly obtainedfrom Greenrsquos function solution [27] whereby

120595119911(x119887) = minus

1

2120587

119873119901

sum

119901=1

Γ119901ln (10038161003816100381610038161003816x119887 minus x119901

10038161003816100381610038161003816) 119887 = 1 119872 (8)

where x119887isin 120597Ω x

119901isin Ω and x

119892= x119901119872 denotes the number

of domain boundary points and the circulation strength ofeach vortex particle Γ

119901= 120596119911(x119901)1205982 We note here that all

particle vorticity values should be considered Fast evaluationof the stream functions at the domain boundaries is furtherdiscussed in Section 33

22 Brinkman PenalizationMethod for Diffusion Thepenal-ization method was initially designed to take into accountsolid obstacles in fluid flows [16] The main point of thepenalty term is to replace the usual vorticity creation algo-rithm for enforcing the no-slip condition for a solid bodyBy adding the penalization term to (1) the vorticity transportequation becomes

119863120596119911

119863119905= ]nabla2120596

119911+ nabla times [120582120594 (u

119904minus u)] (9)

where u119904is the velocity of the solid body and 120594 denotes a

mask function that yields 0 in a fluid and 1 in a solid [16 18]

120582 indicates the penalization parameter with the dimension[119904minus1] To avoid too small time step the penalization term is

evaluated by the implicit expression [16 18 30]Δ120596119911

Δ119905= nabla times [120582120594 (u119899

119904minus u119899+1)] (10)

where

u119899+1 =u119899 + 120582Δ119905120594u119899

119904

1 + 120582Δ119905120594 (11)

This scheme is unconditionally stable [19] Equation (9) isrewritten by replacing its terms with those in (10) and (11) asfollows

120596119899+1

119911minus 120596119899

119911

Δ119905= ]nabla2120596119899

119911+ nabla times [

120582120594

1 + 120582Δ119905120594(u119899119904minus u119899)] (12)

The penalization and diffusion terms of (12) are consecutivelyevaluated on a temporary grid To reduce any error due tothe penalization term we used a mask function with second-order accuracy for which the boundary is located at themidpoint of the grid where the mask function jumps from0 to 1 [18 31]

23 Smagorinsky Turbulence Model The filtered vorticitytransport equation for a Lagrangian vorticity-based large-eddy simulation (LES) in two-dimensions can be expressedas [32]

119863120596119911

119863119905= ]nabla2120596

119911+ nabla sdot Φ (13)

where the bar denotes spatial filtering at length scale 120598 Thesubgrid-scale vorticity stress term Φ which accounts for theeffect of unresolved velocity and vorticity fluctuations can bedefined as Φ = ]

119879nabla120596119911 Smagorinsky eddy-viscosity model is

one of most commonly used subgrid-scale (SGS) turbulencemodels In spite of its simplicity the Smagorinsky model isstill extensively used in practice and has served as a basic toolfor the developments of LES modeling [33] In this modeleddy viscosity is defined by

]119879= (119862119878Δ)2 1003816100381610038161003816100381611987810038161003816100381610038161003816 (14)

where 119862119878is a nondimensional Smagorinsky constant and Δ

is a subgrid characteristic length scale 119878119894119895= (120597119906

119894120597119909119895+

120597119906119895120597119909119894)2 is the resolved strain-rate tensor and |119878| =

(2119878119894119895119878119895119894)12 is its magnitude Here |119878|2 = |120596

119911|2 in incom-

pressible flows (nabla sdot u = 0) using the Caswell formula [34]Finally the vorticity transport equation for the LES in twodimensions is rewritten as

119863120596119911

119863119905= (] + (119862

119878Δ)2 1003816100381610038161003816120596119911

1003816100381610038161003816) nabla2120596119911 (15)

In 2D LES models the filtering process is applied in only twodirections so Δ = (ℎ

119909ℎ119910)12 is defined by the grid spacing

The Smagorinsky constant 119862119878may vary with the type of

flow and the Reynolds number Values ranging from 01 to024 have been suggested in the literatures [33 35 36] Theconstant 119862

119878is prescribed and its standard value is typically

015 [37 38]

4 Modelling and Simulation in Engineering

3 Efficient Implementation ofNumerical Methods

31 Particle-BasedDomainDecomposition Parallelmachinesand algorithms enable significant reduction in computingtime and greater amount of available memory For high-performance computing we used the Message Passing Inter-face (MPI) for our distributed systems in which each proces-sor has its own localmemoryThe appropriate decompositionof particles andor grids should be employed In domainor spatial decomposition a given domain Ω is geometri-cally decomposed by splitting it into several subdomainsTypically the subdomains are uniform and particles are notevenly distributed among the subdomains Differences inthe number of particles assigned to each processor causecomputational load imbalances As some processors willsit idly while waiting for others poor performance canresult

We introduce a simple idea to achieve a better loadbalance during the domain decomposition In vortex particlemethods fluid particles must periodically be redistributedsince their accumulation leads to inaccuracies in numericalsolutions This process is called particle remeshing or redis-tribution The remeshing step creates a new set of particleson a uniform Cartesian grid and then the randomly spacedold particles are removed All the new particles are reindexedby 119894 = 1 2 119873

119901from upstream to downstream that

is higher-indexed particles are located further downstreamDue to the position-dependent index of the particles theentire spatial domain can be easily split to ensure an equalamount of particles in each subdomain (each processor)As a result the horizontal extent of the domain is unequalbut the particles are evenly distributed in subdomains ofdifferent sizes In every remeshing step the entire domainis dynamically partitioned In this way each subdomaincontains an equal number of particles and better load balanceismaintained during simulationThis approach is very simpleand effective No additional algorithms are required to evenlydistribute the particles or to identify the position of theparticles As shown in Figure 1 we obtained an approxi-mately 30 reduction in computation time simply by loadbalancing during parallel computation This technique isalso valuable for the efficient use of available computationalmemory

32 Multidomain Approach To avoid an excessive domainsize we consideredmultiple domains In this study the entiredomainΩ is defined as the union of all the domains coveringall the fluid particles that is x

119901isin Ω where Ω = Ω

1cup

Ω2cup sdot sdot sdot cup Ω

119873119889 The domain number119873

119889is not constant and

is dependent on a spatially evolving flow When the domainsize exceeds a certain limit a new domain is created situateddownstream from the domain As a result the first domainΩ1includes a solid body and the others are sequentially

located downstream from the body For the total circulationconservation the grid spacing of the 119894th domain can bedefined as ℎ

119894= 120598 (single-resolution) or ℎ

119894= 2ℎ119894minus1

(multi-resolution) where ℎ

1= 120598 Here 120598 denotes the particle size To

T

Com

puta

tion

time (

h)

250

200

150

100

50

00 2 4 6 8 10

Equal-sized subdomainsUnequal-sized subdomain

Figure 1 Computation times for equal- and unequal-sized subdo-mains

interpolate the vorticity values of the particles into the nodeson a grid with different grid spacing (6) is rewritten as

120596119911(x119892) =

1205982

ℎ2

119894

119873

sum

119901=1

120596119911(x119901)119882(

x119901minus x119892

ℎ119894

) (16)

where x119901isin Ω119894and x119892isin Ω119894The nodes are declared to be a 2D

array so that the position of each node can be found straight-forwardly from its index The velocity and vorticity of eachdomain are sequentially and independently evaluatedHencerelatively small amounts of working memory are requiredThis multidomain approach also allows more freedom inthe choice of variable arrangements in the domains Theversatility of this approach was demonstrated in our previousstudy [26] It can be achieved at the expense of a significantincrease in computation time This issue is further discussedin the next section

33 Approximation of Far-Field Conditions A substantialpart of the overall computation time is spent on the calcu-lation of the boundary stream functions to solve Poissonrsquosequation for the stream function This requires the order ofO(119872119873

119901119875) operations to obtain boundary values of domain

Ω119894 Here 119872 denotes the number of nodes at the sides of

domainΩ119894and119873

119901119875 is the number of particles per processor

To achieve a nearly perfect load balance particles are evenlydistributed among the processors as discussed in Section 31and each processor has all the boundary nodes 119872 It mustalso be very useful to avoid all unnecessary communications119873119901119875 is reduced by increasing the number of processors

involved in the computation whereas 119872 is not This is whythe number of boundary nodes 119872 is the most critical fordetermining the computation time

In this study we attempted to reduce the number ofboundary nodes 119872 using the cubic spline approximationThis is made possible by the smooth variation of the streamfunctions along the boundary Only one-fourth of the bound-ary nodes were chosen as equidistant points for the directcalculations That is the stream functions at the selected

Modelling and Simulation in Engineering 5

boundary nodes were directly computed using (8) and thevalues at boundary nodes not selected were evaluated byinterpolation using cubic splines [39] In this study theerror between the interpolated value and the true valueis typically less than 001 We have observed that thereduction in computational time is inversely proportionalto the number of nodes selected for direct computationWe believe that this approach can be implemented withmuch lower computational effort because a complex datastructure such as a quad-tree used in the fast multipolemethod [40] is not required A rigorous comparison of theaccuracy and computational costs is planned in the nearfuture

4 Computational Procedure

First the entire simulation domain Ω covering all theparticles is divided into small domains Ω

119894(119894 = 1 119873

119889)

as discussed in Section 32 Each domain Ω119894is consecutively

computed according to the following numerical procedure

(1) Particle-based domain decomposition as discussedin Section 31 domain Ω

119894is decomposed into sub-

domains to ensure that the particles are evenly dis-tributed among the processors If redistribution hasnot been done prior to the time step particles thathave been removed from a subdomain are assignedto an adjacent processor without performing domaindecomposition

(2) Particle-to-grid interpolation each processor inter-polates the vorticity 120596

119911of its own particles into the

grid nodes through the11987210158404interpolation kernel

(3) Calculation of the boundary condition each proces-sor computes the boundary stream functions by directcalculation and interpolation using cubic splines asdiscussed in Section 33

(4) Evaluation of the velocity field the stream functionon the grid is computed with the FFT-based Poissonsolver from an open-source library called FFTW(fastest Fourier transform in the west) [41 42]There-after the rotational velocities on the grid nodes arecomputed from the definitions 119906

119909= 120597120595120597119910 and

119906119910= minus120597120595120597119909 using the fourth order central finite

difference scheme(5) Evaluation of the vorticity field each processor eval-

uates the evolution of the vorticity field in time asfollows

(a) The penalization term that is the second termon the right hand side of (12) is computedwith the velocities on the grid The spatialderivatives of the curl operators are evaluatedusing a centered difference approximation witha fourth-order error

(b) The diffusion term that is the first term onthe right hand side of (12) is evaluated on thegrid using a classical 9-point finite differencemethod

(c) The turbulent viscosity (119862119878Δ)2|120596119911| in (15) is

approximated by the vorticity strength of thegrid Finally the vorticity field 120596119899+1

119911is updated

(6) Grid-to-particle interpolation each processor inter-polates the velocity and vorticity on the grid back to itsown particle positions through the1198721015840

4interpolation

kernel

Vortex particles are advanced in time using a midpointpredictor-corrector method as follows particle positions arepredicted by their velocities xlowast = x119899 + Δ119905u(x119899 120596119899

119911) then the

particle positions are corrected by using combinations of thepredicted and previous positions and velocity values x119899+1 =x119899 + 05Δ119905[u(x119899 120596119899z ) + u(xlowast 120596119899+1

119911)] Particle redistribution is

conducted every three time steps to ensure particle overlapIf a new particle has a small vorticity magnitude that is|Γ119894| lt 00001|Γ|max it is deleted to avoid having too many

particles Loss of circulation is equally distributed among theremaining particles to ensure the conservation of the totalvorticity

5 Numerical Results and Discussions

51 Model Description We selected a NACA 0009 hydrofoilwith a truncated trailing edge for the numerical simulationsThis hydrofoil has the same cross-section as the experimentalmodel used by Ausoni et al [9] Ausoni [10] and Zobeiriet al [11] Ten percent of the original chord 119888

119900was removed

from the trailing-edge region of the NACA 0009 hydrofoilThe hydrofoil thickness distribution is written as

00 le119909

119888119900

le 05

119910

119888119900

= 01737 (119909

119888119900

)

12

minus 02422 (119909

119888119900

) + 03046 (119909

119888119900

)

2

minus 02657 (119909

119888119900

)

3

05 lt119909

119888119900

le 10

119910

119888119900

= 00004 + 01737 (1 minus119909

119888119900

) minus 01898 (1 minus119909

119888119900

)

2

+ 00387 (1 minus119909

119888119900

)

3

(17)

The hydrofoil geometry is further detailed in Ausoni [10]Themaximum thickness as normalized by the chord length 119888 is119905max119888 = 01 and the thickness at the trailing edge 119905TE119888 is00322 The bevel angle 120573 is defined in Figure 2 The trailingedge with 120573 = 0∘ denotes that the trailing edge is truncated inthe vertical direction

52 Numerical Setup Numerical parameters can be deter-mined using the stability condition Δ119905 = 120598

24] [18] The

grid convergence was achieved for 120598 = 000030 and

6 Modelling and Simulation in Engineering

xc

yc

045 05

minus002

0

002

A

BC

00054c

00322c120573

Figure 2 Trailing-edge shape Note that the points A B and C indicate the knuckles of the chamfer and trailing-edge wedge and the bevelangle is defined as 120573

Δ119905 = 000015 was chosen in order to guarantee numericalaccuracy for a Reynolds number of 106 The grid spacingof the 119894th domain Ω

119894is defined as ℎ

119894= 1205982119894minus1 to reduce

computation timeThe penalization parameter 120582 in (12) is setto be 108 The hydrofoil at a zero angle of attack is immersedin a Cartesian grid (the first domain) that does not conformto the surface of the bodyThe first domain with 4096times 1024lattice nodes is fixed and the other domains are dynamicallydetermined depending on the spatially evolving flow In thisstudy numerical simulations were conducted until 119879 = 10A new domain is created downstream of a body once thenumber of horizontal grid points exceeds 2048 Althoughthere is no limit on the number of vertical grid points it wastypically less than or equal to 1024 During simulation lessthan 08 million particles and four computational domainswere created The entire computation in each case includingstart-up and printing times took approximately 170 hourson 16 Intel Xeon64 33 GHz CPUs with 6GB memory perprocessor

53 Truncated Trailing-Edge For an impulsively started flowpast a hydrofoil at a chord-based Reynolds number of 106the wake pattern can be classified into three regimes basedon the time evolutions shown in Figure 3 In the very earlystages the wake remained very symmetrical and consistedprimarily of two large vortices in the near wake as shown inFigure 3(a) At that time the drag force gradually decreasedAs time went on the wake became asymmetrical as shown inFigure 3(b) and then shed a vortex as shown in Figure 3(c)After a number of shedding periods vortices were shedregularly from the symmetric trailing edge in the form of twotrains of oppositezsign but equal-strength vortices as shownin Figure 3(d) Such regular vortex shedding induces periodicloading on the structure as shown in Figure 4 It is wellknown that drag-force oscillation during vortex shedding ismuch smaller than the lift force Numerical results show thatoscillations in the drag force occurred at twice the vortex-shedding frequency due to the fact that two vortices wereshed from alternate sides during one full period of wakeoscillation

From our simulation the Strouhal number based on thetrailing-edge thickness St

119905= 119891119904119905TE119880infin is about 022 for

Re = 106 Ausoni [10] (also refer to Zobeiri et al [11])

had a similar finding and experimentally gave St119905= 025

which may be approximately read from their figure Theyused a hydrofoil with a 100mm chord length and 150mmspan length The aspect ratio defined as the ratio of the spanlength to the chord length was 15 A lower aspect ratioleads to a three-dimensional (3D) effect even though thehydrofoil is 2D Ausoni [10] also observed experimentallythat the shedding process was not induced in phase alongthe span It was therefore speculated in [25] that the slightunderestimation of the Strouhal number is due to 3D effectswhich were absent in our simulation

54 Oblique Trailing-Edges We conducted numerical simu-lations to observe vortex shedding from trailing edges withsix different bevel angles of 120573 = 0 10 20 40 60 and 70degrees Figure 5 shows the instantaneous vorticity distribu-tions at 119879 = 10 for Re = 106 At 120573 = 0∘ the vortex sheddingis regular and periodic As the bevel angle 120573 increasesvortex shedding becomes increasingly disorganized Suchirregular shedding of vortices demonstrates the periodicity ofoscillatory forces Figure 6 shows the power spectral densityfunction (PSD) of the drag and lift coefficients for differentbevel angles Compared with 120573 = 0∘ the amplitude of the liftforce oscillation for asymmetric trailing edges is significantlyreduced compared with the symmetric trailing edge at 120573 =0∘ The lift force amplitude is gradually decreased as the bevelangle increases while the drag force amplitude increases andthen decreases to between 120573 = 10∘ and 60∘The periodicity ofboth forces is almost lost at 120573 = 60∘ and there is no definiteperiodicity of the induced force components at 120573 = 70∘

At a bevel angle of 10∘ as shown in Figure 6(b) it is inter-esting that the dominant peak of the drag force is identical tothat of the lift force From a vibration standpoint the forcingfrequency is important for the prevention of resonance Thedistance between the two vortices in Figure 5(b) does not staythe same but the difference between the shedding vorticesbetween 120573 = 0

∘ and 10∘ (Figures 5(a) and 5(b)) is not

Modelling and Simulation in Engineering 7

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Symmetric wake at 119879 = 0396

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Symmetry-breaking transition at 119879 = 0900

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Nonperiodic vortex shedding at 119879 = 1098

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Fully developed periodic shedding at 119879 = 3500

Figure 3 Wake patterns of the NACA 0009 hydrofoil with a truncated trailing-edge (120573 = 0∘) for Re = 106

CL

CL

002

001

0

minus001

minus002I II III

T

0 05 1 15 2 25 3 35

CD

CD

0015

001

0005

0

Figure 4 Time evolution of the lift and drag coefficients for theimpulsively started NACA 0009 hydrofoil with a truncated trailingedge (120573 = 0∘) at Re = 106 Regime I denotes the developing regimeof a symmetric wake II a symmetry-breaking transition and IIIperiodic vortex shedding

statistically significant A possible reason for this is the phasedifference between the vortices shed from the upper andlower sides In the wake formation region the two vorticesshed from the symmetric trailing edge at 120573 = 0∘ are almostexactly out of phase (180 degrees) whereas at 120573 = 10

∘

there is a phase difference of approximately 30 degrees Thismeans that before one vortex is fully shed from the bodyanother vortex is created At higher bevel angles 120573 gt 20∘ thenegative vortices from the upper side of the hydrofoil becomestronger than the positive ones from the lower side Theoscillation force is induced mostly by the vortices shed fromthe acute (upper) side As a result the dominant frequency

of the drag force oscillation appears to be the same as thevortex shedding frequency We numerically observed thatdifferences in strength andor phase between two adjacentvortices cause nonperiodic and irregular vortex shedding

Figure 7 shows the shedding frequency of vortices fromdifferent beveled trailing edges compared with experimentalresults by Zobeiri et al [11] The numerically simulatedshedding frequencies were approximately 68 and 64 for120573 = 0

∘ and 60∘ respectively Numerical results showgood qualitative agreements with the experimental resultsbut the numerical simulations underestimated the vortex-shedding frequency by about 10 Figure 8 shows the meanvalue and standard deviation of the velocity components Wecompared our numerical results with the experimental resultsmeasured by Zobeiri et al [11] The truncated and obliquetrailing edges in Figure 8 indicate bevel angles of 120573 = 0

∘

and 60∘ respectively For the truncated trailing edge themean velocity profiles are more symmetric than those in theexperimental data The results from numerical simulationare in good agreement with the experimental results Weconclude that the vortex shedding phenomenon from thetrailing edge of a 2D hydrofoil can be reasonably simulatedusing 2D LES which considerably reduces the computationalcost

55 Effect of a Periodically Varying Flow on Vortex SheddingThe propeller is generally located behind the shiprsquos hull andis subjected to a nonaxisymmetric wake field Consideringa 2D flow the propeller section effectively experiences peri-odic variation in flow incidence We conducted numericalsimulations to investigate vortex shedding with respect tothe sinusoidal motions of the free-stream flow The hydrofoilwas at rest and the flow became the oscillator under theseconditions In the current study the angle of attack varied

8 Modelling and Simulation in Engineering

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Bevel angle 120573 = 0∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Bevel angle 120573 = 10∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Bevel angle 120573 = 20∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Bevel angle 120573 = 40∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(e) Bevel angle 120573 = 60∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(f) Bevel angle 120573 = 70∘

Figure 5 Instantaneous vorticity distributions at 119879 = 10 for Re = 106

sinusoidally with time 119905 120572 = 120572119900sin(2120587119891

infin119905) where 119891

infin

was the frequency of the free-stream flow oscillation andits oscillation amplitude was restricted at 120572

119900= 2∘ The

magnitude of the free-stream velocity was kept constant Thetested model is the NACA 0009 hydrofoil with a beveledtrailing edge of 120573 = 60∘

Figure 9 shows the instantaneous vorticity distributionsat119879 = 10 and the PSD of the drag coefficients at four differentfrequencies of the free-stream flow that is 119891

infin= 15 20

25 and 30 There are two peaks in the PSD functions thefirst peak appears around 119891lowast asymp 6 and the second one isidentical to the frequency of the free-stream flow oscillationThe obvious feature recognized in all the PSD functions asshown in Figure 9 is that the first peak frequency is nearlyidentical to the vortex-shedding frequency from the hydrofoilin the uniform free-stream flow Vortex formation in the nearwake is thought to be less affected by small inflow oscillationsMost interestingly at 119891

infin= 20 vortices are regularly shed

from the trailing edge as shown in Figure 9(c) In the PSDfunctions drag oscillation induced by vortex shedding isclearly observed This means that a particular oscillationfrequency in the free-stream velocity can cause regular andperiodic vortex shedding This is significant with respect tochanging a marine propellerrsquos angle of attack over time

6 Concluding Remarks

We simulated impulsively started flows past a modifiedNACA 0009 hydrofoil at a Reynolds number of 106 usingthe hybrid particle-mesh method in conjunction with thevorticity-based subgrid scale model We have achieved sig-nificant savings in both memory usage and computationtime by making improvements in the numerical schemesthe particle-based domain decomposition the use ofmultipledomains and the approximation of domain boundary con-ditions These techniques are simple and easy to implementbut also effective in reducing the computational costs Theparticle-based domain decomposition scheme is suitable fora distributed system in which each processor has its ownprivate memory (distributed memory)

In this study the numerical simulation results nearlymatch the experimental resultsThis demonstrates that vortexshedding from the trailing edge of a 2D hydrofoil can bereasonably simulated with a 2D LES It is due to the fact thatthe flow is characterized by quasi-two dimensionality withvortex shedding from the trailing edge of the hydrofoil Wehave considered that 2D vortex shedding simulations shouldnot be dismissed so easily With two-dimensionality it ispossible to solve the problem much less expensively since

Modelling and Simulation in Engineering 9

0 5 10 15 20 250

10

0 5 10 15 20 250

05times10minus5

PSD

ofC

L

PSD

ofC

D

times10minus6

flowast flowast

(a) Truncated trailing edge 120573 = 0∘

0

08

0

08

0

08

0

08

0

08

0

04

0

04

0

04

0

04

0

04

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

(b) Beveled trailing edge 120573 gt 0∘

Figure 6 Power spectral density (PSD) of the lift and drag coefficients 119891lowast denotes the frequency normalized by the chord length and theundisturbed flow velocity

3D problems require significant computational resourcesHowever the LES should be 3D since turbulence is inherently3D While we accept that more realistic 3D simulation isrequired physically a 2D LES can be used as a basis forcomparison prior to 3D implementation Once the capability

of a numerical algorithm is evaluated in two dimensionsit can be easily extended into three dimensions This paperis intended as a guide for researchers and designers whoare working on a similar problem vortex shedding from ahydrofoil at high Reynolds numbers

10 Modelling and Simulation in Engineering

0 10 20 30 40 50 600

2

4

6

8

10

Zobeiri et al (2012) expPresent cal

flowast

120573 (deg)

Figure 7 Shedding frequency of vortices from the different beveled trailing edges for Re = 106 The experimental data by Zobeiri et al [11]were derived from LDVmeasurements and the approximate shedding frequency as read from their figure is normalized by the chord lengthand the undisturbed flow velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02 04 06 08 1uxmeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(a) Normalized mean of streamwise velocity

minus06

minus04

minus02

0

02

04

06yt

max

0 01 02 03 04uxstdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(b) Normalized mean of transverse velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02uymeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(c) Normalized standard deviation of streamwise velocity

minus06

minus04

minus02

0

02

04

06

yt

max

0 01 02 03 04 05uystdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(d) Normalized standard deviation of transverse velocity

Figure 8 Mean and standard deviation of velocities at 119909119888 = 0516 for Re = 106 The experimental data by Zobeiri et al [11] were measuredat 119909119888 = 052 and 0528 for truncated (120573 = 0∘) and oblique (120573 = 60∘) trailing edges respectively and the reported Reynolds number of theflow is about 2 times 106

Modelling and Simulation in Engineering 11

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(a) Steady flow 119891infin = 0

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(b) Incoming flow frequency 119891infin = 15

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(c) Incoming flow frequency 119891infin = 20

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(d) Incoming flow frequency 119891infin = 25

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(e) Incoming flow frequency 119891infin = 30

Figure 9 Effect of the incoming flow frequency on vortex shedding from a beveled trailing edge of 120573 = 60∘ at 119891infin= 15 20 25 and 30

The contour indicates the instantaneous vorticity distribution at 119879 = 10 and the power spectral density (PSD) is calculated from the dragcoefficients in the range 119879 = [3 10]

12 Modelling and Simulation in Engineering

We found in this study that the vortex formation char-acteristic in periodically varying inflows remains unchangedA certain frequency of free-stream flow oscillation tends tomake vortex shedding more regular again This means thatthe periodicity of flow incidence can stimulate an inherentfrequency of vortex shedding from a hydrofoil Howeverthe effect of inflow oscillation on vortex shedding is notyet fully understood Future work will include experimentalinvestigations as well as a 3D flow simulation to betterunderstand the relationship between inflow oscillation andvortex shedding

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Korean Institute ofOcean Science amp Technology sponsored by the Ministry ofTrade Industry amp Energy (no 10033668) and the IndustrialConvergence Strategic Technology Development Program(no 10044499) funded by the Ministry of Trade Industry ampEnergy (MI Korea)

References

[1] E Berger and R Wille ldquoPeriodic flow phenomenardquo AnnualReview of Fluid Mechanics vol 4 no 1 pp 313ndash340 1972

[2] P W Bearman ldquoVortex shedding from oscillating bluff bodiesrdquoAnnual Review of Fluid Mechanics vol 16 pp 195ndash222 1984

[3] C H K Williamson ldquoVortex dynamics in the cylinder wakerdquoAnnual Review of Fluid Mechanics vol 28 pp 477ndash539 1996

[4] C H K Williamson ldquoVortex-induced vibrationsrdquo AnnualReview of Fluid Mechanics vol 36 pp 413ndash455 2004

[5] R Fischer ldquoSinging propellersmdashsolutions and case historiesrdquoMarine Technology vol 45 no 4 pp 221ndash227 2008

[6] D A Bourgoyne J M Hamel S L Ceccio and D R DowlingldquoTime-averaged flow over a hydrofoil at high reynolds numberrdquoJournal of Fluid Mechanics vol 496 pp 365ndash404 2003

[7] D A Bourgoyne S L Ceccio and D R Dowling ldquoVortexshedding from a hydrofoil at high Reynolds numberrdquo Journalof Fluid Mechanics vol 531 pp 293ndash324 2005

[8] N Mulvany J Tu L Chen and B Anderson ldquoAssessment oftwo-equation turbulence modelling for high Reynolds numberhydrofoil flowsrdquo International Journal for Numerical Methods inFluids vol 45 no 3 pp 275ndash299 2004

[9] P Ausoni M Farhat X Escaler E Egusquiza and F AvellanldquoCavitation influence on von Karman vortex shedding andinduced hydrofoil vibrationsrdquo Journal of Fluids EngineeringTransactions vol 129 no 8 pp 966ndash973 2007

[10] P Ausoni Turbulent vortex shedding from a blunt trailingedge hydrofoil [PhD thesis] Ecole Polytechnique Federale deLausanne Lausanne Switzerland 2009

[11] A Zobeiri P Ausoni F Avellan and M Farhat ldquoHow obliquetrailing edge of a hydrofoil reduces the vortex-induced vibra-tionrdquo Journal of Fluids and Structures vol 32 pp 78ndash89 2012

[12] G-H Cottet and P D Koumoutsakos Vortex Methods Theoryand Practice Cambridge University Press 2000

[13] G Cottet and P Poncet ldquoParticle methods for direct numericalsimulations of three-dimensional wakesrdquo Journal of Turbulencevol 3 article N38 2002

[14] P Chatelain A Curioni M Bergdorf D Rossinelli WAndreoni and P Koumoutsakos ldquoBillion vortex particle directnumerical simulations of aircraft wakesrdquo Computer Methods inAppliedMechanics and Engineering vol 197 no 13ndash16 pp 1296ndash1304 2008

[15] Y-C Kim J-C Suh and K-J Lee ldquoVortex-in-cell methodcombined with a boundary element method for incompressibleviscous flow analysisrdquo International Journal forNumericalMeth-ods in Fluids vol 69 no 10 pp 1567ndash1583 2012

[16] P Angot C-H Bruneau and P Fabrie ldquoA penalization methodto take into account obstacles in incompressible viscous flowsrdquoNumerische Mathematik vol 81 no 4 pp 497ndash520 1999

[17] M El Ossmani and P Poncet ldquoEfficiency of multiscale hybridgrid-particle vortex methodsrdquo Multiscale Modeling amp Simula-tion vol 8 no 5 pp 1671ndash1690 2010

[18] J T Rasmussen G-H Cottet and J HWalther ldquoAmultiresolu-tion remeshed vortex-in-cell algorithm using patchesrdquo Journalof Computational Physics vol 230 no 17 pp 6742ndash6755 2011

[19] M Coquerelle and G-H Cottet ldquoA vortex level set method forthe two-way coupling of an incompressible fluid with collidingrigid bodiesrdquo Journal of Computational Physics vol 227 no 21pp 9121ndash9137 2008

[20] F Morency H Beaugendre F Gallizio and S Laurens ldquoCom-putation of ice shedding trajectories using cartesian gridspenalization and level setsrdquo Modelling and Simulation in Engi-neering vol 2011 Article ID 274947 15 pages 2011

[21] M M Hejlesen P Koumoutsakos A Leonard and J HWalther ldquoIterative Brinkman penalization for remeshed vortexmethodsrdquo Journal of Computational Physics vol 280 pp 547ndash562 2015

[22] M Gazzola P Chatelain W M van Rees and P Koumout-sakos ldquoSimulations of single and multiple swimmers with non-divergence free deforming geometriesrdquo Journal of Computa-tional Physics vol 230 no 19 pp 7093ndash7114 2011

[23] R Chatelin and P Poncet ldquoA hybrid grid-particle method formoving bodies in 3D stokes flow with variable viscosityrdquo SIAMJournal on Scientific Computing vol 35 no 4 pp B925ndashB9492013

[24] R Chatelin and P Poncet ldquoHybrid grid-particle methodsand penalization a Sherman-Morrison-Woodbury approach tocompute 3D viscous flows using FFTrdquo Journal of ComputationalPhysics vol 269 pp 314ndash328 2014

[25] S J Lee J H Lee and J C Suh ldquoVortex shedding frequencyfor a 2D hydrofoil with a truncated trailing edgerdquo Journal of theSociety of Naval Architects of Korea vol 51 no 6 pp 480ndash4882014 (Korean)

[26] S-J Lee and J C Suh ldquoA multi-domain approach for ahybrid particle-mesh methodrdquo Journal of Computational FluidsEngineering vol 19 no 2 pp 72ndash78 2014

[27] S-J Lee J-H Lee and J-C Suh ldquoComputation of pressurefields around a two-dimensional circular cylinder using thevortex-in-cell and penalization methodsrdquo Modelling and Sim-ulation in Engineering vol 2014 Article ID 708372 13 pages2014

[28] J J Monaghan ldquoExtrapolating 119861-splines for interpolationrdquoJournal of Computational Physics vol 60 no 2 pp 253ndash2621985

[29] J J Monaghan ldquoParticle methods for hydrodynamicsrdquo Com-puter Physics Reports vol 3 no 2 pp 71ndash124 1985

Modelling and Simulation in Engineering 13

[30] O V Vasilyev and N K-R Kevlahan ldquoHybrid waveletcollocation-brinkman penalization method for complex geom-etry flowsrdquo International Journal for Numerical Methods inFluids vol 40 no 3-4 pp 531ndash538 2002

[31] W Iwakami Y Yatagai N Hatakeyama and Y Hattori ldquoAnew approach for error reduction in the volume penalizationmethodrdquoCommunications in Computational Physics vol 16 no5 pp 1181ndash1200 2014

[32] J R Mansfield O M Knio and C Meneveau ldquoTowardslagrangian large vortex simulationrdquo ESAIM Proceedings vol 1pp 49ndash64 1996

[33] G-H Cottet and P Poncet ldquoAdvances in direct numerical sim-ulations of 3D wall-bounded flows by vortex-in-cell methodsrdquoJournal of Computational Physics vol 193 no 1 pp 136ndash1582004

[34] B Caswell ldquoKinematics and stress on a surface of restrdquo Archivefor Rational Mechanics and Analysis vol 26 no 5 pp 385ndash3991967

[35] R S Rogallo and P Moin ldquoNumerical simulation of turbulentflowsrdquo Annual Review of Fluid Mechanics vol 16 pp 99ndash1371984

