modeling of the wave breaking with cicsam and hric …

European Conference on Computational Fluid DynamicsECCOMAS CFD 2006

P. Wesseling, E. Onate and J. Periaux (Eds)c© TU Delft, The Netherlands, 2006

MODELING OF THE WAVE BREAKING WITH CICSAMAND HRIC HIGH-RESOLUTION SCHEMES

Tomasz Wac lawczyk∗, Tadeusz Koronowicz†

∗†The Szewalski Institute of Fluid Flow MachineryPolish Academy of Sciences,

Centre for Mechanics of Liquid,ul. J. Fiszera 14, Gdansk 80-952, Poland,

e-mail: [email protected] page: http://www.imp.gda.pl

Key words: Free surface flow, Volume of Fluid method, High-Resolution schemes

Abstract. The paper concerns modeling of two-phase flow with the volume of fluid method(VOF) and two high-resolution advection schemes based on the normalized variable dia-gram (NVD). Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM)and High Resolution Interface Capturing scheme (HRIC). Both considered schemes areused to discretize convective term in the scalar equation for the transport of the volumefraction. High-resolution schemes are employed to minimize influence of the artificialnumerical dissipation and to keep the shape of the step interface profile. Original contri-bution of this work is a detailed comparison of the two high-resolution schemes CICSAMand HRIC in the case of the breaking wave phenomenon. It is shown that using relativelysimple to apply, high-resolution schemes it is possible to obtain good agreement with anexperimental evidence and other authors results. However, some difficulties connectedwith optimal size of the local Courant number are adressed. Additionally, since HRICscheme is used in commercial software (e.g. Comet, Fluent) it is important to understandits capabilities and deficiencies when compared to other schemes.

1 INTRODUCTION

Numerical modelling of the multiphase flows is challenging field of the computationalfluid dynamics, which experiences rapid growth. Recently, several interesting numer-ical approaches to the problem were introduced and developed, among them: level-set methods1,2, particle finite-element method3,4 or smoothed particle hydrodynamics5,6.Moreover, the well established Volume of Fluid7 method gained a large number of im-provements to the interface reconstruction techniques8,9 and to the modelling of the surfacetension10,11.

In this work, Volume of Fluid (VOF) interface capturing method is employed to modelthe flow of the set of fluids, which are immiscible on the molecular level. The interface

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Tomasz Wac lawczyk, Tadeusz Koronowicz

position is captured using two high-resolution advection schemes CICSAM12 and HRIC13,based on the normalised variable diagram NVD14. Unlike geometric interface reconstruc-tion methods, high-resolution schemes do not introduce geometric representation of theinterface. Here to avoid smearing of the step interface profile and preserve boundednesscriterion, properly designed discretization scheme is applied.

To model dynamics of the system of immiscible fluids, the solution of the equationfor transport of the volume fraction is coupled with an in-house code solving Navier-Stokes equations. Implicit finite-volume second order accurate code employs SIMPLEprocedure15,16 for pressure-velocity coupling.

2 EQUATIONS GOVERNING THE FLOW OF IMMISCIBLE FLUIDS

To model dynamics of the set of immiscible, incompressible and viscous fluids one needsto solve Navier-Stokes equation (1) and continuity equation (2):

∂ρui∂t

+∂ρujui∂xj

= − ∂p

∂xi+

∂

∂xj

[µ

(∂ui∂xj

+∂uj∂xi

)]+ ρgi + σκniδs, (1)

∂ui∂xi

= 0, (2)

where ui denotes i-th velocity component, ρ is density, p is pressure, µ is dynamic viscosity,gi denotes i-th component of the gravitational acceleration. Additional term that appearsin Eq. (1) is inherited from one fluid formulation and describes influence of the surfacetension. When surface tension coefficient σ is constant the resulting force acts in thedirection normal to the interface ni. κ = ∇~n denotes mean Gaussian curvature, δs isDirac delta function that indicates local character of the surface tension term, since it isnon-zero only at the interface.

In this paper so called one-fluid formulation is used i.e. one treats the whole system ofimmiscible fluids as a single continuum medium. This assumption leads to the sharing ofthe same velocity in one control volume, between all fluids that are modelled. This lastfact is clearly a disadvantage of the one-fluid formulation since one is not able to modeldirectly effects of the slip phenomenon.

