towards a numerical hydrodynamics laboratory by developing an overlapping mesh solver based on a...

Applied Ocean Research 32 (2010) 308–320

Contents lists available at ScienceDirect

Applied Ocean Research

journal homepage: www.elsevier.com/locate/apor

Towards a numerical hydrodynamics laboratory by developing an overlappingmesh solver based on a moving mesh solver; verification and applicationRoozbeh Panahi, Mehdi Shafieefar ∗Civil Engineering Department, Tarbiat Modares University, Tehran, Iran

a r t i c l e i n f o

Article history:Received 20 April 2009Received in revised form19 December 2009Accepted 19 December 2009Available online 10 February 2010

Keywords:Overlapping meshMoving meshFinite volumeFluid-structure interaction

a b s t r a c t

Simple extension of an available finite volume moving mesh algorithm to an overlapping mesh solveris discussed. The resultant solver can then overcome difficulties with relative/large motions as well asmesh generation in comparison to those when using a moving mesh. However, it handles hydrodynamicproblems including viscous free surface flow interaction with a free rigid body. Changes arise from usingmore than a mesh in the computational domain by an overlapping mesh system which accompaniesdata transfer and solving discrete sets of equations. By developing a two-dimensional code, differentaspects of the proposed solver have been discussed by simulating a cavity flow, an unsymmetrical currentaround a cylinder in a channel,wave generation using an analytical solution and awedge-typewavemakerand finally, free falling of a cylinder into calm water. Comparison of results to available data shows thecapabilities of the solver in spite of its simplicity, as well.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

To assess a hydrodynamic problem, one encounters twomain solutions including experimental and numerical analyses.Although model tests are still useful, their inherent drawbackssuch as time, expense and scale effects have motivated the use ofComputational Fluid Dynamics (CFD) as the best choice in manycases.However, three independent components of a numerical

method to solve a common hydrodynamic problem are couplingof the pressure and the velocity fields, capturing large and/orcomplex free surface deformations and also catching rigid bodymotions. It must be remembered that there are a wide varietyof alternatives in such sub-problems with different assumptionswhich can potentially form many solvers.Now, suppose that a robust finite volume solver to deal

with a two-phase flow of incompressible viscous fluids withlarge/arbitrary interface deformations has been developed. Then,it can be further extended to take into account the effect ofrigid body motions. Such an issue just necessitates the use of anappropriate strategy to model the dynamics and to update thecomputational domain at each time step. Fundamentally, thereare a wide variety of strategies to capture body motions in acomputational domain such as body-attached/moving mesh [1],deformable mesh [2], re-mesh [3], sliding mesh [4], overlapping

∗ Corresponding author. Tel.: +98 21 88883318; fax: +98 21 82883538.E-mail address: [email protected] (M. Shafieefar).

0141-1187/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.apor.2009.12.006

mesh [5] and Cartesian mesh [6]. Nonetheless, they comprisebenefits as well as weaknesses to deal with body motions indifferent situations.A three-dimensional fluid-structure interaction solver was

previously developed by the authors using a Volume of Fluid (VoF)— fractional step method to capture a viscous two-phase flow [7]and a boundary-fitted body-attached mesh to simulate 6- Degreesof Freedom (DoF) motions of an object [1]. This is probably thesimplest approach among the abovementioned motion modelingstrategies while covering an acceptable level of hydrodynamicproblems. It uses a single structured/unstructured mesh of rigidcells in the whole domain following all motions of the body, asshown in Fig. 1 at successive time steps.On the other hand, capturing of large amplitude motions,

relative motions and also complex geometries are difficultiesof such a solver which could be simply overcome using anoverlapping mesh motion modeling strategy. Implementation ofan overlapping mesh system is straightforward when a movingmesh solver is already developed, while it introduces manybenefits. As shown in Fig. 1 an overlapping mesh motion modelingstrategy includes a number of meshes (mesh components)which are partly coincident. They cover different regions of thecomputational domain, although there is always a stationarybackground mesh which lies over the whole domain. Meshes areusually geometrically simple and allow for independent meshingof higher quality as well as their motions than would be possiblein the case of a single mesh.A moving mesh solver is addressed in the next section, in

brief. Then, the overlapping mesh solver is presented. Finally,it is completely verified using different test cases. Although

http://www.elsevier.com/locate/apor

http://www.elsevier.com/locate/apor

mailto:[email protected]

http://dx.doi.org/10.1016/j.apor.2009.12.006

R. Panahi, M. Shafieefar / Applied Ocean Research 32 (2010) 308–320 309

Fig. 1. Motion modeling strategies at successive time steps tn and tn+1 .

all problems are two-dimensional, the algorithm has a three-dimensional basis.