[36] J Meyers and P Sagaut ldquoOn the model coefficients for thestandard and the variational multi-scale Smagorinsky modelrdquoJournal of Fluid Mechanics vol 569 pp 287ndash319 2006

[37] J R Mansfield O M Knio and C Meneveau ldquoA dynamic LESscheme for the vorticity transport equation formulation and apriori testsrdquo Journal of Computational Physics vol 145 no 2 pp693ndash730 1998

[38] S B Pope Turbulent Flows Cambridge University Press 2000[39] WH Press S A TeukolskyW T Vetterling and B P Flannery

Numerical Recipes in C++ The Art of Scientific ComputingCambridge University Press 2nd edition 2002

[40] L Greengard and W D Gropp ldquoA parallel verison of the fastmultipoe methodrdquo Computers amp Mathematics with Applica-tions vol 20 no 7 pp 63ndash71 1990

[41] M Frigo and S G Johnson ldquoFFTW an adaptive software archi-tecture for the FFTrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo98) pp 1381ndash1384 May 1998

[42] M Frigo and S G Johnson ldquoThe design and implementationof FFTW3rdquo Proceedings of the IEEE vol 93 no 2 pp 216ndash2312005

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

2 Modelling and Simulation in Engineering

results showed that a lock-in phenomenon occurs wherethe vortex shedding frequency is locked onto the naturalfrequency of a hydrofoil Interestingly it has been reportedthat cavitation has a minor effect on the wake dynamicRecently Zobeiri et al [11] investigated two NACA 0009hydrofoils with blunt and oblique trailing edges respectivelyfor Reynolds numbers ranging from 50 times 105 to 29 times 106and conducted high-speed visualization and flow-inducedvibrationmeasurementsThey confirmed experimentally thatflow-induced vibration was significantly reduced with anoblique trailing edge compared with a truncated edge Theyalso concluded that the collision between upper and lowervortices and the resulting vorticity redistribution were themain reasons for the obtained vibration reduction with theoblique trailing edge

This study numerically investigated vortex shedding fromtruncated and oblique trailing edges of a hydrofoil to betterunderstand the vortex shedding mechanism and to assess thecapability of our numerical methodology We employed thehybrid particle-meshmethod and the vorticity-based subgridscale model to undertake a numerical flow simulation Thehybrid particle-mesh method is a combination of the vortex-in-cell (VIC) method (see [12ndash15] and references therein)and the penalization technique [16] which has been devel-oped in recent years (see [17ndash24] for a review) The hybridparticle-mesh method enables the use of fast and efficienttechniques for computing differential operators therebyenabling large scale simulations However techniques tosave computational memory and CPU time consumptionare still required Thus we have introduced the implementa-tions of numerical schemes to more efficiently use availablecomputational resources [25 26] With these numericalschemes we numerically investigate vortex shedding fromvarious beveled trailing edges of a NACA 0009 hydrofoil ata Reynolds number of 106 We then compare our numericalresults with experimental data [10 11] to assess the capabilityof our numerical methodology In addition we investigatethe influence of periodically varying free-stream flows onvortex shedding to better understand the vortex sheddingmechanism

The organization of the remainder of this paper is asfollows we present a brief description of the numerical meth-ods used for the vortex shedding simulation in Section 2We describe our implementation of numerical schemes toefficiently use available computational resources in Section 3and the computational procedure followed in Section 4 Wepresent out vortex shedding results from a variety of beveledtrailing edges and discuss the influence of periodic free-streamflows in Section 5 In Section 6 we provide a summaryof this study and some perspectives for future work

2 Governing Equations andNumerical Methods

The vorticity-velocity formulation of the Navier-Stokes equa-tions allows a purely kinematical problem to be decoupledfrom the pressure term which is eliminated by applying thecurl operator Pressure which can be evaluated in an explicit

manner using the identified vorticity and velocity fields [27]is not part of the solution algorithm For a 2D flow parallel tothe 119909119910-plane the vorticity transport equation in terms of thevorticity can be expressed as

120597120596119911

120597119905+ u sdot nabla120596

119911= ]nabla2120596

119911 (1)

where120596119911is the scalar plane component of the vorticity vector

that is 120596 equiv 120596119911in two dimensions The evolution of a flow

is considered in discrete time steps The viscous splittingalgorithm can be expressed in a Lagrangian framework as

119863x119863119905= u (2a)

119863120596119911

119863119905= ]nabla2120596

119911 (2b)

where 119863119863119905 = 120597120597119905 + u sdot nabla is the material derivative 119873119901

discrete Lagrangian fluid particles with a finite core size 120598 arelinearly superposed to approximate the vorticity field as

120596119911 (x 119905) =

119873119901

sum

119901=1

Γ119901120577120598(x minus x

119901) (3)

where 120577120598is a mollification of the Dirac-delta function [12]

Finite core size vortices are used instead of point vorticesEach particle is characterized by its position x

119901and its

strength Γ119901 For a given vortex particle the circulation Γ

119901

is identical to the product of the vorticity and the area ofthe vortex particle 1205982 which also represents the contributionof the vortex particle to the vorticity field Γ

119901= int120596

119911119889119878 asymp

1205961199111205982 The vorticity-carrying fluid particle is advanced with

the velocity u and is gradually diffused because of viscouseffects

21 Vortex-in-Cell (VIC) Method for Convection Since thecontinuity equation and the definition of vorticity lead tonablasdotu = 0 andnablatimesu = 120596 the velocity u of (2a) can be expressedas the gradients of the stream-function120595 The velocity vectoris expressed as

u = Uinfin+ u120596= Uinfin+ nabla times 120595 (4)

where Uinfinis the free-stream velocity and u

120596is the rotational

velocity The Poisson equation for the stream-function isnabla2120595 = minus120596 In two dimensions the stream function is scalar

120595119911 and only the vorticity component perpendicular to the

plane is nonzero 120596 = 120596119911119890119896 The vector Poisson equation

reduces to a scalar Poisson equation

nabla2120595119911= minus120596119911 (5)

Rotational velocity is defined by 119906119909= 120597120595

119911120597119910 and 119906

119910=

minus120597120595119911120597119909 In the VIC method stream functions to evaluate

velocities are computed using a uniform Cartesian meshThefast Fourier transform- (FFT-) based Poisson solver reducesthe computational cost of obtaining the velocity field toO(119873log119873) where 119873 is the number of grid points [12] To

Modelling and Simulation in Engineering 3

compute the stream functions using the FFT-based Poissonsolver the vorticity 120596

119911is interpolated onto equally spaced

Cartesian grid using

120596119911(x119892) =

119873119901

sum

119901=1

120596119911(x119901)119882(

x119901minus x119892

ℎ) (6)

where ℎ is the mesh spacing using the following third orderinterpolation kernel [28 29]

119882(119909) = 1198721015840

4(119909)

=

0 for |119909| gt 21

2(2 minus |119909|)

2(1 minus |119909|) for 1 lt |119909| le 2

1 minus5

2|119909|2+3

2|119909|3 for |119909| le 1

(7)

in each coordinate direction This kernel conserves the 0th1st and 2nd order moments The subscripts

119892and119901denote

the grid and particle quantities respectively A vortex particlecontributes to the nearest 16 nodes through the1198721015840

4scheme

and the total vorticity at each node is obtained by summingthe vorticity contributions of all the vortex particles

Boundary conditions for the stream function are requiredto solve the Poisson equation If the computational domainboundaries are far enough from the particles homogeneousDirichlet boundary conditions (120595 = 0) may be usedHowever using a larger domain is inefficient since it requiresa regular grid with too many points To compactly place asquare FFT-domain Ω we used nonhomogeneous Dirichletboundary conditions in this study Unknown stream func-tions at the domain boundaries (120597Ω) can be directly obtainedfrom Greenrsquos function solution [27] whereby

120595119911(x119887) = minus

1

2120587

119873119901

sum

119901=1

Γ119901ln (10038161003816100381610038161003816x119887 minus x119901

10038161003816100381610038161003816) 119887 = 1 119872 (8)

where x119887isin 120597Ω x

119901isin Ω and x

119892= x119901119872 denotes the number

of domain boundary points and the circulation strength ofeach vortex particle Γ

119901= 120596119911(x119901)1205982 We note here that all

particle vorticity values should be considered Fast evaluationof the stream functions at the domain boundaries is furtherdiscussed in Section 33

22 Brinkman PenalizationMethod for Diffusion Thepenal-ization method was initially designed to take into accountsolid obstacles in fluid flows [16] The main point of thepenalty term is to replace the usual vorticity creation algo-rithm for enforcing the no-slip condition for a solid bodyBy adding the penalization term to (1) the vorticity transportequation becomes

119863120596119911

119863119905= ]nabla2120596

119911+ nabla times [120582120594 (u

119904minus u)] (9)

where u119904is the velocity of the solid body and 120594 denotes a

mask function that yields 0 in a fluid and 1 in a solid [16 18]

120582 indicates the penalization parameter with the dimension[119904minus1] To avoid too small time step the penalization term is

evaluated by the implicit expression [16 18 30]Δ120596119911

Δ119905= nabla times [120582120594 (u119899

119904minus u119899+1)] (10)

where

u119899+1 =u119899 + 120582Δ119905120594u119899

119904

1 + 120582Δ119905120594 (11)

This scheme is unconditionally stable [19] Equation (9) isrewritten by replacing its terms with those in (10) and (11) asfollows

120596119899+1

119911minus 120596119899

119911

Δ119905= ]nabla2120596119899

119911+ nabla times [

120582120594

1 + 120582Δ119905120594(u119899119904minus u119899)] (12)

The penalization and diffusion terms of (12) are consecutivelyevaluated on a temporary grid To reduce any error due tothe penalization term we used a mask function with second-order accuracy for which the boundary is located at themidpoint of the grid where the mask function jumps from0 to 1 [18 31]

23 Smagorinsky Turbulence Model The filtered vorticitytransport equation for a Lagrangian vorticity-based large-eddy simulation (LES) in two-dimensions can be expressedas [32]

119863120596119911

119863119905= ]nabla2120596

119911+ nabla sdot Φ (13)

where the bar denotes spatial filtering at length scale 120598 Thesubgrid-scale vorticity stress term Φ which accounts for theeffect of unresolved velocity and vorticity fluctuations can bedefined as Φ = ]

119879nabla120596119911 Smagorinsky eddy-viscosity model is

one of most commonly used subgrid-scale (SGS) turbulencemodels In spite of its simplicity the Smagorinsky model isstill extensively used in practice and has served as a basic toolfor the developments of LES modeling [33] In this modeleddy viscosity is defined by

]119879= (119862119878Δ)2 1003816100381610038161003816100381611987810038161003816100381610038161003816 (14)

where 119862119878is a nondimensional Smagorinsky constant and Δ

is a subgrid characteristic length scale 119878119894119895= (120597119906

119894120597119909119895+

120597119906119895120597119909119894)2 is the resolved strain-rate tensor and |119878| =

(2119878119894119895119878119895119894)12 is its magnitude Here |119878|2 = |120596

119911|2 in incom-

pressible flows (nabla sdot u = 0) using the Caswell formula [34]Finally the vorticity transport equation for the LES in twodimensions is rewritten as

119863120596119911

119863119905= (] + (119862

119878Δ)2 1003816100381610038161003816120596119911

1003816100381610038161003816) nabla2120596119911 (15)

In 2D LES models the filtering process is applied in only twodirections so Δ = (ℎ

119909ℎ119910)12 is defined by the grid spacing

The Smagorinsky constant 119862119878may vary with the type of

flow and the Reynolds number Values ranging from 01 to024 have been suggested in the literatures [33 35 36] Theconstant 119862

119878is prescribed and its standard value is typically

015 [37 38]

4 Modelling and Simulation in Engineering

3 Efficient Implementation ofNumerical Methods

31 Particle-BasedDomainDecomposition Parallelmachinesand algorithms enable significant reduction in computingtime and greater amount of available memory For high-performance computing we used the Message Passing Inter-face (MPI) for our distributed systems in which each proces-sor has its own localmemoryThe appropriate decompositionof particles andor grids should be employed In domainor spatial decomposition a given domain Ω is geometri-cally decomposed by splitting it into several subdomainsTypically the subdomains are uniform and particles are notevenly distributed among the subdomains Differences inthe number of particles assigned to each processor causecomputational load imbalances As some processors willsit idly while waiting for others poor performance canresult

We introduce a simple idea to achieve a better loadbalance during the domain decomposition In vortex particlemethods fluid particles must periodically be redistributedsince their accumulation leads to inaccuracies in numericalsolutions This process is called particle remeshing or redis-tribution The remeshing step creates a new set of particleson a uniform Cartesian grid and then the randomly spacedold particles are removed All the new particles are reindexedby 119894 = 1 2 119873

119901from upstream to downstream that

is higher-indexed particles are located further downstreamDue to the position-dependent index of the particles theentire spatial domain can be easily split to ensure an equalamount of particles in each subdomain (each processor)As a result the horizontal extent of the domain is unequalbut the particles are evenly distributed in subdomains ofdifferent sizes In every remeshing step the entire domainis dynamically partitioned In this way each subdomaincontains an equal number of particles and better load balanceismaintained during simulationThis approach is very simpleand effective No additional algorithms are required to evenlydistribute the particles or to identify the position of theparticles As shown in Figure 1 we obtained an approxi-mately 30 reduction in computation time simply by loadbalancing during parallel computation This technique isalso valuable for the efficient use of available computationalmemory

32 Multidomain Approach To avoid an excessive domainsize we consideredmultiple domains In this study the entiredomainΩ is defined as the union of all the domains coveringall the fluid particles that is x

119901isin Ω where Ω = Ω

1cup

Ω2cup sdot sdot sdot cup Ω

119873119889 The domain number119873

119889is not constant and

is dependent on a spatially evolving flow When the domainsize exceeds a certain limit a new domain is created situateddownstream from the domain As a result the first domainΩ1includes a solid body and the others are sequentially

located downstream from the body For the total circulationconservation the grid spacing of the 119894th domain can bedefined as ℎ

119894= 120598 (single-resolution) or ℎ

119894= 2ℎ119894minus1

(multi-resolution) where ℎ

1= 120598 Here 120598 denotes the particle size To

T

Com

puta

tion

time (

h)

250

200

150

100

50

00 2 4 6 8 10

Equal-sized subdomainsUnequal-sized subdomain

Figure 1 Computation times for equal- and unequal-sized subdo-mains

interpolate the vorticity values of the particles into the nodeson a grid with different grid spacing (6) is rewritten as

120596119911(x119892) =

1205982

ℎ2

119894

119873

sum

119901=1

120596119911(x119901)119882(

x119901minus x119892

ℎ119894

) (16)

where x119901isin Ω119894and x119892isin Ω119894The nodes are declared to be a 2D

array so that the position of each node can be found straight-forwardly from its index The velocity and vorticity of eachdomain are sequentially and independently evaluatedHencerelatively small amounts of working memory are requiredThis multidomain approach also allows more freedom inthe choice of variable arrangements in the domains Theversatility of this approach was demonstrated in our previousstudy [26] It can be achieved at the expense of a significantincrease in computation time This issue is further discussedin the next section

33 Approximation of Far-Field Conditions A substantialpart of the overall computation time is spent on the calcu-lation of the boundary stream functions to solve Poissonrsquosequation for the stream function This requires the order ofO(119872119873

119901119875) operations to obtain boundary values of domain

Ω119894 Here 119872 denotes the number of nodes at the sides of

domainΩ119894and119873

119901119875 is the number of particles per processor

To achieve a nearly perfect load balance particles are evenlydistributed among the processors as discussed in Section 31and each processor has all the boundary nodes 119872 It mustalso be very useful to avoid all unnecessary communications119873119901119875 is reduced by increasing the number of processors

involved in the computation whereas 119872 is not This is whythe number of boundary nodes 119872 is the most critical fordetermining the computation time

In this study we attempted to reduce the number ofboundary nodes 119872 using the cubic spline approximationThis is made possible by the smooth variation of the streamfunctions along the boundary Only one-fourth of the bound-ary nodes were chosen as equidistant points for the directcalculations That is the stream functions at the selected

Modelling and Simulation in Engineering 5

boundary nodes were directly computed using (8) and thevalues at boundary nodes not selected were evaluated byinterpolation using cubic splines [39] In this study theerror between the interpolated value and the true valueis typically less than 001 We have observed that thereduction in computational time is inversely proportionalto the number of nodes selected for direct computationWe believe that this approach can be implemented withmuch lower computational effort because a complex datastructure such as a quad-tree used in the fast multipolemethod [40] is not required A rigorous comparison of theaccuracy and computational costs is planned in the nearfuture

4 Computational Procedure

First the entire simulation domain Ω covering all theparticles is divided into small domains Ω

119894(119894 = 1 119873

119889)

as discussed in Section 32 Each domain Ω119894is consecutively

computed according to the following numerical procedure

(1) Particle-based domain decomposition as discussedin Section 31 domain Ω

119894is decomposed into sub-

domains to ensure that the particles are evenly dis-tributed among the processors If redistribution hasnot been done prior to the time step particles thathave been removed from a subdomain are assignedto an adjacent processor without performing domaindecomposition

(2) Particle-to-grid interpolation each processor inter-polates the vorticity 120596

119911of its own particles into the

grid nodes through the11987210158404interpolation kernel

(3) Calculation of the boundary condition each proces-sor computes the boundary stream functions by directcalculation and interpolation using cubic splines asdiscussed in Section 33

(4) Evaluation of the velocity field the stream functionon the grid is computed with the FFT-based Poissonsolver from an open-source library called FFTW(fastest Fourier transform in the west) [41 42]There-after the rotational velocities on the grid nodes arecomputed from the definitions 119906

119909= 120597120595120597119910 and

119906119910= minus120597120595120597119909 using the fourth order central finite

difference scheme(5) Evaluation of the vorticity field each processor eval-

uates the evolution of the vorticity field in time asfollows

(a) The penalization term that is the second termon the right hand side of (12) is computedwith the velocities on the grid The spatialderivatives of the curl operators are evaluatedusing a centered difference approximation witha fourth-order error

(b) The diffusion term that is the first term onthe right hand side of (12) is evaluated on thegrid using a classical 9-point finite differencemethod

(c) The turbulent viscosity (119862119878Δ)2|120596119911| in (15) is

approximated by the vorticity strength of thegrid Finally the vorticity field 120596119899+1

119911is updated

(6) Grid-to-particle interpolation each processor inter-polates the velocity and vorticity on the grid back to itsown particle positions through the1198721015840

4interpolation

kernel

Vortex particles are advanced in time using a midpointpredictor-corrector method as follows particle positions arepredicted by their velocities xlowast = x119899 + Δ119905u(x119899 120596119899

119911) then the

particle positions are corrected by using combinations of thepredicted and previous positions and velocity values x119899+1 =x119899 + 05Δ119905[u(x119899 120596119899z ) + u(xlowast 120596119899+1

119911)] Particle redistribution is

conducted every three time steps to ensure particle overlapIf a new particle has a small vorticity magnitude that is|Γ119894| lt 00001|Γ|max it is deleted to avoid having too many

particles Loss of circulation is equally distributed among theremaining particles to ensure the conservation of the totalvorticity

5 Numerical Results and Discussions

51 Model Description We selected a NACA 0009 hydrofoilwith a truncated trailing edge for the numerical simulationsThis hydrofoil has the same cross-section as the experimentalmodel used by Ausoni et al [9] Ausoni [10] and Zobeiriet al [11] Ten percent of the original chord 119888

119900was removed

from the trailing-edge region of the NACA 0009 hydrofoilThe hydrofoil thickness distribution is written as

00 le119909

119888119900

le 05

119910

119888119900

= 01737 (119909

119888119900

)

12

minus 02422 (119909

119888119900

) + 03046 (119909

119888119900

)

2

minus 02657 (119909

119888119900

)

3

05 lt119909

119888119900

le 10

119910

119888119900

= 00004 + 01737 (1 minus119909

119888119900

) minus 01898 (1 minus119909

119888119900

)

2

+ 00387 (1 minus119909

119888119900

)

3

(17)

The hydrofoil geometry is further detailed in Ausoni [10]Themaximum thickness as normalized by the chord length 119888 is119905max119888 = 01 and the thickness at the trailing edge 119905TE119888 is00322 The bevel angle 120573 is defined in Figure 2 The trailingedge with 120573 = 0∘ denotes that the trailing edge is truncated inthe vertical direction

52 Numerical Setup Numerical parameters can be deter-mined using the stability condition Δ119905 = 120598

24] [18] The

grid convergence was achieved for 120598 = 000030 and

6 Modelling and Simulation in Engineering

xc

yc

045 05

minus002

0

002

A

BC

00054c

00322c120573

Figure 2 Trailing-edge shape Note that the points A B and C indicate the knuckles of the chamfer and trailing-edge wedge and the bevelangle is defined as 120573

Δ119905 = 000015 was chosen in order to guarantee numericalaccuracy for a Reynolds number of 106 The grid spacingof the 119894th domain Ω

119894is defined as ℎ

119894= 1205982119894minus1 to reduce

computation timeThe penalization parameter 120582 in (12) is setto be 108 The hydrofoil at a zero angle of attack is immersedin a Cartesian grid (the first domain) that does not conformto the surface of the bodyThe first domain with 4096times 1024lattice nodes is fixed and the other domains are dynamicallydetermined depending on the spatially evolving flow In thisstudy numerical simulations were conducted until 119879 = 10A new domain is created downstream of a body once thenumber of horizontal grid points exceeds 2048 Althoughthere is no limit on the number of vertical grid points it wastypically less than or equal to 1024 During simulation lessthan 08 million particles and four computational domainswere created The entire computation in each case includingstart-up and printing times took approximately 170 hourson 16 Intel Xeon64 33 GHz CPUs with 6GB memory perprocessor

53 Truncated Trailing-Edge For an impulsively started flowpast a hydrofoil at a chord-based Reynolds number of 106the wake pattern can be classified into three regimes basedon the time evolutions shown in Figure 3 In the very earlystages the wake remained very symmetrical and consistedprimarily of two large vortices in the near wake as shown inFigure 3(a) At that time the drag force gradually decreasedAs time went on the wake became asymmetrical as shown inFigure 3(b) and then shed a vortex as shown in Figure 3(c)After a number of shedding periods vortices were shedregularly from the symmetric trailing edge in the form of twotrains of oppositezsign but equal-strength vortices as shownin Figure 3(d) Such regular vortex shedding induces periodicloading on the structure as shown in Figure 4 It is wellknown that drag-force oscillation during vortex shedding ismuch smaller than the lift force Numerical results show thatoscillations in the drag force occurred at twice the vortex-shedding frequency due to the fact that two vortices wereshed from alternate sides during one full period of wakeoscillation

From our simulation the Strouhal number based on thetrailing-edge thickness St

119905= 119891119904119905TE119880infin is about 022 for

Re = 106 Ausoni [10] (also refer to Zobeiri et al [11])

had a similar finding and experimentally gave St119905= 025

which may be approximately read from their figure Theyused a hydrofoil with a 100mm chord length and 150mmspan length The aspect ratio defined as the ratio of the spanlength to the chord length was 15 A lower aspect ratioleads to a three-dimensional (3D) effect even though thehydrofoil is 2D Ausoni [10] also observed experimentallythat the shedding process was not induced in phase alongthe span It was therefore speculated in [25] that the slightunderestimation of the Strouhal number is due to 3D effectswhich were absent in our simulation

54 Oblique Trailing-Edges We conducted numerical simu-lations to observe vortex shedding from trailing edges withsix different bevel angles of 120573 = 0 10 20 40 60 and 70degrees Figure 5 shows the instantaneous vorticity distribu-tions at 119879 = 10 for Re = 106 At 120573 = 0∘ the vortex sheddingis regular and periodic As the bevel angle 120573 increasesvortex shedding becomes increasingly disorganized Suchirregular shedding of vortices demonstrates the periodicity ofoscillatory forces Figure 6 shows the power spectral densityfunction (PSD) of the drag and lift coefficients for differentbevel angles Compared with 120573 = 0∘ the amplitude of the liftforce oscillation for asymmetric trailing edges is significantlyreduced compared with the symmetric trailing edge at 120573 =0∘ The lift force amplitude is gradually decreased as the bevelangle increases while the drag force amplitude increases andthen decreases to between 120573 = 10∘ and 60∘The periodicity ofboth forces is almost lost at 120573 = 60∘ and there is no definiteperiodicity of the induced force components at 120573 = 70∘

At a bevel angle of 10∘ as shown in Figure 6(b) it is inter-esting that the dominant peak of the drag force is identical tothat of the lift force From a vibration standpoint the forcingfrequency is important for the prevention of resonance Thedistance between the two vortices in Figure 5(b) does not staythe same but the difference between the shedding vorticesbetween 120573 = 0

∘ and 10∘ (Figures 5(a) and 5(b)) is not

Modelling and Simulation in Engineering 7

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Symmetric wake at 119879 = 0396

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Symmetry-breaking transition at 119879 = 0900

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Nonperiodic vortex shedding at 119879 = 1098

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Fully developed periodic shedding at 119879 = 3500

Figure 3 Wake patterns of the NACA 0009 hydrofoil with a truncated trailing-edge (120573 = 0∘) for Re = 106

CL

CL

002

001

0

minus001

minus002I II III

T

0 05 1 15 2 25 3 35

CD

CD

0015

001

0005

0

Figure 4 Time evolution of the lift and drag coefficients for theimpulsively started NACA 0009 hydrofoil with a truncated trailingedge (120573 = 0∘) at Re = 106 Regime I denotes the developing regimeof a symmetric wake II a symmetry-breaking transition and IIIperiodic vortex shedding

statistically significant A possible reason for this is the phasedifference between the vortices shed from the upper andlower sides In the wake formation region the two vorticesshed from the symmetric trailing edge at 120573 = 0∘ are almostexactly out of phase (180 degrees) whereas at 120573 = 10

∘

there is a phase difference of approximately 30 degrees Thismeans that before one vortex is fully shed from the bodyanother vortex is created At higher bevel angles 120573 gt 20∘ thenegative vortices from the upper side of the hydrofoil becomestronger than the positive ones from the lower side Theoscillation force is induced mostly by the vortices shed fromthe acute (upper) side As a result the dominant frequency

of the drag force oscillation appears to be the same as thevortex shedding frequency We numerically observed thatdifferences in strength andor phase between two adjacentvortices cause nonperiodic and irregular vortex shedding

Figure 7 shows the shedding frequency of vortices fromdifferent beveled trailing edges compared with experimentalresults by Zobeiri et al [11] The numerically simulatedshedding frequencies were approximately 68 and 64 for120573 = 0

∘ and 60∘ respectively Numerical results showgood qualitative agreements with the experimental resultsbut the numerical simulations underestimated the vortex-shedding frequency by about 10 Figure 8 shows the meanvalue and standard deviation of the velocity components Wecompared our numerical results with the experimental resultsmeasured by Zobeiri et al [11] The truncated and obliquetrailing edges in Figure 8 indicate bevel angles of 120573 = 0

∘

and 60∘ respectively For the truncated trailing edge themean velocity profiles are more symmetric than those in theexperimental data The results from numerical simulationare in good agreement with the experimental results Weconclude that the vortex shedding phenomenon from thetrailing edge of a 2D hydrofoil can be reasonably simulatedusing 2D LES which considerably reduces the computationalcost

55 Effect of a Periodically Varying Flow on Vortex SheddingThe propeller is generally located behind the shiprsquos hull andis subjected to a nonaxisymmetric wake field Consideringa 2D flow the propeller section effectively experiences peri-odic variation in flow incidence We conducted numericalsimulations to investigate vortex shedding with respect tothe sinusoidal motions of the free-stream flow The hydrofoilwas at rest and the flow became the oscillator under theseconditions In the current study the angle of attack varied

8 Modelling and Simulation in Engineering

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Bevel angle 120573 = 0∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Bevel angle 120573 = 10∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Bevel angle 120573 = 20∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Bevel angle 120573 = 40∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(e) Bevel angle 120573 = 60∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(f) Bevel angle 120573 = 70∘

Figure 5 Instantaneous vorticity distributions at 119879 = 10 for Re = 106

sinusoidally with time 119905 120572 = 120572119900sin(2120587119891

infin119905) where 119891

infin

was the frequency of the free-stream flow oscillation andits oscillation amplitude was restricted at 120572

119900= 2∘ The

magnitude of the free-stream velocity was kept constant Thetested model is the NACA 0009 hydrofoil with a beveledtrailing edge of 120573 = 60∘

Figure 9 shows the instantaneous vorticity distributionsat119879 = 10 and the PSD of the drag coefficients at four differentfrequencies of the free-stream flow that is 119891

infin= 15 20

25 and 30 There are two peaks in the PSD functions thefirst peak appears around 119891lowast asymp 6 and the second one isidentical to the frequency of the free-stream flow oscillationThe obvious feature recognized in all the PSD functions asshown in Figure 9 is that the first peak frequency is nearlyidentical to the vortex-shedding frequency from the hydrofoilin the uniform free-stream flow Vortex formation in the nearwake is thought to be less affected by small inflow oscillationsMost interestingly at 119891

infin= 20 vortices are regularly shed

from the trailing edge as shown in Figure 9(c) In the PSDfunctions drag oscillation induced by vortex shedding isclearly observed This means that a particular oscillationfrequency in the free-stream velocity can cause regular andperiodic vortex shedding This is significant with respect tochanging a marine propellerrsquos angle of attack over time

6 Concluding Remarks

We simulated impulsively started flows past a modifiedNACA 0009 hydrofoil at a Reynolds number of 106 usingthe hybrid particle-mesh method in conjunction with thevorticity-based subgrid scale model We have achieved sig-nificant savings in both memory usage and computationtime by making improvements in the numerical schemesthe particle-based domain decomposition the use ofmultipledomains and the approximation of domain boundary con-ditions These techniques are simple and easy to implementbut also effective in reducing the computational costs Theparticle-based domain decomposition scheme is suitable fora distributed system in which each processor has its ownprivate memory (distributed memory)

In this study the numerical simulation results nearlymatch the experimental resultsThis demonstrates that vortexshedding from the trailing edge of a 2D hydrofoil can bereasonably simulated with a 2D LES It is due to the fact thatthe flow is characterized by quasi-two dimensionality withvortex shedding from the trailing edge of the hydrofoil Wehave considered that 2D vortex shedding simulations shouldnot be dismissed so easily With two-dimensionality it ispossible to solve the problem much less expensively since

Modelling and Simulation in Engineering 9

0 5 10 15 20 250

10

0 5 10 15 20 250

05times10minus5

PSD

ofC

L

PSD

ofC

D

times10minus6

flowast flowast

(a) Truncated trailing edge 120573 = 0∘

0

08

0

08

0

08

0

08

0

08

0

04

0

04

0

04

0

04

0

04

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

(b) Beveled trailing edge 120573 gt 0∘

Figure 6 Power spectral density (PSD) of the lift and drag coefficients 119891lowast denotes the frequency normalized by the chord length and theundisturbed flow velocity

3D problems require significant computational resourcesHowever the LES should be 3D since turbulence is inherently3D While we accept that more realistic 3D simulation isrequired physically a 2D LES can be used as a basis forcomparison prior to 3D implementation Once the capability

of a numerical algorithm is evaluated in two dimensionsit can be easily extended into three dimensions This paperis intended as a guide for researchers and designers whoare working on a similar problem vortex shedding from ahydrofoil at high Reynolds numbers

10 Modelling and Simulation in Engineering

0 10 20 30 40 50 600

2

4

6

8

10

Zobeiri et al (2012) expPresent cal

flowast

120573 (deg)

Figure 7 Shedding frequency of vortices from the different beveled trailing edges for Re = 106 The experimental data by Zobeiri et al [11]were derived from LDVmeasurements and the approximate shedding frequency as read from their figure is normalized by the chord lengthand the undisturbed flow velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02 04 06 08 1uxmeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(a) Normalized mean of streamwise velocity

minus06

minus04

minus02

0

02

04

06yt

max

0 01 02 03 04uxstdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(b) Normalized mean of transverse velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02uymeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(c) Normalized standard deviation of streamwise velocity

minus06

minus04

minus02

0

02

04

06

yt

max

0 01 02 03 04 05uystdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(d) Normalized standard deviation of transverse velocity

Figure 8 Mean and standard deviation of velocities at 119909119888 = 0516 for Re = 106 The experimental data by Zobeiri et al [11] were measuredat 119909119888 = 052 and 0528 for truncated (120573 = 0∘) and oblique (120573 = 60∘) trailing edges respectively and the reported Reynolds number of theflow is about 2 times 106

Modelling and Simulation in Engineering 11

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(a) Steady flow 119891infin = 0

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(b) Incoming flow frequency 119891infin = 15

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(c) Incoming flow frequency 119891infin = 20

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(d) Incoming flow frequency 119891infin = 25

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(e) Incoming flow frequency 119891infin = 30

Figure 9 Effect of the incoming flow frequency on vortex shedding from a beveled trailing edge of 120573 = 60∘ at 119891infin= 15 20 25 and 30

The contour indicates the instantaneous vorticity distribution at 119879 = 10 and the power spectral density (PSD) is calculated from the dragcoefficients in the range 119879 = [3 10]

12 Modelling and Simulation in Engineering

We found in this study that the vortex formation char-acteristic in periodically varying inflows remains unchangedA certain frequency of free-stream flow oscillation tends tomake vortex shedding more regular again This means thatthe periodicity of flow incidence can stimulate an inherentfrequency of vortex shedding from a hydrofoil Howeverthe effect of inflow oscillation on vortex shedding is notyet fully understood Future work will include experimentalinvestigations as well as a 3D flow simulation to betterunderstand the relationship between inflow oscillation andvortex shedding

