IMPROVING THE NUMERICAL ROBUSTNESS OF BUOYANCY MODIFIED k ...

Improving the Numerical Robustness of Buoyancy modified k-ω SST Turbulence ModelC. Liu, W. Zhao, J. Wang and D. Wan

VIII International Conference on Computational Methods in Marine Engineering MARINE 2019

IMPROVING THE NUMERICAL ROBUSTNESS OF BUOYANCY MODIFIED k-ω SST TURBULENCE MODEL

CONG LIU, WEIWEN ZHAO, JIANHUA WANG AND DECHENG WAN*

State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering,

Shanghai Jiao Tong University, Shanghai, China *Corresponding author, E-mail: [email protected]

Key words: Numerical instability, Wave decay, Buoyancy modification, Over-predicted turbulence, OpenFOAM

Abstract. In most previous computational fluid dynamics (CFD) studies involving free surfaces progressive wave with RANS model, they always suffered from over-predicted turbulence near the interface especially for the short wave with high steepness. This over-prediction would lead to severe wave damping and make the results invalid. To overcome this problem, many researchers has modified the RANS model to improve the free surface simulation. The over-prediction is suppressed by: 1) adding a buoyancy term in the turbulent kinetic energy (TKE) equation and 2) redefining the eddy-viscosity to stabilize the model in nearly potential flow region. The new stabilized turbulence model prevents from the exponential growth of the eddy-viscosity and subsequent wave decay exhibited by original model. In this study, the modified models mentioned above are implemented in OpenFOAM. But our practice on modified k-ω SST model encountered server numerical instability. The numerical instability is caused by the sudden increase of turbulent viscosity. Careful inspection on the results show that this erroneous spikes in the eddy-viscosity is related to the dissipation rate, ω, going to zero in some grid cell near the free surface. To prevent this from happening, a new limiter is simply imposed on the buoyancy term in the TKE equation. This new limiter is validated by conducting 2D wave simulations. The numerical results show good agreement with the experimental measurements for the surface elevations, undertow profiles of the horizontal velocity and turbulent kinetic energy profiles. Meanwhile, the numerical stability is also guaranteed.

1. INTRODUCTION

Among the most common engineering problem related to aquatic environment, free-surface wave has to be reckoned. Usually, these problems include the process of wave propagation, their interaction with structures and wave breaking, which require solving the two-phase flow and turbulence in the same time. Through many years of research and development, volume of fluid (VOF) method has been proved to be reliable to deal with the multiphase problem. The two equation Reynolds-averaged Navier-Stokes (RANS) model has sufficient accuracy to handle most of the turbulence from the view of engineering. Therefore, their combination is widely used by a lot of researchers.[1–3] But one of the most commonly mentioned problems

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Cong Liu, Weiwen Zhao, Jianhua Wang and Decheng Wan