2. Moving mesh solver

Navier–Stokes and continuity equations govern on the encoun-tered two-phase flow of incompressible viscous fluids. To capturethe interface with a complex geometry between two phases, a vol-ume fraction (α) transport equation is also included.After implementing the Gauss theorem, the equations will

become as follow:

ddt

∫v

ρeff dV +∫Aρeff urel.ndA = 0 (1)

ddt

∫v

uidV +∫Aui(urel.n)dA

= −

∫A

1ρeffPnid A+

∫Aveff E∇ui.n dA+

∫v

gidV (2)

ddt

∫v

αdV +∫Aαurel.n dA = 0 (3)

where ρeff and νeff are the effective phase properties and urel =u − um is the fluid velocity vector relative to the mesh velocityvector (um). ui and gi are the i Cartesian direction component of thefluid velocity and the gravity field, respectively. Also, n representsa unit vector normal to a Control Volume (CV) face and ni is its iCartesian direction component.However, the simplest approximation for the spatial discretiza-

tion of the unsteady term in the Navier–Stokes equations is to re-place it by the product of the value of the integrand at the CVcenter and the volume of the CV. The convection term is also dis-cretized using Gamma interpolation [8]. Here, the convective fluxis estimated based on Rhie and Chow interpolation because of the

co-located approach used in saving main variables [9]. Besides,the Piecewise Linear Interpolation (PLI) is implemented to dis-cretize the pressure termwhile there is no free surface reconstruc-tionmethod, recently introduced by Jahanbakhsh [7]. The diffusionterm is treated by over-relaxed interpolation [8]. Finally, the grav-ity term is discretized as the unsteady term.In the case of the volume fraction transport equation, the

spatial discretization of the unsteady term is done similar to thatof the Navier–Stokes equations. For its temporal discretization,the second-order three-time-levels interpolation [10] is applieddue to its performance in the case of wave generation andtransportation, as will be discussed in the results. The CompressiveInterface Capturing Scheme for Arbitrary Meshes (CICSAM) [11] isused for spatial discretization of the convection term as well asthe Crank–Nicholson interpolation for its temporal discretizationaccording to an investigation conducted by Panahi [12].To compute the pressure and the velocity fields, the fractional

step method of Kim [13] with small changes to improve its capa-bilities as presented by Panahi as a flowchart is implemented [1].However, when there is a moving body in the computational

domain, its movement has to be considered irrespective of beingforced or free. Here, the motions are applied on the body-attachedmeshposition and the velocity vector is inserted into the governingequations in each time step. It must be remembered that, in thecase of a free tomove body,movements have to be calculated basedon loads acting on the structure, by solving the linear and angularmomentum equations. Such loads can be raised from the effects offlow field, body weight and probably external components.The general algorithm showing the main interconnections in

the moving mesh solver is presented in Fig. 2 with part (a).After developing the code based on themovingmesh algorithm,

it has been completely verified as presented in Table 1 by compari-son to available experimental and numerical results [1,7,12,14]. So,it can be strongly used for any new problem under the scope of thedeveloped solver.

310 R. Panahi, M. Shafieefar / Applied Ocean Research 32 (2010) 308–320

Fig. 2. General algorithm of the developed solver to simulate rigid body motions in the context of a two-phase flow where the bold box is the start point; moving meshsolver with hatched box (a), overlapping mesh solver with hatched box (b).

Table 1Verification of a moving mesh solver to simulate two-phase flow of viscous incompressible fluids interaction with a six degree of freedom rigid body.

Issue for verification Implemented test case

Pressure–velocity coupling Orthogonal cavity flow

Non-orthogonality Non-orthogonal cavity flow

Volume fraction transport equation discretizationScalar transport in a predefined:– Constant oblique velocity field– Shear flow

Two-phase flow– Raleigh–Taylor instability– Dam breaking with and without obstacle– Sloshing in a fixed tank

Forced fluid-structure interaction (zero degree of freedommotion)– Sloshing in a forced oscillating tank– Wigley hull resistance– Cylinder water-exit

Free fluid-structure interaction (up to six degree of freedommotions)

– Symmetric and asymmetric free falling of a wedge– Free falling of a cylinder– Barge resistance and maneuvering– Catamaran resistance and maneuvering– Trimaran resistance and maneuvering

3. Overlapping mesh solver

Here, flow variables just have to be interpolated between theoverlapped meshes to exchange the information which is notnecessary when using one moving mesh in the whole domain.However, the main drawback of the overlapping mesh strategy isthe difficulty to ensure conservation of the computed variables,which can be neglected in many cases [15,10] as presented in thisstudy by capturing appropriate results. The present overlapping

mesh motion modeling strategy includes three distinct stepswhich will be discussed in the following sub-section for atwo-dimensional case. It consists of a non-uniform Cartesianbackground mesh as well as an overset mesh of quad cells.

3.1. Identification of CVs

When two mesh components necessary to appropriately coverthe computational domain are generated, the next step is to


Fig. 3. An algorithm to identify CVs in the background mesh after assume an appropriate width for the overlap zone; gray squares present when a decision is made aboutthe identity of a CV.

identify the characteristic of all CVs according to their role in thesolution process [10]:(a) discretization cells which are used to discretize the governingequations

(b) interpolation cells which receive the solution from the overlapmesh component by interpolation

(c) inactive cells which are disregarded during the solutionprocess.

The main activity toward marking cells in this step is then called‘‘hole cutting’’. This is just implemented for the background meshand has to be repeated in each time step in the case of a movingstructure as presented in Fig. 3 in a tripartite procedure. Here, thewidth of the overlap zone δ0, measured from the boundary of theoverset mesh, determines a band for communication between thebackground mesh and the overset mesh. It depends on the meshspacing in such an area and all interpolation cells lie in this region.It has to be large enough to provide sufficient overlap between themeshes as an essential element to have an appropriate inter-meshcoupling.About the overset mesh, type classification is very simple.