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Korean Institute ofOcean Science amp Technology sponsored by the Ministry ofTrade Industry amp Energy (no 10033668) and the IndustrialConvergence Strategic Technology Development Program(no 10044499) funded by the Ministry of Trade Industry ampEnergy (MI Korea)

References

[1] E Berger and R Wille ldquoPeriodic flow phenomenardquo AnnualReview of Fluid Mechanics vol 4 no 1 pp 313ndash340 1972

[2] P W Bearman ldquoVortex shedding from oscillating bluff bodiesrdquoAnnual Review of Fluid Mechanics vol 16 pp 195ndash222 1984

[3] C H K Williamson ldquoVortex dynamics in the cylinder wakerdquoAnnual Review of Fluid Mechanics vol 28 pp 477ndash539 1996

[4] C H K Williamson ldquoVortex-induced vibrationsrdquo AnnualReview of Fluid Mechanics vol 36 pp 413ndash455 2004

[5] R Fischer ldquoSinging propellersmdashsolutions and case historiesrdquoMarine Technology vol 45 no 4 pp 221ndash227 2008

[6] D A Bourgoyne J M Hamel S L Ceccio and D R DowlingldquoTime-averaged flow over a hydrofoil at high reynolds numberrdquoJournal of Fluid Mechanics vol 496 pp 365ndash404 2003

[7] D A Bourgoyne S L Ceccio and D R Dowling ldquoVortexshedding from a hydrofoil at high Reynolds numberrdquo Journalof Fluid Mechanics vol 531 pp 293ndash324 2005

[8] N Mulvany J Tu L Chen and B Anderson ldquoAssessment oftwo-equation turbulence modelling for high Reynolds numberhydrofoil flowsrdquo International Journal for Numerical Methods inFluids vol 45 no 3 pp 275ndash299 2004

[9] P Ausoni M Farhat X Escaler E Egusquiza and F AvellanldquoCavitation influence on von Karman vortex shedding andinduced hydrofoil vibrationsrdquo Journal of Fluids EngineeringTransactions vol 129 no 8 pp 966ndash973 2007

[10] P Ausoni Turbulent vortex shedding from a blunt trailingedge hydrofoil [PhD thesis] Ecole Polytechnique Federale deLausanne Lausanne Switzerland 2009

[11] A Zobeiri P Ausoni F Avellan and M Farhat ldquoHow obliquetrailing edge of a hydrofoil reduces the vortex-induced vibra-tionrdquo Journal of Fluids and Structures vol 32 pp 78ndash89 2012

[12] G-H Cottet and P D Koumoutsakos Vortex Methods Theoryand Practice Cambridge University Press 2000

[13] G Cottet and P Poncet ldquoParticle methods for direct numericalsimulations of three-dimensional wakesrdquo Journal of Turbulencevol 3 article N38 2002

[14] P Chatelain A Curioni M Bergdorf D Rossinelli WAndreoni and P Koumoutsakos ldquoBillion vortex particle directnumerical simulations of aircraft wakesrdquo Computer Methods inAppliedMechanics and Engineering vol 197 no 13ndash16 pp 1296ndash1304 2008

[15] Y-C Kim J-C Suh and K-J Lee ldquoVortex-in-cell methodcombined with a boundary element method for incompressibleviscous flow analysisrdquo International Journal forNumericalMeth-ods in Fluids vol 69 no 10 pp 1567ndash1583 2012

[16] P Angot C-H Bruneau and P Fabrie ldquoA penalization methodto take into account obstacles in incompressible viscous flowsrdquoNumerische Mathematik vol 81 no 4 pp 497ndash520 1999

[17] M El Ossmani and P Poncet ldquoEfficiency of multiscale hybridgrid-particle vortex methodsrdquo Multiscale Modeling amp Simula-tion vol 8 no 5 pp 1671ndash1690 2010

[18] J T Rasmussen G-H Cottet and J HWalther ldquoAmultiresolu-tion remeshed vortex-in-cell algorithm using patchesrdquo Journalof Computational Physics vol 230 no 17 pp 6742ndash6755 2011

[19] M Coquerelle and G-H Cottet ldquoA vortex level set method forthe two-way coupling of an incompressible fluid with collidingrigid bodiesrdquo Journal of Computational Physics vol 227 no 21pp 9121ndash9137 2008

[20] F Morency H Beaugendre F Gallizio and S Laurens ldquoCom-putation of ice shedding trajectories using cartesian gridspenalization and level setsrdquo Modelling and Simulation in Engi-neering vol 2011 Article ID 274947 15 pages 2011

[21] M M Hejlesen P Koumoutsakos A Leonard and J HWalther ldquoIterative Brinkman penalization for remeshed vortexmethodsrdquo Journal of Computational Physics vol 280 pp 547ndash562 2015

[22] M Gazzola P Chatelain W M van Rees and P Koumout-sakos ldquoSimulations of single and multiple swimmers with non-divergence free deforming geometriesrdquo Journal of Computa-tional Physics vol 230 no 19 pp 7093ndash7114 2011

[23] R Chatelin and P Poncet ldquoA hybrid grid-particle method formoving bodies in 3D stokes flow with variable viscosityrdquo SIAMJournal on Scientific Computing vol 35 no 4 pp B925ndashB9492013

[24] R Chatelin and P Poncet ldquoHybrid grid-particle methodsand penalization a Sherman-Morrison-Woodbury approach tocompute 3D viscous flows using FFTrdquo Journal of ComputationalPhysics vol 269 pp 314ndash328 2014

[25] S J Lee J H Lee and J C Suh ldquoVortex shedding frequencyfor a 2D hydrofoil with a truncated trailing edgerdquo Journal of theSociety of Naval Architects of Korea vol 51 no 6 pp 480ndash4882014 (Korean)

[26] S-J Lee and J C Suh ldquoA multi-domain approach for ahybrid particle-mesh methodrdquo Journal of Computational FluidsEngineering vol 19 no 2 pp 72ndash78 2014

[27] S-J Lee J-H Lee and J-C Suh ldquoComputation of pressurefields around a two-dimensional circular cylinder using thevortex-in-cell and penalization methodsrdquo Modelling and Sim-ulation in Engineering vol 2014 Article ID 708372 13 pages2014

[28] J J Monaghan ldquoExtrapolating 119861-splines for interpolationrdquoJournal of Computational Physics vol 60 no 2 pp 253ndash2621985

[29] J J Monaghan ldquoParticle methods for hydrodynamicsrdquo Com-puter Physics Reports vol 3 no 2 pp 71ndash124 1985

Modelling and Simulation in Engineering 13

[30] O V Vasilyev and N K-R Kevlahan ldquoHybrid waveletcollocation-brinkman penalization method for complex geom-etry flowsrdquo International Journal for Numerical Methods inFluids vol 40 no 3-4 pp 531ndash538 2002

[31] W Iwakami Y Yatagai N Hatakeyama and Y Hattori ldquoAnew approach for error reduction in the volume penalizationmethodrdquoCommunications in Computational Physics vol 16 no5 pp 1181ndash1200 2014

[32] J R Mansfield O M Knio and C Meneveau ldquoTowardslagrangian large vortex simulationrdquo ESAIM Proceedings vol 1pp 49ndash64 1996

[33] G-H Cottet and P Poncet ldquoAdvances in direct numerical sim-ulations of 3D wall-bounded flows by vortex-in-cell methodsrdquoJournal of Computational Physics vol 193 no 1 pp 136ndash1582004

[34] B Caswell ldquoKinematics and stress on a surface of restrdquo Archivefor Rational Mechanics and Analysis vol 26 no 5 pp 385ndash3991967

[35] R S Rogallo and P Moin ldquoNumerical simulation of turbulentflowsrdquo Annual Review of Fluid Mechanics vol 16 pp 99ndash1371984

[36] J Meyers and P Sagaut ldquoOn the model coefficients for thestandard and the variational multi-scale Smagorinsky modelrdquoJournal of Fluid Mechanics vol 569 pp 287ndash319 2006

[37] J R Mansfield O M Knio and C Meneveau ldquoA dynamic LESscheme for the vorticity transport equation formulation and apriori testsrdquo Journal of Computational Physics vol 145 no 2 pp693ndash730 1998

[38] S B Pope Turbulent Flows Cambridge University Press 2000[39] WH Press S A TeukolskyW T Vetterling and B P Flannery

Numerical Recipes in C++ The Art of Scientific ComputingCambridge University Press 2nd edition 2002

[40] L Greengard and W D Gropp ldquoA parallel verison of the fastmultipoe methodrdquo Computers amp Mathematics with Applica-tions vol 20 no 7 pp 63ndash71 1990

[41] M Frigo and S G Johnson ldquoFFTW an adaptive software archi-tecture for the FFTrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo98) pp 1381ndash1384 May 1998

[42] M Frigo and S G Johnson ldquoThe design and implementationof FFTW3rdquo Proceedings of the IEEE vol 93 no 2 pp 216ndash2312005

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

Modelling and Simulation in Engineering 3

compute the stream functions using the FFT-based Poissonsolver the vorticity 120596

119911is interpolated onto equally spaced

Cartesian grid using

120596119911(x119892) =

119873119901

sum

119901=1

120596119911(x119901)119882(

x119901minus x119892

ℎ) (6)

where ℎ is the mesh spacing using the following third orderinterpolation kernel [28 29]

119882(119909) = 1198721015840

4(119909)

=

0 for |119909| gt 21

2(2 minus |119909|)

2(1 minus |119909|) for 1 lt |119909| le 2

1 minus5

2|119909|2+3

2|119909|3 for |119909| le 1

(7)

in each coordinate direction This kernel conserves the 0th1st and 2nd order moments The subscripts

119892and119901denote

the grid and particle quantities respectively A vortex particlecontributes to the nearest 16 nodes through the1198721015840

4scheme

and the total vorticity at each node is obtained by summingthe vorticity contributions of all the vortex particles

Boundary conditions for the stream function are requiredto solve the Poisson equation If the computational domainboundaries are far enough from the particles homogeneousDirichlet boundary conditions (120595 = 0) may be usedHowever using a larger domain is inefficient since it requiresa regular grid with too many points To compactly place asquare FFT-domain Ω we used nonhomogeneous Dirichletboundary conditions in this study Unknown stream func-tions at the domain boundaries (120597Ω) can be directly obtainedfrom Greenrsquos function solution [27] whereby

120595119911(x119887) = minus

1

2120587

119873119901

sum

119901=1

Γ119901ln (10038161003816100381610038161003816x119887 minus x119901

10038161003816100381610038161003816) 119887 = 1 119872 (8)

where x119887isin 120597Ω x

119901isin Ω and x

119892= x119901119872 denotes the number

of domain boundary points and the circulation strength ofeach vortex particle Γ

119901= 120596119911(x119901)1205982 We note here that all

particle vorticity values should be considered Fast evaluationof the stream functions at the domain boundaries is furtherdiscussed in Section 33

22 Brinkman PenalizationMethod for Diffusion Thepenal-ization method was initially designed to take into accountsolid obstacles in fluid flows [16] The main point of thepenalty term is to replace the usual vorticity creation algo-rithm for enforcing the no-slip condition for a solid bodyBy adding the penalization term to (1) the vorticity transportequation becomes

119863120596119911

119863119905= ]nabla2120596

119911+ nabla times [120582120594 (u

119904minus u)] (9)

where u119904is the velocity of the solid body and 120594 denotes a

mask function that yields 0 in a fluid and 1 in a solid [16 18]

120582 indicates the penalization parameter with the dimension[119904minus1] To avoid too small time step the penalization term is

evaluated by the implicit expression [16 18 30]Δ120596119911

Δ119905= nabla times [120582120594 (u119899

119904minus u119899+1)] (10)

where

u119899+1 =u119899 + 120582Δ119905120594u119899

119904

1 + 120582Δ119905120594 (11)

This scheme is unconditionally stable [19] Equation (9) isrewritten by replacing its terms with those in (10) and (11) asfollows

120596119899+1

119911minus 120596119899

119911

Δ119905= ]nabla2120596119899

119911+ nabla times [

120582120594

1 + 120582Δ119905120594(u119899119904minus u119899)] (12)

The penalization and diffusion terms of (12) are consecutivelyevaluated on a temporary grid To reduce any error due tothe penalization term we used a mask function with second-order accuracy for which the boundary is located at themidpoint of the grid where the mask function jumps from0 to 1 [18 31]

23 Smagorinsky Turbulence Model The filtered vorticitytransport equation for a Lagrangian vorticity-based large-eddy simulation (LES) in two-dimensions can be expressedas [32]

119863120596119911

119863119905= ]nabla2120596

119911+ nabla sdot Φ (13)

where the bar denotes spatial filtering at length scale 120598 Thesubgrid-scale vorticity stress term Φ which accounts for theeffect of unresolved velocity and vorticity fluctuations can bedefined as Φ = ]

119879nabla120596119911 Smagorinsky eddy-viscosity model is

one of most commonly used subgrid-scale (SGS) turbulencemodels In spite of its simplicity the Smagorinsky model isstill extensively used in practice and has served as a basic toolfor the developments of LES modeling [33] In this modeleddy viscosity is defined by

]119879= (119862119878Δ)2 1003816100381610038161003816100381611987810038161003816100381610038161003816 (14)

where 119862119878is a nondimensional Smagorinsky constant and Δ

is a subgrid characteristic length scale 119878119894119895= (120597119906

119894120597119909119895+

120597119906119895120597119909119894)2 is the resolved strain-rate tensor and |119878| =

(2119878119894119895119878119895119894)12 is its magnitude Here |119878|2 = |120596

119911|2 in incom-

pressible flows (nabla sdot u = 0) using the Caswell formula [34]Finally the vorticity transport equation for the LES in twodimensions is rewritten as

119863120596119911

119863119905= (] + (119862

119878Δ)2 1003816100381610038161003816120596119911

1003816100381610038161003816) nabla2120596119911 (15)

In 2D LES models the filtering process is applied in only twodirections so Δ = (ℎ

119909ℎ119910)12 is defined by the grid spacing

The Smagorinsky constant 119862119878may vary with the type of

flow and the Reynolds number Values ranging from 01 to024 have been suggested in the literatures [33 35 36] Theconstant 119862

119878is prescribed and its standard value is typically

015 [37 38]

4 Modelling and Simulation in Engineering

3 Efficient Implementation ofNumerical Methods

31 Particle-BasedDomainDecomposition Parallelmachinesand algorithms enable significant reduction in computingtime and greater amount of available memory For high-performance computing we used the Message Passing Inter-face (MPI) for our distributed systems in which each proces-sor has its own localmemoryThe appropriate decompositionof particles andor grids should be employed In domainor spatial decomposition a given domain Ω is geometri-cally decomposed by splitting it into several subdomainsTypically the subdomains are uniform and particles are notevenly distributed among the subdomains Differences inthe number of particles assigned to each processor causecomputational load imbalances As some processors willsit idly while waiting for others poor performance canresult

We introduce a simple idea to achieve a better loadbalance during the domain decomposition In vortex particlemethods fluid particles must periodically be redistributedsince their accumulation leads to inaccuracies in numericalsolutions This process is called particle remeshing or redis-tribution The remeshing step creates a new set of particleson a uniform Cartesian grid and then the randomly spacedold particles are removed All the new particles are reindexedby 119894 = 1 2 119873

119901from upstream to downstream that

is higher-indexed particles are located further downstreamDue to the position-dependent index of the particles theentire spatial domain can be easily split to ensure an equalamount of particles in each subdomain (each processor)As a result the horizontal extent of the domain is unequalbut the particles are evenly distributed in subdomains ofdifferent sizes In every remeshing step the entire domainis dynamically partitioned In this way each subdomaincontains an equal number of particles and better load balanceismaintained during simulationThis approach is very simpleand effective No additional algorithms are required to evenlydistribute the particles or to identify the position of theparticles As shown in Figure 1 we obtained an approxi-mately 30 reduction in computation time simply by loadbalancing during parallel computation This technique isalso valuable for the efficient use of available computationalmemory

32 Multidomain Approach To avoid an excessive domainsize we consideredmultiple domains In this study the entiredomainΩ is defined as the union of all the domains coveringall the fluid particles that is x

119901isin Ω where Ω = Ω

1cup

Ω2cup sdot sdot sdot cup Ω

119873119889 The domain number119873

119889is not constant and

is dependent on a spatially evolving flow When the domainsize exceeds a certain limit a new domain is created situateddownstream from the domain As a result the first domainΩ1includes a solid body and the others are sequentially

located downstream from the body For the total circulationconservation the grid spacing of the 119894th domain can bedefined as ℎ

119894= 120598 (single-resolution) or ℎ

119894= 2ℎ119894minus1

(multi-resolution) where ℎ

1= 120598 Here 120598 denotes the particle size To

T

Com

puta

tion

time (

h)

250

200

150

100

50

00 2 4 6 8 10

Equal-sized subdomainsUnequal-sized subdomain

Figure 1 Computation times for equal- and unequal-sized subdo-mains

interpolate the vorticity values of the particles into the nodeson a grid with different grid spacing (6) is rewritten as

120596119911(x119892) =

1205982

ℎ2

119894

119873

sum

119901=1

120596119911(x119901)119882(

x119901minus x119892

ℎ119894

) (16)

where x119901isin Ω119894and x119892isin Ω119894The nodes are declared to be a 2D

array so that the position of each node can be found straight-forwardly from its index The velocity and vorticity of eachdomain are sequentially and independently evaluatedHencerelatively small amounts of working memory are requiredThis multidomain approach also allows more freedom inthe choice of variable arrangements in the domains Theversatility of this approach was demonstrated in our previousstudy [26] It can be achieved at the expense of a significantincrease in computation time This issue is further discussedin the next section

33 Approximation of Far-Field Conditions A substantialpart of the overall computation time is spent on the calcu-lation of the boundary stream functions to solve Poissonrsquosequation for the stream function This requires the order ofO(119872119873

119901119875) operations to obtain boundary values of domain

Ω119894 Here 119872 denotes the number of nodes at the sides of

domainΩ119894and119873

119901119875 is the number of particles per processor

To achieve a nearly perfect load balance particles are evenlydistributed among the processors as discussed in Section 31and each processor has all the boundary nodes 119872 It mustalso be very useful to avoid all unnecessary communications119873119901119875 is reduced by increasing the number of processors

involved in the computation whereas 119872 is not This is whythe number of boundary nodes 119872 is the most critical fordetermining the computation time

In this study we attempted to reduce the number ofboundary nodes 119872 using the cubic spline approximationThis is made possible by the smooth variation of the streamfunctions along the boundary Only one-fourth of the bound-ary nodes were chosen as equidistant points for the directcalculations That is the stream functions at the selected

Modelling and Simulation in Engineering 5

boundary nodes were directly computed using (8) and thevalues at boundary nodes not selected were evaluated byinterpolation using cubic splines [39] In this study theerror between the interpolated value and the true valueis typically less than 001 We have observed that thereduction in computational time is inversely proportionalto the number of nodes selected for direct computationWe believe that this approach can be implemented withmuch lower computational effort because a complex datastructure such as a quad-tree used in the fast multipolemethod [40] is not required A rigorous comparison of theaccuracy and computational costs is planned in the nearfuture

4 Computational Procedure

First the entire simulation domain Ω covering all theparticles is divided into small domains Ω

119894(119894 = 1 119873

119889)

as discussed in Section 32 Each domain Ω119894is consecutively

computed according to the following numerical procedure

(1) Particle-based domain decomposition as discussedin Section 31 domain Ω

119894is decomposed into sub-

domains to ensure that the particles are evenly dis-tributed among the processors If redistribution hasnot been done prior to the time step particles thathave been removed from a subdomain are assignedto an adjacent processor without performing domaindecomposition

(2) Particle-to-grid interpolation each processor inter-polates the vorticity 120596

119911of its own particles into the

grid nodes through the11987210158404interpolation kernel

(3) Calculation of the boundary condition each proces-sor computes the boundary stream functions by directcalculation and interpolation using cubic splines asdiscussed in Section 33

(4) Evaluation of the velocity field the stream functionon the grid is computed with the FFT-based Poissonsolver from an open-source library called FFTW(fastest Fourier transform in the west) [41 42]There-after the rotational velocities on the grid nodes arecomputed from the definitions 119906

119909= 120597120595120597119910 and

119906119910= minus120597120595120597119909 using the fourth order central finite

difference scheme(5) Evaluation of the vorticity field each processor eval-

uates the evolution of the vorticity field in time asfollows

(a) The penalization term that is the second termon the right hand side of (12) is computedwith the velocities on the grid The spatialderivatives of the curl operators are evaluatedusing a centered difference approximation witha fourth-order error

(b) The diffusion term that is the first term onthe right hand side of (12) is evaluated on thegrid using a classical 9-point finite differencemethod

(c) The turbulent viscosity (119862119878Δ)2|120596119911| in (15) is

approximated by the vorticity strength of thegrid Finally the vorticity field 120596119899+1

119911is updated

(6) Grid-to-particle interpolation each processor inter-polates the velocity and vorticity on the grid back to itsown particle positions through the1198721015840

4interpolation

kernel

Vortex particles are advanced in time using a midpointpredictor-corrector method as follows particle positions arepredicted by their velocities xlowast = x119899 + Δ119905u(x119899 120596119899

119911) then the

particle positions are corrected by using combinations of thepredicted and previous positions and velocity values x119899+1 =x119899 + 05Δ119905[u(x119899 120596119899z ) + u(xlowast 120596119899+1

119911)] Particle redistribution is

conducted every three time steps to ensure particle overlapIf a new particle has a small vorticity magnitude that is|Γ119894| lt 00001|Γ|max it is deleted to avoid having too many

particles Loss of circulation is equally distributed among theremaining particles to ensure the conservation of the totalvorticity

5 Numerical Results and Discussions

51 Model Description We selected a NACA 0009 hydrofoilwith a truncated trailing edge for the numerical simulationsThis hydrofoil has the same cross-section as the experimentalmodel used by Ausoni et al [9] Ausoni [10] and Zobeiriet al [11] Ten percent of the original chord 119888

119900was removed

from the trailing-edge region of the NACA 0009 hydrofoilThe hydrofoil thickness distribution is written as

00 le119909

119888119900

le 05

119910

119888119900

= 01737 (119909

119888119900

)

12

minus 02422 (119909

119888119900

) + 03046 (119909

119888119900

)

2

minus 02657 (119909

119888119900

)

3

05 lt119909

119888119900

le 10

119910

119888119900

= 00004 + 01737 (1 minus119909

119888119900

) minus 01898 (1 minus119909

119888119900

)

2

+ 00387 (1 minus119909

119888119900

)

3

(17)

The hydrofoil geometry is further detailed in Ausoni [10]Themaximum thickness as normalized by the chord length 119888 is119905max119888 = 01 and the thickness at the trailing edge 119905TE119888 is00322 The bevel angle 120573 is defined in Figure 2 The trailingedge with 120573 = 0∘ denotes that the trailing edge is truncated inthe vertical direction

52 Numerical Setup Numerical parameters can be deter-mined using the stability condition Δ119905 = 120598

24] [18] The

grid convergence was achieved for 120598 = 000030 and

6 Modelling and Simulation in Engineering

xc

yc

045 05

minus002

0

002

A

BC

00054c

00322c120573

Figure 2 Trailing-edge shape Note that the points A B and C indicate the knuckles of the chamfer and trailing-edge wedge and the bevelangle is defined as 120573

Δ119905 = 000015 was chosen in order to guarantee numericalaccuracy for a Reynolds number of 106 The grid spacingof the 119894th domain Ω

119894is defined as ℎ

119894= 1205982119894minus1 to reduce

computation timeThe penalization parameter 120582 in (12) is setto be 108 The hydrofoil at a zero angle of attack is immersedin a Cartesian grid (the first domain) that does not conformto the surface of the bodyThe first domain with 4096times 1024lattice nodes is fixed and the other domains are dynamicallydetermined depending on the spatially evolving flow In thisstudy numerical simulations were conducted until 119879 = 10A new domain is created downstream of a body once thenumber of horizontal grid points exceeds 2048 Althoughthere is no limit on the number of vertical grid points it wastypically less than or equal to 1024 During simulation lessthan 08 million particles and four computational domainswere created The entire computation in each case includingstart-up and printing times took approximately 170 hourson 16 Intel Xeon64 33 GHz CPUs with 6GB memory perprocessor

53 Truncated Trailing-Edge For an impulsively started flowpast a hydrofoil at a chord-based Reynolds number of 106the wake pattern can be classified into three regimes basedon the time evolutions shown in Figure 3 In the very earlystages the wake remained very symmetrical and consistedprimarily of two large vortices in the near wake as shown inFigure 3(a) At that time the drag force gradually decreasedAs time went on the wake became asymmetrical as shown inFigure 3(b) and then shed a vortex as shown in Figure 3(c)After a number of shedding periods vortices were shedregularly from the symmetric trailing edge in the form of twotrains of oppositezsign but equal-strength vortices as shownin Figure 3(d) Such regular vortex shedding induces periodicloading on the structure as shown in Figure 4 It is wellknown that drag-force oscillation during vortex shedding ismuch smaller than the lift force Numerical results show thatoscillations in the drag force occurred at twice the vortex-shedding frequency due to the fact that two vortices wereshed from alternate sides during one full period of wakeoscillation

From our simulation the Strouhal number based on thetrailing-edge thickness St

119905= 119891119904119905TE119880infin is about 022 for

Re = 106 Ausoni [10] (also refer to Zobeiri et al [11])

had a similar finding and experimentally gave St119905= 025

which may be approximately read from their figure Theyused a hydrofoil with a 100mm chord length and 150mmspan length The aspect ratio defined as the ratio of the spanlength to the chord length was 15 A lower aspect ratioleads to a three-dimensional (3D) effect even though thehydrofoil is 2D Ausoni [10] also observed experimentallythat the shedding process was not induced in phase alongthe span It was therefore speculated in [25] that the slightunderestimation of the Strouhal number is due to 3D effectswhich were absent in our simulation

54 Oblique Trailing-Edges We conducted numerical simu-lations to observe vortex shedding from trailing edges withsix different bevel angles of 120573 = 0 10 20 40 60 and 70degrees Figure 5 shows the instantaneous vorticity distribu-tions at 119879 = 10 for Re = 106 At 120573 = 0∘ the vortex sheddingis regular and periodic As the bevel angle 120573 increasesvortex shedding becomes increasingly disorganized Suchirregular shedding of vortices demonstrates the periodicity ofoscillatory forces Figure 6 shows the power spectral densityfunction (PSD) of the drag and lift coefficients for differentbevel angles Compared with 120573 = 0∘ the amplitude of the liftforce oscillation for asymmetric trailing edges is significantlyreduced compared with the symmetric trailing edge at 120573 =0∘ The lift force amplitude is gradually decreased as the bevelangle increases while the drag force amplitude increases andthen decreases to between 120573 = 10∘ and 60∘The periodicity ofboth forces is almost lost at 120573 = 60∘ and there is no definiteperiodicity of the induced force components at 120573 = 70∘

At a bevel angle of 10∘ as shown in Figure 6(b) it is inter-esting that the dominant peak of the drag force is identical tothat of the lift force From a vibration standpoint the forcingfrequency is important for the prevention of resonance Thedistance between the two vortices in Figure 5(b) does not staythe same but the difference between the shedding vorticesbetween 120573 = 0

∘ and 10∘ (Figures 5(a) and 5(b)) is not

Modelling and Simulation in Engineering 7

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Symmetric wake at 119879 = 0396

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Symmetry-breaking transition at 119879 = 0900

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Nonperiodic vortex shedding at 119879 = 1098

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Fully developed periodic shedding at 119879 = 3500

Figure 3 Wake patterns of the NACA 0009 hydrofoil with a truncated trailing-edge (120573 = 0∘) for Re = 106

CL

CL

002

001

0

minus001

minus002I II III

T

0 05 1 15 2 25 3 35

CD

CD

0015

001

0005

0

Figure 4 Time evolution of the lift and drag coefficients for theimpulsively started NACA 0009 hydrofoil with a truncated trailingedge (120573 = 0∘) at Re = 106 Regime I denotes the developing regimeof a symmetric wake II a symmetry-breaking transition and IIIperiodic vortex shedding

statistically significant A possible reason for this is the phasedifference between the vortices shed from the upper andlower sides In the wake formation region the two vorticesshed from the symmetric trailing edge at 120573 = 0∘ are almostexactly out of phase (180 degrees) whereas at 120573 = 10

∘

there is a phase difference of approximately 30 degrees Thismeans that before one vortex is fully shed from the bodyanother vortex is created At higher bevel angles 120573 gt 20∘ thenegative vortices from the upper side of the hydrofoil becomestronger than the positive ones from the lower side Theoscillation force is induced mostly by the vortices shed fromthe acute (upper) side As a result the dominant frequency

of the drag force oscillation appears to be the same as thevortex shedding frequency We numerically observed thatdifferences in strength andor phase between two adjacentvortices cause nonperiodic and irregular vortex shedding

Figure 7 shows the shedding frequency of vortices fromdifferent beveled trailing edges compared with experimentalresults by Zobeiri et al [11] The numerically simulatedshedding frequencies were approximately 68 and 64 for120573 = 0

∘ and 60∘ respectively Numerical results showgood qualitative agreements with the experimental resultsbut the numerical simulations underestimated the vortex-shedding frequency by about 10 Figure 8 shows the meanvalue and standard deviation of the velocity components Wecompared our numerical results with the experimental resultsmeasured by Zobeiri et al [11] The truncated and obliquetrailing edges in Figure 8 indicate bevel angles of 120573 = 0

∘

and 60∘ respectively For the truncated trailing edge themean velocity profiles are more symmetric than those in theexperimental data The results from numerical simulationare in good agreement with the experimental results Weconclude that the vortex shedding phenomenon from thetrailing edge of a 2D hydrofoil can be reasonably simulatedusing 2D LES which considerably reduces the computationalcost

55 Effect of a Periodically Varying Flow on Vortex SheddingThe propeller is generally located behind the shiprsquos hull andis subjected to a nonaxisymmetric wake field Consideringa 2D flow the propeller section effectively experiences peri-odic variation in flow incidence We conducted numericalsimulations to investigate vortex shedding with respect tothe sinusoidal motions of the free-stream flow The hydrofoilwas at rest and the flow became the oscillator under theseconditions In the current study the angle of attack varied

8 Modelling and Simulation in Engineering

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Bevel angle 120573 = 0∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Bevel angle 120573 = 10∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Bevel angle 120573 = 20∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Bevel angle 120573 = 40∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(e) Bevel angle 120573 = 60∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(f) Bevel angle 120573 = 70∘

Figure 5 Instantaneous vorticity distributions at 119879 = 10 for Re = 106

sinusoidally with time 119905 120572 = 120572119900sin(2120587119891

infin119905) where 119891

infin

was the frequency of the free-stream flow oscillation andits oscillation amplitude was restricted at 120572

119900= 2∘ The

magnitude of the free-stream velocity was kept constant Thetested model is the NACA 0009 hydrofoil with a beveledtrailing edge of 120573 = 60∘

Figure 9 shows the instantaneous vorticity distributionsat119879 = 10 and the PSD of the drag coefficients at four differentfrequencies of the free-stream flow that is 119891

infin= 15 20

25 and 30 There are two peaks in the PSD functions thefirst peak appears around 119891lowast asymp 6 and the second one isidentical to the frequency of the free-stream flow oscillationThe obvious feature recognized in all the PSD functions asshown in Figure 9 is that the first peak frequency is nearlyidentical to the vortex-shedding frequency from the hydrofoilin the uniform free-stream flow Vortex formation in the nearwake is thought to be less affected by small inflow oscillationsMost interestingly at 119891

infin= 20 vortices are regularly shed

from the trailing edge as shown in Figure 9(c) In the PSDfunctions drag oscillation induced by vortex shedding isclearly observed This means that a particular oscillationfrequency in the free-stream velocity can cause regular andperiodic vortex shedding This is significant with respect tochanging a marine propellerrsquos angle of attack over time

6 Concluding Remarks

We simulated impulsively started flows past a modifiedNACA 0009 hydrofoil at a Reynolds number of 106 usingthe hybrid particle-mesh method in conjunction with thevorticity-based subgrid scale model We have achieved sig-nificant savings in both memory usage and computationtime by making improvements in the numerical schemesthe particle-based domain decomposition the use ofmultipledomains and the approximation of domain boundary con-ditions These techniques are simple and easy to implementbut also effective in reducing the computational costs Theparticle-based domain decomposition scheme is suitable fora distributed system in which each processor has its ownprivate memory (distributed memory)

In this study the numerical simulation results nearlymatch the experimental resultsThis demonstrates that vortexshedding from the trailing edge of a 2D hydrofoil can bereasonably simulated with a 2D LES It is due to the fact thatthe flow is characterized by quasi-two dimensionality withvortex shedding from the trailing edge of the hydrofoil Wehave considered that 2D vortex shedding simulations shouldnot be dismissed so easily With two-dimensionality it ispossible to solve the problem much less expensively since

Modelling and Simulation in Engineering 9

0 5 10 15 20 250

10

0 5 10 15 20 250

05times10minus5

PSD

ofC

L

PSD

ofC

D

times10minus6

flowast flowast

(a) Truncated trailing edge 120573 = 0∘

0

08

0

08

0

08

0

08

0

08

0

04

0

04

0

04

0

04

0

04

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

(b) Beveled trailing edge 120573 gt 0∘

Figure 6 Power spectral density (PSD) of the lift and drag coefficients 119891lowast denotes the frequency normalized by the chord length and theundisturbed flow velocity