To introduce material properties of the fluids that built aforementioned system, oneneeds to consider additional constitutive relations. Since in the case of the present workonly dynamics of the set of two fluids (water and air) is modelled, it is enough to chooseone fluid as the background (water) and to consider only its volume fraction φ. Takingthis into account, one obtains:

ρ = (1− φ)ρ1 + φρ2, (3)

µ = (1− φ)µ1 + φµ2, (4)

where ρj, µj denotes density and dynamic viscosity of the j-th fluid respectively. Finally,using assumption about incompressible character of the considered fluids one can notice

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that substitution of the Eq. (3) to the conservative form of the continuity equation leadsto the equation for transport of the volume fraction that writes:

∂φ

∂t+ uj

∂φ

∂xj= 0, (5)

where, as mentioned before, φ denotes volume fraction of the background fluid. In thecase of the VOF framework, value of the volume fraction φ in the control volume indicatesits presence φ = 1 or absence φ = 0. When 0 < φ < 1 volume fraction distribution carriesinformation about position of the interface.

The crucial issue for the modelling of the multiphase flow is a proper solution of Eq. (5)i.e. the discretization of time derivative and nonlinear convective term. The discretizationhave to avoid smearing of the step interface profile and assures that the boundednesscriterion is satisfied. Here, Eq. (5) is discretized using finite volume method, employingCrank-Nicolson method for integration in time. This discretization procedure gives theimplicit, unsplit time discretization scheme12:

VP∆tφt+∆tP +

1

2

n∑f=1

φt+∆tf Sfu

ti,f =

VP∆tφtP −

1

2

n∑f=1

φtfSfuti,f , (6)

where VP denotes the volume of the control volume P , n is the number of faces of thecontrol volume, Sf is the face area and finally ∆t denotes the time step size. Convectiveflux (value of the volume fraction at the face of the control volume φf ) is obtained fromtwo high-resolution schemes CICSAM12 and HRIC13 that are compared in this work.

3 HIGH-RESOLUTION SCHEMES

Discretization of the convective term in Eq. (5) is critical for proper capturing of theinterface position. One can show17 that commonly used schemes e.g. upwind (UDS),central (CDS) differencing schemes, introduce effects of artificial diffusion or dispersionrespectively to their order of accuracy. Also other higher order schemes result in localoscillations18 of the variable, here volume fraction, that is unacceptable when one wantsto employ them for the discretization of convective term in Eq. (5). To avoid this artificialeffects high-resolution schemes are built that assure both lack of the numerical diffusionand compressive character i.e. sharpening of the step interface profile12,13.

3.1 Normalised Variable Diagram NVD

Normalised Variable Diagram14 (NVD) provides the foundation for both CICSAM12

and HRIC13 schemes. It is based on the convective boundedness criterion12,13,14 (CBC)that states that the variable distribution between the centers of the neighbourhood controlvolumes e.g. D and A should remain smooth φD ≤ φf ≤ φA, see Fig. 1a. Using thisconstraint, and information about value of the variable in the upwind control volume φU ,normalised variables are introduced:

3


f ADU

V

A

φU

φD

φφf

φ~D

UD

1

0.5

1 DD

0.5

~φf

a) b)

Figure 1: a) Boundedness criterion, U upwind, D donor, A acceptor cells, b) Normalized Variable DiagramNVD, shaded region indicates where CBC is satisfied, notice that UDS satisfies CBC criterion in the wholerange of φD

φf =φf − φUφA − φU

, (7)

φD =φD − φUφA − φU

. (8)

Equations (7), (8) allow to rewrite the boundedness criterion basing on normalised vari-ables φD ≤ φf ≤ 1, what can be depicted on a diagram, cf. Fig. 1b, that shows regionwhere the boundedness criterion is satisfied by any differential scheme. One needs tonotice that when using Eq. (7)-(8) it is possible to find the value of the volume fractionat the control volume face φf :

φf =(1− βf

)φD + βfφA, (9)

βf =φf − φD1− φD

, (10)

which is used for the calculation of the volume fraction flux in Eq. (6).