During themesh generation process, outer boundary of the oversetmesh is assigned a special boundary type (oversetmesh boundary).Then, the CVs that lie along such a boundary are recognized asinterpolation cells. Other CVs in the oversetmesh are discretizationcells and there is no inactive cell.

3.2. Coupling of mesh components

Here, a variable at an interpolation cell of a mesh component,identified in the previous step for both background and oversetmeshes, is obtained by a non-conservative interpolation fromthe overlapped mesh. Therefore, an interpolation stencil must beconstructed for each interpolation cell of the consideredmesh fromCVs of the corresponding coincident mesh (donor mesh). A CVin the donor mesh whose center is closest to the center of theinterpolation cell is the base of this interpolation stencil. It is calledthe host cell. Any additional cells in the donormesh contributing tothe interpolation formula come from the immediate neighborhoodof the host cell.After finding the host cells for all interpolation cells using

the neighbor-to-neighbor searching algorithm [16], suitable forunstructured meshes, it is time to construct interpolation stencils.In this study, an interpolation formula consists of the host cell andits first neighbor CVs for all flow variables. A fully implicit algebraicequation for an interpolation cell is created as below for variableϕ which is velocity components (u and w), pressure (P) and alsovolume fraction (α):

ϕI = ϕH +(⇀∇ ϕ

)H. (rI − rH) (4)

where subscripts I and H indicate the interpolation cell and the

host cell, respectively. Also, r is the position vector.(⇀∇ ϕ

)His

calculated as:(⇀∇ ϕ

)H=1VH

∑f=faces of the host cell on the donor mesh

ϕf Af (5)

here, VH is the volume and Af is the face area vector. However, ϕat the face center (ϕf ) is approximated using linear interpolationexcept in the case of pressure, where ϕf is approximated usingPLI [7] as below:

ϕf = ϕAκ + ϕB (1− κ) (6)

where the weighting factor κ is calculated as follows:

κ =ρBδB

ρAδA + ρBδB(7)

here, indices A and B show two neighbor CVs, δ is the distance fromthe face center f to CV center and ρ is the density. It is evident that,without ρ such an interpolation would be equal to a simple.Actually, using the common linear interpolation between

two neighbor CVs to calculate the face pressure (Pf ), results inoscillations in the velocity field in the case of two-phase flow. Suchoscillations force the solution to the divergence, especially whenthere are two phases with a high density ratio; e.g. water and air.So, the PLI is implemented including the effect of the availability ofphases in CVs [7].

3.3. Solution of discrete equations

Here, all equations extracted frommesh components have beensolved simultaneously, while they can be also solved by going backand forth between mesh components [17].To express the procedure implemented in this study, assume

that there are a background mesh (A) and an overset mesh (B) inthe computational domain.An equation for any variable (ϕ) at a discretization cell of A can

be generally rearranged as below:

aD−AϕD−A =∑

ngb=nearest neighbors

angb−Aϕngb−A + SD−A (8)

where a and S areϕ coefficients and source term, respectively. Also,the subscripts are as follows:D: discretization cellA: in the Amesh.The equation for an interpolation cell of A (Eq. (4)) can be

rewritten to represent a form similar to that of a discretization cell(Eq. (8)) based on its interpolation stencil in B as below:

aI−AϕI−A =∑

ngb=nearest neighbors

angb−Bϕngb−B + SI−A (9)

where the subscripts are as follows:I: interpolation cellB: in the Bmesh.So, interpolation cells play an implicit rule in the solution

procedure.Equation for an inactive cell of A is also prepared:

aIA−AϕAI−A = ϕ∗IA−A (10)

where the subscript IA denotes an inactive cell, aIA−A = 1 and ϕ∗IA−Ais the last known value of ϕ.


After constructing analogous equations for ϕ in B which isan overset mesh and surely has no inactive cell, it is time toassemble the global matrix for variable ϕ using a new continuouscell numbering extracted by a simple shift of the overset cellsnumbers. In other words, cell number 1 in the overset mesh willbe n + 1 where n is the total number of cells in the backgroundmesh.

3.4. Solution algorithm

Fig. 2 with part (b) shows the algorithm based on theoverlapping mesh motion modeling strategy. Using the solutionalgorithm, one can solve a wide variety of problems but, a mostcommon case consists of free rigid bodies in an interfacial flowfield.Here, an internal loop between the solution of Navier–Stokes

and rigid body motion equations has a vital role in the procedure,especially in the case of the overlapping mesh. This providesa strongly coupled solution in the domain in addition tocompensation of data lack for fresh cells. These are cellswhich were inactive in the previous time step and becomeinterpolation/discretization cells in the present time step due tomesh movements. Subsequently, they have no information whilethey are needed in the temporal discretization.

4. Numerical results

By developing a code based on the explained algorithm,different problems are solved to assess all aspects of the methodas well as its capabilities. At the first step, a cavity flow isimplemented to show that the method is independent of theoverset mesh position relative to the background mesh. Here,convergence of the solution is also analyzed showing no differencebetween a single mesh and an overlapping mesh of the samequality. The next problem is a cylinder in a steady unsymmetricalcurrent. Although an interpolation scheme is included in thesolution, it clearly shows the second-order accuracy of thespatial discretization. The effect of overlapping zone width isalso investigated. The third case is the ability to generate andtransport a wave. The importance of a temporal discretizationis shown by generating an Airy wave. So, the capability of thesolver to represent a high quality wave is proofed. Then, a plungerwavemaker is simulated. This problem indicates the accuracy ofa developed solver in the presence of structures with relativemotions. Finally, slamming of a cylinder as a result of free fallingis examined, which includes complex deformations of free surfaceas well as large and free motion of a structure.