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among these researches is the over-prediction of turbulence and subsequent unavoidable wave decay during its propagation. In the case of breaking wave study using RANS model, there has been a clear tendency to predict turbulence levels that are much higher than have been measured[4][5]. This over-prediction shows itself not only in the surf zone where the wave breaking happened, but also in the region prior to breaking. The over-predicted turbulence both on and beneath the surface wave. On the surface, this over-prediction is induced by the large production of turbulent kinetic energy, k, which is related to the large velocity gradient around the interface between water and air. This issue is suppressed by implementing a buoyancy-modified turbulence model, where an additional sink term is added to the TKE equation[6][7]. The justification of this additional term is given by Van Maele and Merci[8]. For the computation of two-phase flow using VOF, the density is not constant near the air-water interface. But the turbulence model is developed for the incompressible flow. The density variation at the interface has a non-negligible effect on the turbulent nature. For the compressible turbulence flow, the instantaneous density can be expressed as the sum of time average value and a fluctuating term, just like the instantaneous velocity. An extra source term related to the correlation of density-velocity fluctuation will appear in the TKE for compressible turbulence flows, which accounts for the effect of buoyancy on the turbulence. This buoyancy term can be simply calculated using standard gradient diffusion hypothesis (SGDH), which related this density-velocity correlation to the production of density gradient of and turbulence viscosity. This process is similar with the Boussinesq approximation which relates the Reynolds Stress (the correlation of velocity-velocity fluctuation) to the production of mean velocity gradient and turbulence viscosity. Devolder et al. [6][7] developed the buoyancy modified k-ω and k-ω SST model and his code is available on the Github[9]. Following this research, the author also modified the code of k-ω and k-ω SST turbulence model in OpenFOAM, but the authors’ numerical experiments show that the Devolder’s k-ω SST implementation suffers from the numerical instability. This numerical instability is characterized by the sudden increase of turbulent viscosity. Careful inspection on the results show that this erroneous spikes in the eddy-viscosity is related to the dissipation rate, ω, going to zero in some grid cell near the free surface. According to the study of Menter on k-ω SST model[10], a small disturbances in the source term of TKE can lead to erroneous spikes in the eddy-viscosity. Similarly, our additional buoyancy term in the TKE could also lead to this instability. To prevent this from happening, a limiter is simply added to the buoyancy term in the TKE equation.

Besides, the wave damping crisis still exists beneath the wave surface because of the shear production term in TKE. This issue was originally described by Mayer and Madsen [11] who strictly proved the conditional instability of the k-ω closure model. Substituting the velocity field from the potential wave theory into the production term of the TKE equation results in a no-zero value by which the k will increase exponentially. They also tried to solve this problem by using the mean rotation instead of the original strain rate in the production terms of k. This alteration greatly suppresses the unphysical growth of the eddy-viscosity and hence improved predictions of the wave breaking point. But the following research by Larsen and Fuhrman [12] proved that there are several fundamental deficiencies with Mayer and Madsen’s modification [11]. This modification actually delays the growth of the eddy-viscosity, rather than completely avoids it. And this replacement is a direct violation of the Boussinesq approximation and has a negative effect on performance of model in the shear flow region. To eliminate this anomaly

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fundamentally, Larsen and Fuhrman [12] introduced a limiter on turbulent viscosity provided by the ratio of rotation and strain rate. This modification will be activated to restrain the turbulence in the potential flow region while remains passive in sheared flow region. Through this way, the wave decay can be prevented and the capability of turbulence model for shear flow is preserved in the same time. Larsen’s code is developed in the frame of OpenFOAM and available on the Github[13] named as stable model.

In this study, k-ω SST model are improved by both adding the buoyancy term and introducing the limiter on turbulent viscosity following the methods mentioned above. The main contribution of the present study is to eliminate the numerical instability caused by the buoyancy term. The present paper is organized as follows: In section 2, the theories of buoyancy modified model and Larsen’s stable model will be briefly reviewed and the cause of numerical instability is presented. A new limiter imposed on buoyancy term is proposed to remedy this instability. In section 3, the robust model is validated by conducting two kinds of simulations: simple progressive wave trains and spilling breaking waves. Finally, a brief conclusion is given.

2. MATHEMATICAL AND NUMERICAL MODELLING

2.1 The standard k-ω SST model and production limiter. The equation of incompressible k-ω SST model [10] originally implemented in OpenFOAM

are:

( )*ikk t

i ii

u kk kP kt x x x

+ = − + + (1)

( ) ( )221

12 1

+ = − + + + − i

k ti i i i it

u kFPt x x x x x (2)

where the turbulent eddy-viscosity is defined as follows:

( )1

1 2max ;

=t

kS F

(3)

The detailed parameters definition can be find in[10]. The only term should be paid attention here is the production term k

P defined as： *=min( ,10 ) k kP P k where = ( )

+

ji i

k tj j i

UU UPx x x

(4)