3D problems require significant computational resourcesHowever the LES should be 3D since turbulence is inherently3D While we accept that more realistic 3D simulation isrequired physically a 2D LES can be used as a basis forcomparison prior to 3D implementation Once the capability

of a numerical algorithm is evaluated in two dimensionsit can be easily extended into three dimensions This paperis intended as a guide for researchers and designers whoare working on a similar problem vortex shedding from ahydrofoil at high Reynolds numbers

10 Modelling and Simulation in Engineering

0 10 20 30 40 50 600

2

4

6

8

10

Zobeiri et al (2012) expPresent cal

flowast

120573 (deg)

Figure 7 Shedding frequency of vortices from the different beveled trailing edges for Re = 106 The experimental data by Zobeiri et al [11]were derived from LDVmeasurements and the approximate shedding frequency as read from their figure is normalized by the chord lengthand the undisturbed flow velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02 04 06 08 1uxmeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(a) Normalized mean of streamwise velocity

minus06

minus04

minus02

0

02

04

06yt

max

0 01 02 03 04uxstdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(b) Normalized mean of transverse velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02uymeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(c) Normalized standard deviation of streamwise velocity

minus06

minus04

minus02

0

02

04

06

yt

max

0 01 02 03 04 05uystdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(d) Normalized standard deviation of transverse velocity

Figure 8 Mean and standard deviation of velocities at 119909119888 = 0516 for Re = 106 The experimental data by Zobeiri et al [11] were measuredat 119909119888 = 052 and 0528 for truncated (120573 = 0∘) and oblique (120573 = 60∘) trailing edges respectively and the reported Reynolds number of theflow is about 2 times 106

Modelling and Simulation in Engineering 11

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(a) Steady flow 119891infin = 0

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(b) Incoming flow frequency 119891infin = 15

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(c) Incoming flow frequency 119891infin = 20

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(d) Incoming flow frequency 119891infin = 25

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(e) Incoming flow frequency 119891infin = 30

Figure 9 Effect of the incoming flow frequency on vortex shedding from a beveled trailing edge of 120573 = 60∘ at 119891infin= 15 20 25 and 30

The contour indicates the instantaneous vorticity distribution at 119879 = 10 and the power spectral density (PSD) is calculated from the dragcoefficients in the range 119879 = [3 10]

12 Modelling and Simulation in Engineering

We found in this study that the vortex formation char-acteristic in periodically varying inflows remains unchangedA certain frequency of free-stream flow oscillation tends tomake vortex shedding more regular again This means thatthe periodicity of flow incidence can stimulate an inherentfrequency of vortex shedding from a hydrofoil Howeverthe effect of inflow oscillation on vortex shedding is notyet fully understood Future work will include experimentalinvestigations as well as a 3D flow simulation to betterunderstand the relationship between inflow oscillation andvortex shedding

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Korean Institute ofOcean Science amp Technology sponsored by the Ministry ofTrade Industry amp Energy (no 10033668) and the IndustrialConvergence Strategic Technology Development Program(no 10044499) funded by the Ministry of Trade Industry ampEnergy (MI Korea)

References

[1] E Berger and R Wille ldquoPeriodic flow phenomenardquo AnnualReview of Fluid Mechanics vol 4 no 1 pp 313ndash340 1972

[2] P W Bearman ldquoVortex shedding from oscillating bluff bodiesrdquoAnnual Review of Fluid Mechanics vol 16 pp 195ndash222 1984

[3] C H K Williamson ldquoVortex dynamics in the cylinder wakerdquoAnnual Review of Fluid Mechanics vol 28 pp 477ndash539 1996

[4] C H K Williamson ldquoVortex-induced vibrationsrdquo AnnualReview of Fluid Mechanics vol 36 pp 413ndash455 2004

[5] R Fischer ldquoSinging propellersmdashsolutions and case historiesrdquoMarine Technology vol 45 no 4 pp 221ndash227 2008

[6] D A Bourgoyne J M Hamel S L Ceccio and D R DowlingldquoTime-averaged flow over a hydrofoil at high reynolds numberrdquoJournal of Fluid Mechanics vol 496 pp 365ndash404 2003

[7] D A Bourgoyne S L Ceccio and D R Dowling ldquoVortexshedding from a hydrofoil at high Reynolds numberrdquo Journalof Fluid Mechanics vol 531 pp 293ndash324 2005

[8] N Mulvany J Tu L Chen and B Anderson ldquoAssessment oftwo-equation turbulence modelling for high Reynolds numberhydrofoil flowsrdquo International Journal for Numerical Methods inFluids vol 45 no 3 pp 275ndash299 2004

[9] P Ausoni M Farhat X Escaler E Egusquiza and F AvellanldquoCavitation influence on von Karman vortex shedding andinduced hydrofoil vibrationsrdquo Journal of Fluids EngineeringTransactions vol 129 no 8 pp 966ndash973 2007

[10] P Ausoni Turbulent vortex shedding from a blunt trailingedge hydrofoil [PhD thesis] Ecole Polytechnique Federale deLausanne Lausanne Switzerland 2009

[11] A Zobeiri P Ausoni F Avellan and M Farhat ldquoHow obliquetrailing edge of a hydrofoil reduces the vortex-induced vibra-tionrdquo Journal of Fluids and Structures vol 32 pp 78ndash89 2012

[12] G-H Cottet and P D Koumoutsakos Vortex Methods Theoryand Practice Cambridge University Press 2000

[13] G Cottet and P Poncet ldquoParticle methods for direct numericalsimulations of three-dimensional wakesrdquo Journal of Turbulencevol 3 article N38 2002

[14] P Chatelain A Curioni M Bergdorf D Rossinelli WAndreoni and P Koumoutsakos ldquoBillion vortex particle directnumerical simulations of aircraft wakesrdquo Computer Methods inAppliedMechanics and Engineering vol 197 no 13ndash16 pp 1296ndash1304 2008

[15] Y-C Kim J-C Suh and K-J Lee ldquoVortex-in-cell methodcombined with a boundary element method for incompressibleviscous flow analysisrdquo International Journal forNumericalMeth-ods in Fluids vol 69 no 10 pp 1567ndash1583 2012

[16] P Angot C-H Bruneau and P Fabrie ldquoA penalization methodto take into account obstacles in incompressible viscous flowsrdquoNumerische Mathematik vol 81 no 4 pp 497ndash520 1999

[17] M El Ossmani and P Poncet ldquoEfficiency of multiscale hybridgrid-particle vortex methodsrdquo Multiscale Modeling amp Simula-tion vol 8 no 5 pp 1671ndash1690 2010

[18] J T Rasmussen G-H Cottet and J HWalther ldquoAmultiresolu-tion remeshed vortex-in-cell algorithm using patchesrdquo Journalof Computational Physics vol 230 no 17 pp 6742ndash6755 2011

[19] M Coquerelle and G-H Cottet ldquoA vortex level set method forthe two-way coupling of an incompressible fluid with collidingrigid bodiesrdquo Journal of Computational Physics vol 227 no 21pp 9121ndash9137 2008

[20] F Morency H Beaugendre F Gallizio and S Laurens ldquoCom-putation of ice shedding trajectories using cartesian gridspenalization and level setsrdquo Modelling and Simulation in Engi-neering vol 2011 Article ID 274947 15 pages 2011

[21] M M Hejlesen P Koumoutsakos A Leonard and J HWalther ldquoIterative Brinkman penalization for remeshed vortexmethodsrdquo Journal of Computational Physics vol 280 pp 547ndash562 2015

[22] M Gazzola P Chatelain W M van Rees and P Koumout-sakos ldquoSimulations of single and multiple swimmers with non-divergence free deforming geometriesrdquo Journal of Computa-tional Physics vol 230 no 19 pp 7093ndash7114 2011

[23] R Chatelin and P Poncet ldquoA hybrid grid-particle method formoving bodies in 3D stokes flow with variable viscosityrdquo SIAMJournal on Scientific Computing vol 35 no 4 pp B925ndashB9492013

[24] R Chatelin and P Poncet ldquoHybrid grid-particle methodsand penalization a Sherman-Morrison-Woodbury approach tocompute 3D viscous flows using FFTrdquo Journal of ComputationalPhysics vol 269 pp 314ndash328 2014

[25] S J Lee J H Lee and J C Suh ldquoVortex shedding frequencyfor a 2D hydrofoil with a truncated trailing edgerdquo Journal of theSociety of Naval Architects of Korea vol 51 no 6 pp 480ndash4882014 (Korean)

[26] S-J Lee and J C Suh ldquoA multi-domain approach for ahybrid particle-mesh methodrdquo Journal of Computational FluidsEngineering vol 19 no 2 pp 72ndash78 2014

[27] S-J Lee J-H Lee and J-C Suh ldquoComputation of pressurefields around a two-dimensional circular cylinder using thevortex-in-cell and penalization methodsrdquo Modelling and Sim-ulation in Engineering vol 2014 Article ID 708372 13 pages2014

[28] J J Monaghan ldquoExtrapolating 119861-splines for interpolationrdquoJournal of Computational Physics vol 60 no 2 pp 253ndash2621985

[29] J J Monaghan ldquoParticle methods for hydrodynamicsrdquo Com-puter Physics Reports vol 3 no 2 pp 71ndash124 1985

Modelling and Simulation in Engineering 13

[30] O V Vasilyev and N K-R Kevlahan ldquoHybrid waveletcollocation-brinkman penalization method for complex geom-etry flowsrdquo International Journal for Numerical Methods inFluids vol 40 no 3-4 pp 531ndash538 2002

[31] W Iwakami Y Yatagai N Hatakeyama and Y Hattori ldquoAnew approach for error reduction in the volume penalizationmethodrdquoCommunications in Computational Physics vol 16 no5 pp 1181ndash1200 2014

[32] J R Mansfield O M Knio and C Meneveau ldquoTowardslagrangian large vortex simulationrdquo ESAIM Proceedings vol 1pp 49ndash64 1996

[33] G-H Cottet and P Poncet ldquoAdvances in direct numerical sim-ulations of 3D wall-bounded flows by vortex-in-cell methodsrdquoJournal of Computational Physics vol 193 no 1 pp 136ndash1582004

[34] B Caswell ldquoKinematics and stress on a surface of restrdquo Archivefor Rational Mechanics and Analysis vol 26 no 5 pp 385ndash3991967

[35] R S Rogallo and P Moin ldquoNumerical simulation of turbulentflowsrdquo Annual Review of Fluid Mechanics vol 16 pp 99ndash1371984

[36] J Meyers and P Sagaut ldquoOn the model coefficients for thestandard and the variational multi-scale Smagorinsky modelrdquoJournal of Fluid Mechanics vol 569 pp 287ndash319 2006

[37] J R Mansfield O M Knio and C Meneveau ldquoA dynamic LESscheme for the vorticity transport equation formulation and apriori testsrdquo Journal of Computational Physics vol 145 no 2 pp693ndash730 1998

[38] S B Pope Turbulent Flows Cambridge University Press 2000[39] WH Press S A TeukolskyW T Vetterling and B P Flannery

Numerical Recipes in C++ The Art of Scientific ComputingCambridge University Press 2nd edition 2002

[40] L Greengard and W D Gropp ldquoA parallel verison of the fastmultipoe methodrdquo Computers amp Mathematics with Applica-tions vol 20 no 7 pp 63ndash71 1990

[41] M Frigo and S G Johnson ldquoFFTW an adaptive software archi-tecture for the FFTrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo98) pp 1381ndash1384 May 1998

[42] M Frigo and S G Johnson ldquoThe design and implementationof FFTW3rdquo Proceedings of the IEEE vol 93 no 2 pp 216ndash2312005

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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4 Modelling and Simulation in Engineering

3 Efficient Implementation ofNumerical Methods

31 Particle-BasedDomainDecomposition Parallelmachinesand algorithms enable significant reduction in computingtime and greater amount of available memory For high-performance computing we used the Message Passing Inter-face (MPI) for our distributed systems in which each proces-sor has its own localmemoryThe appropriate decompositionof particles andor grids should be employed In domainor spatial decomposition a given domain Ω is geometri-cally decomposed by splitting it into several subdomainsTypically the subdomains are uniform and particles are notevenly distributed among the subdomains Differences inthe number of particles assigned to each processor causecomputational load imbalances As some processors willsit idly while waiting for others poor performance canresult

We introduce a simple idea to achieve a better loadbalance during the domain decomposition In vortex particlemethods fluid particles must periodically be redistributedsince their accumulation leads to inaccuracies in numericalsolutions This process is called particle remeshing or redis-tribution The remeshing step creates a new set of particleson a uniform Cartesian grid and then the randomly spacedold particles are removed All the new particles are reindexedby 119894 = 1 2 119873

119901from upstream to downstream that

is higher-indexed particles are located further downstreamDue to the position-dependent index of the particles theentire spatial domain can be easily split to ensure an equalamount of particles in each subdomain (each processor)As a result the horizontal extent of the domain is unequalbut the particles are evenly distributed in subdomains ofdifferent sizes In every remeshing step the entire domainis dynamically partitioned In this way each subdomaincontains an equal number of particles and better load balanceismaintained during simulationThis approach is very simpleand effective No additional algorithms are required to evenlydistribute the particles or to identify the position of theparticles As shown in Figure 1 we obtained an approxi-mately 30 reduction in computation time simply by loadbalancing during parallel computation This technique isalso valuable for the efficient use of available computationalmemory

32 Multidomain Approach To avoid an excessive domainsize we consideredmultiple domains In this study the entiredomainΩ is defined as the union of all the domains coveringall the fluid particles that is x

119901isin Ω where Ω = Ω

1cup

Ω2cup sdot sdot sdot cup Ω

119873119889 The domain number119873

119889is not constant and

is dependent on a spatially evolving flow When the domainsize exceeds a certain limit a new domain is created situateddownstream from the domain As a result the first domainΩ1includes a solid body and the others are sequentially

located downstream from the body For the total circulationconservation the grid spacing of the 119894th domain can bedefined as ℎ

119894= 120598 (single-resolution) or ℎ

119894= 2ℎ119894minus1

(multi-resolution) where ℎ

1= 120598 Here 120598 denotes the particle size To

T

Com

puta

tion

time (

h)

250

200

150

100

50

00 2 4 6 8 10

Equal-sized subdomainsUnequal-sized subdomain

Figure 1 Computation times for equal- and unequal-sized subdo-mains

interpolate the vorticity values of the particles into the nodeson a grid with different grid spacing (6) is rewritten as

120596119911(x119892) =

1205982

ℎ2

119894

119873

sum

119901=1

120596119911(x119901)119882(

x119901minus x119892

ℎ119894

) (16)

where x119901isin Ω119894and x119892isin Ω119894The nodes are declared to be a 2D

array so that the position of each node can be found straight-forwardly from its index The velocity and vorticity of eachdomain are sequentially and independently evaluatedHencerelatively small amounts of working memory are requiredThis multidomain approach also allows more freedom inthe choice of variable arrangements in the domains Theversatility of this approach was demonstrated in our previousstudy [26] It can be achieved at the expense of a significantincrease in computation time This issue is further discussedin the next section

33 Approximation of Far-Field Conditions A substantialpart of the overall computation time is spent on the calcu-lation of the boundary stream functions to solve Poissonrsquosequation for the stream function This requires the order ofO(119872119873

119901119875) operations to obtain boundary values of domain

Ω119894 Here 119872 denotes the number of nodes at the sides of

domainΩ119894and119873

119901119875 is the number of particles per processor

To achieve a nearly perfect load balance particles are evenlydistributed among the processors as discussed in Section 31and each processor has all the boundary nodes 119872 It mustalso be very useful to avoid all unnecessary communications119873119901119875 is reduced by increasing the number of processors

involved in the computation whereas 119872 is not This is whythe number of boundary nodes 119872 is the most critical fordetermining the computation time

In this study we attempted to reduce the number ofboundary nodes 119872 using the cubic spline approximationThis is made possible by the smooth variation of the streamfunctions along the boundary Only one-fourth of the bound-ary nodes were chosen as equidistant points for the directcalculations That is the stream functions at the selected

Modelling and Simulation in Engineering 5

boundary nodes were directly computed using (8) and thevalues at boundary nodes not selected were evaluated byinterpolation using cubic splines [39] In this study theerror between the interpolated value and the true valueis typically less than 001 We have observed that thereduction in computational time is inversely proportionalto the number of nodes selected for direct computationWe believe that this approach can be implemented withmuch lower computational effort because a complex datastructure such as a quad-tree used in the fast multipolemethod [40] is not required A rigorous comparison of theaccuracy and computational costs is planned in the nearfuture

4 Computational Procedure

First the entire simulation domain Ω covering all theparticles is divided into small domains Ω

119894(119894 = 1 119873

119889)

as discussed in Section 32 Each domain Ω119894is consecutively

computed according to the following numerical procedure

(1) Particle-based domain decomposition as discussedin Section 31 domain Ω

119894is decomposed into sub-

domains to ensure that the particles are evenly dis-tributed among the processors If redistribution hasnot been done prior to the time step particles thathave been removed from a subdomain are assignedto an adjacent processor without performing domaindecomposition

(2) Particle-to-grid interpolation each processor inter-polates the vorticity 120596

119911of its own particles into the

grid nodes through the11987210158404interpolation kernel

(3) Calculation of the boundary condition each proces-sor computes the boundary stream functions by directcalculation and interpolation using cubic splines asdiscussed in Section 33

(4) Evaluation of the velocity field the stream functionon the grid is computed with the FFT-based Poissonsolver from an open-source library called FFTW(fastest Fourier transform in the west) [41 42]There-after the rotational velocities on the grid nodes arecomputed from the definitions 119906

119909= 120597120595120597119910 and

119906119910= minus120597120595120597119909 using the fourth order central finite

difference scheme(5) Evaluation of the vorticity field each processor eval-

uates the evolution of the vorticity field in time asfollows

(a) The penalization term that is the second termon the right hand side of (12) is computedwith the velocities on the grid The spatialderivatives of the curl operators are evaluatedusing a centered difference approximation witha fourth-order error

(b) The diffusion term that is the first term onthe right hand side of (12) is evaluated on thegrid using a classical 9-point finite differencemethod

(c) The turbulent viscosity (119862119878Δ)2|120596119911| in (15) is

approximated by the vorticity strength of thegrid Finally the vorticity field 120596119899+1

119911is updated

(6) Grid-to-particle interpolation each processor inter-polates the velocity and vorticity on the grid back to itsown particle positions through the1198721015840

4interpolation

kernel

Vortex particles are advanced in time using a midpointpredictor-corrector method as follows particle positions arepredicted by their velocities xlowast = x119899 + Δ119905u(x119899 120596119899

119911) then the

particle positions are corrected by using combinations of thepredicted and previous positions and velocity values x119899+1 =x119899 + 05Δ119905[u(x119899 120596119899z ) + u(xlowast 120596119899+1

119911)] Particle redistribution is

conducted every three time steps to ensure particle overlapIf a new particle has a small vorticity magnitude that is|Γ119894| lt 00001|Γ|max it is deleted to avoid having too many

particles Loss of circulation is equally distributed among theremaining particles to ensure the conservation of the totalvorticity

5 Numerical Results and Discussions

51 Model Description We selected a NACA 0009 hydrofoilwith a truncated trailing edge for the numerical simulationsThis hydrofoil has the same cross-section as the experimentalmodel used by Ausoni et al [9] Ausoni [10] and Zobeiriet al [11] Ten percent of the original chord 119888

119900was removed

from the trailing-edge region of the NACA 0009 hydrofoilThe hydrofoil thickness distribution is written as

00 le119909

119888119900

le 05

119910

119888119900

= 01737 (119909

119888119900

)

12

minus 02422 (119909

119888119900

) + 03046 (119909

119888119900

)

2

minus 02657 (119909

119888119900

)

3

05 lt119909

119888119900

le 10

119910

119888119900

= 00004 + 01737 (1 minus119909

119888119900

) minus 01898 (1 minus119909

119888119900

)

2

+ 00387 (1 minus119909

119888119900

)

3

(17)

The hydrofoil geometry is further detailed in Ausoni [10]Themaximum thickness as normalized by the chord length 119888 is119905max119888 = 01 and the thickness at the trailing edge 119905TE119888 is00322 The bevel angle 120573 is defined in Figure 2 The trailingedge with 120573 = 0∘ denotes that the trailing edge is truncated inthe vertical direction

52 Numerical Setup Numerical parameters can be deter-mined using the stability condition Δ119905 = 120598

24] [18] The

grid convergence was achieved for 120598 = 000030 and

6 Modelling and Simulation in Engineering

xc

yc

045 05

minus002

0

002

A

BC

00054c

00322c120573

Figure 2 Trailing-edge shape Note that the points A B and C indicate the knuckles of the chamfer and trailing-edge wedge and the bevelangle is defined as 120573

Δ119905 = 000015 was chosen in order to guarantee numericalaccuracy for a Reynolds number of 106 The grid spacingof the 119894th domain Ω

119894is defined as ℎ

119894= 1205982119894minus1 to reduce

computation timeThe penalization parameter 120582 in (12) is setto be 108 The hydrofoil at a zero angle of attack is immersedin a Cartesian grid (the first domain) that does not conformto the surface of the bodyThe first domain with 4096times 1024lattice nodes is fixed and the other domains are dynamicallydetermined depending on the spatially evolving flow In thisstudy numerical simulations were conducted until 119879 = 10A new domain is created downstream of a body once thenumber of horizontal grid points exceeds 2048 Althoughthere is no limit on the number of vertical grid points it wastypically less than or equal to 1024 During simulation lessthan 08 million particles and four computational domainswere created The entire computation in each case includingstart-up and printing times took approximately 170 hourson 16 Intel Xeon64 33 GHz CPUs with 6GB memory perprocessor

53 Truncated Trailing-Edge For an impulsively started flowpast a hydrofoil at a chord-based Reynolds number of 106the wake pattern can be classified into three regimes basedon the time evolutions shown in Figure 3 In the very earlystages the wake remained very symmetrical and consistedprimarily of two large vortices in the near wake as shown inFigure 3(a) At that time the drag force gradually decreasedAs time went on the wake became asymmetrical as shown inFigure 3(b) and then shed a vortex as shown in Figure 3(c)After a number of shedding periods vortices were shedregularly from the symmetric trailing edge in the form of twotrains of oppositezsign but equal-strength vortices as shownin Figure 3(d) Such regular vortex shedding induces periodicloading on the structure as shown in Figure 4 It is wellknown that drag-force oscillation during vortex shedding ismuch smaller than the lift force Numerical results show thatoscillations in the drag force occurred at twice the vortex-shedding frequency due to the fact that two vortices wereshed from alternate sides during one full period of wakeoscillation

From our simulation the Strouhal number based on thetrailing-edge thickness St

119905= 119891119904119905TE119880infin is about 022 for

Re = 106 Ausoni [10] (also refer to Zobeiri et al [11])

had a similar finding and experimentally gave St119905= 025

which may be approximately read from their figure Theyused a hydrofoil with a 100mm chord length and 150mmspan length The aspect ratio defined as the ratio of the spanlength to the chord length was 15 A lower aspect ratioleads to a three-dimensional (3D) effect even though thehydrofoil is 2D Ausoni [10] also observed experimentallythat the shedding process was not induced in phase alongthe span It was therefore speculated in [25] that the slightunderestimation of the Strouhal number is due to 3D effectswhich were absent in our simulation

54 Oblique Trailing-Edges We conducted numerical simu-lations to observe vortex shedding from trailing edges withsix different bevel angles of 120573 = 0 10 20 40 60 and 70degrees Figure 5 shows the instantaneous vorticity distribu-tions at 119879 = 10 for Re = 106 At 120573 = 0∘ the vortex sheddingis regular and periodic As the bevel angle 120573 increasesvortex shedding becomes increasingly disorganized Suchirregular shedding of vortices demonstrates the periodicity ofoscillatory forces Figure 6 shows the power spectral densityfunction (PSD) of the drag and lift coefficients for differentbevel angles Compared with 120573 = 0∘ the amplitude of the liftforce oscillation for asymmetric trailing edges is significantlyreduced compared with the symmetric trailing edge at 120573 =0∘ The lift force amplitude is gradually decreased as the bevelangle increases while the drag force amplitude increases andthen decreases to between 120573 = 10∘ and 60∘The periodicity ofboth forces is almost lost at 120573 = 60∘ and there is no definiteperiodicity of the induced force components at 120573 = 70∘

At a bevel angle of 10∘ as shown in Figure 6(b) it is inter-esting that the dominant peak of the drag force is identical tothat of the lift force From a vibration standpoint the forcingfrequency is important for the prevention of resonance Thedistance between the two vortices in Figure 5(b) does not staythe same but the difference between the shedding vorticesbetween 120573 = 0

∘ and 10∘ (Figures 5(a) and 5(b)) is not

Modelling and Simulation in Engineering 7

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Symmetric wake at 119879 = 0396

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Symmetry-breaking transition at 119879 = 0900

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Nonperiodic vortex shedding at 119879 = 1098

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Fully developed periodic shedding at 119879 = 3500

Figure 3 Wake patterns of the NACA 0009 hydrofoil with a truncated trailing-edge (120573 = 0∘) for Re = 106

CL

CL

002

001

0

minus001

minus002I II III

T

0 05 1 15 2 25 3 35

CD

CD

0015

001

0005

0

Figure 4 Time evolution of the lift and drag coefficients for theimpulsively started NACA 0009 hydrofoil with a truncated trailingedge (120573 = 0∘) at Re = 106 Regime I denotes the developing regimeof a symmetric wake II a symmetry-breaking transition and IIIperiodic vortex shedding

statistically significant A possible reason for this is the phasedifference between the vortices shed from the upper andlower sides In the wake formation region the two vorticesshed from the symmetric trailing edge at 120573 = 0∘ are almostexactly out of phase (180 degrees) whereas at 120573 = 10

∘

there is a phase difference of approximately 30 degrees Thismeans that before one vortex is fully shed from the bodyanother vortex is created At higher bevel angles 120573 gt 20∘ thenegative vortices from the upper side of the hydrofoil becomestronger than the positive ones from the lower side Theoscillation force is induced mostly by the vortices shed fromthe acute (upper) side As a result the dominant frequency

of the drag force oscillation appears to be the same as thevortex shedding frequency We numerically observed thatdifferences in strength andor phase between two adjacentvortices cause nonperiodic and irregular vortex shedding

Figure 7 shows the shedding frequency of vortices fromdifferent beveled trailing edges compared with experimentalresults by Zobeiri et al [11] The numerically simulatedshedding frequencies were approximately 68 and 64 for120573 = 0

∘ and 60∘ respectively Numerical results showgood qualitative agreements with the experimental resultsbut the numerical simulations underestimated the vortex-shedding frequency by about 10 Figure 8 shows the meanvalue and standard deviation of the velocity components Wecompared our numerical results with the experimental resultsmeasured by Zobeiri et al [11] The truncated and obliquetrailing edges in Figure 8 indicate bevel angles of 120573 = 0

∘

and 60∘ respectively For the truncated trailing edge themean velocity profiles are more symmetric than those in theexperimental data The results from numerical simulationare in good agreement with the experimental results Weconclude that the vortex shedding phenomenon from thetrailing edge of a 2D hydrofoil can be reasonably simulatedusing 2D LES which considerably reduces the computationalcost

55 Effect of a Periodically Varying Flow on Vortex SheddingThe propeller is generally located behind the shiprsquos hull andis subjected to a nonaxisymmetric wake field Consideringa 2D flow the propeller section effectively experiences peri-odic variation in flow incidence We conducted numericalsimulations to investigate vortex shedding with respect tothe sinusoidal motions of the free-stream flow The hydrofoilwas at rest and the flow became the oscillator under theseconditions In the current study the angle of attack varied

8 Modelling and Simulation in Engineering

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Bevel angle 120573 = 0∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Bevel angle 120573 = 10∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Bevel angle 120573 = 20∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Bevel angle 120573 = 40∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(e) Bevel angle 120573 = 60∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(f) Bevel angle 120573 = 70∘

Figure 5 Instantaneous vorticity distributions at 119879 = 10 for Re = 106

sinusoidally with time 119905 120572 = 120572119900sin(2120587119891

infin119905) where 119891

infin

was the frequency of the free-stream flow oscillation andits oscillation amplitude was restricted at 120572

119900= 2∘ The

magnitude of the free-stream velocity was kept constant Thetested model is the NACA 0009 hydrofoil with a beveledtrailing edge of 120573 = 60∘

Figure 9 shows the instantaneous vorticity distributionsat119879 = 10 and the PSD of the drag coefficients at four differentfrequencies of the free-stream flow that is 119891

infin= 15 20

25 and 30 There are two peaks in the PSD functions thefirst peak appears around 119891lowast asymp 6 and the second one isidentical to the frequency of the free-stream flow oscillationThe obvious feature recognized in all the PSD functions asshown in Figure 9 is that the first peak frequency is nearlyidentical to the vortex-shedding frequency from the hydrofoilin the uniform free-stream flow Vortex formation in the nearwake is thought to be less affected by small inflow oscillationsMost interestingly at 119891

infin= 20 vortices are regularly shed

from the trailing edge as shown in Figure 9(c) In the PSDfunctions drag oscillation induced by vortex shedding isclearly observed This means that a particular oscillationfrequency in the free-stream velocity can cause regular andperiodic vortex shedding This is significant with respect tochanging a marine propellerrsquos angle of attack over time

6 Concluding Remarks

We simulated impulsively started flows past a modifiedNACA 0009 hydrofoil at a Reynolds number of 106 usingthe hybrid particle-mesh method in conjunction with thevorticity-based subgrid scale model We have achieved sig-nificant savings in both memory usage and computationtime by making improvements in the numerical schemesthe particle-based domain decomposition the use ofmultipledomains and the approximation of domain boundary con-ditions These techniques are simple and easy to implementbut also effective in reducing the computational costs Theparticle-based domain decomposition scheme is suitable fora distributed system in which each processor has its ownprivate memory (distributed memory)

In this study the numerical simulation results nearlymatch the experimental resultsThis demonstrates that vortexshedding from the trailing edge of a 2D hydrofoil can bereasonably simulated with a 2D LES It is due to the fact thatthe flow is characterized by quasi-two dimensionality withvortex shedding from the trailing edge of the hydrofoil Wehave considered that 2D vortex shedding simulations shouldnot be dismissed so easily With two-dimensionality it ispossible to solve the problem much less expensively since

Modelling and Simulation in Engineering 9

0 5 10 15 20 250

10

0 5 10 15 20 250

05times10minus5

PSD

ofC

L

PSD

ofC

D

times10minus6

flowast flowast

(a) Truncated trailing edge 120573 = 0∘

0

08

0

08

0

08

0

08

0

08

0

04

0

04

0

04

0

04

0

04

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

(b) Beveled trailing edge 120573 gt 0∘

Figure 6 Power spectral density (PSD) of the lift and drag coefficients 119891lowast denotes the frequency normalized by the chord length and theundisturbed flow velocity

3D problems require significant computational resourcesHowever the LES should be 3D since turbulence is inherently3D While we accept that more realistic 3D simulation isrequired physically a 2D LES can be used as a basis forcomparison prior to 3D implementation Once the capability

of a numerical algorithm is evaluated in two dimensionsit can be easily extended into three dimensions This paperis intended as a guide for researchers and designers whoare working on a similar problem vortex shedding from ahydrofoil at high Reynolds numbers

10 Modelling and Simulation in Engineering

0 10 20 30 40 50 600

2

4

6

8

10

Zobeiri et al (2012) expPresent cal

flowast

120573 (deg)

Figure 7 Shedding frequency of vortices from the different beveled trailing edges for Re = 106 The experimental data by Zobeiri et al [11]were derived from LDVmeasurements and the approximate shedding frequency as read from their figure is normalized by the chord lengthand the undisturbed flow velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02 04 06 08 1uxmeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(a) Normalized mean of streamwise velocity

minus06

minus04

minus02

0

02

04

06yt

max

0 01 02 03 04uxstdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(b) Normalized mean of transverse velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02uymeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(c) Normalized standard deviation of streamwise velocity

minus06

minus04

minus02

0

02

04

06

yt

max

0 01 02 03 04 05uystdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(d) Normalized standard deviation of transverse velocity

Figure 8 Mean and standard deviation of velocities at 119909119888 = 0516 for Re = 106 The experimental data by Zobeiri et al [11] were measuredat 119909119888 = 052 and 0528 for truncated (120573 = 0∘) and oblique (120573 = 60∘) trailing edges respectively and the reported Reynolds number of theflow is about 2 times 106

Modelling and Simulation in Engineering 11

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(a) Steady flow 119891infin = 0

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(b) Incoming flow frequency 119891infin = 15

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(c) Incoming flow frequency 119891infin = 20

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(d) Incoming flow frequency 119891infin = 25

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(e) Incoming flow frequency 119891infin = 30

Figure 9 Effect of the incoming flow frequency on vortex shedding from a beveled trailing edge of 120573 = 60∘ at 119891infin= 15 20 25 and 30

The contour indicates the instantaneous vorticity distribution at 119879 = 10 and the power spectral density (PSD) is calculated from the dragcoefficients in the range 119879 = [3 10]

12 Modelling and Simulation in Engineering

We found in this study that the vortex formation char-acteristic in periodically varying inflows remains unchangedA certain frequency of free-stream flow oscillation tends tomake vortex shedding more regular again This means thatthe periodicity of flow incidence can stimulate an inherentfrequency of vortex shedding from a hydrofoil Howeverthe effect of inflow oscillation on vortex shedding is notyet fully understood Future work will include experimentalinvestigations as well as a 3D flow simulation to betterunderstand the relationship between inflow oscillation andvortex shedding

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Korean Institute ofOcean Science amp Technology sponsored by the Ministry ofTrade Industry amp Energy (no 10033668) and the IndustrialConvergence Strategic Technology Development Program(no 10044499) funded by the Ministry of Trade Industry ampEnergy (MI Korea)