4.1. Cavity flow

As a common problem to verify a CFD flow solver, the wellknown lid-driven cavity, a zero-gravity flow, is considered. Thistest case has been studied by many authors and accurate resultsare available in literature; see e.g. [18]. The computational domainis a two-dimensional square cavity. Here, the Reynolds number isdefined as Re = ρUH/µ = 1000 where ρ and µ are fluid densityand viscosity, respectively. U is the velocity of the lid and H is thesquare dimension. The velocity is zero at all boundaries except themoving wall where the velocity is equal to the velocity of the lid.The normal pressure gradient is also zero at all boundaries.For the solution of this problem a single Cartesian mesh would

be the most suitable. However, we are mainly interested in thesolution on an overlapping mesh system. So, there are two meshcomponents: a backgroundmesh, which covers the entire domain,and another mesh embedded within the background mesh, whichcovers a part of the computational domain in the central region

Fig. 4. Overlapping meshes used for the computation of lid-driven cavity flowconsisting of a background mesh of 40 × 40 CVs and an overset mesh of 24 × 24CVs in four positions. White CVs: discretization cells, red CVs: interpolation cells ofthe overset mesh and green CVs: interpolation cells of the background mesh. (Forinterpretation of the references to colour in this figure legend, the reader is referredto the web version of this article.)

of the cavity. A number of cells in the background mesh weredeactivated, thus making the solution on both meshes dependenton each other. Fig. 4 shows four overlapping mesh systemsconsisting of two uniform Cartesian meshes. Here, the spacingon both meshes is the same (0.025 in non-dimension form withrespect to the H) and the width of the overlap zone (δ0) is equalto 0.04 in the non-dimension form. They are used to assess thesensitivity of results to the relative position of mesh components.Because when there is a moving body, the position of the oversetmesh changes in each time step. It must be remembered that thestudy is comparative to examine the performance of presentedoverlapping mesh in contrast to that of the moving mesh. So, justa mesh of the same quality has to be implemented and the authorsdo not insist on 0.025 as particular mesh spacing. The width ofthe overlap zone is selected according to the implemented meshresolution and also the interpolation scheme. Such a width has tobe adjusted in away to avoid any interpolation cell of amesh beingincluded in interpolation stencils of the othermesh. This is a simpletask when there is no moving object in the computational domain,so thewidth of the overlap zone (δ0) can be adjusted to aminimumvalue. But, when there is a moving object resulting in the changeof cells identity, δ0 is defined in the safe side (larger than usual) toavoid the aforementioned inclusion.Fig. 5 shows the profiles of u velocity component taken along

the vertical symmetry line of the cavity on single mesh with40 × 40 CVs and four overlapping mesh systems. The profilesare also compared to the benchmark results from [18]. Thereis a discrepancy between the research data and those of thebenchmark, but obviously the results on the overlapping meshsystems are not degraded and approximately the same accuracyis achieved on both single and overlapping meshes by differentoverset mesh angles. However, such a discrepancy can be easilyrefined using a finemesh to produce amesh-independent solution.This has been investigated in Fig. 6 where the profiles of v velocitycomponent taken along the horizontal symmetry line of the cavityare shown. There are four levels of refined mesh in both single andoverlapping mesh systems where the single mesh is equal to thebackground mesh of the overlapping mesh; see Fig. 6. Also, theoverset mesh is at α = 45◦ in all cases. The profile is comparedwith the benchmark results from [18], showing a very goodagreement on the finest system. Here, a monotonic convergenceof approximately second-order to a mesh-independent solutioncan be observed for single as well as for overlapping meshes.Obviously, the results on overlappingmeshes are not degraded andapproximately the same accuracy is achieved on both single andoverlapping meshes.

4.2. Cylinder in a steady unsymmetrical current

In this section, the steady laminar flow around a circularcylinder asymmetrically placed in a channel is considered. Thecylinder diameter is D, the channel length is 22D and the channelheight is 4.1D. The cylinder center is at 2D from the inlet as


Fig. 5. Profiles of u velocity component at the vertical centerline of the cavitycalculated on a single mesh and four overlapping mesh systems with sameresolutions; positions of the overset meshes are also shown in the figure withsquares.

well as the lower wall of the channel. The xz coordinate systemis positioned at the center of the cylinder, where x and z axesare directed toward channel length and upward, respectively.The parabolic velocity profile corresponding to a fully developedlaminar flow in a channel is prescribed at the inlet:

u =6UH2

[(z + 2D)H − (z + 2D)2

];w = 0 (11)

where Uand H = 4.1D are mean inlet velocity and channel height,respectively.The velocity gradient is equal to zero at the outlet and its

magnitude is equal to zero at the cylinder surface aswell as channelwalls. The normal pressure gradient is also equal to zero at allboundaries. The Reynolds number based on themean inlet velocityand the cylinder diameter isRe = ρUD/µ = 20. The flow is slightlyasymmetric since the cylinder center is not on the horizontal

symmetry plane of the channel. Due to the asymmetry, differentflow rates and different pressures appear above and below thecylinder, resulting in a small lift force.For the analysis of spatial discretization errors, the computation