3.2 Compressive Interface Capturing Scheme for Arbitrary Meshes CICSAM

In the case of the CICSAM scheme, additional assumption about the dependence ofthe region where the CBC is satisfied on the CFL condition is used12. The local value ofthe Courant number, defined at the face of the control volume Cf = ufSf∆t/V together

with CBC gives the following formula φD ≤ φf ≤ min(1, φD/Cf

), which can be plotted

at NVD, see Fig. 2. One needs to notice, that for explicit schemes, if local value of theCourant number is equal to Cf = 1, only the UDS satisfies the CBC criterion. On theother hand, the smaller the value of the local Cf the lager the domain where the CBCcriterion is satisfied.

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C f φ~D

UD

1

0.5

1 DD

0.5

~φf

Figure 2: Dependence of the CBC region on local CFL condition

The combining of the donor-acceptor7 scheme (DAS) with NVD formulation depen-dent on the CFL condition, results in the first component of the CICSAM known as theHYPER-C12 scheme:

φfCBC =

φf = φD : 0 < φD, φD > 1

min(

1, φDCf

): 0 ≤ φD ≤ 1

. (11)

The HYPER-C scheme Eq. (11) satisfies the CBC criterion and is shown to be compres-sive, which means that it changes any gradient to a step profile due to the downwinddifferencing scheme employed (DDS).

However, compressive character of the HYPER-C is not always desirable. When inter-face is tangential to the flow direction it is shown that aforementioned scheme tends toartificially deform its shape. For this reason it is found to be necessary to switch betweenscheme Eq. (11) and other less compressive formulation. In the case of the CICSAM,the ULTIMATE-QUICKEST14 (UQ) scheme is employed, that is bounded version of thethird order accurate QUICK:

φfUQ =

φD : 0 < φD, φD > 1

min{

8Cf φD+(1−Cf )(6φD+3)

8, φfCBC

}: 0 ≤ φD ≤ 1

. (12)

To switch smoothly between both schemes, linear blending is used, with blending factor0 ≤ γf ≤ 1. Value of the γf depends on the angle θf Eq. (14) between unit vector normalto the interface ~n = ∇φD/|∇φD| and unit vector parallel to the line between centers of

the donor D and acceptor A cells ~d = ~DA/| ~DA|, cf. Fig. 3a. When interface position isnormal to the direction of the flow γf = 1 and Eq. (11) is used. In the case of tangentialorientation of the interface γf = 0 and Eq. (12) is employed.

φf = γf φfCBC + (1− γf )φfUQ , 0 ≤ γf ≤ 1, (13)

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θf = arccos∣∣∣~d~n∣∣∣ , (14)

γf = min

{1 + cos 2θf

2, 1

}. (15)

It should be noticed that above derivation of the high-resolution CICSAM scheme wascarried out only in one-dimension. To extend above formulation to multiply dimensionscell Courant number is introduced CD =

∑nf=1 max(Cf , 0), where n denotes number of

the control volume faces.

3.3 High Resolution Interface Capturing scheme HRIC

To simplify above procedure an get rid of the explicit dependence on the CFL conditionthe HRIC13 scheme was introduced. As it was mentioned this scheme also relies on theNVD and normalised variables. Application of the HRIC scheme can be divided in to

d

n

θf

D A

φf~∗∗

φf~*

Cf0.70.3

φ~D

1

a) b)

Figure 3: a) Definition of the vector normal to the interface ~n = ∇φ/|∇φ| and vector parallel to theline between centers of D, A control volumes ~d = ~DA/| ~DA|, b) Continuity of the HRIC scheme in timedomainthree steps. Firstly, normalised cell face value will be estimated from a scheme thatcontinuously connects upwind and downwind schemes on the NVD diagram, cf. Fig. 1b:

φf =

φD : φD < 0, φD > 1

2φD : 0 ≤ φD < 0.5

1 : 0.5 ≤ φD ≤ 1

. (16)

Secondly, since DDS can cause alignment of the interface with the mesh and its artificialdeformation (as in the case of the HYPER-C scheme) one needs other scheme that satisfiesthe CBC. In the case of the HRIC, as the most straightforward choice the UDS schemeis employed. Again the blending factor γf connected with angle θf is introduced, see Eq.(18), to switch smoothly between the schemes.

