Applied Mathematics and Computation 215 (2009) 591–598

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Applied Mathematics and Computation

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

A large-eddy-based lattice Boltzmann modelfor turbulent flow simulation

Sheng ChenState Key Lab of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, ChinaInstitute for Computational Modeling in Civil Engineering, Technical University, Braunschweig 38106, Germany

a r t i c l e i n f o a b s t r a c t

Keywords:TurbulenceLattice Boltzmann methodLarge-eddy simulation (LES)

0096-3003/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.amc.2009.05.040

E-mail address: chen@irmb.tu-bs.de

In this paper a novel and simple large-eddy-based lattice Boltzmann model is proposed tosimulate two-dimensional turbulence. Unlike existing lattice Boltzmann models for turbu-lent flow simulation, which were based on primitive-variables Navier–Stokes equations,the target macroscopic equations of the present model are vorticity-streamfunction equa-tions. Thanks to the intrinsic features of vorticity-streamfunction equations, the presentmodel is efficient, stable and simple for two-dimensional turbulence simulation. Theadvantages of the present model are validated by numerical experiments.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

Turbulence is still a big challenge not only for scientists but also for engineers though numerous efforts have been madeon it from more than one hundred years ago [1]. At present the most promising tool for progress in understanding the phys-ical phenomena and for guiding theoretical development is probably computer simulation [2,3]. The realistic turbulence al-ways is three-dimensional, which usually too expensive for available computer capability to simulate. Two-dimensionalturbulence has the special distinction that it is nowhere realized in nature or the laboratory but only in computer simula-tions. Its importance is twofold [4]: first, that it idealizes geophysical phenomena in the atmosphere, oceans and magneto-sphere and provides a starting point for modeling these phenomena ; second, that it presents a bizarre and instructivestatistical mechanics. Phenomena characteristic of two-dimensional turbulence also play essential roles in the confinementof thermonuclear plasmas and in superfluid and superconductive behavior of thin films. Consequently until now many stud-ies still focus on two-dimensional turbulence [5–7], to cite only a few.

In the last two decades, the lattice Boltzmann (LB) model has matured as an efficient alternative for simulating and mod-eling complicated physical, chemical and social systems [8–25]. The implementation of a LB procedure is much easier thanthat of traditional CFD methods. Parallelization of the LB method is natural since the relaxation is local and the communi-cation pattern in propagation is one way, and the performance increases nearly linearly with the number of CPUs. Moreover,the LB method has been compared favourably with spectral methods [26], artificial compressibility methods [27], finite vol-ume methods [28,29] and finite difference methods [30], all quantitative results further validate excellent performance ofthe LB method not only in computational efficiency but also in numerical accuracy. Due to these advantages, the LB methodhas been extended to simulate turbulent flows [31–35], to cite only a few. For turbulence simulation, the LB method can beused as a direct numerical simulation tool or be combined with the large-eddy simulation (LES) [10]. The latter is more pop-ular due to the balance between accuracy and efficiency. The spirit of large-eddy-based LB models is to split the effectiveviscosity me into two parts m0 and mt . m0 is the molecular viscosity and mt the eddy viscosity [31]. The effective lattice

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592 S. Chen / Applied Mathematics and Computation 215 (2009) 591–598

relaxation time se, which relates to me, will avoid approaching too closely to 0:5 due to the additional eddy viscosity mt . Con-sequently the numerical stability of the LB method for high Reynolds number flow is ensured [31,35].

However, all existing large-eddy-based LB models for turbulence simulations are based on primitive-variables Navier–Stokes equations. Though they have been successfully used in many applications, the disadvantages of the primitive-vari-ables large-eddy-based LB models are obvious: First, for most geophysical turbulence, usually the vorticity-streamfunctionequations, instead of primitive-variables Navier–Stokes equations, serve as the governing equations [4,36]. Therefore theprimitive-variables large-eddy-based LB models cannot be used for such geophysical turbulence. Second, the calculationof se of the primitive-variables large-eddy-based LB models is generally complicated, not only for the Bhatnagar–Gross–Krook approximation [31] but also for multiple-relaxation-time models [35], and in some cases to obtain the exact valueof se is extremely difficult [37].