has been performed on four systematically refined meshes. Here,two meshes are used to cover the computational domain: acylindrical mesh in the vicinity of the cylinder and an orthogonalnon-uniform mesh covering the whole channel. The cylindricalmesh extends one diameter from the cylinder wall and is stretchedto resolve the flow close to the cylinder surface. The first leveloversetmesh has 32 uniformly distributed CVs around the cylinderand 10 CVs in the radial direction. However, finer meshes areobtained by doubling the number of cells in each direction. Thethicknesses of the cell next to the wall, in the direction normal tothe cylinder surface are 0.3, 0.0142, 0.0069 and 0.00335 in fourlevels of mesh refinement, in non-dimension form with respect tothe D. In addition, the first level background mesh has 20 CVs inz direction and 46 CVs in x direction. The mesh is stretched in zdirection to get a better resolution near the channel walls. In the xdirection, the mesh is uniform in front of the cylinder and up to 2Dbehind the cylinder. Thereafter, themesh is coarsened towards theoutlet boundary. In addition, δ0 is 0.62, 0.34, 0.18 and 0.1 for fouraforementioned levels, in the non-dimension form.Cell identity in the vicinity of the oversetmesh is shown in Fig. 7.

It is so important to tune δ0 as no interpolation cell of the oversetmesh is included in the interpolation stencil constructed for aninterpolation cell of the background mesh and vice versa.Here, drag and lift coefficients are defined as:

CD =Fx

12ρU

2D; CL =

Fz12ρU

2D(12)

where, Fx and Fzare the total forces on the cylinder in x and zdirections per unit depth, as the problem is two-dimensional.They are obtained to investigate the accuracy of themethod and

summarized in Table 2 in addition to a benchmark [19]. All resultsare very close to each other and they are in very good agreementwith the benchmark data. The difference between solutions on

Fig. 6. Profiles of v velocity component at the horizontal centerline of the cavity; calculated on three levels of single and overlapping mesh refinement where the oversetmesh is at α = 45◦ .


Fig. 7. Four overlapping meshes used to simulate flow around a cylinder; white CVs: discretization cells, red CVs: interpolation cells of the overset mesh and green CVs:interpolation cells of the background mesh. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 2Drag coefficient CD and lift coefficient CLas a function of mesh fineness.

Current numerical simulation Benchmark [19] Error in comparison tothe fine mesh

Mesh Number of CVs CD CL CD CL CD CL

grid 1 1240 5.40077 0.01183grid 2 4960 5.53412 0.01161 5.5800 0.0107 0.04% 0.18%grid 3 19840 5.56939 0.01072grid 4 79360 5.57779 0.01068

consecutive meshes reduces by about a factor of four, which isin accordance with expectations of a second-order discretization.This is good news while interpolation cells are also included in thesolution. In other words, the interpolation scheme has no negativeeffect on the total accuracy of the solution and the accuracyof spatial discretization is maintained. It must be rememberedthat the extracted results from the code have one more digit incomparison to the available test case [19], see Table 2. Althoughthis is not convenient, it can be helpful to clarify the errors.A more detailed view of the flow field at the first 11D of the

channel length is given in Fig. 8 which shows the pressure andthe velocity fields in the channel using grid 3; see Fig. 7. Theoverset mesh boundary is also shown in this figure by a blackline. Although two different meshes are used in the overlap zone,there is almost no difference between the contours on two meshcomponents. The slight difference that appears near the oversetmesh boundary is due to the lack of the flow information atthe boundary points during the post-processing (presentation ofresults). Smooth representation of the flow field in the overlapzone confirms that the interpolation scheme introduced in thisstudy provides a correct coupling between the mesh componentsand leads to a unique solution over the whole domain.One of the factors which might influence the results obtained

with an overlapping mesh system is the width of the overlapping

region. This issue is also investigated here. To this end, four systemsare used,with different oversetmeshes around the cylinder. So, thearea around the cylinder covered by the cylindrical mesh is variedfrom 0.5D to 1.25D as shown in Fig. 9. Their resolutions are similarto the grid 3 in Fig. 7.It is obvious that the change in the width of the overset

mesh affects the ratio between the interpolation cells of theoverset mesh and their host cell in the background mesh; seeFig. 9. This ratio is of great importance while the previousstudies insisted on the similarity of background and overset cellsin the area of information exchange. In addition, they suggestkeeping the overset mesh boundary away from the area ofstrong gradients [10]. Now, by extracting the drag coefficients aspresented in Table 3, it is obvious that the differences betweensolutions with different location of the overset outer boundary arevery small. However, by decreasing the radius of the overset meshto cover 0.25D from the cylinder wall, where the aforementionedratio would be upper than four, a large error occurs. Then, it canbe reported that one of the thresholds of this overlapping meshstrategy is probably the ratio of four between the interpolationcells of the overset mesh and their host cell. However, it mustbe mentioned that more investigation is still necessary to cite ageneral issue about such an extreme ratio. It is evident that the


Fig. 8. Pressure, u velocity field and w velocity fields at first 11D of the channel length obtained by an overlapping mesh system of 19840 CVs; circle shows boundary ofthe overset mesh around the cylinder; continuity of contours across the overset mesh boundary obviously shows the performance of implemented interpolation scheme forboth pressure and velocity fields.

larger value of this ratio helps one to capture the object witha finer overset mesh. To clearly show the mesh-independenceof the algorithm, the pressure distribution along the cylindersurface versus a counterclockwise angle beginning at the rearstagnation point of the cylinder is presented in Fig. 10. Very smalldifference between results on the two finest grids suggests that thediscretization errors on the finest grid are very small. The pressuredifference between the front and rear stagnation points can also beextracted from this figure. The front stagnation point is at phi =180◦ and phi = 0◦ corresponds to the rear stagnation point. So,the difference in pressure using the result of finest grid (grid 4) is0.1174. This is in good accordance with the data of the availablebenchmark [19].