φ∗f = γf φf + φD(1− γf ), (17)

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γf =√| cos θ|. (18)

Blending of the UDS and the DDS schemes is dynamic and takes into account localdistribution of the volume fraction. In the case when the CFL condition is not satisfiedthe dynamic character of this scheme can cause stability problems. Therefore, φ∗f , see Eq.(17), is corrected with respect to the local Courant number Cf , cf. Eq. (19). The goal ofthis correction is to force continuous switching between schemes also in time domain, cf.Fig. 3b.

φ∗∗f =

φ∗f : Cf < 0.3

φD : Cf > 0.7

φD + (φ∗f − φD)0.7−Cf0.7−0.3

: 0.3 ≤ Cf ≤ 0.7

. (19)

When using this scheme in multiple dimensions the local Courant number Cf is againreplaced by its cell definition CD.

One can notice that main difference between the CICSAM and the HRIC are the orderof accuracy of the component schemes and differently included CFL condition. Nextseveral tests will be performed to asses how those differences influence results obtainedwith two presented high-resolution schemes.

4 ADVECTION TEST CASES

It is known that when convection is the only transport phenomenon, distance betweentwo points advected in the uniform flow field remains constant. Therefore, properties ofthe high-resolution schemes are assessed using known in literature advection test cases(only Eq. (5) is solved, velocity field is assumed to be known). Firstly, advection of thecircular spot in uniform, oblique velocity field is investigated, cf. Fig. 4. During this testcase, orders of accuracy of the CICSAM and the HRIC are estimated.

r = [0.15,0.15]mI

r = [1.55,0.85]m

u = [0.1, 0.05]m/s

2m

1m

F

φ=0

φ=1

outlet

inlet

Figure 4: Computational domain for circular spot advection test case

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Moreover, since both formulations depend on the value of the cell Courant number itsinfluence is checked.

The second solid body revolution test case, see Fig. 5, is performed to check propertiesof both schemes concerning conservation of the shape of the interface in more compli-cated rotational velocity field. Results obtained with the HRIC and the CICSAM high-resolution schemes are compared with two versions of constrained interpolation profile(CIP) semi-Lagrangian scheme18.

symmetryplane

ω

s=0.1

dy=

1

R=0.2

dx=1

Figure 5: Revolution of the solid body, advection test case

4.1 Circle advection test

The first, circle advection test case is carried out to check shape preserving propertiesof both the CICSAM and the HRIC. Since both schemes are convolved with the localvalue of the cell Courant number, see Eq. (11)-(12), Eq. (19), advection test is carriedout using four different cell Courant numbers CD, see Fig. 6. The values of CD arechosen to check the characteristic points in the range 0 < CD < 1, cf. Fig. 3b. Figure 4,depicts computational domain and applied boundary conditions: inlet at the south andwest boundaries and outlet on the east and north boundaries, initial ~rI = [0.15, 0.15] andfinal ~rF = [1.55, 0.85] position of the spot. Let us notice that in the case of the largestvalue of the cell Courant number CD = 0.7, the position of the bubble is captured at~rF = [0.73, 0.43]. Because it was the last position where the circular spot was observedon the coarsest 64× 32 grid, compare Fig. 6a and Fig. 6d.

On the first grid 64×32, results obtained with the CICSAM and the HRIC both sufferfrom an artificial cell alignment, cf. Fig. 6a-d, i.e. initial circular spot shape deform tothe square one. However, one can notice that when CD = 0.1, shape reconstruction isperformed better with the CICSAM scheme. When Courant number becomes larger, itresults in more visible effects of the numerical diffusion that changes initial volume of thecircular spot at the final position ~rF . One can notice that for CD = 0.7, see Fig. 6d, the

8


CICSAM scheme dependence on large CD values is much stronger than in the case of theHRIC. This effect might be explained by implicit dependence of the CICSAM scheme onthe CFL condition, while the HRIC includes it only in the one corrector step. In the caseof the second 128× 64 grid the superiority of the CICSAM, in circle shape reconstructionover the HRIC is clearly visible, see CD = 0.1, cf. Fig. 6e.