References

[1] E Berger and R Wille ldquoPeriodic flow phenomenardquo AnnualReview of Fluid Mechanics vol 4 no 1 pp 313ndash340 1972

[2] P W Bearman ldquoVortex shedding from oscillating bluff bodiesrdquoAnnual Review of Fluid Mechanics vol 16 pp 195ndash222 1984

[3] C H K Williamson ldquoVortex dynamics in the cylinder wakerdquoAnnual Review of Fluid Mechanics vol 28 pp 477ndash539 1996

[4] C H K Williamson ldquoVortex-induced vibrationsrdquo AnnualReview of Fluid Mechanics vol 36 pp 413ndash455 2004

[5] R Fischer ldquoSinging propellersmdashsolutions and case historiesrdquoMarine Technology vol 45 no 4 pp 221ndash227 2008

[6] D A Bourgoyne J M Hamel S L Ceccio and D R DowlingldquoTime-averaged flow over a hydrofoil at high reynolds numberrdquoJournal of Fluid Mechanics vol 496 pp 365ndash404 2003

[7] D A Bourgoyne S L Ceccio and D R Dowling ldquoVortexshedding from a hydrofoil at high Reynolds numberrdquo Journalof Fluid Mechanics vol 531 pp 293ndash324 2005

[8] N Mulvany J Tu L Chen and B Anderson ldquoAssessment oftwo-equation turbulence modelling for high Reynolds numberhydrofoil flowsrdquo International Journal for Numerical Methods inFluids vol 45 no 3 pp 275ndash299 2004

[9] P Ausoni M Farhat X Escaler E Egusquiza and F AvellanldquoCavitation influence on von Karman vortex shedding andinduced hydrofoil vibrationsrdquo Journal of Fluids EngineeringTransactions vol 129 no 8 pp 966ndash973 2007

[10] P Ausoni Turbulent vortex shedding from a blunt trailingedge hydrofoil [PhD thesis] Ecole Polytechnique Federale deLausanne Lausanne Switzerland 2009

[11] A Zobeiri P Ausoni F Avellan and M Farhat ldquoHow obliquetrailing edge of a hydrofoil reduces the vortex-induced vibra-tionrdquo Journal of Fluids and Structures vol 32 pp 78ndash89 2012

[12] G-H Cottet and P D Koumoutsakos Vortex Methods Theoryand Practice Cambridge University Press 2000

[13] G Cottet and P Poncet ldquoParticle methods for direct numericalsimulations of three-dimensional wakesrdquo Journal of Turbulencevol 3 article N38 2002

[14] P Chatelain A Curioni M Bergdorf D Rossinelli WAndreoni and P Koumoutsakos ldquoBillion vortex particle directnumerical simulations of aircraft wakesrdquo Computer Methods inAppliedMechanics and Engineering vol 197 no 13ndash16 pp 1296ndash1304 2008

[15] Y-C Kim J-C Suh and K-J Lee ldquoVortex-in-cell methodcombined with a boundary element method for incompressibleviscous flow analysisrdquo International Journal forNumericalMeth-ods in Fluids vol 69 no 10 pp 1567ndash1583 2012

[16] P Angot C-H Bruneau and P Fabrie ldquoA penalization methodto take into account obstacles in incompressible viscous flowsrdquoNumerische Mathematik vol 81 no 4 pp 497ndash520 1999

[17] M El Ossmani and P Poncet ldquoEfficiency of multiscale hybridgrid-particle vortex methodsrdquo Multiscale Modeling amp Simula-tion vol 8 no 5 pp 1671ndash1690 2010

[18] J T Rasmussen G-H Cottet and J HWalther ldquoAmultiresolu-tion remeshed vortex-in-cell algorithm using patchesrdquo Journalof Computational Physics vol 230 no 17 pp 6742ndash6755 2011

[19] M Coquerelle and G-H Cottet ldquoA vortex level set method forthe two-way coupling of an incompressible fluid with collidingrigid bodiesrdquo Journal of Computational Physics vol 227 no 21pp 9121ndash9137 2008

[20] F Morency H Beaugendre F Gallizio and S Laurens ldquoCom-putation of ice shedding trajectories using cartesian gridspenalization and level setsrdquo Modelling and Simulation in Engi-neering vol 2011 Article ID 274947 15 pages 2011

[21] M M Hejlesen P Koumoutsakos A Leonard and J HWalther ldquoIterative Brinkman penalization for remeshed vortexmethodsrdquo Journal of Computational Physics vol 280 pp 547ndash562 2015

[22] M Gazzola P Chatelain W M van Rees and P Koumout-sakos ldquoSimulations of single and multiple swimmers with non-divergence free deforming geometriesrdquo Journal of Computa-tional Physics vol 230 no 19 pp 7093ndash7114 2011

[23] R Chatelin and P Poncet ldquoA hybrid grid-particle method formoving bodies in 3D stokes flow with variable viscosityrdquo SIAMJournal on Scientific Computing vol 35 no 4 pp B925ndashB9492013

[24] R Chatelin and P Poncet ldquoHybrid grid-particle methodsand penalization a Sherman-Morrison-Woodbury approach tocompute 3D viscous flows using FFTrdquo Journal of ComputationalPhysics vol 269 pp 314ndash328 2014

[25] S J Lee J H Lee and J C Suh ldquoVortex shedding frequencyfor a 2D hydrofoil with a truncated trailing edgerdquo Journal of theSociety of Naval Architects of Korea vol 51 no 6 pp 480ndash4882014 (Korean)

[26] S-J Lee and J C Suh ldquoA multi-domain approach for ahybrid particle-mesh methodrdquo Journal of Computational FluidsEngineering vol 19 no 2 pp 72ndash78 2014

[27] S-J Lee J-H Lee and J-C Suh ldquoComputation of pressurefields around a two-dimensional circular cylinder using thevortex-in-cell and penalization methodsrdquo Modelling and Sim-ulation in Engineering vol 2014 Article ID 708372 13 pages2014

[28] J J Monaghan ldquoExtrapolating 119861-splines for interpolationrdquoJournal of Computational Physics vol 60 no 2 pp 253ndash2621985

[29] J J Monaghan ldquoParticle methods for hydrodynamicsrdquo Com-puter Physics Reports vol 3 no 2 pp 71ndash124 1985

Modelling and Simulation in Engineering 13

[30] O V Vasilyev and N K-R Kevlahan ldquoHybrid waveletcollocation-brinkman penalization method for complex geom-etry flowsrdquo International Journal for Numerical Methods inFluids vol 40 no 3-4 pp 531ndash538 2002

[31] W Iwakami Y Yatagai N Hatakeyama and Y Hattori ldquoAnew approach for error reduction in the volume penalizationmethodrdquoCommunications in Computational Physics vol 16 no5 pp 1181ndash1200 2014

[32] J R Mansfield O M Knio and C Meneveau ldquoTowardslagrangian large vortex simulationrdquo ESAIM Proceedings vol 1pp 49ndash64 1996

[33] G-H Cottet and P Poncet ldquoAdvances in direct numerical sim-ulations of 3D wall-bounded flows by vortex-in-cell methodsrdquoJournal of Computational Physics vol 193 no 1 pp 136ndash1582004

[34] B Caswell ldquoKinematics and stress on a surface of restrdquo Archivefor Rational Mechanics and Analysis vol 26 no 5 pp 385ndash3991967

[35] R S Rogallo and P Moin ldquoNumerical simulation of turbulentflowsrdquo Annual Review of Fluid Mechanics vol 16 pp 99ndash1371984

[36] J Meyers and P Sagaut ldquoOn the model coefficients for thestandard and the variational multi-scale Smagorinsky modelrdquoJournal of Fluid Mechanics vol 569 pp 287ndash319 2006

[37] J R Mansfield O M Knio and C Meneveau ldquoA dynamic LESscheme for the vorticity transport equation formulation and apriori testsrdquo Journal of Computational Physics vol 145 no 2 pp693ndash730 1998

[38] S B Pope Turbulent Flows Cambridge University Press 2000[39] WH Press S A TeukolskyW T Vetterling and B P Flannery

Numerical Recipes in C++ The Art of Scientific ComputingCambridge University Press 2nd edition 2002

[40] L Greengard and W D Gropp ldquoA parallel verison of the fastmultipoe methodrdquo Computers amp Mathematics with Applica-tions vol 20 no 7 pp 63ndash71 1990

[41] M Frigo and S G Johnson ldquoFFTW an adaptive software archi-tecture for the FFTrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo98) pp 1381ndash1384 May 1998

[42] M Frigo and S G Johnson ldquoThe design and implementationof FFTW3rdquo Proceedings of the IEEE vol 93 no 2 pp 216ndash2312005

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Modelling and Simulation in Engineering 5

boundary nodes were directly computed using (8) and thevalues at boundary nodes not selected were evaluated byinterpolation using cubic splines [39] In this study theerror between the interpolated value and the true valueis typically less than 001 We have observed that thereduction in computational time is inversely proportionalto the number of nodes selected for direct computationWe believe that this approach can be implemented withmuch lower computational effort because a complex datastructure such as a quad-tree used in the fast multipolemethod [40] is not required A rigorous comparison of theaccuracy and computational costs is planned in the nearfuture

4 Computational Procedure

First the entire simulation domain Ω covering all theparticles is divided into small domains Ω

119894(119894 = 1 119873

119889)

as discussed in Section 32 Each domain Ω119894is consecutively

computed according to the following numerical procedure

(1) Particle-based domain decomposition as discussedin Section 31 domain Ω

119894is decomposed into sub-

domains to ensure that the particles are evenly dis-tributed among the processors If redistribution hasnot been done prior to the time step particles thathave been removed from a subdomain are assignedto an adjacent processor without performing domaindecomposition

(2) Particle-to-grid interpolation each processor inter-polates the vorticity 120596

119911of its own particles into the

grid nodes through the11987210158404interpolation kernel

(3) Calculation of the boundary condition each proces-sor computes the boundary stream functions by directcalculation and interpolation using cubic splines asdiscussed in Section 33

(4) Evaluation of the velocity field the stream functionon the grid is computed with the FFT-based Poissonsolver from an open-source library called FFTW(fastest Fourier transform in the west) [41 42]There-after the rotational velocities on the grid nodes arecomputed from the definitions 119906

119909= 120597120595120597119910 and

119906119910= minus120597120595120597119909 using the fourth order central finite

difference scheme(5) Evaluation of the vorticity field each processor eval-

uates the evolution of the vorticity field in time asfollows

(a) The penalization term that is the second termon the right hand side of (12) is computedwith the velocities on the grid The spatialderivatives of the curl operators are evaluatedusing a centered difference approximation witha fourth-order error

(b) The diffusion term that is the first term onthe right hand side of (12) is evaluated on thegrid using a classical 9-point finite differencemethod

(c) The turbulent viscosity (119862119878Δ)2|120596119911| in (15) is

approximated by the vorticity strength of thegrid Finally the vorticity field 120596119899+1

119911is updated

(6) Grid-to-particle interpolation each processor inter-polates the velocity and vorticity on the grid back to itsown particle positions through the1198721015840

4interpolation

kernel

Vortex particles are advanced in time using a midpointpredictor-corrector method as follows particle positions arepredicted by their velocities xlowast = x119899 + Δ119905u(x119899 120596119899

119911) then the

particle positions are corrected by using combinations of thepredicted and previous positions and velocity values x119899+1 =x119899 + 05Δ119905[u(x119899 120596119899z ) + u(xlowast 120596119899+1

119911)] Particle redistribution is

conducted every three time steps to ensure particle overlapIf a new particle has a small vorticity magnitude that is|Γ119894| lt 00001|Γ|max it is deleted to avoid having too many

particles Loss of circulation is equally distributed among theremaining particles to ensure the conservation of the totalvorticity

5 Numerical Results and Discussions

51 Model Description We selected a NACA 0009 hydrofoilwith a truncated trailing edge for the numerical simulationsThis hydrofoil has the same cross-section as the experimentalmodel used by Ausoni et al [9] Ausoni [10] and Zobeiriet al [11] Ten percent of the original chord 119888

119900was removed

from the trailing-edge region of the NACA 0009 hydrofoilThe hydrofoil thickness distribution is written as

00 le119909

119888119900

le 05

119910

119888119900

= 01737 (119909

119888119900

)

12

minus 02422 (119909

119888119900

) + 03046 (119909

119888119900

)

2

minus 02657 (119909

119888119900

)

3

05 lt119909

119888119900

le 10

119910

119888119900

= 00004 + 01737 (1 minus119909

119888119900

) minus 01898 (1 minus119909

119888119900

)

2

+ 00387 (1 minus119909

119888119900

)

3

(17)

The hydrofoil geometry is further detailed in Ausoni [10]Themaximum thickness as normalized by the chord length 119888 is119905max119888 = 01 and the thickness at the trailing edge 119905TE119888 is00322 The bevel angle 120573 is defined in Figure 2 The trailingedge with 120573 = 0∘ denotes that the trailing edge is truncated inthe vertical direction

52 Numerical Setup Numerical parameters can be deter-mined using the stability condition Δ119905 = 120598

24] [18] The

grid convergence was achieved for 120598 = 000030 and

6 Modelling and Simulation in Engineering

xc

yc

045 05

minus002

0

002

A

BC

00054c

00322c120573

Figure 2 Trailing-edge shape Note that the points A B and C indicate the knuckles of the chamfer and trailing-edge wedge and the bevelangle is defined as 120573

Δ119905 = 000015 was chosen in order to guarantee numericalaccuracy for a Reynolds number of 106 The grid spacingof the 119894th domain Ω

119894is defined as ℎ

119894= 1205982119894minus1 to reduce

computation timeThe penalization parameter 120582 in (12) is setto be 108 The hydrofoil at a zero angle of attack is immersedin a Cartesian grid (the first domain) that does not conformto the surface of the bodyThe first domain with 4096times 1024lattice nodes is fixed and the other domains are dynamicallydetermined depending on the spatially evolving flow In thisstudy numerical simulations were conducted until 119879 = 10A new domain is created downstream of a body once thenumber of horizontal grid points exceeds 2048 Althoughthere is no limit on the number of vertical grid points it wastypically less than or equal to 1024 During simulation lessthan 08 million particles and four computational domainswere created The entire computation in each case includingstart-up and printing times took approximately 170 hourson 16 Intel Xeon64 33 GHz CPUs with 6GB memory perprocessor

53 Truncated Trailing-Edge For an impulsively started flowpast a hydrofoil at a chord-based Reynolds number of 106the wake pattern can be classified into three regimes basedon the time evolutions shown in Figure 3 In the very earlystages the wake remained very symmetrical and consistedprimarily of two large vortices in the near wake as shown inFigure 3(a) At that time the drag force gradually decreasedAs time went on the wake became asymmetrical as shown inFigure 3(b) and then shed a vortex as shown in Figure 3(c)After a number of shedding periods vortices were shedregularly from the symmetric trailing edge in the form of twotrains of oppositezsign but equal-strength vortices as shownin Figure 3(d) Such regular vortex shedding induces periodicloading on the structure as shown in Figure 4 It is wellknown that drag-force oscillation during vortex shedding ismuch smaller than the lift force Numerical results show thatoscillations in the drag force occurred at twice the vortex-shedding frequency due to the fact that two vortices wereshed from alternate sides during one full period of wakeoscillation

From our simulation the Strouhal number based on thetrailing-edge thickness St

119905= 119891119904119905TE119880infin is about 022 for

Re = 106 Ausoni [10] (also refer to Zobeiri et al [11])

had a similar finding and experimentally gave St119905= 025

which may be approximately read from their figure Theyused a hydrofoil with a 100mm chord length and 150mmspan length The aspect ratio defined as the ratio of the spanlength to the chord length was 15 A lower aspect ratioleads to a three-dimensional (3D) effect even though thehydrofoil is 2D Ausoni [10] also observed experimentallythat the shedding process was not induced in phase alongthe span It was therefore speculated in [25] that the slightunderestimation of the Strouhal number is due to 3D effectswhich were absent in our simulation

54 Oblique Trailing-Edges We conducted numerical simu-lations to observe vortex shedding from trailing edges withsix different bevel angles of 120573 = 0 10 20 40 60 and 70degrees Figure 5 shows the instantaneous vorticity distribu-tions at 119879 = 10 for Re = 106 At 120573 = 0∘ the vortex sheddingis regular and periodic As the bevel angle 120573 increasesvortex shedding becomes increasingly disorganized Suchirregular shedding of vortices demonstrates the periodicity ofoscillatory forces Figure 6 shows the power spectral densityfunction (PSD) of the drag and lift coefficients for differentbevel angles Compared with 120573 = 0∘ the amplitude of the liftforce oscillation for asymmetric trailing edges is significantlyreduced compared with the symmetric trailing edge at 120573 =0∘ The lift force amplitude is gradually decreased as the bevelangle increases while the drag force amplitude increases andthen decreases to between 120573 = 10∘ and 60∘The periodicity ofboth forces is almost lost at 120573 = 60∘ and there is no definiteperiodicity of the induced force components at 120573 = 70∘

At a bevel angle of 10∘ as shown in Figure 6(b) it is inter-esting that the dominant peak of the drag force is identical tothat of the lift force From a vibration standpoint the forcingfrequency is important for the prevention of resonance Thedistance between the two vortices in Figure 5(b) does not staythe same but the difference between the shedding vorticesbetween 120573 = 0

∘ and 10∘ (Figures 5(a) and 5(b)) is not

Modelling and Simulation in Engineering 7

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Symmetric wake at 119879 = 0396

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Symmetry-breaking transition at 119879 = 0900

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Nonperiodic vortex shedding at 119879 = 1098

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Fully developed periodic shedding at 119879 = 3500

Figure 3 Wake patterns of the NACA 0009 hydrofoil with a truncated trailing-edge (120573 = 0∘) for Re = 106

CL

CL

002

001

0

minus001

minus002I II III

T

0 05 1 15 2 25 3 35

CD

CD

0015

001

0005

0

Figure 4 Time evolution of the lift and drag coefficients for theimpulsively started NACA 0009 hydrofoil with a truncated trailingedge (120573 = 0∘) at Re = 106 Regime I denotes the developing regimeof a symmetric wake II a symmetry-breaking transition and IIIperiodic vortex shedding

statistically significant A possible reason for this is the phasedifference between the vortices shed from the upper andlower sides In the wake formation region the two vorticesshed from the symmetric trailing edge at 120573 = 0∘ are almostexactly out of phase (180 degrees) whereas at 120573 = 10

∘

there is a phase difference of approximately 30 degrees Thismeans that before one vortex is fully shed from the bodyanother vortex is created At higher bevel angles 120573 gt 20∘ thenegative vortices from the upper side of the hydrofoil becomestronger than the positive ones from the lower side Theoscillation force is induced mostly by the vortices shed fromthe acute (upper) side As a result the dominant frequency

of the drag force oscillation appears to be the same as thevortex shedding frequency We numerically observed thatdifferences in strength andor phase between two adjacentvortices cause nonperiodic and irregular vortex shedding

Figure 7 shows the shedding frequency of vortices fromdifferent beveled trailing edges compared with experimentalresults by Zobeiri et al [11] The numerically simulatedshedding frequencies were approximately 68 and 64 for120573 = 0

∘ and 60∘ respectively Numerical results showgood qualitative agreements with the experimental resultsbut the numerical simulations underestimated the vortex-shedding frequency by about 10 Figure 8 shows the meanvalue and standard deviation of the velocity components Wecompared our numerical results with the experimental resultsmeasured by Zobeiri et al [11] The truncated and obliquetrailing edges in Figure 8 indicate bevel angles of 120573 = 0

∘

and 60∘ respectively For the truncated trailing edge themean velocity profiles are more symmetric than those in theexperimental data The results from numerical simulationare in good agreement with the experimental results Weconclude that the vortex shedding phenomenon from thetrailing edge of a 2D hydrofoil can be reasonably simulatedusing 2D LES which considerably reduces the computationalcost

55 Effect of a Periodically Varying Flow on Vortex SheddingThe propeller is generally located behind the shiprsquos hull andis subjected to a nonaxisymmetric wake field Consideringa 2D flow the propeller section effectively experiences peri-odic variation in flow incidence We conducted numericalsimulations to investigate vortex shedding with respect tothe sinusoidal motions of the free-stream flow The hydrofoilwas at rest and the flow became the oscillator under theseconditions In the current study the angle of attack varied

8 Modelling and Simulation in Engineering

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Bevel angle 120573 = 0∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Bevel angle 120573 = 10∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Bevel angle 120573 = 20∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Bevel angle 120573 = 40∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(e) Bevel angle 120573 = 60∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(f) Bevel angle 120573 = 70∘

Figure 5 Instantaneous vorticity distributions at 119879 = 10 for Re = 106

sinusoidally with time 119905 120572 = 120572119900sin(2120587119891

infin119905) where 119891

infin

was the frequency of the free-stream flow oscillation andits oscillation amplitude was restricted at 120572

119900= 2∘ The

magnitude of the free-stream velocity was kept constant Thetested model is the NACA 0009 hydrofoil with a beveledtrailing edge of 120573 = 60∘

Figure 9 shows the instantaneous vorticity distributionsat119879 = 10 and the PSD of the drag coefficients at four differentfrequencies of the free-stream flow that is 119891

infin= 15 20

25 and 30 There are two peaks in the PSD functions thefirst peak appears around 119891lowast asymp 6 and the second one isidentical to the frequency of the free-stream flow oscillationThe obvious feature recognized in all the PSD functions asshown in Figure 9 is that the first peak frequency is nearlyidentical to the vortex-shedding frequency from the hydrofoilin the uniform free-stream flow Vortex formation in the nearwake is thought to be less affected by small inflow oscillationsMost interestingly at 119891

infin= 20 vortices are regularly shed

from the trailing edge as shown in Figure 9(c) In the PSDfunctions drag oscillation induced by vortex shedding isclearly observed This means that a particular oscillationfrequency in the free-stream velocity can cause regular andperiodic vortex shedding This is significant with respect tochanging a marine propellerrsquos angle of attack over time

6 Concluding Remarks

We simulated impulsively started flows past a modifiedNACA 0009 hydrofoil at a Reynolds number of 106 usingthe hybrid particle-mesh method in conjunction with thevorticity-based subgrid scale model We have achieved sig-nificant savings in both memory usage and computationtime by making improvements in the numerical schemesthe particle-based domain decomposition the use ofmultipledomains and the approximation of domain boundary con-ditions These techniques are simple and easy to implementbut also effective in reducing the computational costs Theparticle-based domain decomposition scheme is suitable fora distributed system in which each processor has its ownprivate memory (distributed memory)

In this study the numerical simulation results nearlymatch the experimental resultsThis demonstrates that vortexshedding from the trailing edge of a 2D hydrofoil can bereasonably simulated with a 2D LES It is due to the fact thatthe flow is characterized by quasi-two dimensionality withvortex shedding from the trailing edge of the hydrofoil Wehave considered that 2D vortex shedding simulations shouldnot be dismissed so easily With two-dimensionality it ispossible to solve the problem much less expensively since

Modelling and Simulation in Engineering 9

0 5 10 15 20 250

10

0 5 10 15 20 250

05times10minus5

PSD

ofC

L

PSD

ofC

D

times10minus6

flowast flowast

(a) Truncated trailing edge 120573 = 0∘

0

08

0

08

0

08

0

08

0

08

0

04

0

04

0

04

0

04

0

04

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

(b) Beveled trailing edge 120573 gt 0∘

Figure 6 Power spectral density (PSD) of the lift and drag coefficients 119891lowast denotes the frequency normalized by the chord length and theundisturbed flow velocity

3D problems require significant computational resourcesHowever the LES should be 3D since turbulence is inherently3D While we accept that more realistic 3D simulation isrequired physically a 2D LES can be used as a basis forcomparison prior to 3D implementation Once the capability

of a numerical algorithm is evaluated in two dimensionsit can be easily extended into three dimensions This paperis intended as a guide for researchers and designers whoare working on a similar problem vortex shedding from ahydrofoil at high Reynolds numbers

10 Modelling and Simulation in Engineering

0 10 20 30 40 50 600

2

4

6

8

10

Zobeiri et al (2012) expPresent cal

flowast

120573 (deg)

Figure 7 Shedding frequency of vortices from the different beveled trailing edges for Re = 106 The experimental data by Zobeiri et al [11]were derived from LDVmeasurements and the approximate shedding frequency as read from their figure is normalized by the chord lengthand the undisturbed flow velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02 04 06 08 1uxmeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(a) Normalized mean of streamwise velocity

minus06

minus04

minus02

0

02

04

06yt

max

0 01 02 03 04uxstdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(b) Normalized mean of transverse velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02uymeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(c) Normalized standard deviation of streamwise velocity

minus06

minus04

minus02

0

02

04

06

yt

max

0 01 02 03 04 05uystdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(d) Normalized standard deviation of transverse velocity

Figure 8 Mean and standard deviation of velocities at 119909119888 = 0516 for Re = 106 The experimental data by Zobeiri et al [11] were measuredat 119909119888 = 052 and 0528 for truncated (120573 = 0∘) and oblique (120573 = 60∘) trailing edges respectively and the reported Reynolds number of theflow is about 2 times 106

Modelling and Simulation in Engineering 11

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(a) Steady flow 119891infin = 0

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(b) Incoming flow frequency 119891infin = 15

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(c) Incoming flow frequency 119891infin = 20

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(d) Incoming flow frequency 119891infin = 25

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(e) Incoming flow frequency 119891infin = 30

Figure 9 Effect of the incoming flow frequency on vortex shedding from a beveled trailing edge of 120573 = 60∘ at 119891infin= 15 20 25 and 30

The contour indicates the instantaneous vorticity distribution at 119879 = 10 and the power spectral density (PSD) is calculated from the dragcoefficients in the range 119879 = [3 10]

12 Modelling and Simulation in Engineering

We found in this study that the vortex formation char-acteristic in periodically varying inflows remains unchangedA certain frequency of free-stream flow oscillation tends tomake vortex shedding more regular again This means thatthe periodicity of flow incidence can stimulate an inherentfrequency of vortex shedding from a hydrofoil Howeverthe effect of inflow oscillation on vortex shedding is notyet fully understood Future work will include experimentalinvestigations as well as a 3D flow simulation to betterunderstand the relationship between inflow oscillation andvortex shedding

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Korean Institute ofOcean Science amp Technology sponsored by the Ministry ofTrade Industry amp Energy (no 10033668) and the IndustrialConvergence Strategic Technology Development Program(no 10044499) funded by the Ministry of Trade Industry ampEnergy (MI Korea)

References

[1] E Berger and R Wille ldquoPeriodic flow phenomenardquo AnnualReview of Fluid Mechanics vol 4 no 1 pp 313ndash340 1972

[2] P W Bearman ldquoVortex shedding from oscillating bluff bodiesrdquoAnnual Review of Fluid Mechanics vol 16 pp 195ndash222 1984

[3] C H K Williamson ldquoVortex dynamics in the cylinder wakerdquoAnnual Review of Fluid Mechanics vol 28 pp 477ndash539 1996

[4] C H K Williamson ldquoVortex-induced vibrationsrdquo AnnualReview of Fluid Mechanics vol 36 pp 413ndash455 2004

[5] R Fischer ldquoSinging propellersmdashsolutions and case historiesrdquoMarine Technology vol 45 no 4 pp 221ndash227 2008

[6] D A Bourgoyne J M Hamel S L Ceccio and D R DowlingldquoTime-averaged flow over a hydrofoil at high reynolds numberrdquoJournal of Fluid Mechanics vol 496 pp 365ndash404 2003

[7] D A Bourgoyne S L Ceccio and D R Dowling ldquoVortexshedding from a hydrofoil at high Reynolds numberrdquo Journalof Fluid Mechanics vol 531 pp 293ndash324 2005

[8] N Mulvany J Tu L Chen and B Anderson ldquoAssessment oftwo-equation turbulence modelling for high Reynolds numberhydrofoil flowsrdquo International Journal for Numerical Methods inFluids vol 45 no 3 pp 275ndash299 2004

[9] P Ausoni M Farhat X Escaler E Egusquiza and F AvellanldquoCavitation influence on von Karman vortex shedding andinduced hydrofoil vibrationsrdquo Journal of Fluids EngineeringTransactions vol 129 no 8 pp 966ndash973 2007

[10] P Ausoni Turbulent vortex shedding from a blunt trailingedge hydrofoil [PhD thesis] Ecole Polytechnique Federale deLausanne Lausanne Switzerland 2009

[11] A Zobeiri P Ausoni F Avellan and M Farhat ldquoHow obliquetrailing edge of a hydrofoil reduces the vortex-induced vibra-tionrdquo Journal of Fluids and Structures vol 32 pp 78ndash89 2012

[12] G-H Cottet and P D Koumoutsakos Vortex Methods Theoryand Practice Cambridge University Press 2000

[13] G Cottet and P Poncet ldquoParticle methods for direct numericalsimulations of three-dimensional wakesrdquo Journal of Turbulencevol 3 article N38 2002

[14] P Chatelain A Curioni M Bergdorf D Rossinelli WAndreoni and P Koumoutsakos ldquoBillion vortex particle directnumerical simulations of aircraft wakesrdquo Computer Methods inAppliedMechanics and Engineering vol 197 no 13ndash16 pp 1296ndash1304 2008

[15] Y-C Kim J-C Suh and K-J Lee ldquoVortex-in-cell methodcombined with a boundary element method for incompressibleviscous flow analysisrdquo International Journal forNumericalMeth-ods in Fluids vol 69 no 10 pp 1567ndash1583 2012

[16] P Angot C-H Bruneau and P Fabrie ldquoA penalization methodto take into account obstacles in incompressible viscous flowsrdquoNumerische Mathematik vol 81 no 4 pp 497ndash520 1999

[17] M El Ossmani and P Poncet ldquoEfficiency of multiscale hybridgrid-particle vortex methodsrdquo Multiscale Modeling amp Simula-tion vol 8 no 5 pp 1671ndash1690 2010

[18] J T Rasmussen G-H Cottet and J HWalther ldquoAmultiresolu-tion remeshed vortex-in-cell algorithm using patchesrdquo Journalof Computational Physics vol 230 no 17 pp 6742ndash6755 2011

[19] M Coquerelle and G-H Cottet ldquoA vortex level set method forthe two-way coupling of an incompressible fluid with collidingrigid bodiesrdquo Journal of Computational Physics vol 227 no 21pp 9121ndash9137 2008

[20] F Morency H Beaugendre F Gallizio and S Laurens ldquoCom-putation of ice shedding trajectories using cartesian gridspenalization and level setsrdquo Modelling and Simulation in Engi-neering vol 2011 Article ID 274947 15 pages 2011

[21] M M Hejlesen P Koumoutsakos A Leonard and J HWalther ldquoIterative Brinkman penalization for remeshed vortexmethodsrdquo Journal of Computational Physics vol 280 pp 547ndash562 2015

[22] M Gazzola P Chatelain W M van Rees and P Koumout-sakos ldquoSimulations of single and multiple swimmers with non-divergence free deforming geometriesrdquo Journal of Computa-tional Physics vol 230 no 19 pp 7093ndash7114 2011

[23] R Chatelin and P Poncet ldquoA hybrid grid-particle method formoving bodies in 3D stokes flow with variable viscosityrdquo SIAMJournal on Scientific Computing vol 35 no 4 pp B925ndashB9492013

[24] R Chatelin and P Poncet ldquoHybrid grid-particle methodsand penalization a Sherman-Morrison-Woodbury approach tocompute 3D viscous flows using FFTrdquo Journal of ComputationalPhysics vol 269 pp 314ndash328 2014

[25] S J Lee J H Lee and J C Suh ldquoVortex shedding frequencyfor a 2D hydrofoil with a truncated trailing edgerdquo Journal of theSociety of Naval Architects of Korea vol 51 no 6 pp 480ndash4882014 (Korean)

[26] S-J Lee and J C Suh ldquoA multi-domain approach for ahybrid particle-mesh methodrdquo Journal of Computational FluidsEngineering vol 19 no 2 pp 72ndash78 2014

[27] S-J Lee J-H Lee and J-C Suh ldquoComputation of pressurefields around a two-dimensional circular cylinder using thevortex-in-cell and penalization methodsrdquo Modelling and Sim-ulation in Engineering vol 2014 Article ID 708372 13 pages2014

[28] J J Monaghan ldquoExtrapolating 119861-splines for interpolationrdquoJournal of Computational Physics vol 60 no 2 pp 253ndash2621985

[29] J J Monaghan ldquoParticle methods for hydrodynamicsrdquo Com-puter Physics Reports vol 3 no 2 pp 71ndash124 1985

Modelling and Simulation in Engineering 13

[30] O V Vasilyev and N K-R Kevlahan ldquoHybrid waveletcollocation-brinkman penalization method for complex geom-etry flowsrdquo International Journal for Numerical Methods inFluids vol 40 no 3-4 pp 531ndash538 2002

[31] W Iwakami Y Yatagai N Hatakeyama and Y Hattori ldquoAnew approach for error reduction in the volume penalizationmethodrdquoCommunications in Computational Physics vol 16 no5 pp 1181ndash1200 2014

[32] J R Mansfield O M Knio and C Meneveau ldquoTowardslagrangian large vortex simulationrdquo ESAIM Proceedings vol 1pp 49ndash64 1996

[33] G-H Cottet and P Poncet ldquoAdvances in direct numerical sim-ulations of 3D wall-bounded flows by vortex-in-cell methodsrdquoJournal of Computational Physics vol 193 no 1 pp 136ndash1582004

[34] B Caswell ldquoKinematics and stress on a surface of restrdquo Archivefor Rational Mechanics and Analysis vol 26 no 5 pp 385ndash3991967

[35] R S Rogallo and P Moin ldquoNumerical simulation of turbulentflowsrdquo Annual Review of Fluid Mechanics vol 16 pp 99ndash1371984