4.3. Wave generation

4.3.1. Wave generation by inlet boundary conditionBefore using the proposed overlapping mesh solver to generate

a wave, it is very useful to assess the quality of wave generationand transportation in a single mesh using the same formulation.For small amplitude waves, the velocities can be derived from

the linear wave theory based on the potential flow. By assumingthat the water depth is larger than the half of the wave length,therewill be a deepwater condition [20]with velocity componentsu = ωaekz cos (kx− ωt + ε) and w = ωaekz sin (kx− ωt + ε).

Table 3Drag coefficient obtained on four grids given in Fig. 9.

Overset mesh radius 0.5D 0.75D 1.0D 1.25D

CD 5.57212 5.56969 5.56939 5.57003

Here, the wave travels in positive x-direction and z points positiveupwards with z = 0 at the undisturbed free surface. a is thewave amplitude, k the wave number, ω the circular frequencyand ε an arbitrary phase angle which can be set equal to zero bya suitable choice of the origin x = 0. The corresponding watersurface elevation is therefore expressed as η = a cos(kx−ωt+ ε).To generate such anAirywave anormal zero-gradient boundary

condition is used for all quantities except at the inlet wherethe velocities are conducted by a Dirichlet boundary conditionbased on aforementioned u and w. Also, the numerical grid usedto resolve the wave profile has 20 CVs per wavelength and16 CVs in wave height after conducting a mesh-independencestudy. It must be mentioned that using a numerical beach withgradually coarsened cells is very important to prevent probablewave reflection from the outlet boundary [21].If air flow plays a minor role it might be ignored at most

parts of the inlet boundaries except the region just above the freesurface, where interpolation should be incorporated to avoid alarge gradient of the imposed velocity acrosswater and air. If the air


a b

c d

Fig. 9. Overlapping mesh system to simulate flow around a cylinder with four different overset mesh circle diameter, (a): 0.5D, (b): 0.75D, (c): 1D and (d): 1.25D.

Fig. 10. Pressure distribution along cylinder wall where phi = 0 corresponds to the rear stagnation point of the cylinder and increases in a counterclockwise direction.

flow is to be specified at the inflow, its velocity is given in a similarmanner as for water beneath the calm surface with an opposite x-direction component.Small amplitudewaves of a = 0.001mandω = 17.73 s−1 have

been generated first in the numerical tank taking into account thatω2 = gk in this kind of wave [20]. Fig. 11 shows the comparisonof wave profile between computation and the linear theory forone wavelength. As can be seen, the sinusoidal wave profile agreesvery well with the analytical solution according to the linear wavetheory. So, it has been shown that potential wave theory canbe used in specifying the velocities at the inlet boundaries forwave generation in a numerical water tank with a satisfactoryaccuracy. However, there is still a slight difference in the wave

height as well as the wave length. It must be remembered that theflow is viscous and the effect of parameters like surface tensionis not included in the solver. So, there are differences betweenassumptions of developing the Airy wave solution and those of thecurrent solver. That is, to investigate the effect of such parameterson the computed wave, more studies have to be conducted.Now, attention has to be paid to the damping ofwave amplitude

in space and time during propagation ofwaves. Thewave-dampingcannot be avoided though it can be decreased by carefully selectingthe numerical parameters. In practice, one can try to estimatethe wave-damping in advance and then apply higher amplitudeat the inlet to generate a wave with expected amplitude aroundthe position of a structure, but experience and care are needed


Fig. 11. Wave profile comparison in one wavelength for a small amplitude linear wave.

Fig. 12. Effect of the temporal discretization of the volume fraction equation unsteady term in wave damping.

since the damping factor is mesh/case-dependent. The wave-damping factor can be influenced by many factors such as meshresolution. Here, the great importance of temporal discretizationof the unsteady term in the volume fraction transport equation isshown. A small amplitudewave of a = 0.001m and λ = 0.2mhasbeen generated using the aforementioned mesh by two differenttemporal discretization schemes; see Fig. 12. The result from thesecond-order three-time-level scheme ismore satisfactory relativeto those captured by implementing a first-order Euler implicitscheme. Also, according to such an investigation, the temporaldiscretization of other equations does not have a significant rolein this case.

4.3.2. Wedge-type wave generatorNow, it is time to deal with the wave generation using the

overlapping mesh solver. Here, a plunger wavemaker [22] issimulated to validate the method in the case of a forced bodymotion; see Fig. 13. The wedge has a sinusoidal vertical motion ofz = sin(

√9.81t) where the overset mesh also follows its motion.