64× 32, CD = 0.1 64× 32, CD = 0.3 64× 32, CD = 0.5 64× 32, CD = 0.7

a) b) c) d)

128× 64, CD = 0.1 128× 64, CD = 0.3 128× 64, CD = 0.5 128× 64, CD = 0.7

e) f) g) h)

256× 128, CD = 0.1 256× 128, CD = 0.3 256× 128, CD = 0.5 256× 128, CD = 0.7

i) j) k) l)

512× 256, CD = 0.1 512× 256, CD = 0.3 512× 256, CD = 0.5 512× 256, CD = 0.7

m) n) o) p)

Figure 6: Comparison of the CICSAM and the HRIC shape preserving properties on four graduallyrefined grids and using four values of the cell Courant number CD, figures depicts contours of the volumefraction φ = 0.5

The circle shape is much better preserved by the first aforementioned scheme, while thelatter still suffers from mesh alignment problems. Again, with increasing CD shape pre-serving properties of both schemes become similar, see Figs.6f-h, finally showing smallerdependence of the HRIC on the CFL condition, see Fig. 6h.

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Solution obtained on the third grid 256× 128 shows that both high-resolution schemesstart to fit the exact shape of the circular spot, when CD = 0.1 and CD = 0.3. It isinteresting to notice that for CD = 0.5, locally, mesh alignment phenomenon can beobserved in the case of both formulations, cf. Fig. 6k. For the CD = 0.7 this phenomenonis not visible at Fig. 6l but one needs to keep in mind that this figure is made half wayfrom the final position of the spot ~rF . The results obtained on the last grid 512 × 256,show that both schemes converge to the exact solution for both CD = 0.1 and CD = 0.3.Additionally, one can notice that the error in reconstruction of the shape is almost thesame, see Fig. 7.

CD = 0.1

a)

CD = 0.3

b)

CD = 0.5

c)

CD = 0.7

d)

Figure 7: Order of accuracy of the CICSAM and the HRIC high-resolution schemes calculated fromresults on four gradually refined grids a) CD = 0.1, b) CD = 0.3, c) CD = 0.5, d) CD = 0.7

As it was mentioned, since exact solution of this problem is known, results of the testcase where used to estimate order of accuracy of both high-resolution schemes. The errorof the numerical solution was calculated from the following average error formula15:

E2 =1

N

N∑i=1

|φexacti − φi|, (20)

10


where φexacti , φi are exact and numerical solutions, respectively, N is a number of volumefraction contour samples. From Fig. 7a-c it can be seen that both the CICSAM and theHRIC are first order accurate12. In the case of the CD = 0.7 one can notice that order ofaccuracy of the method (approximately, since this error estimation is made for differentfinal position of the spot) is smaller then one. Moreover, in the case of the largest CD,average errors of the CICSAM scheme are larger then the average errors of the HRIC, cf.Fig. 7d, on each considered grid. When the mesh is gradually refined, the difference inerror of the solution between both high-resolution schemes becomes smaller independentlyon the value of the Courant number CD. Finally for the finest grid the average error ofthe solution is almost the same for all values of CD.

4.2 Revolution of the solid body

The second advection test case was performed to compare the shape preserving prop-erties of the CICSAM and the HRIC high-resolution schemes with the CIP18 semi-Lagrangian scheme in rotational velocity field.

a) b) c)

d) e)

Figure 8: Volume fraction distribution after one revolution of the solid body, CD ≈ 0.15 at the circlecontour: a) initial condition, b) CICSAM, c) HRIC, d) rational CIP18, e) tangent-transformed CIP18

In this test case, circular solid body with a slot is revolved with constant angularvelocity ω that is chosen to satisfy CD ≈ 0.15 at the circle edges. Final solution wasobtained after 1360 time steps, with ∆t = 0.005s on 64 × 64 grid. Figure 8 depicts

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initial shape of the solid body and solutions obtained with aforementioned high-resolutionschemes. Comparing results obtained with the CICSAM and the HRIC, one can notice,that the first one preserves the shape of the interface better then the latter, compare Fig.8b and Fig. 8c. The interface calculated with the HRIC is more smeared, non-zero volumefraction values are visible in the slot after one revolution. Conclusion from this test caseis that the CICSAM does better work, what is in agreement with the results obtained inthe first circular spot advection test for small CD values.

Finally, the CICSAM and the HRIC results can be compared with results obtainedusing two CIP18 formulations. It is noticeable, that the CICSAM shape preserving prop-erties lie between capabilities of the rational CIP18 and the tangent-transformed CIP18

formulations, however, closer to the last mentioned scheme. Unlike the CICSAM results,the HRIC schemes shape preserving properties are closer to rational CIP formulation,however one needs to notice that interface deformation is smaller in the case of the HRICscheme.