[36] J Meyers and P Sagaut ldquoOn the model coefficients for thestandard and the variational multi-scale Smagorinsky modelrdquoJournal of Fluid Mechanics vol 569 pp 287ndash319 2006

[37] J R Mansfield O M Knio and C Meneveau ldquoA dynamic LESscheme for the vorticity transport equation formulation and apriori testsrdquo Journal of Computational Physics vol 145 no 2 pp693ndash730 1998

[38] S B Pope Turbulent Flows Cambridge University Press 2000[39] WH Press S A TeukolskyW T Vetterling and B P Flannery

Numerical Recipes in C++ The Art of Scientific ComputingCambridge University Press 2nd edition 2002

[40] L Greengard and W D Gropp ldquoA parallel verison of the fastmultipoe methodrdquo Computers amp Mathematics with Applica-tions vol 20 no 7 pp 63ndash71 1990

[41] M Frigo and S G Johnson ldquoFFTW an adaptive software archi-tecture for the FFTrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo98) pp 1381ndash1384 May 1998

[42] M Frigo and S G Johnson ldquoThe design and implementationof FFTW3rdquo Proceedings of the IEEE vol 93 no 2 pp 216ndash2312005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Control Scienceand Engineering

Journal of

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International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Shock and Vibration

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

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Navigation and Observation

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DistributedSensor Networks

International Journal of

6 Modelling and Simulation in Engineering

xc

yc

045 05

minus002

0

002

A

BC

00054c

00322c120573

Figure 2 Trailing-edge shape Note that the points A B and C indicate the knuckles of the chamfer and trailing-edge wedge and the bevelangle is defined as 120573

Δ119905 = 000015 was chosen in order to guarantee numericalaccuracy for a Reynolds number of 106 The grid spacingof the 119894th domain Ω

119894is defined as ℎ

119894= 1205982119894minus1 to reduce

computation timeThe penalization parameter 120582 in (12) is setto be 108 The hydrofoil at a zero angle of attack is immersedin a Cartesian grid (the first domain) that does not conformto the surface of the bodyThe first domain with 4096times 1024lattice nodes is fixed and the other domains are dynamicallydetermined depending on the spatially evolving flow In thisstudy numerical simulations were conducted until 119879 = 10A new domain is created downstream of a body once thenumber of horizontal grid points exceeds 2048 Althoughthere is no limit on the number of vertical grid points it wastypically less than or equal to 1024 During simulation lessthan 08 million particles and four computational domainswere created The entire computation in each case includingstart-up and printing times took approximately 170 hourson 16 Intel Xeon64 33 GHz CPUs with 6GB memory perprocessor

53 Truncated Trailing-Edge For an impulsively started flowpast a hydrofoil at a chord-based Reynolds number of 106the wake pattern can be classified into three regimes basedon the time evolutions shown in Figure 3 In the very earlystages the wake remained very symmetrical and consistedprimarily of two large vortices in the near wake as shown inFigure 3(a) At that time the drag force gradually decreasedAs time went on the wake became asymmetrical as shown inFigure 3(b) and then shed a vortex as shown in Figure 3(c)After a number of shedding periods vortices were shedregularly from the symmetric trailing edge in the form of twotrains of oppositezsign but equal-strength vortices as shownin Figure 3(d) Such regular vortex shedding induces periodicloading on the structure as shown in Figure 4 It is wellknown that drag-force oscillation during vortex shedding ismuch smaller than the lift force Numerical results show thatoscillations in the drag force occurred at twice the vortex-shedding frequency due to the fact that two vortices wereshed from alternate sides during one full period of wakeoscillation

From our simulation the Strouhal number based on thetrailing-edge thickness St

119905= 119891119904119905TE119880infin is about 022 for

Re = 106 Ausoni [10] (also refer to Zobeiri et al [11])

had a similar finding and experimentally gave St119905= 025

which may be approximately read from their figure Theyused a hydrofoil with a 100mm chord length and 150mmspan length The aspect ratio defined as the ratio of the spanlength to the chord length was 15 A lower aspect ratioleads to a three-dimensional (3D) effect even though thehydrofoil is 2D Ausoni [10] also observed experimentallythat the shedding process was not induced in phase alongthe span It was therefore speculated in [25] that the slightunderestimation of the Strouhal number is due to 3D effectswhich were absent in our simulation

54 Oblique Trailing-Edges We conducted numerical simu-lations to observe vortex shedding from trailing edges withsix different bevel angles of 120573 = 0 10 20 40 60 and 70degrees Figure 5 shows the instantaneous vorticity distribu-tions at 119879 = 10 for Re = 106 At 120573 = 0∘ the vortex sheddingis regular and periodic As the bevel angle 120573 increasesvortex shedding becomes increasingly disorganized Suchirregular shedding of vortices demonstrates the periodicity ofoscillatory forces Figure 6 shows the power spectral densityfunction (PSD) of the drag and lift coefficients for differentbevel angles Compared with 120573 = 0∘ the amplitude of the liftforce oscillation for asymmetric trailing edges is significantlyreduced compared with the symmetric trailing edge at 120573 =0∘ The lift force amplitude is gradually decreased as the bevelangle increases while the drag force amplitude increases andthen decreases to between 120573 = 10∘ and 60∘The periodicity ofboth forces is almost lost at 120573 = 60∘ and there is no definiteperiodicity of the induced force components at 120573 = 70∘

At a bevel angle of 10∘ as shown in Figure 6(b) it is inter-esting that the dominant peak of the drag force is identical tothat of the lift force From a vibration standpoint the forcingfrequency is important for the prevention of resonance Thedistance between the two vortices in Figure 5(b) does not staythe same but the difference between the shedding vorticesbetween 120573 = 0

∘ and 10∘ (Figures 5(a) and 5(b)) is not

Modelling and Simulation in Engineering 7

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Symmetric wake at 119879 = 0396

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Symmetry-breaking transition at 119879 = 0900

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Nonperiodic vortex shedding at 119879 = 1098

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Fully developed periodic shedding at 119879 = 3500

Figure 3 Wake patterns of the NACA 0009 hydrofoil with a truncated trailing-edge (120573 = 0∘) for Re = 106

CL

CL

002

001

0

minus001

minus002I II III

T

0 05 1 15 2 25 3 35

CD

CD

0015

001

0005

0

Figure 4 Time evolution of the lift and drag coefficients for theimpulsively started NACA 0009 hydrofoil with a truncated trailingedge (120573 = 0∘) at Re = 106 Regime I denotes the developing regimeof a symmetric wake II a symmetry-breaking transition and IIIperiodic vortex shedding

statistically significant A possible reason for this is the phasedifference between the vortices shed from the upper andlower sides In the wake formation region the two vorticesshed from the symmetric trailing edge at 120573 = 0∘ are almostexactly out of phase (180 degrees) whereas at 120573 = 10

∘

there is a phase difference of approximately 30 degrees Thismeans that before one vortex is fully shed from the bodyanother vortex is created At higher bevel angles 120573 gt 20∘ thenegative vortices from the upper side of the hydrofoil becomestronger than the positive ones from the lower side Theoscillation force is induced mostly by the vortices shed fromthe acute (upper) side As a result the dominant frequency

of the drag force oscillation appears to be the same as thevortex shedding frequency We numerically observed thatdifferences in strength andor phase between two adjacentvortices cause nonperiodic and irregular vortex shedding

Figure 7 shows the shedding frequency of vortices fromdifferent beveled trailing edges compared with experimentalresults by Zobeiri et al [11] The numerically simulatedshedding frequencies were approximately 68 and 64 for120573 = 0

∘ and 60∘ respectively Numerical results showgood qualitative agreements with the experimental resultsbut the numerical simulations underestimated the vortex-shedding frequency by about 10 Figure 8 shows the meanvalue and standard deviation of the velocity components Wecompared our numerical results with the experimental resultsmeasured by Zobeiri et al [11] The truncated and obliquetrailing edges in Figure 8 indicate bevel angles of 120573 = 0

∘

and 60∘ respectively For the truncated trailing edge themean velocity profiles are more symmetric than those in theexperimental data The results from numerical simulationare in good agreement with the experimental results Weconclude that the vortex shedding phenomenon from thetrailing edge of a 2D hydrofoil can be reasonably simulatedusing 2D LES which considerably reduces the computationalcost

55 Effect of a Periodically Varying Flow on Vortex SheddingThe propeller is generally located behind the shiprsquos hull andis subjected to a nonaxisymmetric wake field Consideringa 2D flow the propeller section effectively experiences peri-odic variation in flow incidence We conducted numericalsimulations to investigate vortex shedding with respect tothe sinusoidal motions of the free-stream flow The hydrofoilwas at rest and the flow became the oscillator under theseconditions In the current study the angle of attack varied

8 Modelling and Simulation in Engineering

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Bevel angle 120573 = 0∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Bevel angle 120573 = 10∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Bevel angle 120573 = 20∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Bevel angle 120573 = 40∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(e) Bevel angle 120573 = 60∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(f) Bevel angle 120573 = 70∘

Figure 5 Instantaneous vorticity distributions at 119879 = 10 for Re = 106

sinusoidally with time 119905 120572 = 120572119900sin(2120587119891

infin119905) where 119891

infin

was the frequency of the free-stream flow oscillation andits oscillation amplitude was restricted at 120572

119900= 2∘ The

magnitude of the free-stream velocity was kept constant Thetested model is the NACA 0009 hydrofoil with a beveledtrailing edge of 120573 = 60∘

Figure 9 shows the instantaneous vorticity distributionsat119879 = 10 and the PSD of the drag coefficients at four differentfrequencies of the free-stream flow that is 119891

infin= 15 20

25 and 30 There are two peaks in the PSD functions thefirst peak appears around 119891lowast asymp 6 and the second one isidentical to the frequency of the free-stream flow oscillationThe obvious feature recognized in all the PSD functions asshown in Figure 9 is that the first peak frequency is nearlyidentical to the vortex-shedding frequency from the hydrofoilin the uniform free-stream flow Vortex formation in the nearwake is thought to be less affected by small inflow oscillationsMost interestingly at 119891

infin= 20 vortices are regularly shed

from the trailing edge as shown in Figure 9(c) In the PSDfunctions drag oscillation induced by vortex shedding isclearly observed This means that a particular oscillationfrequency in the free-stream velocity can cause regular andperiodic vortex shedding This is significant with respect tochanging a marine propellerrsquos angle of attack over time

6 Concluding Remarks

We simulated impulsively started flows past a modifiedNACA 0009 hydrofoil at a Reynolds number of 106 usingthe hybrid particle-mesh method in conjunction with thevorticity-based subgrid scale model We have achieved sig-nificant savings in both memory usage and computationtime by making improvements in the numerical schemesthe particle-based domain decomposition the use ofmultipledomains and the approximation of domain boundary con-ditions These techniques are simple and easy to implementbut also effective in reducing the computational costs Theparticle-based domain decomposition scheme is suitable fora distributed system in which each processor has its ownprivate memory (distributed memory)

In this study the numerical simulation results nearlymatch the experimental resultsThis demonstrates that vortexshedding from the trailing edge of a 2D hydrofoil can bereasonably simulated with a 2D LES It is due to the fact thatthe flow is characterized by quasi-two dimensionality withvortex shedding from the trailing edge of the hydrofoil Wehave considered that 2D vortex shedding simulations shouldnot be dismissed so easily With two-dimensionality it ispossible to solve the problem much less expensively since

Modelling and Simulation in Engineering 9

0 5 10 15 20 250

10

0 5 10 15 20 250

05times10minus5

PSD

ofC

L

PSD

ofC

D

times10minus6

flowast flowast

(a) Truncated trailing edge 120573 = 0∘

0

08

0

08

0

08

0

08

0

08

0

04

0

04

0

04

0

04

0

04

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

(b) Beveled trailing edge 120573 gt 0∘

Figure 6 Power spectral density (PSD) of the lift and drag coefficients 119891lowast denotes the frequency normalized by the chord length and theundisturbed flow velocity

3D problems require significant computational resourcesHowever the LES should be 3D since turbulence is inherently3D While we accept that more realistic 3D simulation isrequired physically a 2D LES can be used as a basis forcomparison prior to 3D implementation Once the capability

of a numerical algorithm is evaluated in two dimensionsit can be easily extended into three dimensions This paperis intended as a guide for researchers and designers whoare working on a similar problem vortex shedding from ahydrofoil at high Reynolds numbers

10 Modelling and Simulation in Engineering

0 10 20 30 40 50 600

2

4

6

8

10

Zobeiri et al (2012) expPresent cal

flowast

120573 (deg)

Figure 7 Shedding frequency of vortices from the different beveled trailing edges for Re = 106 The experimental data by Zobeiri et al [11]were derived from LDVmeasurements and the approximate shedding frequency as read from their figure is normalized by the chord lengthand the undisturbed flow velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02 04 06 08 1uxmeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(a) Normalized mean of streamwise velocity

minus06

minus04

minus02

0

02

04

06yt

max

0 01 02 03 04uxstdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(b) Normalized mean of transverse velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02uymeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(c) Normalized standard deviation of streamwise velocity

minus06

minus04

minus02

0

02

04

06

yt

max

0 01 02 03 04 05uystdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(d) Normalized standard deviation of transverse velocity

Figure 8 Mean and standard deviation of velocities at 119909119888 = 0516 for Re = 106 The experimental data by Zobeiri et al [11] were measuredat 119909119888 = 052 and 0528 for truncated (120573 = 0∘) and oblique (120573 = 60∘) trailing edges respectively and the reported Reynolds number of theflow is about 2 times 106

Modelling and Simulation in Engineering 11

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(a) Steady flow 119891infin = 0

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(b) Incoming flow frequency 119891infin = 15

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(c) Incoming flow frequency 119891infin = 20

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(d) Incoming flow frequency 119891infin = 25

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(e) Incoming flow frequency 119891infin = 30

Figure 9 Effect of the incoming flow frequency on vortex shedding from a beveled trailing edge of 120573 = 60∘ at 119891infin= 15 20 25 and 30

The contour indicates the instantaneous vorticity distribution at 119879 = 10 and the power spectral density (PSD) is calculated from the dragcoefficients in the range 119879 = [3 10]

12 Modelling and Simulation in Engineering

We found in this study that the vortex formation char-acteristic in periodically varying inflows remains unchangedA certain frequency of free-stream flow oscillation tends tomake vortex shedding more regular again This means thatthe periodicity of flow incidence can stimulate an inherentfrequency of vortex shedding from a hydrofoil Howeverthe effect of inflow oscillation on vortex shedding is notyet fully understood Future work will include experimentalinvestigations as well as a 3D flow simulation to betterunderstand the relationship between inflow oscillation andvortex shedding

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Korean Institute ofOcean Science amp Technology sponsored by the Ministry ofTrade Industry amp Energy (no 10033668) and the IndustrialConvergence Strategic Technology Development Program(no 10044499) funded by the Ministry of Trade Industry ampEnergy (MI Korea)

References

[1] E Berger and R Wille ldquoPeriodic flow phenomenardquo AnnualReview of Fluid Mechanics vol 4 no 1 pp 313ndash340 1972

[2] P W Bearman ldquoVortex shedding from oscillating bluff bodiesrdquoAnnual Review of Fluid Mechanics vol 16 pp 195ndash222 1984

[3] C H K Williamson ldquoVortex dynamics in the cylinder wakerdquoAnnual Review of Fluid Mechanics vol 28 pp 477ndash539 1996

[4] C H K Williamson ldquoVortex-induced vibrationsrdquo AnnualReview of Fluid Mechanics vol 36 pp 413ndash455 2004

[5] R Fischer ldquoSinging propellersmdashsolutions and case historiesrdquoMarine Technology vol 45 no 4 pp 221ndash227 2008

[6] D A Bourgoyne J M Hamel S L Ceccio and D R DowlingldquoTime-averaged flow over a hydrofoil at high reynolds numberrdquoJournal of Fluid Mechanics vol 496 pp 365ndash404 2003

[7] D A Bourgoyne S L Ceccio and D R Dowling ldquoVortexshedding from a hydrofoil at high Reynolds numberrdquo Journalof Fluid Mechanics vol 531 pp 293ndash324 2005

[8] N Mulvany J Tu L Chen and B Anderson ldquoAssessment oftwo-equation turbulence modelling for high Reynolds numberhydrofoil flowsrdquo International Journal for Numerical Methods inFluids vol 45 no 3 pp 275ndash299 2004

[9] P Ausoni M Farhat X Escaler E Egusquiza and F AvellanldquoCavitation influence on von Karman vortex shedding andinduced hydrofoil vibrationsrdquo Journal of Fluids EngineeringTransactions vol 129 no 8 pp 966ndash973 2007

[10] P Ausoni Turbulent vortex shedding from a blunt trailingedge hydrofoil [PhD thesis] Ecole Polytechnique Federale deLausanne Lausanne Switzerland 2009

[11] A Zobeiri P Ausoni F Avellan and M Farhat ldquoHow obliquetrailing edge of a hydrofoil reduces the vortex-induced vibra-tionrdquo Journal of Fluids and Structures vol 32 pp 78ndash89 2012

[12] G-H Cottet and P D Koumoutsakos Vortex Methods Theoryand Practice Cambridge University Press 2000

[13] G Cottet and P Poncet ldquoParticle methods for direct numericalsimulations of three-dimensional wakesrdquo Journal of Turbulencevol 3 article N38 2002

[14] P Chatelain A Curioni M Bergdorf D Rossinelli WAndreoni and P Koumoutsakos ldquoBillion vortex particle directnumerical simulations of aircraft wakesrdquo Computer Methods inAppliedMechanics and Engineering vol 197 no 13ndash16 pp 1296ndash1304 2008

[15] Y-C Kim J-C Suh and K-J Lee ldquoVortex-in-cell methodcombined with a boundary element method for incompressibleviscous flow analysisrdquo International Journal forNumericalMeth-ods in Fluids vol 69 no 10 pp 1567ndash1583 2012

[16] P Angot C-H Bruneau and P Fabrie ldquoA penalization methodto take into account obstacles in incompressible viscous flowsrdquoNumerische Mathematik vol 81 no 4 pp 497ndash520 1999

[17] M El Ossmani and P Poncet ldquoEfficiency of multiscale hybridgrid-particle vortex methodsrdquo Multiscale Modeling amp Simula-tion vol 8 no 5 pp 1671ndash1690 2010

[18] J T Rasmussen G-H Cottet and J HWalther ldquoAmultiresolu-tion remeshed vortex-in-cell algorithm using patchesrdquo Journalof Computational Physics vol 230 no 17 pp 6742ndash6755 2011

[19] M Coquerelle and G-H Cottet ldquoA vortex level set method forthe two-way coupling of an incompressible fluid with collidingrigid bodiesrdquo Journal of Computational Physics vol 227 no 21pp 9121ndash9137 2008

[20] F Morency H Beaugendre F Gallizio and S Laurens ldquoCom-putation of ice shedding trajectories using cartesian gridspenalization and level setsrdquo Modelling and Simulation in Engi-neering vol 2011 Article ID 274947 15 pages 2011

[21] M M Hejlesen P Koumoutsakos A Leonard and J HWalther ldquoIterative Brinkman penalization for remeshed vortexmethodsrdquo Journal of Computational Physics vol 280 pp 547ndash562 2015

[22] M Gazzola P Chatelain W M van Rees and P Koumout-sakos ldquoSimulations of single and multiple swimmers with non-divergence free deforming geometriesrdquo Journal of Computa-tional Physics vol 230 no 19 pp 7093ndash7114 2011

[23] R Chatelin and P Poncet ldquoA hybrid grid-particle method formoving bodies in 3D stokes flow with variable viscosityrdquo SIAMJournal on Scientific Computing vol 35 no 4 pp B925ndashB9492013

[24] R Chatelin and P Poncet ldquoHybrid grid-particle methodsand penalization a Sherman-Morrison-Woodbury approach tocompute 3D viscous flows using FFTrdquo Journal of ComputationalPhysics vol 269 pp 314ndash328 2014

[25] S J Lee J H Lee and J C Suh ldquoVortex shedding frequencyfor a 2D hydrofoil with a truncated trailing edgerdquo Journal of theSociety of Naval Architects of Korea vol 51 no 6 pp 480ndash4882014 (Korean)

[26] S-J Lee and J C Suh ldquoA multi-domain approach for ahybrid particle-mesh methodrdquo Journal of Computational FluidsEngineering vol 19 no 2 pp 72ndash78 2014

[27] S-J Lee J-H Lee and J-C Suh ldquoComputation of pressurefields around a two-dimensional circular cylinder using thevortex-in-cell and penalization methodsrdquo Modelling and Sim-ulation in Engineering vol 2014 Article ID 708372 13 pages2014

[28] J J Monaghan ldquoExtrapolating 119861-splines for interpolationrdquoJournal of Computational Physics vol 60 no 2 pp 253ndash2621985

[29] J J Monaghan ldquoParticle methods for hydrodynamicsrdquo Com-puter Physics Reports vol 3 no 2 pp 71ndash124 1985

Modelling and Simulation in Engineering 13

[30] O V Vasilyev and N K-R Kevlahan ldquoHybrid waveletcollocation-brinkman penalization method for complex geom-etry flowsrdquo International Journal for Numerical Methods inFluids vol 40 no 3-4 pp 531ndash538 2002

[31] W Iwakami Y Yatagai N Hatakeyama and Y Hattori ldquoAnew approach for error reduction in the volume penalizationmethodrdquoCommunications in Computational Physics vol 16 no5 pp 1181ndash1200 2014

[32] J R Mansfield O M Knio and C Meneveau ldquoTowardslagrangian large vortex simulationrdquo ESAIM Proceedings vol 1pp 49ndash64 1996

[33] G-H Cottet and P Poncet ldquoAdvances in direct numerical sim-ulations of 3D wall-bounded flows by vortex-in-cell methodsrdquoJournal of Computational Physics vol 193 no 1 pp 136ndash1582004

[34] B Caswell ldquoKinematics and stress on a surface of restrdquo Archivefor Rational Mechanics and Analysis vol 26 no 5 pp 385ndash3991967

[35] R S Rogallo and P Moin ldquoNumerical simulation of turbulentflowsrdquo Annual Review of Fluid Mechanics vol 16 pp 99ndash1371984

[36] J Meyers and P Sagaut ldquoOn the model coefficients for thestandard and the variational multi-scale Smagorinsky modelrdquoJournal of Fluid Mechanics vol 569 pp 287ndash319 2006

[37] J R Mansfield O M Knio and C Meneveau ldquoA dynamic LESscheme for the vorticity transport equation formulation and apriori testsrdquo Journal of Computational Physics vol 145 no 2 pp693ndash730 1998

[38] S B Pope Turbulent Flows Cambridge University Press 2000[39] WH Press S A TeukolskyW T Vetterling and B P Flannery

Numerical Recipes in C++ The Art of Scientific ComputingCambridge University Press 2nd edition 2002

[40] L Greengard and W D Gropp ldquoA parallel verison of the fastmultipoe methodrdquo Computers amp Mathematics with Applica-tions vol 20 no 7 pp 63ndash71 1990

[41] M Frigo and S G Johnson ldquoFFTW an adaptive software archi-tecture for the FFTrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo98) pp 1381ndash1384 May 1998

[42] M Frigo and S G Johnson ldquoThe design and implementationof FFTW3rdquo Proceedings of the IEEE vol 93 no 2 pp 216ndash2312005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Modelling and Simulation in Engineering 7

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Symmetric wake at 119879 = 0396

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Symmetry-breaking transition at 119879 = 0900

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Nonperiodic vortex shedding at 119879 = 1098

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Fully developed periodic shedding at 119879 = 3500

Figure 3 Wake patterns of the NACA 0009 hydrofoil with a truncated trailing-edge (120573 = 0∘) for Re = 106

CL

CL

002

001

0

minus001

minus002I II III

T

0 05 1 15 2 25 3 35

CD

CD

0015

001

0005

0

Figure 4 Time evolution of the lift and drag coefficients for theimpulsively started NACA 0009 hydrofoil with a truncated trailingedge (120573 = 0∘) at Re = 106 Regime I denotes the developing regimeof a symmetric wake II a symmetry-breaking transition and IIIperiodic vortex shedding

statistically significant A possible reason for this is the phasedifference between the vortices shed from the upper andlower sides In the wake formation region the two vorticesshed from the symmetric trailing edge at 120573 = 0∘ are almostexactly out of phase (180 degrees) whereas at 120573 = 10

∘

there is a phase difference of approximately 30 degrees Thismeans that before one vortex is fully shed from the bodyanother vortex is created At higher bevel angles 120573 gt 20∘ thenegative vortices from the upper side of the hydrofoil becomestronger than the positive ones from the lower side Theoscillation force is induced mostly by the vortices shed fromthe acute (upper) side As a result the dominant frequency

of the drag force oscillation appears to be the same as thevortex shedding frequency We numerically observed thatdifferences in strength andor phase between two adjacentvortices cause nonperiodic and irregular vortex shedding

Figure 7 shows the shedding frequency of vortices fromdifferent beveled trailing edges compared with experimentalresults by Zobeiri et al [11] The numerically simulatedshedding frequencies were approximately 68 and 64 for120573 = 0

∘ and 60∘ respectively Numerical results showgood qualitative agreements with the experimental resultsbut the numerical simulations underestimated the vortex-shedding frequency by about 10 Figure 8 shows the meanvalue and standard deviation of the velocity components Wecompared our numerical results with the experimental resultsmeasured by Zobeiri et al [11] The truncated and obliquetrailing edges in Figure 8 indicate bevel angles of 120573 = 0

∘

and 60∘ respectively For the truncated trailing edge themean velocity profiles are more symmetric than those in theexperimental data The results from numerical simulationare in good agreement with the experimental results Weconclude that the vortex shedding phenomenon from thetrailing edge of a 2D hydrofoil can be reasonably simulatedusing 2D LES which considerably reduces the computationalcost

55 Effect of a Periodically Varying Flow on Vortex SheddingThe propeller is generally located behind the shiprsquos hull andis subjected to a nonaxisymmetric wake field Consideringa 2D flow the propeller section effectively experiences peri-odic variation in flow incidence We conducted numericalsimulations to investigate vortex shedding with respect tothe sinusoidal motions of the free-stream flow The hydrofoilwas at rest and the flow became the oscillator under theseconditions In the current study the angle of attack varied

8 Modelling and Simulation in Engineering

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Bevel angle 120573 = 0∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Bevel angle 120573 = 10∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Bevel angle 120573 = 20∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Bevel angle 120573 = 40∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(e) Bevel angle 120573 = 60∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(f) Bevel angle 120573 = 70∘

Figure 5 Instantaneous vorticity distributions at 119879 = 10 for Re = 106

sinusoidally with time 119905 120572 = 120572119900sin(2120587119891

infin119905) where 119891

infin

was the frequency of the free-stream flow oscillation andits oscillation amplitude was restricted at 120572

119900= 2∘ The

magnitude of the free-stream velocity was kept constant Thetested model is the NACA 0009 hydrofoil with a beveledtrailing edge of 120573 = 60∘

Figure 9 shows the instantaneous vorticity distributionsat119879 = 10 and the PSD of the drag coefficients at four differentfrequencies of the free-stream flow that is 119891

infin= 15 20

25 and 30 There are two peaks in the PSD functions thefirst peak appears around 119891lowast asymp 6 and the second one isidentical to the frequency of the free-stream flow oscillationThe obvious feature recognized in all the PSD functions asshown in Figure 9 is that the first peak frequency is nearlyidentical to the vortex-shedding frequency from the hydrofoilin the uniform free-stream flow Vortex formation in the nearwake is thought to be less affected by small inflow oscillationsMost interestingly at 119891

infin= 20 vortices are regularly shed

from the trailing edge as shown in Figure 9(c) In the PSDfunctions drag oscillation induced by vortex shedding isclearly observed This means that a particular oscillationfrequency in the free-stream velocity can cause regular andperiodic vortex shedding This is significant with respect tochanging a marine propellerrsquos angle of attack over time

6 Concluding Remarks

We simulated impulsively started flows past a modifiedNACA 0009 hydrofoil at a Reynolds number of 106 usingthe hybrid particle-mesh method in conjunction with thevorticity-based subgrid scale model We have achieved sig-nificant savings in both memory usage and computationtime by making improvements in the numerical schemesthe particle-based domain decomposition the use ofmultipledomains and the approximation of domain boundary con-ditions These techniques are simple and easy to implementbut also effective in reducing the computational costs Theparticle-based domain decomposition scheme is suitable fora distributed system in which each processor has its ownprivate memory (distributed memory)

In this study the numerical simulation results nearlymatch the experimental resultsThis demonstrates that vortexshedding from the trailing edge of a 2D hydrofoil can bereasonably simulated with a 2D LES It is due to the fact thatthe flow is characterized by quasi-two dimensionality withvortex shedding from the trailing edge of the hydrofoil Wehave considered that 2D vortex shedding simulations shouldnot be dismissed so easily With two-dimensionality it ispossible to solve the problem much less expensively since

Modelling and Simulation in Engineering 9

0 5 10 15 20 250

10

0 5 10 15 20 250

05times10minus5

PSD

ofC

L

PSD

ofC

D

times10minus6

flowast flowast

(a) Truncated trailing edge 120573 = 0∘

0

08

0

08

0

08

0

08

0

08

0

04

0

04

0

04

0

04

0

04

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

(b) Beveled trailing edge 120573 gt 0∘

Figure 6 Power spectral density (PSD) of the lift and drag coefficients 119891lowast denotes the frequency normalized by the chord length and theundisturbed flow velocity

3D problems require significant computational resourcesHowever the LES should be 3D since turbulence is inherently3D While we accept that more realistic 3D simulation isrequired physically a 2D LES can be used as a basis forcomparison prior to 3D implementation Once the capability

of a numerical algorithm is evaluated in two dimensionsit can be easily extended into three dimensions This paperis intended as a guide for researchers and designers whoare working on a similar problem vortex shedding from ahydrofoil at high Reynolds numbers

10 Modelling and Simulation in Engineering

0 10 20 30 40 50 600

2

4

6

8

10

Zobeiri et al (2012) expPresent cal

flowast

120573 (deg)

Figure 7 Shedding frequency of vortices from the different beveled trailing edges for Re = 106 The experimental data by Zobeiri et al [11]were derived from LDVmeasurements and the approximate shedding frequency as read from their figure is normalized by the chord lengthand the undisturbed flow velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02 04 06 08 1uxmeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(a) Normalized mean of streamwise velocity

minus06

minus04

minus02

0

02

04

06yt

max

0 01 02 03 04uxstdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(b) Normalized mean of transverse velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02uymeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(c) Normalized standard deviation of streamwise velocity

minus06

minus04

minus02

0

02

04

06

yt

max

0 01 02 03 04 05uystdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(d) Normalized standard deviation of transverse velocity

Figure 8 Mean and standard deviation of velocities at 119909119888 = 0516 for Re = 106 The experimental data by Zobeiri et al [11] were measuredat 119909119888 = 052 and 0528 for truncated (120573 = 0∘) and oblique (120573 = 60∘) trailing edges respectively and the reported Reynolds number of theflow is about 2 times 106

Modelling and Simulation in Engineering 11

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(a) Steady flow 119891infin = 0

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(b) Incoming flow frequency 119891infin = 15

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(c) Incoming flow frequency 119891infin = 20

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(d) Incoming flow frequency 119891infin = 25

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(e) Incoming flow frequency 119891infin = 30

Figure 9 Effect of the incoming flow frequency on vortex shedding from a beveled trailing edge of 120573 = 60∘ at 119891infin= 15 20 25 and 30

The contour indicates the instantaneous vorticity distribution at 119879 = 10 and the power spectral density (PSD) is calculated from the dragcoefficients in the range 119879 = [3 10]

12 Modelling and Simulation in Engineering

We found in this study that the vortex formation char-acteristic in periodically varying inflows remains unchangedA certain frequency of free-stream flow oscillation tends tomake vortex shedding more regular again This means thatthe periodicity of flow incidence can stimulate an inherentfrequency of vortex shedding from a hydrofoil Howeverthe effect of inflow oscillation on vortex shedding is notyet fully understood Future work will include experimentalinvestigations as well as a 3D flow simulation to betterunderstand the relationship between inflow oscillation andvortex shedding

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Korean Institute ofOcean Science amp Technology sponsored by the Ministry ofTrade Industry amp Energy (no 10033668) and the IndustrialConvergence Strategic Technology Development Program(no 10044499) funded by the Ministry of Trade Industry ampEnergy (MI Korea)

References

[1] E Berger and R Wille ldquoPeriodic flow phenomenardquo AnnualReview of Fluid Mechanics vol 4 no 1 pp 313ndash340 1972

[2] P W Bearman ldquoVortex shedding from oscillating bluff bodiesrdquoAnnual Review of Fluid Mechanics vol 16 pp 195ndash222 1984

[3] C H K Williamson ldquoVortex dynamics in the cylinder wakerdquoAnnual Review of Fluid Mechanics vol 28 pp 477ndash539 1996

[4] C H K Williamson ldquoVortex-induced vibrationsrdquo AnnualReview of Fluid Mechanics vol 36 pp 413ndash455 2004

[5] R Fischer ldquoSinging propellersmdashsolutions and case historiesrdquoMarine Technology vol 45 no 4 pp 221ndash227 2008

[6] D A Bourgoyne J M Hamel S L Ceccio and D R DowlingldquoTime-averaged flow over a hydrofoil at high reynolds numberrdquoJournal of Fluid Mechanics vol 496 pp 365ndash404 2003

[7] D A Bourgoyne S L Ceccio and D R Dowling ldquoVortexshedding from a hydrofoil at high Reynolds numberrdquo Journalof Fluid Mechanics vol 531 pp 293ndash324 2005