From the definition, it is clear that kP is a production limiter in the SST model. This limiter

is used to prevent the erroneous spikes in the eddy-viscosity [10]. In this work, this limiter is extended to limit the buoyancy term which will be introduced in the TKE equation. The justification of this production limiter is given by Menter [10]. Through the experience with two-equation turbulence models, he found that in regions of low values of ω, small disturbances in the shear strain rate can lead to erroneous spikes in the eddy-viscosity in the freestream or near the boundary layer edge. In order to understand the effect, the transport equation for the

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eddy-viscosity is derived from the Eqn. 1-3(In this discussion, the production term in Eqn. 1 and 2 should be Pk instead of the production limiter kP ):

2 + =1 =(1- ) (1- ( ))

= −

+ +

t t jk i i

j j i

UP U Ux

D Dkx x

k DDt Dt Dt

(5)

If ω goes to zero and t is finite, the production term for the eddy-viscosity goes to infinity for small disturbances in the strain rate. A simple way to prevent this from happening is to limit the production term by the following formula:

min( ,10 )k k kP P D= (6)

where kD is dissipation term in Eqn.1 define as * k .

This limiter has been very carefully tested by Menter[14] and it was found that even for complex flows the ratio of Pk/Dk reaches in maximum levels of only about two inside shear layers. This production limiter does not change the solution but only eliminates the occurrence of spikes in the eddy-viscosity due to numerical "wiggles" in the shear-strain tensor.

In our followed study, the buoyancy term will induce the spikes in the eddy-viscosity just like the shear-strain tensor did.

2.2 The buoyancy modification for k-ω SST model and its robustness According to the Van Maele and Merci [8], a buoyancy term is needed in order to take the

varying density around the air-water interface into account. The buoyancy term is only included in the TKE equation based on the SGDH. Following Devolder et al. [6], the buoyancy term is:

= −

t

jt j

uGb gx

(7)

where is density, g is gravitational acceleration and t is constant.

Adding this buoyancy term to the right hand side of Eqn. 1, The TKE equation of buoyancy-modified k-ω SST model becomes:

( )*ikk b t

i ii

u kk kP G kt x x x

+ = + − + + (8)

which is the same with Eqn. 19 in [7]. But this buoyancy term will induce the numerical instability. To indicate this effect, Eqn .5

is rewritten by Eqn. 8:

2

1 = ( )- 1(1- ) ji it t tj k

t jj j i

D Dk k D g pDt

UU Ux xDt Dt xx

= − =

+

(9)

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Similar to the Eqn.5, if goes to zero, eddy-viscosity goes to infinity for small disturbances in the kp .

To prevent this from happening, Eqn.6 is modified in this work to limit the production term by the following formula:

min( - ,10 )k k b kP P G D= (10)

This new limiter should be further tested and ensure that the ratio of ( - ) /10k b kP G D is far less than 1.

2.3 Larsen’s stabilized k-ω SST model Simply adding the buoyancy term does not solve the over-prediction of turbulence level

fundamentally. Rather, Mayer and Madsen [11] attributed the widespread over-prediction of turbulence in RANS models of surface waves to its unstable when applied to a region of potential flow beneath surface waves. Inspired by this research, Larsen and Fuhrman [12] further proved that k-ω SST model is unconditionally unstable for the conditions of potential flow progressive waves. Inserting velocity given by Stokes first-order wave theory into source term of TKE equation, after period averaging, and depth averaging, the source term is not zero. Subsequently, k and t growing exponentially at certain growth rate. To prevent it from happening, not only the buoyance term should be added in TKE equation, but also the eddy-viscosity should be redefined by:

1

01 2 0 1 2max( , , )

*

ta k

pa F p ap

= (11)

where 01=2 = ( )2

jiij ij ij

j i

uup S S Sx x

−

，，

1=2 = ( )2

jiij ij ij

j i

uupx x

−

， and λ2=0.05

Thus, in a region of nearly potential flow where 0p p ,the new addition to the limiter in (16) will be active. Larsen and Fuhrman [12] suggested that the new closure model will be formally stable as 0 2p p . Instead, in other more complicated sheared flow regions, p and 0p will be the same order of magnitude and the new limiter will remain inactive.