Since the constrained interpolation profile CIP18 formulation is a semi-Lagrangian onei.e. additionally three equations are solved to trace advection of each component of thegradient of the volume fraction, with step interface profile constrained by higher-orderpolynomial function. In opinion of the authors comparison of the CICSAM with the lastvery good result obtained with tangent-transformed CIP18 should be found satisfactory.

5 BREAKING OF THE WATER WAVE

Next simulation is performed to check how both considered schemes can manage withthe real physical example of the multiphase flow. In this case Eq. (5) is coupled withEqs.(1)-(4), thus the whole set of equations that describe flow of the two fluids is solved.Here, the water column with height H = 600mm and width 2H, see Fig. 9, collapsesin the gravitational field. Results of the simulations are compared with an experimentaldata6 consisting of height of the water surface recorded at two positions H1, H2 andthe pressure history that was measured at point P1, cf. Fig. 9. Moreover, flow fieldparameters obtained with the CICSAM and the HRIC schemes are compared with theSPH6 (Smoothed Particle Hydrodynamics) method and with results of the LS19 (Level-Set) method.

Initial volume fraction distribution is shown in Fig. 9, initial pressure field is set accord-ing to the hydrostatic pressure connected with density distribution, initial velocity field isset to zero. Material properties of fluids are for water: ρ1 = 998 kg/m3, µ1 = 0.99× 10−5

kg/(ms) and for air: ρ2 = 1.205 kg/m3, µ2 = 1.81 × 10−3 kg/(ms). To trace behaviourof the water-air interface a non-dimensional time T , based on initial water column heightH is introduced:

T = t√g/H. (21)

After mesh independence study it was found out that simulations should be performedon 512 × 256 control volumes grid. Time step size value is set to assure that maximum

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slip wallno

g=9.8 m/s2

H

H2H1

5.366H

3H

3.713H

4.541H

φ=0

0.26H

P1

2H

φ=1

Figure 9: Computational domain for the breaking wave test case, H denotes initial height of the watercolumn, H1, H2 positions of the water height probes, P1 position of the pressure probe

value of the cell Courant number is CD = 0.2.

5.1 Comparison with water height history from H1, H2 probes

Initially water column collapses in the gravitational field colliding with the oppositewall (see Fig. 12, time T1, T2). After collision, it climbs up across the west wall of thetank (see Fig. 12, time T3, T4), afterwards, when its whole kinetic energy changes intopotential energy it starts to move downward. During descending motion the water waveis formed that finally overturns and breaks, (cf. Fig. 12, time T5).

Results obtained during the simulation compare well with experimental evidence andresults of the SPH method6, when T < 6.3 in the case of height probe H1 and for timesT < 5.7 for probe H2, cf. Fig. 10. One can notice that water height predicted with theCICSAM and the HRIC schemes fit experiment better in the case of H1 probe when waterinitially changes its height, see Fig. 10, (H1 probe about T = 1.5). The differences betweenexperiment and numerical results obtained with high-resolution schemes is noticeable, butcharacter of the flow is preserved. The results start to differ more after the water waveoverturns and breaks. The sudden peak in pressure history might be observed, see Fig.11 about T = 6 when broken wave hits the surface of the water. The same phenomenonis recorded at Fig. 10 in rapid increase of water level first noticed at H2 probe andthen observed at H1 probe. The data recorded by the probe H1 indicate that both theCICSAM and the HRIC schemes predict correctly maximal height of the water wave,slightly better then the SPH method6. However, change of the water level when T > 6.7is better modelled with the referred SPH implementation, where motion of the two fluids

13


is taken into account and compressibility effects are partially included. Despite the factthat in current implementation of the VOF method both fluids are incompressible, resultsobtained are closer to experimental evidence then SPH results6 where only dynamics ofthe water is taken in to account, compare Fig. 10. It proves that it is necessary includeboth the motion of the water and air, when wave braking phenomena are modelled.Additionally, it might be noticed that change of the water height reconstructed with

Figure 10: Water height obtained with the CICSAM and the HRIC schemes (solid line), compared withexperiment (black dots) and SPH method6 (dashed-line - model for water-air system with compressibilityincluded, doted line - model for water)

the HRIC compares better with experimental data than this obtained with the CICSAM.Comparison of the results from the H2 probe shows that both high-resolution schemes andthe SPH6 only qualitatively predict behaviour of the interface, compare 5.6 < T < 7.5.However, for T > 7.5 solution obtained with the CICSAM and the HRIC is closer tothe experiment than results from the SPH6 method, cf. Fig. 10. The differences betweenexperimental evidence and calculation after wave break, are justified by experimentaldifficulties in measurement of the water level6.19.