[8] N Mulvany J Tu L Chen and B Anderson ldquoAssessment oftwo-equation turbulence modelling for high Reynolds numberhydrofoil flowsrdquo International Journal for Numerical Methods inFluids vol 45 no 3 pp 275ndash299 2004

[9] P Ausoni M Farhat X Escaler E Egusquiza and F AvellanldquoCavitation influence on von Karman vortex shedding andinduced hydrofoil vibrationsrdquo Journal of Fluids EngineeringTransactions vol 129 no 8 pp 966ndash973 2007

[10] P Ausoni Turbulent vortex shedding from a blunt trailingedge hydrofoil [PhD thesis] Ecole Polytechnique Federale deLausanne Lausanne Switzerland 2009

[11] A Zobeiri P Ausoni F Avellan and M Farhat ldquoHow obliquetrailing edge of a hydrofoil reduces the vortex-induced vibra-tionrdquo Journal of Fluids and Structures vol 32 pp 78ndash89 2012

[12] G-H Cottet and P D Koumoutsakos Vortex Methods Theoryand Practice Cambridge University Press 2000

[13] G Cottet and P Poncet ldquoParticle methods for direct numericalsimulations of three-dimensional wakesrdquo Journal of Turbulencevol 3 article N38 2002

[14] P Chatelain A Curioni M Bergdorf D Rossinelli WAndreoni and P Koumoutsakos ldquoBillion vortex particle directnumerical simulations of aircraft wakesrdquo Computer Methods inAppliedMechanics and Engineering vol 197 no 13ndash16 pp 1296ndash1304 2008

[15] Y-C Kim J-C Suh and K-J Lee ldquoVortex-in-cell methodcombined with a boundary element method for incompressibleviscous flow analysisrdquo International Journal forNumericalMeth-ods in Fluids vol 69 no 10 pp 1567ndash1583 2012

[16] P Angot C-H Bruneau and P Fabrie ldquoA penalization methodto take into account obstacles in incompressible viscous flowsrdquoNumerische Mathematik vol 81 no 4 pp 497ndash520 1999

[17] M El Ossmani and P Poncet ldquoEfficiency of multiscale hybridgrid-particle vortex methodsrdquo Multiscale Modeling amp Simula-tion vol 8 no 5 pp 1671ndash1690 2010

[18] J T Rasmussen G-H Cottet and J HWalther ldquoAmultiresolu-tion remeshed vortex-in-cell algorithm using patchesrdquo Journalof Computational Physics vol 230 no 17 pp 6742ndash6755 2011

[19] M Coquerelle and G-H Cottet ldquoA vortex level set method forthe two-way coupling of an incompressible fluid with collidingrigid bodiesrdquo Journal of Computational Physics vol 227 no 21pp 9121ndash9137 2008

[20] F Morency H Beaugendre F Gallizio and S Laurens ldquoCom-putation of ice shedding trajectories using cartesian gridspenalization and level setsrdquo Modelling and Simulation in Engi-neering vol 2011 Article ID 274947 15 pages 2011

[21] M M Hejlesen P Koumoutsakos A Leonard and J HWalther ldquoIterative Brinkman penalization for remeshed vortexmethodsrdquo Journal of Computational Physics vol 280 pp 547ndash562 2015

[22] M Gazzola P Chatelain W M van Rees and P Koumout-sakos ldquoSimulations of single and multiple swimmers with non-divergence free deforming geometriesrdquo Journal of Computa-tional Physics vol 230 no 19 pp 7093ndash7114 2011

[23] R Chatelin and P Poncet ldquoA hybrid grid-particle method formoving bodies in 3D stokes flow with variable viscosityrdquo SIAMJournal on Scientific Computing vol 35 no 4 pp B925ndashB9492013

[24] R Chatelin and P Poncet ldquoHybrid grid-particle methodsand penalization a Sherman-Morrison-Woodbury approach tocompute 3D viscous flows using FFTrdquo Journal of ComputationalPhysics vol 269 pp 314ndash328 2014

[25] S J Lee J H Lee and J C Suh ldquoVortex shedding frequencyfor a 2D hydrofoil with a truncated trailing edgerdquo Journal of theSociety of Naval Architects of Korea vol 51 no 6 pp 480ndash4882014 (Korean)

[26] S-J Lee and J C Suh ldquoA multi-domain approach for ahybrid particle-mesh methodrdquo Journal of Computational FluidsEngineering vol 19 no 2 pp 72ndash78 2014

[27] S-J Lee J-H Lee and J-C Suh ldquoComputation of pressurefields around a two-dimensional circular cylinder using thevortex-in-cell and penalization methodsrdquo Modelling and Sim-ulation in Engineering vol 2014 Article ID 708372 13 pages2014

[28] J J Monaghan ldquoExtrapolating 119861-splines for interpolationrdquoJournal of Computational Physics vol 60 no 2 pp 253ndash2621985

[29] J J Monaghan ldquoParticle methods for hydrodynamicsrdquo Com-puter Physics Reports vol 3 no 2 pp 71ndash124 1985

Modelling and Simulation in Engineering 13

[30] O V Vasilyev and N K-R Kevlahan ldquoHybrid waveletcollocation-brinkman penalization method for complex geom-etry flowsrdquo International Journal for Numerical Methods inFluids vol 40 no 3-4 pp 531ndash538 2002

[31] W Iwakami Y Yatagai N Hatakeyama and Y Hattori ldquoAnew approach for error reduction in the volume penalizationmethodrdquoCommunications in Computational Physics vol 16 no5 pp 1181ndash1200 2014

[32] J R Mansfield O M Knio and C Meneveau ldquoTowardslagrangian large vortex simulationrdquo ESAIM Proceedings vol 1pp 49ndash64 1996

[33] G-H Cottet and P Poncet ldquoAdvances in direct numerical sim-ulations of 3D wall-bounded flows by vortex-in-cell methodsrdquoJournal of Computational Physics vol 193 no 1 pp 136ndash1582004

[34] B Caswell ldquoKinematics and stress on a surface of restrdquo Archivefor Rational Mechanics and Analysis vol 26 no 5 pp 385ndash3991967

[35] R S Rogallo and P Moin ldquoNumerical simulation of turbulentflowsrdquo Annual Review of Fluid Mechanics vol 16 pp 99ndash1371984

[36] J Meyers and P Sagaut ldquoOn the model coefficients for thestandard and the variational multi-scale Smagorinsky modelrdquoJournal of Fluid Mechanics vol 569 pp 287ndash319 2006

[37] J R Mansfield O M Knio and C Meneveau ldquoA dynamic LESscheme for the vorticity transport equation formulation and apriori testsrdquo Journal of Computational Physics vol 145 no 2 pp693ndash730 1998

[38] S B Pope Turbulent Flows Cambridge University Press 2000[39] WH Press S A TeukolskyW T Vetterling and B P Flannery

Numerical Recipes in C++ The Art of Scientific ComputingCambridge University Press 2nd edition 2002

[40] L Greengard and W D Gropp ldquoA parallel verison of the fastmultipoe methodrdquo Computers amp Mathematics with Applica-tions vol 20 no 7 pp 63ndash71 1990

[41] M Frigo and S G Johnson ldquoFFTW an adaptive software archi-tecture for the FFTrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo98) pp 1381ndash1384 May 1998

[42] M Frigo and S G Johnson ldquoThe design and implementationof FFTW3rdquo Proceedings of the IEEE vol 93 no 2 pp 216ndash2312005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Modelling and Simulation in Engineering

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(a) Bevel angle 120573 = 0∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(b) Bevel angle 120573 = 10∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(c) Bevel angle 120573 = 20∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(d) Bevel angle 120573 = 40∘

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(e) Bevel angle 120573 = 60∘xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

(f) Bevel angle 120573 = 70∘

Figure 5 Instantaneous vorticity distributions at 119879 = 10 for Re = 106

sinusoidally with time 119905 120572 = 120572119900sin(2120587119891

infin119905) where 119891

infin

was the frequency of the free-stream flow oscillation andits oscillation amplitude was restricted at 120572

119900= 2∘ The

magnitude of the free-stream velocity was kept constant Thetested model is the NACA 0009 hydrofoil with a beveledtrailing edge of 120573 = 60∘

Figure 9 shows the instantaneous vorticity distributionsat119879 = 10 and the PSD of the drag coefficients at four differentfrequencies of the free-stream flow that is 119891

infin= 15 20

25 and 30 There are two peaks in the PSD functions thefirst peak appears around 119891lowast asymp 6 and the second one isidentical to the frequency of the free-stream flow oscillationThe obvious feature recognized in all the PSD functions asshown in Figure 9 is that the first peak frequency is nearlyidentical to the vortex-shedding frequency from the hydrofoilin the uniform free-stream flow Vortex formation in the nearwake is thought to be less affected by small inflow oscillationsMost interestingly at 119891

infin= 20 vortices are regularly shed

from the trailing edge as shown in Figure 9(c) In the PSDfunctions drag oscillation induced by vortex shedding isclearly observed This means that a particular oscillationfrequency in the free-stream velocity can cause regular andperiodic vortex shedding This is significant with respect tochanging a marine propellerrsquos angle of attack over time

6 Concluding Remarks

We simulated impulsively started flows past a modifiedNACA 0009 hydrofoil at a Reynolds number of 106 usingthe hybrid particle-mesh method in conjunction with thevorticity-based subgrid scale model We have achieved sig-nificant savings in both memory usage and computationtime by making improvements in the numerical schemesthe particle-based domain decomposition the use ofmultipledomains and the approximation of domain boundary con-ditions These techniques are simple and easy to implementbut also effective in reducing the computational costs Theparticle-based domain decomposition scheme is suitable fora distributed system in which each processor has its ownprivate memory (distributed memory)

In this study the numerical simulation results nearlymatch the experimental resultsThis demonstrates that vortexshedding from the trailing edge of a 2D hydrofoil can bereasonably simulated with a 2D LES It is due to the fact thatthe flow is characterized by quasi-two dimensionality withvortex shedding from the trailing edge of the hydrofoil Wehave considered that 2D vortex shedding simulations shouldnot be dismissed so easily With two-dimensionality it ispossible to solve the problem much less expensively since

Modelling and Simulation in Engineering 9

0 5 10 15 20 250

10

0 5 10 15 20 250

05times10minus5

PSD

ofC

L

PSD

ofC

D

times10minus6

flowast flowast

(a) Truncated trailing edge 120573 = 0∘

0

08

0

08

0

08

0

08

0

08

0

04

0

04

0

04

0

04

0

04

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

(b) Beveled trailing edge 120573 gt 0∘

Figure 6 Power spectral density (PSD) of the lift and drag coefficients 119891lowast denotes the frequency normalized by the chord length and theundisturbed flow velocity

3D problems require significant computational resourcesHowever the LES should be 3D since turbulence is inherently3D While we accept that more realistic 3D simulation isrequired physically a 2D LES can be used as a basis forcomparison prior to 3D implementation Once the capability

of a numerical algorithm is evaluated in two dimensionsit can be easily extended into three dimensions This paperis intended as a guide for researchers and designers whoare working on a similar problem vortex shedding from ahydrofoil at high Reynolds numbers

10 Modelling and Simulation in Engineering

0 10 20 30 40 50 600

2

4

6

8

10

Zobeiri et al (2012) expPresent cal

flowast

120573 (deg)

Figure 7 Shedding frequency of vortices from the different beveled trailing edges for Re = 106 The experimental data by Zobeiri et al [11]were derived from LDVmeasurements and the approximate shedding frequency as read from their figure is normalized by the chord lengthand the undisturbed flow velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02 04 06 08 1uxmeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(a) Normalized mean of streamwise velocity

minus06

minus04

minus02

0

02

04

06yt

max

0 01 02 03 04uxstdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(b) Normalized mean of transverse velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02uymeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(c) Normalized standard deviation of streamwise velocity

minus06

minus04

minus02

0

02

04

06

yt

max

0 01 02 03 04 05uystdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(d) Normalized standard deviation of transverse velocity

Figure 8 Mean and standard deviation of velocities at 119909119888 = 0516 for Re = 106 The experimental data by Zobeiri et al [11] were measuredat 119909119888 = 052 and 0528 for truncated (120573 = 0∘) and oblique (120573 = 60∘) trailing edges respectively and the reported Reynolds number of theflow is about 2 times 106

Modelling and Simulation in Engineering 11

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(a) Steady flow 119891infin = 0

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(b) Incoming flow frequency 119891infin = 15

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(c) Incoming flow frequency 119891infin = 20

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(d) Incoming flow frequency 119891infin = 25

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(e) Incoming flow frequency 119891infin = 30

Figure 9 Effect of the incoming flow frequency on vortex shedding from a beveled trailing edge of 120573 = 60∘ at 119891infin= 15 20 25 and 30

The contour indicates the instantaneous vorticity distribution at 119879 = 10 and the power spectral density (PSD) is calculated from the dragcoefficients in the range 119879 = [3 10]

12 Modelling and Simulation in Engineering

We found in this study that the vortex formation char-acteristic in periodically varying inflows remains unchangedA certain frequency of free-stream flow oscillation tends tomake vortex shedding more regular again This means thatthe periodicity of flow incidence can stimulate an inherentfrequency of vortex shedding from a hydrofoil Howeverthe effect of inflow oscillation on vortex shedding is notyet fully understood Future work will include experimentalinvestigations as well as a 3D flow simulation to betterunderstand the relationship between inflow oscillation andvortex shedding

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Korean Institute ofOcean Science amp Technology sponsored by the Ministry ofTrade Industry amp Energy (no 10033668) and the IndustrialConvergence Strategic Technology Development Program(no 10044499) funded by the Ministry of Trade Industry ampEnergy (MI Korea)

References

[1] E Berger and R Wille ldquoPeriodic flow phenomenardquo AnnualReview of Fluid Mechanics vol 4 no 1 pp 313ndash340 1972

[2] P W Bearman ldquoVortex shedding from oscillating bluff bodiesrdquoAnnual Review of Fluid Mechanics vol 16 pp 195ndash222 1984

[3] C H K Williamson ldquoVortex dynamics in the cylinder wakerdquoAnnual Review of Fluid Mechanics vol 28 pp 477ndash539 1996

[4] C H K Williamson ldquoVortex-induced vibrationsrdquo AnnualReview of Fluid Mechanics vol 36 pp 413ndash455 2004

[5] R Fischer ldquoSinging propellersmdashsolutions and case historiesrdquoMarine Technology vol 45 no 4 pp 221ndash227 2008

[6] D A Bourgoyne J M Hamel S L Ceccio and D R DowlingldquoTime-averaged flow over a hydrofoil at high reynolds numberrdquoJournal of Fluid Mechanics vol 496 pp 365ndash404 2003

[7] D A Bourgoyne S L Ceccio and D R Dowling ldquoVortexshedding from a hydrofoil at high Reynolds numberrdquo Journalof Fluid Mechanics vol 531 pp 293ndash324 2005

[8] N Mulvany J Tu L Chen and B Anderson ldquoAssessment oftwo-equation turbulence modelling for high Reynolds numberhydrofoil flowsrdquo International Journal for Numerical Methods inFluids vol 45 no 3 pp 275ndash299 2004

[9] P Ausoni M Farhat X Escaler E Egusquiza and F AvellanldquoCavitation influence on von Karman vortex shedding andinduced hydrofoil vibrationsrdquo Journal of Fluids EngineeringTransactions vol 129 no 8 pp 966ndash973 2007

[10] P Ausoni Turbulent vortex shedding from a blunt trailingedge hydrofoil [PhD thesis] Ecole Polytechnique Federale deLausanne Lausanne Switzerland 2009

[11] A Zobeiri P Ausoni F Avellan and M Farhat ldquoHow obliquetrailing edge of a hydrofoil reduces the vortex-induced vibra-tionrdquo Journal of Fluids and Structures vol 32 pp 78ndash89 2012

[12] G-H Cottet and P D Koumoutsakos Vortex Methods Theoryand Practice Cambridge University Press 2000

[13] G Cottet and P Poncet ldquoParticle methods for direct numericalsimulations of three-dimensional wakesrdquo Journal of Turbulencevol 3 article N38 2002

[14] P Chatelain A Curioni M Bergdorf D Rossinelli WAndreoni and P Koumoutsakos ldquoBillion vortex particle directnumerical simulations of aircraft wakesrdquo Computer Methods inAppliedMechanics and Engineering vol 197 no 13ndash16 pp 1296ndash1304 2008

[15] Y-C Kim J-C Suh and K-J Lee ldquoVortex-in-cell methodcombined with a boundary element method for incompressibleviscous flow analysisrdquo International Journal forNumericalMeth-ods in Fluids vol 69 no 10 pp 1567ndash1583 2012

[16] P Angot C-H Bruneau and P Fabrie ldquoA penalization methodto take into account obstacles in incompressible viscous flowsrdquoNumerische Mathematik vol 81 no 4 pp 497ndash520 1999

[17] M El Ossmani and P Poncet ldquoEfficiency of multiscale hybridgrid-particle vortex methodsrdquo Multiscale Modeling amp Simula-tion vol 8 no 5 pp 1671ndash1690 2010

[18] J T Rasmussen G-H Cottet and J HWalther ldquoAmultiresolu-tion remeshed vortex-in-cell algorithm using patchesrdquo Journalof Computational Physics vol 230 no 17 pp 6742ndash6755 2011

[19] M Coquerelle and G-H Cottet ldquoA vortex level set method forthe two-way coupling of an incompressible fluid with collidingrigid bodiesrdquo Journal of Computational Physics vol 227 no 21pp 9121ndash9137 2008

[20] F Morency H Beaugendre F Gallizio and S Laurens ldquoCom-putation of ice shedding trajectories using cartesian gridspenalization and level setsrdquo Modelling and Simulation in Engi-neering vol 2011 Article ID 274947 15 pages 2011

[21] M M Hejlesen P Koumoutsakos A Leonard and J HWalther ldquoIterative Brinkman penalization for remeshed vortexmethodsrdquo Journal of Computational Physics vol 280 pp 547ndash562 2015

[22] M Gazzola P Chatelain W M van Rees and P Koumout-sakos ldquoSimulations of single and multiple swimmers with non-divergence free deforming geometriesrdquo Journal of Computa-tional Physics vol 230 no 19 pp 7093ndash7114 2011

[23] R Chatelin and P Poncet ldquoA hybrid grid-particle method formoving bodies in 3D stokes flow with variable viscosityrdquo SIAMJournal on Scientific Computing vol 35 no 4 pp B925ndashB9492013

[24] R Chatelin and P Poncet ldquoHybrid grid-particle methodsand penalization a Sherman-Morrison-Woodbury approach tocompute 3D viscous flows using FFTrdquo Journal of ComputationalPhysics vol 269 pp 314ndash328 2014

[25] S J Lee J H Lee and J C Suh ldquoVortex shedding frequencyfor a 2D hydrofoil with a truncated trailing edgerdquo Journal of theSociety of Naval Architects of Korea vol 51 no 6 pp 480ndash4882014 (Korean)

[26] S-J Lee and J C Suh ldquoA multi-domain approach for ahybrid particle-mesh methodrdquo Journal of Computational FluidsEngineering vol 19 no 2 pp 72ndash78 2014

[27] S-J Lee J-H Lee and J-C Suh ldquoComputation of pressurefields around a two-dimensional circular cylinder using thevortex-in-cell and penalization methodsrdquo Modelling and Sim-ulation in Engineering vol 2014 Article ID 708372 13 pages2014

[28] J J Monaghan ldquoExtrapolating 119861-splines for interpolationrdquoJournal of Computational Physics vol 60 no 2 pp 253ndash2621985

[29] J J Monaghan ldquoParticle methods for hydrodynamicsrdquo Com-puter Physics Reports vol 3 no 2 pp 71ndash124 1985

Modelling and Simulation in Engineering 13

[30] O V Vasilyev and N K-R Kevlahan ldquoHybrid waveletcollocation-brinkman penalization method for complex geom-etry flowsrdquo International Journal for Numerical Methods inFluids vol 40 no 3-4 pp 531ndash538 2002

[31] W Iwakami Y Yatagai N Hatakeyama and Y Hattori ldquoAnew approach for error reduction in the volume penalizationmethodrdquoCommunications in Computational Physics vol 16 no5 pp 1181ndash1200 2014

[32] J R Mansfield O M Knio and C Meneveau ldquoTowardslagrangian large vortex simulationrdquo ESAIM Proceedings vol 1pp 49ndash64 1996

[33] G-H Cottet and P Poncet ldquoAdvances in direct numerical sim-ulations of 3D wall-bounded flows by vortex-in-cell methodsrdquoJournal of Computational Physics vol 193 no 1 pp 136ndash1582004

[34] B Caswell ldquoKinematics and stress on a surface of restrdquo Archivefor Rational Mechanics and Analysis vol 26 no 5 pp 385ndash3991967

[35] R S Rogallo and P Moin ldquoNumerical simulation of turbulentflowsrdquo Annual Review of Fluid Mechanics vol 16 pp 99ndash1371984

[36] J Meyers and P Sagaut ldquoOn the model coefficients for thestandard and the variational multi-scale Smagorinsky modelrdquoJournal of Fluid Mechanics vol 569 pp 287ndash319 2006

[37] J R Mansfield O M Knio and C Meneveau ldquoA dynamic LESscheme for the vorticity transport equation formulation and apriori testsrdquo Journal of Computational Physics vol 145 no 2 pp693ndash730 1998

[38] S B Pope Turbulent Flows Cambridge University Press 2000[39] WH Press S A TeukolskyW T Vetterling and B P Flannery

Numerical Recipes in C++ The Art of Scientific ComputingCambridge University Press 2nd edition 2002

[40] L Greengard and W D Gropp ldquoA parallel verison of the fastmultipoe methodrdquo Computers amp Mathematics with Applica-tions vol 20 no 7 pp 63ndash71 1990

[41] M Frigo and S G Johnson ldquoFFTW an adaptive software archi-tecture for the FFTrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo98) pp 1381ndash1384 May 1998

[42] M Frigo and S G Johnson ldquoThe design and implementationof FFTW3rdquo Proceedings of the IEEE vol 93 no 2 pp 216ndash2312005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Modelling and Simulation in Engineering 9

0 5 10 15 20 250

10

0 5 10 15 20 250

05times10minus5

PSD

ofC

L

PSD

ofC

D

times10minus6

flowast flowast

(a) Truncated trailing edge 120573 = 0∘

0

08

0

08

0

08

0

08

0

08

0

04

0

04

0

04

0

04

0

04

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

PSD

ofC

L

PSD

ofC

D

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

times10minus5 times10minus6

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25

flowast flowast

0 5 10 15 20 25 0 5 10 15 20 25flowast flowast

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

120573 = 10∘

120573 = 20∘

120573 = 40∘

120573 = 60∘

120573 = 70∘

(b) Beveled trailing edge 120573 gt 0∘

Figure 6 Power spectral density (PSD) of the lift and drag coefficients 119891lowast denotes the frequency normalized by the chord length and theundisturbed flow velocity

3D problems require significant computational resourcesHowever the LES should be 3D since turbulence is inherently3D While we accept that more realistic 3D simulation isrequired physically a 2D LES can be used as a basis forcomparison prior to 3D implementation Once the capability

of a numerical algorithm is evaluated in two dimensionsit can be easily extended into three dimensions This paperis intended as a guide for researchers and designers whoare working on a similar problem vortex shedding from ahydrofoil at high Reynolds numbers

10 Modelling and Simulation in Engineering

0 10 20 30 40 50 600

2

4

6

8

10

Zobeiri et al (2012) expPresent cal

flowast

120573 (deg)

Figure 7 Shedding frequency of vortices from the different beveled trailing edges for Re = 106 The experimental data by Zobeiri et al [11]were derived from LDVmeasurements and the approximate shedding frequency as read from their figure is normalized by the chord lengthand the undisturbed flow velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02 04 06 08 1uxmeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(a) Normalized mean of streamwise velocity

minus06

minus04

minus02

0

02

04

06yt

max

0 01 02 03 04uxstdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(b) Normalized mean of transverse velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02uymeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(c) Normalized standard deviation of streamwise velocity

minus06

minus04

minus02

0

02

04

06

yt

max

0 01 02 03 04 05uystdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(d) Normalized standard deviation of transverse velocity

Figure 8 Mean and standard deviation of velocities at 119909119888 = 0516 for Re = 106 The experimental data by Zobeiri et al [11] were measuredat 119909119888 = 052 and 0528 for truncated (120573 = 0∘) and oblique (120573 = 60∘) trailing edges respectively and the reported Reynolds number of theflow is about 2 times 106

Modelling and Simulation in Engineering 11

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(a) Steady flow 119891infin = 0

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(b) Incoming flow frequency 119891infin = 15

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(c) Incoming flow frequency 119891infin = 20

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(d) Incoming flow frequency 119891infin = 25

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(e) Incoming flow frequency 119891infin = 30

Figure 9 Effect of the incoming flow frequency on vortex shedding from a beveled trailing edge of 120573 = 60∘ at 119891infin= 15 20 25 and 30

The contour indicates the instantaneous vorticity distribution at 119879 = 10 and the power spectral density (PSD) is calculated from the dragcoefficients in the range 119879 = [3 10]

12 Modelling and Simulation in Engineering

We found in this study that the vortex formation char-acteristic in periodically varying inflows remains unchangedA certain frequency of free-stream flow oscillation tends tomake vortex shedding more regular again This means thatthe periodicity of flow incidence can stimulate an inherentfrequency of vortex shedding from a hydrofoil Howeverthe effect of inflow oscillation on vortex shedding is notyet fully understood Future work will include experimentalinvestigations as well as a 3D flow simulation to betterunderstand the relationship between inflow oscillation andvortex shedding

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Korean Institute ofOcean Science amp Technology sponsored by the Ministry ofTrade Industry amp Energy (no 10033668) and the IndustrialConvergence Strategic Technology Development Program(no 10044499) funded by the Ministry of Trade Industry ampEnergy (MI Korea)

References

[1] E Berger and R Wille ldquoPeriodic flow phenomenardquo AnnualReview of Fluid Mechanics vol 4 no 1 pp 313ndash340 1972

[2] P W Bearman ldquoVortex shedding from oscillating bluff bodiesrdquoAnnual Review of Fluid Mechanics vol 16 pp 195ndash222 1984

[3] C H K Williamson ldquoVortex dynamics in the cylinder wakerdquoAnnual Review of Fluid Mechanics vol 28 pp 477ndash539 1996

[4] C H K Williamson ldquoVortex-induced vibrationsrdquo AnnualReview of Fluid Mechanics vol 36 pp 413ndash455 2004

[5] R Fischer ldquoSinging propellersmdashsolutions and case historiesrdquoMarine Technology vol 45 no 4 pp 221ndash227 2008

[6] D A Bourgoyne J M Hamel S L Ceccio and D R DowlingldquoTime-averaged flow over a hydrofoil at high reynolds numberrdquoJournal of Fluid Mechanics vol 496 pp 365ndash404 2003

[7] D A Bourgoyne S L Ceccio and D R Dowling ldquoVortexshedding from a hydrofoil at high Reynolds numberrdquo Journalof Fluid Mechanics vol 531 pp 293ndash324 2005

[8] N Mulvany J Tu L Chen and B Anderson ldquoAssessment oftwo-equation turbulence modelling for high Reynolds numberhydrofoil flowsrdquo International Journal for Numerical Methods inFluids vol 45 no 3 pp 275ndash299 2004

[9] P Ausoni M Farhat X Escaler E Egusquiza and F AvellanldquoCavitation influence on von Karman vortex shedding andinduced hydrofoil vibrationsrdquo Journal of Fluids EngineeringTransactions vol 129 no 8 pp 966ndash973 2007

[10] P Ausoni Turbulent vortex shedding from a blunt trailingedge hydrofoil [PhD thesis] Ecole Polytechnique Federale deLausanne Lausanne Switzerland 2009

[11] A Zobeiri P Ausoni F Avellan and M Farhat ldquoHow obliquetrailing edge of a hydrofoil reduces the vortex-induced vibra-tionrdquo Journal of Fluids and Structures vol 32 pp 78ndash89 2012

[12] G-H Cottet and P D Koumoutsakos Vortex Methods Theoryand Practice Cambridge University Press 2000

[13] G Cottet and P Poncet ldquoParticle methods for direct numericalsimulations of three-dimensional wakesrdquo Journal of Turbulencevol 3 article N38 2002

[14] P Chatelain A Curioni M Bergdorf D Rossinelli WAndreoni and P Koumoutsakos ldquoBillion vortex particle directnumerical simulations of aircraft wakesrdquo Computer Methods inAppliedMechanics and Engineering vol 197 no 13ndash16 pp 1296ndash1304 2008

[15] Y-C Kim J-C Suh and K-J Lee ldquoVortex-in-cell methodcombined with a boundary element method for incompressibleviscous flow analysisrdquo International Journal forNumericalMeth-ods in Fluids vol 69 no 10 pp 1567ndash1583 2012

[16] P Angot C-H Bruneau and P Fabrie ldquoA penalization methodto take into account obstacles in incompressible viscous flowsrdquoNumerische Mathematik vol 81 no 4 pp 497ndash520 1999

[17] M El Ossmani and P Poncet ldquoEfficiency of multiscale hybridgrid-particle vortex methodsrdquo Multiscale Modeling amp Simula-tion vol 8 no 5 pp 1671ndash1690 2010

[18] J T Rasmussen G-H Cottet and J HWalther ldquoAmultiresolu-tion remeshed vortex-in-cell algorithm using patchesrdquo Journalof Computational Physics vol 230 no 17 pp 6742ndash6755 2011

[19] M Coquerelle and G-H Cottet ldquoA vortex level set method forthe two-way coupling of an incompressible fluid with collidingrigid bodiesrdquo Journal of Computational Physics vol 227 no 21pp 9121ndash9137 2008

[20] F Morency H Beaugendre F Gallizio and S Laurens ldquoCom-putation of ice shedding trajectories using cartesian gridspenalization and level setsrdquo Modelling and Simulation in Engi-neering vol 2011 Article ID 274947 15 pages 2011

[21] M M Hejlesen P Koumoutsakos A Leonard and J HWalther ldquoIterative Brinkman penalization for remeshed vortexmethodsrdquo Journal of Computational Physics vol 280 pp 547ndash562 2015

[22] M Gazzola P Chatelain W M van Rees and P Koumout-sakos ldquoSimulations of single and multiple swimmers with non-divergence free deforming geometriesrdquo Journal of Computa-tional Physics vol 230 no 19 pp 7093ndash7114 2011

[23] R Chatelin and P Poncet ldquoA hybrid grid-particle method formoving bodies in 3D stokes flow with variable viscosityrdquo SIAMJournal on Scientific Computing vol 35 no 4 pp B925ndashB9492013

[24] R Chatelin and P Poncet ldquoHybrid grid-particle methodsand penalization a Sherman-Morrison-Woodbury approach tocompute 3D viscous flows using FFTrdquo Journal of ComputationalPhysics vol 269 pp 314ndash328 2014

[25] S J Lee J H Lee and J C Suh ldquoVortex shedding frequencyfor a 2D hydrofoil with a truncated trailing edgerdquo Journal of theSociety of Naval Architects of Korea vol 51 no 6 pp 480ndash4882014 (Korean)

[26] S-J Lee and J C Suh ldquoA multi-domain approach for ahybrid particle-mesh methodrdquo Journal of Computational FluidsEngineering vol 19 no 2 pp 72ndash78 2014

[27] S-J Lee J-H Lee and J-C Suh ldquoComputation of pressurefields around a two-dimensional circular cylinder using thevortex-in-cell and penalization methodsrdquo Modelling and Sim-ulation in Engineering vol 2014 Article ID 708372 13 pages2014

[28] J J Monaghan ldquoExtrapolating 119861-splines for interpolationrdquoJournal of Computational Physics vol 60 no 2 pp 253ndash2621985

[29] J J Monaghan ldquoParticle methods for hydrodynamicsrdquo Com-puter Physics Reports vol 3 no 2 pp 71ndash124 1985

Modelling and Simulation in Engineering 13

[30] O V Vasilyev and N K-R Kevlahan ldquoHybrid waveletcollocation-brinkman penalization method for complex geom-etry flowsrdquo International Journal for Numerical Methods inFluids vol 40 no 3-4 pp 531ndash538 2002

[31] W Iwakami Y Yatagai N Hatakeyama and Y Hattori ldquoAnew approach for error reduction in the volume penalizationmethodrdquoCommunications in Computational Physics vol 16 no5 pp 1181ndash1200 2014

[32] J R Mansfield O M Knio and C Meneveau ldquoTowardslagrangian large vortex simulationrdquo ESAIM Proceedings vol 1pp 49ndash64 1996

[33] G-H Cottet and P Poncet ldquoAdvances in direct numerical sim-ulations of 3D wall-bounded flows by vortex-in-cell methodsrdquoJournal of Computational Physics vol 193 no 1 pp 136ndash1582004

[34] B Caswell ldquoKinematics and stress on a surface of restrdquo Archivefor Rational Mechanics and Analysis vol 26 no 5 pp 385ndash3991967

[35] R S Rogallo and P Moin ldquoNumerical simulation of turbulentflowsrdquo Annual Review of Fluid Mechanics vol 16 pp 99ndash1371984

[36] J Meyers and P Sagaut ldquoOn the model coefficients for thestandard and the variational multi-scale Smagorinsky modelrdquoJournal of Fluid Mechanics vol 569 pp 287ndash319 2006

[37] J R Mansfield O M Knio and C Meneveau ldquoA dynamic LESscheme for the vorticity transport equation formulation and apriori testsrdquo Journal of Computational Physics vol 145 no 2 pp693ndash730 1998