3. NUMERICAL TESTS FOR SURFACE WAVES In the following two subsections, both Devolder’s model and Larsen’s model with our new

limiter are tested. Two kinds of tests are conducted: a simple progressive wave train and spilling breaking wave. The first simulation is used to demonstrate the stable performance of new model without the sheared flow involved. The latter is to evaluate performance of the new model in a more physically complex situation. In all simulations, the time step has been adjusted such that a maximum Courant number Co=0.05 is always guaranteed to ensures accurate velocity kinematics and reduce the numerical dissipation. Our in-house solver naoe-FOAM-SJTU[15][16] is used to solve the governing equations combining with waves2FOAM[2]

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toolbox for wave generation and absorption. In our solver, relaxation zones blend the government equation implicitly instead of blending the volume fields explicitly before solving the government equation as suggested by[17]. In our simulations, temporal term is discretized by a second-order backward scheme. Convection terms are discretized using a blend of central difference and upwind schemes. Remaining schemes are second-order accurate central difference schemes.

3.1 Simulation of simple progressive wave train In this subsection, computed results will be presented for the propagation of surface wave.

This is a potential flow dominated case and should ideally be passed by the new models mentioned above. A stokes first-order wave with wave period T=1.2s and wave height H=0.12m is simulated in 2D grids. Mesh resolution guarantees the horizontal and vertical size Δx=Δz=0.01m.

Layout of the computational domain is showed in Figure 1. Two relaxation zones are set near the inlet and outlet to generate/absorb the wave. To demonstrate the performance of the turbulence closure relative to standard approaches, we will compare computed results from the laminar model, standard k-ω SST model[10], Devolder’s buoyancy modified k-ω SST model[7] with our new limiter in Eqn. 10, and Larsen’s stabilized k-ω SST model[12] also with our new limiter.

Figure 1 Layout of the computational domain for simulation of simple progressive wave

To demonstrate the necessity of our new limiter in Eqn. 10, this progressive wave simulation is first conducted using Devolder’s buoyancy model[17] without our new limiter. This calculation collapsed after around 100 time-steps. At several time-steps before the simulation going divergence, a spike value of turbulent viscosity, t , is presented in one cell as showed in Figure 2. In this cell, the value of ω is very small where a small disturbance of kp will leading to a large growth of t according to Eqn. 9. It shows that Devolder’s original buoyancy model cannot complete this simulation.

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Figure 2 The value of turbulent viscosity t , dissipation rate ω and kp

In the following content, both Devolder’s model and Larsen’s model are combined with our new limiter in default. For the sake of brevity, the new limiter will not be mentioned explicitly in the model’s name.

The time history of surface elevation at wave probe is shown in Figure 3. The position of wave probe is 11.17 m away from the inlet boundary as shown in Figure 1. As we can see, the standard k-ω model has a very different performance from other models. The wave height is terribly decayed when the wave approach to wave probe, while the performance of Devolder’s model and Larsen’s model is almost identical to the laminar model. All improved models maintain a nearly constant wave height. But a close look at the peak values of wave height shows that Devolder’s model causes a slight decay comparing to the Larsen’s model (as shown in the enlarged part of Figure 3 ). This result is consistent with our expectation since Larsen’s model not only include the buoyancy term, but also stabilize the turbulent viscosity in the potential flow region in waves. The success of this simple wave simulation proves that our new limiter ensures the numerical stability while maintains the performance of the improved turbulence models in potential flow dominated case.