While the background mesh remains stationary during the wavegeneration, itmust be remembered that, due to the relativemotion

between the wedge and the bottom of the tank, the moving meshstrategy can be used in this case.A no-slip boundary condition is applied for the velocity at all

boundaries. Besides, a normal zero-gradient boundary condition isused for both pressure and volume fraction. In order to minimizethe reflection of the flow from the right wall of the wave tank, adamping zone is considered through the last 16d of its length [20];see Fig. 13. Width of the overlap zone is set to δ0 = 0.25m and thetime step is 0.002 s. Snapshots of the free surface are illustratedafter the beginning of wavemaker harmonic motion in Fig. 14.Besides, Fig. 15 shows comparisons of the results with numericalreference data from the ISOPE Workshop [22].

4.4. Cylinder free falling

To evaluate themethod in the case of a free bodymotion, waterentry of a neutrally-buoyant circular cylinder is studied; see Fig. 16.The cylinder is released from a position just above the still waterlevel. It intersects the water surface with the downward velocityof 4 m/s. Here, a no-slip boundary condition at cylinder wall, zerovalue at down boundary and zero-gradient at other boundaries are


Fig. 13. Plunger wavemaker; schematic view of the computational domain including an overset mesh of 16000 CVs with a vertical sinusoidal motion and a stationarybackground mesh of 75000 CVs; a = 1.

Fig. 14. Snapshots of the free surface after the beginning of plunger verticaloscillations in approximately the first 58 m of the tank.

applied on velocity. Also, the zero-gradient condition is used forpressure at whole boundaries. Width of the overlap zone is set to

δ0 = 0.02 m and time step is 0.0001 s. It must be mentioned thatthe overlapping system used in this study (Fig. 16) is a result ofmesh-independence assessment as will be expressed at the end ofthis section. However, the cells in the background mesh are fourtimes larger than those in the overset mesh in the overlap zone.This announces a low sensitivity to have a similar mesh quality inthe overlap zone as is a common case when using an overlappingmesh system [10]. It is actually an important capability whichfacilitates the use of a high quality mesh for a body irrespectiveof the quality of the background mesh. Besides, it has a high valuein the case of a moving body and helps to reduce the number ofcells in the background mesh, while a desired resolution can beimplemented in the vicinity of the moving body.After the cylinder impacts on the calm water surface, the

velocity of cylinder is decreased significantly due to the effectsof hydrodynamic impact forces. As shown in Fig. 17 for threetime instants, water sprays are thrown up at each side of thecylinder and travel straight upward until they become unstable.The contour of velocity magnitude is also shown in this figure.Fig. 18 shows the time history of vertical displacement of thecylinder. The instantaneous vertical positions of the cylinder arecomparedwith experimental data of Greenhow [23] andnumericalsimulation of Xing-Kaeding [24]. It shows a reasonably goodagreementwith experimental data in comparison to the numericalstudy using a moving mesh. It is probably due to a better quality ofthe mesh in the vicinity of the cylinder and also minimizing theerrors due to CVs motions. However, there is also a discrepancybetween the numerical and experimental data. This may be dueto ignoring the effect of compressibility of flow and flexibility ofcylinder in calculation or 3D aspects in the experimental set up.

Fig. 15. Comparison of wave profiles close to thewedgewithwedge at itsmean positionmoving up; results of a Boundary ElementMethod (BEM) aswell as a Finite ElementMethod (FEM) are extracted from ISOPE workshop [22].


Fig. 16. Free falling cylinder; schematic view of the problem where the half of thedomain is used including an overset mesh of 6000 CVs which follows the cylinderand a stationary background mesh of 20000 CVs.

As mentioned earlier, to capture a mesh-independent solutiona study has been conducted. Here, three overlapping systems havebeen used to solve the problem. The finer mesh is created bydividing each cell of the coarser mesh into four cells in bothoverset and background meshes. The time step is also adjustedfor these three systems as 0.0004 s, 0.0002 s and 0.0001 s forcoarse, medium and fine systems, respectively. Although theresults from the fine mesh have already been presented inFig. 18, vertical motion of cylinder is presented using threeaforementioned systems in Fig. 19. It is obvious that the coarsesystem has difficulties in predicting the trajectory but twoother systems represent a closely identical trend. So, the finesystem is appropriate to solve the problem. The non-dimensionalacceleration which can be used to extract the impact load actingon the cylinder during its sinkage is also reported in Fig. 19. It isevident that two finer systems have an approximately same trendwith a slight difference in the predicted peak of the impact force(non-dimensional). So, there are reasonable behaviors in time-history diagrams of both trajectory and acceleration when refining

Fig. 18. Time history of the cylinder water-entry just after the impact: comparisonto available data.

the system in diagrams which encourage using the fine system tolimit the discretization error.

5. Conclusion

A CFD tool to assess hydrodynamic problems was previouslydeveloped by the authors to deal with six degree of freedomrigid motions of a structure in a two-phase flow of viscousincompressible fluids. The experience of using the moving meshmotion modeling strategy in that solver and its inherit restrictionsin the case of relatively/ large motions as well as mesh generationencouraged the authors to implement a more sophisticatedstrategy. So, an overlappingmesh is simply applied using the sameformulation. It is evident that such a solver has to be completelyassessed in its early step due to the complexities and such studiesare covered in this paper using appropriate test cases.The present study can be very helpful to modify available

moving mesh solvers as well as developing a new robust codebased on the overlapping mesh. For the sake of simplicity butwithout loss of generality, development has been performed in twospatial dimensions. So, the extension to three-dimensional flows isstraightforward.Actually, predicting lift and drag coefficients, implementing a

wavemaker to generate a wave or recording impact load on thecylinder during slamming are just simple applications of the solverin its early stage. In the final step, one can expect such a solver to beused in recording 6-DoF motions of a floating structure in front ofan arbitrary irregular wave as well as forces and moments exertedon it. This yields a robust numerical laboratory in hydrodynamicswhich is now under development by the authors.