[38] S B Pope Turbulent Flows Cambridge University Press 2000[39] WH Press S A TeukolskyW T Vetterling and B P Flannery

Numerical Recipes in C++ The Art of Scientific ComputingCambridge University Press 2nd edition 2002

[40] L Greengard and W D Gropp ldquoA parallel verison of the fastmultipoe methodrdquo Computers amp Mathematics with Applica-tions vol 20 no 7 pp 63ndash71 1990

[41] M Frigo and S G Johnson ldquoFFTW an adaptive software archi-tecture for the FFTrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo98) pp 1381ndash1384 May 1998

[42] M Frigo and S G Johnson ldquoThe design and implementationof FFTW3rdquo Proceedings of the IEEE vol 93 no 2 pp 216ndash2312005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Modelling and Simulation in Engineering

0 10 20 30 40 50 600

2

4

6

8

10

Zobeiri et al (2012) expPresent cal

flowast

120573 (deg)

Figure 7 Shedding frequency of vortices from the different beveled trailing edges for Re = 106 The experimental data by Zobeiri et al [11]were derived from LDVmeasurements and the approximate shedding frequency as read from their figure is normalized by the chord lengthand the undisturbed flow velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02 04 06 08 1uxmeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(a) Normalized mean of streamwise velocity

minus06

minus04

minus02

0

02

04

06yt

max

0 01 02 03 04uxstdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(b) Normalized mean of transverse velocity

minus06

minus04

minus02

0

02

04

06

yt

max

minus02 0 02uymeanUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(c) Normalized standard deviation of streamwise velocity

minus06

minus04

minus02

0

02

04

06

yt

max

0 01 02 03 04 05uystdevUinfin

Truncated TE Zobeiri et al (2012)Oblique TE Zobeiri et al (2012)

Truncated TE present calOblique TE present cal

(d) Normalized standard deviation of transverse velocity

Figure 8 Mean and standard deviation of velocities at 119909119888 = 0516 for Re = 106 The experimental data by Zobeiri et al [11] were measuredat 119909119888 = 052 and 0528 for truncated (120573 = 0∘) and oblique (120573 = 60∘) trailing edges respectively and the reported Reynolds number of theflow is about 2 times 106

Modelling and Simulation in Engineering 11

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(a) Steady flow 119891infin = 0

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(b) Incoming flow frequency 119891infin = 15

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(c) Incoming flow frequency 119891infin = 20

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(d) Incoming flow frequency 119891infin = 25

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(e) Incoming flow frequency 119891infin = 30

Figure 9 Effect of the incoming flow frequency on vortex shedding from a beveled trailing edge of 120573 = 60∘ at 119891infin= 15 20 25 and 30

The contour indicates the instantaneous vorticity distribution at 119879 = 10 and the power spectral density (PSD) is calculated from the dragcoefficients in the range 119879 = [3 10]

12 Modelling and Simulation in Engineering

We found in this study that the vortex formation char-acteristic in periodically varying inflows remains unchangedA certain frequency of free-stream flow oscillation tends tomake vortex shedding more regular again This means thatthe periodicity of flow incidence can stimulate an inherentfrequency of vortex shedding from a hydrofoil Howeverthe effect of inflow oscillation on vortex shedding is notyet fully understood Future work will include experimentalinvestigations as well as a 3D flow simulation to betterunderstand the relationship between inflow oscillation andvortex shedding

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Korean Institute ofOcean Science amp Technology sponsored by the Ministry ofTrade Industry amp Energy (no 10033668) and the IndustrialConvergence Strategic Technology Development Program(no 10044499) funded by the Ministry of Trade Industry ampEnergy (MI Korea)

References

[1] E Berger and R Wille ldquoPeriodic flow phenomenardquo AnnualReview of Fluid Mechanics vol 4 no 1 pp 313ndash340 1972

[2] P W Bearman ldquoVortex shedding from oscillating bluff bodiesrdquoAnnual Review of Fluid Mechanics vol 16 pp 195ndash222 1984

[3] C H K Williamson ldquoVortex dynamics in the cylinder wakerdquoAnnual Review of Fluid Mechanics vol 28 pp 477ndash539 1996

[4] C H K Williamson ldquoVortex-induced vibrationsrdquo AnnualReview of Fluid Mechanics vol 36 pp 413ndash455 2004

[5] R Fischer ldquoSinging propellersmdashsolutions and case historiesrdquoMarine Technology vol 45 no 4 pp 221ndash227 2008

[6] D A Bourgoyne J M Hamel S L Ceccio and D R DowlingldquoTime-averaged flow over a hydrofoil at high reynolds numberrdquoJournal of Fluid Mechanics vol 496 pp 365ndash404 2003

[7] D A Bourgoyne S L Ceccio and D R Dowling ldquoVortexshedding from a hydrofoil at high Reynolds numberrdquo Journalof Fluid Mechanics vol 531 pp 293ndash324 2005

[8] N Mulvany J Tu L Chen and B Anderson ldquoAssessment oftwo-equation turbulence modelling for high Reynolds numberhydrofoil flowsrdquo International Journal for Numerical Methods inFluids vol 45 no 3 pp 275ndash299 2004

[9] P Ausoni M Farhat X Escaler E Egusquiza and F AvellanldquoCavitation influence on von Karman vortex shedding andinduced hydrofoil vibrationsrdquo Journal of Fluids EngineeringTransactions vol 129 no 8 pp 966ndash973 2007

[10] P Ausoni Turbulent vortex shedding from a blunt trailingedge hydrofoil [PhD thesis] Ecole Polytechnique Federale deLausanne Lausanne Switzerland 2009

[11] A Zobeiri P Ausoni F Avellan and M Farhat ldquoHow obliquetrailing edge of a hydrofoil reduces the vortex-induced vibra-tionrdquo Journal of Fluids and Structures vol 32 pp 78ndash89 2012

[12] G-H Cottet and P D Koumoutsakos Vortex Methods Theoryand Practice Cambridge University Press 2000

[13] G Cottet and P Poncet ldquoParticle methods for direct numericalsimulations of three-dimensional wakesrdquo Journal of Turbulencevol 3 article N38 2002

[14] P Chatelain A Curioni M Bergdorf D Rossinelli WAndreoni and P Koumoutsakos ldquoBillion vortex particle directnumerical simulations of aircraft wakesrdquo Computer Methods inAppliedMechanics and Engineering vol 197 no 13ndash16 pp 1296ndash1304 2008

[15] Y-C Kim J-C Suh and K-J Lee ldquoVortex-in-cell methodcombined with a boundary element method for incompressibleviscous flow analysisrdquo International Journal forNumericalMeth-ods in Fluids vol 69 no 10 pp 1567ndash1583 2012

[16] P Angot C-H Bruneau and P Fabrie ldquoA penalization methodto take into account obstacles in incompressible viscous flowsrdquoNumerische Mathematik vol 81 no 4 pp 497ndash520 1999

[17] M El Ossmani and P Poncet ldquoEfficiency of multiscale hybridgrid-particle vortex methodsrdquo Multiscale Modeling amp Simula-tion vol 8 no 5 pp 1671ndash1690 2010

[18] J T Rasmussen G-H Cottet and J HWalther ldquoAmultiresolu-tion remeshed vortex-in-cell algorithm using patchesrdquo Journalof Computational Physics vol 230 no 17 pp 6742ndash6755 2011

[19] M Coquerelle and G-H Cottet ldquoA vortex level set method forthe two-way coupling of an incompressible fluid with collidingrigid bodiesrdquo Journal of Computational Physics vol 227 no 21pp 9121ndash9137 2008

[20] F Morency H Beaugendre F Gallizio and S Laurens ldquoCom-putation of ice shedding trajectories using cartesian gridspenalization and level setsrdquo Modelling and Simulation in Engi-neering vol 2011 Article ID 274947 15 pages 2011

[21] M M Hejlesen P Koumoutsakos A Leonard and J HWalther ldquoIterative Brinkman penalization for remeshed vortexmethodsrdquo Journal of Computational Physics vol 280 pp 547ndash562 2015

[22] M Gazzola P Chatelain W M van Rees and P Koumout-sakos ldquoSimulations of single and multiple swimmers with non-divergence free deforming geometriesrdquo Journal of Computa-tional Physics vol 230 no 19 pp 7093ndash7114 2011

[23] R Chatelin and P Poncet ldquoA hybrid grid-particle method formoving bodies in 3D stokes flow with variable viscosityrdquo SIAMJournal on Scientific Computing vol 35 no 4 pp B925ndashB9492013

[24] R Chatelin and P Poncet ldquoHybrid grid-particle methodsand penalization a Sherman-Morrison-Woodbury approach tocompute 3D viscous flows using FFTrdquo Journal of ComputationalPhysics vol 269 pp 314ndash328 2014

[25] S J Lee J H Lee and J C Suh ldquoVortex shedding frequencyfor a 2D hydrofoil with a truncated trailing edgerdquo Journal of theSociety of Naval Architects of Korea vol 51 no 6 pp 480ndash4882014 (Korean)

[26] S-J Lee and J C Suh ldquoA multi-domain approach for ahybrid particle-mesh methodrdquo Journal of Computational FluidsEngineering vol 19 no 2 pp 72ndash78 2014

[27] S-J Lee J-H Lee and J-C Suh ldquoComputation of pressurefields around a two-dimensional circular cylinder using thevortex-in-cell and penalization methodsrdquo Modelling and Sim-ulation in Engineering vol 2014 Article ID 708372 13 pages2014

[28] J J Monaghan ldquoExtrapolating 119861-splines for interpolationrdquoJournal of Computational Physics vol 60 no 2 pp 253ndash2621985

[29] J J Monaghan ldquoParticle methods for hydrodynamicsrdquo Com-puter Physics Reports vol 3 no 2 pp 71ndash124 1985

Modelling and Simulation in Engineering 13

[30] O V Vasilyev and N K-R Kevlahan ldquoHybrid waveletcollocation-brinkman penalization method for complex geom-etry flowsrdquo International Journal for Numerical Methods inFluids vol 40 no 3-4 pp 531ndash538 2002

[31] W Iwakami Y Yatagai N Hatakeyama and Y Hattori ldquoAnew approach for error reduction in the volume penalizationmethodrdquoCommunications in Computational Physics vol 16 no5 pp 1181ndash1200 2014

[32] J R Mansfield O M Knio and C Meneveau ldquoTowardslagrangian large vortex simulationrdquo ESAIM Proceedings vol 1pp 49ndash64 1996

[33] G-H Cottet and P Poncet ldquoAdvances in direct numerical sim-ulations of 3D wall-bounded flows by vortex-in-cell methodsrdquoJournal of Computational Physics vol 193 no 1 pp 136ndash1582004

[34] B Caswell ldquoKinematics and stress on a surface of restrdquo Archivefor Rational Mechanics and Analysis vol 26 no 5 pp 385ndash3991967

[35] R S Rogallo and P Moin ldquoNumerical simulation of turbulentflowsrdquo Annual Review of Fluid Mechanics vol 16 pp 99ndash1371984

[36] J Meyers and P Sagaut ldquoOn the model coefficients for thestandard and the variational multi-scale Smagorinsky modelrdquoJournal of Fluid Mechanics vol 569 pp 287ndash319 2006

[37] J R Mansfield O M Knio and C Meneveau ldquoA dynamic LESscheme for the vorticity transport equation formulation and apriori testsrdquo Journal of Computational Physics vol 145 no 2 pp693ndash730 1998

[38] S B Pope Turbulent Flows Cambridge University Press 2000[39] WH Press S A TeukolskyW T Vetterling and B P Flannery

Numerical Recipes in C++ The Art of Scientific ComputingCambridge University Press 2nd edition 2002

[40] L Greengard and W D Gropp ldquoA parallel verison of the fastmultipoe methodrdquo Computers amp Mathematics with Applica-tions vol 20 no 7 pp 63ndash71 1990

[41] M Frigo and S G Johnson ldquoFFTW an adaptive software archi-tecture for the FFTrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo98) pp 1381ndash1384 May 1998

[42] M Frigo and S G Johnson ldquoThe design and implementationof FFTW3rdquo Proceedings of the IEEE vol 93 no 2 pp 216ndash2312005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Modelling and Simulation in Engineering 11

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(a) Steady flow 119891infin = 0

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(b) Incoming flow frequency 119891infin = 15

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(c) Incoming flow frequency 119891infin = 20

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(d) Incoming flow frequency 119891infin = 25

xc

yc

025 05 075 1

minus01

0

0120012040minus40minus120minus200

0 5 10 15 20 25 30 350

03

06times10minus6

PSD

ofC

D

flowast

(e) Incoming flow frequency 119891infin = 30

Figure 9 Effect of the incoming flow frequency on vortex shedding from a beveled trailing edge of 120573 = 60∘ at 119891infin= 15 20 25 and 30

The contour indicates the instantaneous vorticity distribution at 119879 = 10 and the power spectral density (PSD) is calculated from the dragcoefficients in the range 119879 = [3 10]

12 Modelling and Simulation in Engineering

We found in this study that the vortex formation char-acteristic in periodically varying inflows remains unchangedA certain frequency of free-stream flow oscillation tends tomake vortex shedding more regular again This means thatthe periodicity of flow incidence can stimulate an inherentfrequency of vortex shedding from a hydrofoil Howeverthe effect of inflow oscillation on vortex shedding is notyet fully understood Future work will include experimentalinvestigations as well as a 3D flow simulation to betterunderstand the relationship between inflow oscillation andvortex shedding

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Korean Institute ofOcean Science amp Technology sponsored by the Ministry ofTrade Industry amp Energy (no 10033668) and the IndustrialConvergence Strategic Technology Development Program(no 10044499) funded by the Ministry of Trade Industry ampEnergy (MI Korea)

References

[1] E Berger and R Wille ldquoPeriodic flow phenomenardquo AnnualReview of Fluid Mechanics vol 4 no 1 pp 313ndash340 1972

[2] P W Bearman ldquoVortex shedding from oscillating bluff bodiesrdquoAnnual Review of Fluid Mechanics vol 16 pp 195ndash222 1984

[3] C H K Williamson ldquoVortex dynamics in the cylinder wakerdquoAnnual Review of Fluid Mechanics vol 28 pp 477ndash539 1996

[4] C H K Williamson ldquoVortex-induced vibrationsrdquo AnnualReview of Fluid Mechanics vol 36 pp 413ndash455 2004

[5] R Fischer ldquoSinging propellersmdashsolutions and case historiesrdquoMarine Technology vol 45 no 4 pp 221ndash227 2008

[6] D A Bourgoyne J M Hamel S L Ceccio and D R DowlingldquoTime-averaged flow over a hydrofoil at high reynolds numberrdquoJournal of Fluid Mechanics vol 496 pp 365ndash404 2003

[7] D A Bourgoyne S L Ceccio and D R Dowling ldquoVortexshedding from a hydrofoil at high Reynolds numberrdquo Journalof Fluid Mechanics vol 531 pp 293ndash324 2005

[8] N Mulvany J Tu L Chen and B Anderson ldquoAssessment oftwo-equation turbulence modelling for high Reynolds numberhydrofoil flowsrdquo International Journal for Numerical Methods inFluids vol 45 no 3 pp 275ndash299 2004

[9] P Ausoni M Farhat X Escaler E Egusquiza and F AvellanldquoCavitation influence on von Karman vortex shedding andinduced hydrofoil vibrationsrdquo Journal of Fluids EngineeringTransactions vol 129 no 8 pp 966ndash973 2007

[10] P Ausoni Turbulent vortex shedding from a blunt trailingedge hydrofoil [PhD thesis] Ecole Polytechnique Federale deLausanne Lausanne Switzerland 2009

[11] A Zobeiri P Ausoni F Avellan and M Farhat ldquoHow obliquetrailing edge of a hydrofoil reduces the vortex-induced vibra-tionrdquo Journal of Fluids and Structures vol 32 pp 78ndash89 2012

[12] G-H Cottet and P D Koumoutsakos Vortex Methods Theoryand Practice Cambridge University Press 2000

[13] G Cottet and P Poncet ldquoParticle methods for direct numericalsimulations of three-dimensional wakesrdquo Journal of Turbulencevol 3 article N38 2002

[14] P Chatelain A Curioni M Bergdorf D Rossinelli WAndreoni and P Koumoutsakos ldquoBillion vortex particle directnumerical simulations of aircraft wakesrdquo Computer Methods inAppliedMechanics and Engineering vol 197 no 13ndash16 pp 1296ndash1304 2008

[15] Y-C Kim J-C Suh and K-J Lee ldquoVortex-in-cell methodcombined with a boundary element method for incompressibleviscous flow analysisrdquo International Journal forNumericalMeth-ods in Fluids vol 69 no 10 pp 1567ndash1583 2012

[16] P Angot C-H Bruneau and P Fabrie ldquoA penalization methodto take into account obstacles in incompressible viscous flowsrdquoNumerische Mathematik vol 81 no 4 pp 497ndash520 1999

[17] M El Ossmani and P Poncet ldquoEfficiency of multiscale hybridgrid-particle vortex methodsrdquo Multiscale Modeling amp Simula-tion vol 8 no 5 pp 1671ndash1690 2010

[18] J T Rasmussen G-H Cottet and J HWalther ldquoAmultiresolu-tion remeshed vortex-in-cell algorithm using patchesrdquo Journalof Computational Physics vol 230 no 17 pp 6742ndash6755 2011

[19] M Coquerelle and G-H Cottet ldquoA vortex level set method forthe two-way coupling of an incompressible fluid with collidingrigid bodiesrdquo Journal of Computational Physics vol 227 no 21pp 9121ndash9137 2008

[20] F Morency H Beaugendre F Gallizio and S Laurens ldquoCom-putation of ice shedding trajectories using cartesian gridspenalization and level setsrdquo Modelling and Simulation in Engi-neering vol 2011 Article ID 274947 15 pages 2011

[21] M M Hejlesen P Koumoutsakos A Leonard and J HWalther ldquoIterative Brinkman penalization for remeshed vortexmethodsrdquo Journal of Computational Physics vol 280 pp 547ndash562 2015

[22] M Gazzola P Chatelain W M van Rees and P Koumout-sakos ldquoSimulations of single and multiple swimmers with non-divergence free deforming geometriesrdquo Journal of Computa-tional Physics vol 230 no 19 pp 7093ndash7114 2011

[23] R Chatelin and P Poncet ldquoA hybrid grid-particle method formoving bodies in 3D stokes flow with variable viscosityrdquo SIAMJournal on Scientific Computing vol 35 no 4 pp B925ndashB9492013

[24] R Chatelin and P Poncet ldquoHybrid grid-particle methodsand penalization a Sherman-Morrison-Woodbury approach tocompute 3D viscous flows using FFTrdquo Journal of ComputationalPhysics vol 269 pp 314ndash328 2014

[25] S J Lee J H Lee and J C Suh ldquoVortex shedding frequencyfor a 2D hydrofoil with a truncated trailing edgerdquo Journal of theSociety of Naval Architects of Korea vol 51 no 6 pp 480ndash4882014 (Korean)

[26] S-J Lee and J C Suh ldquoA multi-domain approach for ahybrid particle-mesh methodrdquo Journal of Computational FluidsEngineering vol 19 no 2 pp 72ndash78 2014

[27] S-J Lee J-H Lee and J-C Suh ldquoComputation of pressurefields around a two-dimensional circular cylinder using thevortex-in-cell and penalization methodsrdquo Modelling and Sim-ulation in Engineering vol 2014 Article ID 708372 13 pages2014

[28] J J Monaghan ldquoExtrapolating 119861-splines for interpolationrdquoJournal of Computational Physics vol 60 no 2 pp 253ndash2621985

[29] J J Monaghan ldquoParticle methods for hydrodynamicsrdquo Com-puter Physics Reports vol 3 no 2 pp 71ndash124 1985

Modelling and Simulation in Engineering 13

[30] O V Vasilyev and N K-R Kevlahan ldquoHybrid waveletcollocation-brinkman penalization method for complex geom-etry flowsrdquo International Journal for Numerical Methods inFluids vol 40 no 3-4 pp 531ndash538 2002

[31] W Iwakami Y Yatagai N Hatakeyama and Y Hattori ldquoAnew approach for error reduction in the volume penalizationmethodrdquoCommunications in Computational Physics vol 16 no5 pp 1181ndash1200 2014

[32] J R Mansfield O M Knio and C Meneveau ldquoTowardslagrangian large vortex simulationrdquo ESAIM Proceedings vol 1pp 49ndash64 1996

[33] G-H Cottet and P Poncet ldquoAdvances in direct numerical sim-ulations of 3D wall-bounded flows by vortex-in-cell methodsrdquoJournal of Computational Physics vol 193 no 1 pp 136ndash1582004

[34] B Caswell ldquoKinematics and stress on a surface of restrdquo Archivefor Rational Mechanics and Analysis vol 26 no 5 pp 385ndash3991967

[35] R S Rogallo and P Moin ldquoNumerical simulation of turbulentflowsrdquo Annual Review of Fluid Mechanics vol 16 pp 99ndash1371984

[36] J Meyers and P Sagaut ldquoOn the model coefficients for thestandard and the variational multi-scale Smagorinsky modelrdquoJournal of Fluid Mechanics vol 569 pp 287ndash319 2006

[37] J R Mansfield O M Knio and C Meneveau ldquoA dynamic LESscheme for the vorticity transport equation formulation and apriori testsrdquo Journal of Computational Physics vol 145 no 2 pp693ndash730 1998

[38] S B Pope Turbulent Flows Cambridge University Press 2000[39] WH Press S A TeukolskyW T Vetterling and B P Flannery

Numerical Recipes in C++ The Art of Scientific ComputingCambridge University Press 2nd edition 2002

[40] L Greengard and W D Gropp ldquoA parallel verison of the fastmultipoe methodrdquo Computers amp Mathematics with Applica-tions vol 20 no 7 pp 63ndash71 1990

[41] M Frigo and S G Johnson ldquoFFTW an adaptive software archi-tecture for the FFTrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo98) pp 1381ndash1384 May 1998

[42] M Frigo and S G Johnson ldquoThe design and implementationof FFTW3rdquo Proceedings of the IEEE vol 93 no 2 pp 216ndash2312005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Modelling and Simulation in Engineering

We found in this study that the vortex formation char-acteristic in periodically varying inflows remains unchangedA certain frequency of free-stream flow oscillation tends tomake vortex shedding more regular again This means thatthe periodicity of flow incidence can stimulate an inherentfrequency of vortex shedding from a hydrofoil Howeverthe effect of inflow oscillation on vortex shedding is notyet fully understood Future work will include experimentalinvestigations as well as a 3D flow simulation to betterunderstand the relationship between inflow oscillation andvortex shedding

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Korean Institute ofOcean Science amp Technology sponsored by the Ministry ofTrade Industry amp Energy (no 10033668) and the IndustrialConvergence Strategic Technology Development Program(no 10044499) funded by the Ministry of Trade Industry ampEnergy (MI Korea)

References

[1] E Berger and R Wille ldquoPeriodic flow phenomenardquo AnnualReview of Fluid Mechanics vol 4 no 1 pp 313ndash340 1972

[2] P W Bearman ldquoVortex shedding from oscillating bluff bodiesrdquoAnnual Review of Fluid Mechanics vol 16 pp 195ndash222 1984

[3] C H K Williamson ldquoVortex dynamics in the cylinder wakerdquoAnnual Review of Fluid Mechanics vol 28 pp 477ndash539 1996

[4] C H K Williamson ldquoVortex-induced vibrationsrdquo AnnualReview of Fluid Mechanics vol 36 pp 413ndash455 2004

[5] R Fischer ldquoSinging propellersmdashsolutions and case historiesrdquoMarine Technology vol 45 no 4 pp 221ndash227 2008

[6] D A Bourgoyne J M Hamel S L Ceccio and D R DowlingldquoTime-averaged flow over a hydrofoil at high reynolds numberrdquoJournal of Fluid Mechanics vol 496 pp 365ndash404 2003

[7] D A Bourgoyne S L Ceccio and D R Dowling ldquoVortexshedding from a hydrofoil at high Reynolds numberrdquo Journalof Fluid Mechanics vol 531 pp 293ndash324 2005

[8] N Mulvany J Tu L Chen and B Anderson ldquoAssessment oftwo-equation turbulence modelling for high Reynolds numberhydrofoil flowsrdquo International Journal for Numerical Methods inFluids vol 45 no 3 pp 275ndash299 2004

[9] P Ausoni M Farhat X Escaler E Egusquiza and F AvellanldquoCavitation influence on von Karman vortex shedding andinduced hydrofoil vibrationsrdquo Journal of Fluids EngineeringTransactions vol 129 no 8 pp 966ndash973 2007

[10] P Ausoni Turbulent vortex shedding from a blunt trailingedge hydrofoil [PhD thesis] Ecole Polytechnique Federale deLausanne Lausanne Switzerland 2009

[11] A Zobeiri P Ausoni F Avellan and M Farhat ldquoHow obliquetrailing edge of a hydrofoil reduces the vortex-induced vibra-tionrdquo Journal of Fluids and Structures vol 32 pp 78ndash89 2012

[12] G-H Cottet and P D Koumoutsakos Vortex Methods Theoryand Practice Cambridge University Press 2000

[13] G Cottet and P Poncet ldquoParticle methods for direct numericalsimulations of three-dimensional wakesrdquo Journal of Turbulencevol 3 article N38 2002

[14] P Chatelain A Curioni M Bergdorf D Rossinelli WAndreoni and P Koumoutsakos ldquoBillion vortex particle directnumerical simulations of aircraft wakesrdquo Computer Methods inAppliedMechanics and Engineering vol 197 no 13ndash16 pp 1296ndash1304 2008

[15] Y-C Kim J-C Suh and K-J Lee ldquoVortex-in-cell methodcombined with a boundary element method for incompressibleviscous flow analysisrdquo International Journal forNumericalMeth-ods in Fluids vol 69 no 10 pp 1567ndash1583 2012

[16] P Angot C-H Bruneau and P Fabrie ldquoA penalization methodto take into account obstacles in incompressible viscous flowsrdquoNumerische Mathematik vol 81 no 4 pp 497ndash520 1999

[17] M El Ossmani and P Poncet ldquoEfficiency of multiscale hybridgrid-particle vortex methodsrdquo Multiscale Modeling amp Simula-tion vol 8 no 5 pp 1671ndash1690 2010

[18] J T Rasmussen G-H Cottet and J HWalther ldquoAmultiresolu-tion remeshed vortex-in-cell algorithm using patchesrdquo Journalof Computational Physics vol 230 no 17 pp 6742ndash6755 2011

[19] M Coquerelle and G-H Cottet ldquoA vortex level set method forthe two-way coupling of an incompressible fluid with collidingrigid bodiesrdquo Journal of Computational Physics vol 227 no 21pp 9121ndash9137 2008

[20] F Morency H Beaugendre F Gallizio and S Laurens ldquoCom-putation of ice shedding trajectories using cartesian gridspenalization and level setsrdquo Modelling and Simulation in Engi-neering vol 2011 Article ID 274947 15 pages 2011

[21] M M Hejlesen P Koumoutsakos A Leonard and J HWalther ldquoIterative Brinkman penalization for remeshed vortexmethodsrdquo Journal of Computational Physics vol 280 pp 547ndash562 2015

[22] M Gazzola P Chatelain W M van Rees and P Koumout-sakos ldquoSimulations of single and multiple swimmers with non-divergence free deforming geometriesrdquo Journal of Computa-tional Physics vol 230 no 19 pp 7093ndash7114 2011

[23] R Chatelin and P Poncet ldquoA hybrid grid-particle method formoving bodies in 3D stokes flow with variable viscosityrdquo SIAMJournal on Scientific Computing vol 35 no 4 pp B925ndashB9492013

[24] R Chatelin and P Poncet ldquoHybrid grid-particle methodsand penalization a Sherman-Morrison-Woodbury approach tocompute 3D viscous flows using FFTrdquo Journal of ComputationalPhysics vol 269 pp 314ndash328 2014

[25] S J Lee J H Lee and J C Suh ldquoVortex shedding frequencyfor a 2D hydrofoil with a truncated trailing edgerdquo Journal of theSociety of Naval Architects of Korea vol 51 no 6 pp 480ndash4882014 (Korean)

[26] S-J Lee and J C Suh ldquoA multi-domain approach for ahybrid particle-mesh methodrdquo Journal of Computational FluidsEngineering vol 19 no 2 pp 72ndash78 2014

[27] S-J Lee J-H Lee and J-C Suh ldquoComputation of pressurefields around a two-dimensional circular cylinder using thevortex-in-cell and penalization methodsrdquo Modelling and Sim-ulation in Engineering vol 2014 Article ID 708372 13 pages2014

[28] J J Monaghan ldquoExtrapolating 119861-splines for interpolationrdquoJournal of Computational Physics vol 60 no 2 pp 253ndash2621985

[29] J J Monaghan ldquoParticle methods for hydrodynamicsrdquo Com-puter Physics Reports vol 3 no 2 pp 71ndash124 1985

Modelling and Simulation in Engineering 13

[30] O V Vasilyev and N K-R Kevlahan ldquoHybrid waveletcollocation-brinkman penalization method for complex geom-etry flowsrdquo International Journal for Numerical Methods inFluids vol 40 no 3-4 pp 531ndash538 2002

[31] W Iwakami Y Yatagai N Hatakeyama and Y Hattori ldquoAnew approach for error reduction in the volume penalizationmethodrdquoCommunications in Computational Physics vol 16 no5 pp 1181ndash1200 2014

[32] J R Mansfield O M Knio and C Meneveau ldquoTowardslagrangian large vortex simulationrdquo ESAIM Proceedings vol 1pp 49ndash64 1996

[33] G-H Cottet and P Poncet ldquoAdvances in direct numerical sim-ulations of 3D wall-bounded flows by vortex-in-cell methodsrdquoJournal of Computational Physics vol 193 no 1 pp 136ndash1582004

[34] B Caswell ldquoKinematics and stress on a surface of restrdquo Archivefor Rational Mechanics and Analysis vol 26 no 5 pp 385ndash3991967

[35] R S Rogallo and P Moin ldquoNumerical simulation of turbulentflowsrdquo Annual Review of Fluid Mechanics vol 16 pp 99ndash1371984

[36] J Meyers and P Sagaut ldquoOn the model coefficients for thestandard and the variational multi-scale Smagorinsky modelrdquoJournal of Fluid Mechanics vol 569 pp 287ndash319 2006

[37] J R Mansfield O M Knio and C Meneveau ldquoA dynamic LESscheme for the vorticity transport equation formulation and apriori testsrdquo Journal of Computational Physics vol 145 no 2 pp693ndash730 1998

[38] S B Pope Turbulent Flows Cambridge University Press 2000[39] WH Press S A TeukolskyW T Vetterling and B P Flannery

Numerical Recipes in C++ The Art of Scientific ComputingCambridge University Press 2nd edition 2002

[40] L Greengard and W D Gropp ldquoA parallel verison of the fastmultipoe methodrdquo Computers amp Mathematics with Applica-tions vol 20 no 7 pp 63ndash71 1990

[41] M Frigo and S G Johnson ldquoFFTW an adaptive software archi-tecture for the FFTrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo98) pp 1381ndash1384 May 1998

[42] M Frigo and S G Johnson ldquoThe design and implementationof FFTW3rdquo Proceedings of the IEEE vol 93 no 2 pp 216ndash2312005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Modelling and Simulation in Engineering 13

[30] O V Vasilyev and N K-R Kevlahan ldquoHybrid waveletcollocation-brinkman penalization method for complex geom-etry flowsrdquo International Journal for Numerical Methods inFluids vol 40 no 3-4 pp 531ndash538 2002

[31] W Iwakami Y Yatagai N Hatakeyama and Y Hattori ldquoAnew approach for error reduction in the volume penalizationmethodrdquoCommunications in Computational Physics vol 16 no5 pp 1181ndash1200 2014

[32] J R Mansfield O M Knio and C Meneveau ldquoTowardslagrangian large vortex simulationrdquo ESAIM Proceedings vol 1pp 49ndash64 1996

[33] G-H Cottet and P Poncet ldquoAdvances in direct numerical sim-ulations of 3D wall-bounded flows by vortex-in-cell methodsrdquoJournal of Computational Physics vol 193 no 1 pp 136ndash1582004

[34] B Caswell ldquoKinematics and stress on a surface of restrdquo Archivefor Rational Mechanics and Analysis vol 26 no 5 pp 385ndash3991967

[35] R S Rogallo and P Moin ldquoNumerical simulation of turbulentflowsrdquo Annual Review of Fluid Mechanics vol 16 pp 99ndash1371984

[36] J Meyers and P Sagaut ldquoOn the model coefficients for thestandard and the variational multi-scale Smagorinsky modelrdquoJournal of Fluid Mechanics vol 569 pp 287ndash319 2006

[37] J R Mansfield O M Knio and C Meneveau ldquoA dynamic LESscheme for the vorticity transport equation formulation and apriori testsrdquo Journal of Computational Physics vol 145 no 2 pp693ndash730 1998

[38] S B Pope Turbulent Flows Cambridge University Press 2000[39] WH Press S A TeukolskyW T Vetterling and B P Flannery

Numerical Recipes in C++ The Art of Scientific ComputingCambridge University Press 2nd edition 2002

[40] L Greengard and W D Gropp ldquoA parallel verison of the fastmultipoe methodrdquo Computers amp Mathematics with Applica-tions vol 20 no 7 pp 63ndash71 1990

[41] M Frigo and S G Johnson ldquoFFTW an adaptive software archi-tecture for the FFTrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo98) pp 1381ndash1384 May 1998

[42] M Frigo and S G Johnson ldquoThe design and implementationof FFTW3rdquo Proceedings of the IEEE vol 93 no 2 pp 216ndash2312005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of