Figure 3 Time histories of surface elevations in the simple progressive wave train simulation using laminar model, k-ω SST model, Devolder’s model and Larsen’s model

3.2 Simulation of spilling breaking wave Given that our new limiter passes the simple wave propagation, the new model should be

tested for more complex situations in the expectation that our limiter can maintain the numerical

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stability and at the same time keep the performance of the improved model. In this subsection, the simulations of the spilling breaking wave experiment[18] is conducted using standard k-ω SST model and Larsen’s model.

The computational domain for these simulations consists of a flat region with water depth D=0.4 m, connected to a 1:35 slope. The waves with wave period T=2 s and wave height H=0.25 m are generated on the flat region and wave will break as it propagates along the slope. Figure 4 demonstrates the layout of the computational domain. The bottom is modelled as a smooth solid wall on which wall functions are activated. A continuous wall function based on Spalding's law wall function is implemented for the turbulent viscosity. The initial values for k and ω in the computational domain are set to 2.28e-7 m2/s2 and 2.1 s-1, respectively.

Figure 4: Layout of the computational domain for simulation of the spilling breaking

The numerically obtained surface elevations, undertow profiles and TKE profiles are presented and compared to experimental measurements from Ting and Kirby[18] for spilling breaking. Each numerical simulation ran for 120 wave periods to establish stable conditions in the computational flume. All the results presented in this section are time averaged using the last 20 waves of the 120 waves simulated.

(a)

(b) Figure 5: Comparison of modelled (lines) and measured (red dot, from Ting and Kirby[18]) surface elevation

envelopes. (a) the standard k-ω SST model (b) Larsen’s k-ω SST model Figure 5 shows the comparison of the computed and experimental surface elevation

envelopes as well. It can be seen that the standard model delays the breaking point around x=6.5 m and overpredicts the surface elevation in the subsequent surf zone. On the contrary, Larsen’s model better captures the horizontal position of the breaking point and the surface elevation at breaking point and its subsequent decay in the surf zone have a well agreement with experiment.

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As a result, the Larsen’s model has better performance in terms of surface elevations for the case of spilling and our new limiter does not have the adverse effect to this advanced model.

x =7.275 m x =7.885 m x =8.495 m x =9.110 m Figure 6: Comparison of modelled (lines) and measured (line with dot) time average turbulent kinetic energy k

profiles and horizontal velocity u at x =7.275 m , x =7.885 m, x =8.495 m, x =9.110 m.

Figure 6 presents both measured and simulated undertow profiles at locations x =7.225m, x = 7.885 m, x = 8.495 m and x = 9.110 m ( the coordinate positions of these locations are same as the experiment from Ting and Kirby[18]). All these locations locate in the surf zone. Along those vertical profiles, the calculated time averaged horizontal velocity u and time averaged turbulent kinetic energy k are analyzed. In Figure 6, the results of Devolder’s model are directly from his paper[7]. As might be expected, the standard k-ω SST models severely over-predict the turbulence. In contrast, Larsen’s model predicts lower turbulent kinetic energy much more in line with the experiments.

In Figure 6, the velocity profiles computed with Larsen’s model remain qualitatively correct in structure, but become exaggerated. In contrast, the standard model actually results in more accurate velocity profiles. This unexpected result is very similar to the calculation obtained from [7] [12] indicating that it is not caused by our new limiter. Larsen and Fuhrman[12] argued that the better performance of standard model on velocity profile should likely be regarded as fortuitous, given that these models did not result in (i) properly spilling waves or (ii) correct turbulence/undertow structure at many positions leading up to the inner surf i.e. a correct qualitative description of the breaking process is lacking with these models. And very similar

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exaggerated velocity results were also obtained by Christensen[19] using more accurate LES model.