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5.2 Comparison with pressure history from P1 probe

Pressure history obtained with the CICSAM and the HRIC schemes compares wellwith experimental evidence and results obtained with the SPH method6, see Fig. 11. Itmight be observed that the moment of the first pressure peak, about T = 6 is betterpredicted with high-resolution schemes. The magnitude of the pressure peak is almostthe same as in the case of SPH results6. Fast oscillations of the pressure from the SPH,see Fig. 11, are explained as an effect of the air oscillations inside the bubble entrapped bythe breaking wave. In the case of the solutions obtained with the CICSAM or the HRICchange in the pressure of an entrapped air is also modelled, however resulting bubble“oscillations” have much longer period. Additionally, one can notice that pressure historyobtained with the CICSAM, is closer to the experimental evidence and the SPH6 resultsthan in the case of the HRIC, see Fig. 11, T > 10.

Figure 11: Pressure history obtained with the CICSAM and the HRIC schemes (solid line) comparedwith experiment (solid dots) and results of the SPH method6 (dashed line - model for water-air systemcompressibility effects included)

5.3 Comparison of the wave profiles

Finally, differences in wave profiles predicted with both high-resolution schemes may becompared. Figure 12 depicts evolution of the free surface in non-dimensional time T . Thedifference between following time moments is constant and equal Tn+1 − Tn = 1.2. Thefirst noticeable differences in the shape of the wave profiles between both high-resolutionschemes are visible at T5 = 5.65 at the wave front. Wave profiles for times larger thenT5 are compared with results obtained using the LS (Level-Set) methods6,19, see Fig.13. One can notice, that wave profile obtained with the CICSAM scheme, see Fig. 13,is more deformed showing larger number of the flow details for time T = 7.14. Unlikethe CICSAM results, the HRIC solution preserves smoothness and is more similar to thesolution obtained with the LS method6,19. In both cases the main features of the flow areproperly reconstructed.

15


Figure 12: Free surface evolution in time obtained with the CICSAM (solid line) and the HRIC (dashedline) schemes, first differences in solution are noticable at T5

Figure 13: Wave profiles in following times obtained with the CICSAM and the HRIC schemes (solidline) compared with solution obtained with the LS method19 (black triangles), contours denote φ = 0.5

16


Differences appear in the volume and shape of the air bubble entrapped by the breakingwave and localisation and shape of the water filament that reflects after collision with thefree surface.

6 CONCLUSIONS

In present paper, the VOF method with high-resolution scheme methodology was testedin two advection test cases and was applied to model dynamics of the system of immisciblefluids in the case of the breaking wave phenomenon. Two high-resolution schemes theCICSAM and the HRIC were employed for discretization of the nonlinear convective termin equation for transport of the volume fraction, cf. Eq. (5). Both high-resolution schemesproved their ability to capture interface position and preserve shape of the step interfaceprofile.

Results obtained with the CICSAM and the HRIC schemes in two advection test casesshow that both formulations are first order accurate and that their shape preserving orinterface capturing properties are dependent on the size of the cell Courant number. Sincethe HRIC dependence on the cell Courant number is explicit, cf. Eq. (18), it seems to beless sensitive on the value of CD then the CICSAM. However, in opinion of the authors forcalculations with both considered high-resolution schemes Courant number value shouldbe chosen smaller then CD < 0.5.

Comparison of both high-resolution schemes with the CIP18 type schemes, shows thatinterface capturing properties of the CICSAM are close to the tangent-transformed CIP18.Taking into consideration complexity of both approaches, the first one seems to be prefer-able. However, this issue needs further investigation.

Finally, solution in the case of the breaking wave problem shows that it is possible toobtain results that compare well with an experimental evidence and other author resultsusing the CICSAM and the HRIC high-resolution schemes.

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modeling of the wave breaking with cicsam and hric …

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