Fig. 17. Free surface deformation in cylinderwater-entry problem; (left): numerical simulation using the overlappingmesh system including the velocitymagnitude (right):experimental data [23].


Fig. 19. Time history of the cylinder trajectory (left) and acceleration (right): a mesh-independence study.

References

[1] Panahi R, Jahanbakhsh E, Seif MS. Development of a VoF fractional stepsolver for floating body motion simulation. Appl Ocean Res 2006;28(3):171–181.

[2] Chentanez N, Goktekin TG, Feldman BE, O’Brien JF.. Simultaneous couplingof fluids and deformable bodies. In: Proceedings of the 2006 ACM SIG-GRAPH/Eurographics Symposium on Computer Animation. 2006.

[3] Tremel U, Sorensen KA, Hitzel S, Rieger H, Hassan O, Weatherill NP. Parallelremeshing of unstructured volumegrids for CFDapplications. Internat J NumerMethods Fluids 2007;53(8):1361–79.

[4] Blades E, Marcum DL. A sliding interface method for unsteady unstructuredflow simulations. Internat J Numer Methods Fluids 2007;53:507–29.

[5] Carrica PM, Wilson RV, Noack RW, Stern F. Ship motions using single-phaselevel set with dynamic overset grids. Comput Fluid 2007;36(9):1415–33.

[6] Mittal M, Iaccarino G. Immersed boundary methods. Annu Rev Fluid Mech2005;37:239–61.

[7] Jahanbakhsh E, Panahi R, Seif MS. Numerical simulation of three-dimensionalinterfacial flows. Interant J Numer Methods Heat Fluid Flow 2007;17(4):384–404.

[8] Jasak H. Error analysis and estimation for finite volume method withapplication to fluid flows. Ph.D. Thesis. University of London; 1996.

[9] Rhie CM, ChowWL. Numerical study of the turbulent flow past an airfoil withtrailing edge separation. AIAA J 1983;21(11):1523–32.

[10] Hadzic H.. Development and application of a finite volume method for thecomputation of flows around moving bodies on unstructured, overlappinggrids. Ph.D. Thesis, Teschnichen Universitat Hamburg, Harburg. 2005.

[11] Ubbink O, Issa RI. A method for capturing sharp fluid interfaces on arbitrarymeshes. J Comput Phys 1999;153(1):26–50.

[12] Panahi R, Jahanbakhsh E, Seif MS. Comparison of interface capturing methodsin two phase flow. Iran J Sci Technol Trans B Technol 2005;29(B6):539–548.

[13] Kim D, Choi H. A second-order time-accurate finite volume method forunsteady incompressible flow on hybrid unstructured grids. J Comput Phys2000;162(2):411–28.

[14] Panahi R, Jahanbakhsh E, Seif MS. Towards simulation of 3D nonlinear high-speed vessels motion. Ocean Eng 2009;36:256–65.

[15] Togashi F, Nakahashi K, Ito Y, Iwamiyata T, Shimbo Y. Flow simulationof NAL experimental supersonic airplane/booster separation using oversetunstructured grids. Comput Fluid 2001;30(6):673–88.

[16] Löhner R. Robust, vectorized search algorithms for interpolation on unstruc-tured grids. J Comput Phys 1995;118(2):380–7.

[17] Drakakis D,Majewski J, Rokichki J, Zoltak J. Investigation of blending-function-based overlapping-grid technique for compressible flows. Comput Meth ApplMech Eng 2001;190:5173–95.

[18] Ghia U, Ghia KN, Shin CT. High-Re solutions for incompressible flow usingthe Navier–Stokes equations and a multigrid method. J Comput Phys 1982;48:387–411.

[19] SchäferM, Turek S. Benchmark computations of laminar flow around cylinder.Hirschel EH, editor. Flow simulation with high-performance computers: DFGpriority research program results 1993–1995, vol. 52. Vieweg. Braunschweig;1996. p. 547–66.

[20] Dean RG, Dalrymple RA. Water wave mechanics for engineers and scientists.World Scientific; 2004.

[21] Park JC, Miyata H. Numerical simulation of fully-nonlinear wave motionsaround arctic and offshore structures. J Soc Naval Arch Japan 2001;189:13–20.

[22] Tanizawa K, Clément AH.. Report of the 2nd workshop of ISOPE numericalwave tank group: Benchmark test cases of radiation problem. In: 10thinternational offshore and polar engineering conference. Seattle (USA); 2000.

[23] Greenhow M, Lin W. Nonlinear free surface effects: Experiments and theory,Report No. 83–19, Massachusetts Institute of Technology; 1983.

[24] Xing-Kaeding Y. Unified approach to ship seakeeping and maneuvering bya RANSE method. Ph.D. Thesis. Teschnichen Universitat Hamburg, Harburg,2004.

towards a numerical hydrodynamics laboratory by developing an overlapping mesh solver based on a...

Documents