To further illustrate the difference between the models. The computed snapshots depicting the surface and non-dimensional eddy-viscosity tu u in one period are showed in Figure 7. As can be seen, the standard SST model gives over-predicted turbulence model both in the pre-breaking and surf zone. According to the experiment[18], these high values are nonphysical and should not be presented prior to breaking, where nearly potential flow is dominated. These artificial high values will in further contaminate the results of surf zone. Conversely, the results using the Larsen’s model surpasses the increase of eddy-viscosity in the nearly protentional region prior to breaking. Meanwhile the eddy-viscosity is rightly predicted in the bottom of boundary layer and surf zone. This demonstrates that Larsen’s model with our new limiter give a more reasonable prediction over standard models while the numerical stability is guaranteed.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(g) Figure 7: Numerically obtained snapshots around the breaking point of the turbulent kinematic viscosity and

free surface at different time phases using the standard k-ω SST (on the upper side) and Larsen’s k-ω SST model (on the lower side).

4. CONCLUSIONS In this study, a new source term limiter is proposed to suppress the numerical instability

induced by buoyancy term in k-ω SST model. The reason caused this instability is analyzed by

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deriving the transport equation for the eddy-viscosity derivation. It shows that a small disturbances of buoyancy term will induce spike value of eddy-viscosity when the dissipation rate ω goes to zero. To stabilize this instability, the buoyance term compared with the original source term is limited to smaller than the 10 times the dissipation term. The effectiveness of coefficient 10 is validated by two kinds of numerical tests: a simple progressive wave train and spilling breaking wave. The first simulation is used to demonstrate the stable performance of new model without the sheared flow involved. The latter is to evaluate performance of the new model in a more physically complex situation. In summary, our new limiter does not affect the advantage of the buoyance term, and in the meantime eliminates the occurrence of spikes in the eddy-viscosity. 5. REFERENCES [1] Xie, Z. (2013) Two-phase flow modelling of spilling and plunging breaking waves. Applied Mathematical

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CFD library: OpenFoam®. International Journal for Numerical Methods in Fluids. 70 (9), 1073–1088. [3] Brown, S.A., Greaves, D.M., Magar, V., and Conley, D.C. (2016) Evaluation of turbulence closure models

under spilling and plunging breakers in the surf zone. Coastal Engineering. 114 177–193. [4] Lin, P. and Liu, P.L.-F. (1998) A numerical study of breaking waves in the surf zone. Journal of Fluid

Mechanics. 359 239–264. [5] Bradford Scott F. (2000) Numerical Simulation of Surf Zone Dynamics. Journal of Waterway, Port,

Coastal, and Ocean Engineering. 126 (1), 1–13. [6] Devolder, B., Rauwoens, P., and Troch, P. (2017) Application of a buoyancy-modified k-ω SST

turbulence model to simulate wave run-up around a monopile subjected to regular waves using OpenFOAM ®. Coastal Engineering. 125 81–94.

[7] Devolder, B., Troch, P., and Rauwoens, P. (2018) Performance of a buoyancy-modified k-ω and k-ω SST turbulence model for simulating wave breaking under regular waves using OpenFOAM ®. Coastal Engineering. 138 49–65.

[8] Van Maele, K. and Merci, B. (2006) Application of two buoyancy-modified turbulence models to different types of buoyant plumes. Fire Safety Journal. 41 (2), 122–138.

[9] https://github.com/BrechtDevolder-UGent-KULeuven/buoyancyModifiedTurbulenceModels [10] Menter, F.R., Kuntz, M., and Langtry, R. (2003) Ten years of industrial experience with the SST

turbulence model. Turbulence, Heat and Mass Transfer. 4 (1), 625–632. [11] Mayer, S. and Madsen, P.A. (2001) Simulation of Breaking Waves in the Surf Zone using a Navier-Stokes

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cylinders by naoe-FOAM-SJTU solver. Applied Ocean Research. 65 327–337. [16] Shen, Z., Wan, D., and Carrica, P.M. (2015) Dynamic overset grids in OpenFOAM with application to

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IMPROVING THE NUMERICAL ROBUSTNESS OF BUOYANCY MODIFIED k ...

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