In order to overcome the above disadvantages, in this paper a novel and simple large-eddy-based lattice Boltzmann mod-el, which is an extension of the model designed in our previous work [38–40], is proposed to simulate two-dimensional tur-bulence. Unlike existing primitive-variables large-eddy-based LB models, the target macroscopic equations of the presentmodel are vorticity-streamfunction equations. Therefore the present model can be employed straightforwardly for geophys-ical turbulence simulation. Moreover, the calculation of se of the present model is much simpler and more efficient than thatof primitive-variables large-eddy-based LB models.

The rest of the paper is organized as follows. Vorticity-streamfunction equations for two-dimensional turbulent flow ispresented in Section 2. In Section 3, a novel large-eddy-based LB model is introduced. In Section 4, numerical experimentsare performed to test the performance of the present model. The comparison of the computational efficiency between thepresent model and another LB model is also made. Conclusion is presented in the last section.

2. Governing equations

The vorticity-streamfunction-based governing equations for two-dimensional turbulent flow read [41–43]:

oxotþ u

oxoxþ v ox

oy¼ o

oxme

oxox

� �þ o

oyme

oxoy

� �; ð1Þ

o2wox2 þ

o2woy2 ¼ �x; ð2Þ

where w and x are the streamfunction and vorticity respectively. The velocities u and v are obtained from:

u ¼ owoy; ð3Þ

v ¼ � owox: ð4Þ

The effective viscosity me ¼ m0 þ mt . The eddy viscosity mt is computed from the local shear rate and a length scale when theSmagorinsky model is used

mt ¼ ðCDÞ2jSj; ð5Þ

where the constant C is called the Smagorinsky constant and is adjustable. In the present simulation C ¼ 0:1. Too small valueof C is not sufficient to damp the high-frequency fluctuations at small scales and may cause numerical instability for turbu-lent flow simulation. However, too big value of C would introduce significant viscosity-dependent error [39,40]. Throughnumerical tests we find that perhaps C ¼ 0:1 is the optimal value for the balance between numerical stability and accuracy[44]. In order to avoid the effect of viscosity-dependent error caused by variable value of C, it is better to fix the value of Cduring the simulation.

In above equation D is the filter width D ¼ Dx. Dx is the lattice grid spacing. The local intensity of the strain tensor is de-fined as

jSj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2SabSab

q; ð6Þ

where Sab is the strain rate tensor.

3. Large-eddy-based lattice Boltzmann model

Because Eq. (1) is nothing but a convection–diffusion equation, there are many matured efficient LB models for this typeof equation [14,44–47]. Recently Rasin et al. developed a multi-relaxation LB model to handle non-isotropic diffusion advec-tion problems and they gave an excellent discussion on the non-isotropy error in LB scheme [48]. In previous studies, thepresent author found that the D2Q4 lattice model is numerical instability for convection with high Reynolds number whichperhaps caused by the non-isotropy error and needed further investigation. In this paper the D2Q5 model proposed in ourprevious work [38] is employed. It reads:

S. Chen / Applied Mathematics and Computation 215 (2009) 591–598 593

gkð~xþ c~ekDt; t þ DtÞ � gkð~x; tÞ ¼ �s�1e ½gkð~x; tÞ � gðeqÞ

k ð~x; tÞ�; ð7Þ

where~ek (k ¼ 0 . . . 4) are the discrete velocity directions:

~ek ¼ð0;0Þ : k ¼ 0;ðcosðk� 1Þp=2; sinðk� 1Þp=2Þ : k ¼ 1;2;3;4

�

and c ¼ Dx=Dt is the fluid particle speed. Dt is the time step.The equilibrium distribution function gðeqÞ

k is defined by

gðeqÞk ¼ x

51þ 2:5

~ek �~uc

� �: ð8Þ

The vorticity is obtained by

x ¼XkP0

gk; ð9Þ

and se is determined by

se ¼ s0 þðCDÞ2jSj

c2s Dt

; ð10Þ

where s0 ¼ 1c2

s ReDtþ 0:5 and c2

s ¼ 25 c2. Re is the Reynolds number.

The complication of calculation of se of the primitive-variables large-eddy-based LB models results from the complicationof calculation of jSj (please see Eq. (9) in [31], Eq. (22) in [35] and Eq. (12) in [37]). Fortunately, thanks to the intrinsic featuresof vorticity-streamfunction equations, the calculation of jSj is very simple in the present model, just

jSj ¼ jxj: ð11Þ

Please see Appendix A for the details. Because the value of vorticity x is already known at every grid point, therefore com-pared with primitive-variables large-eddy-based LB models [31,35,37] no extra computational cost needed in the presentmodel.

Eq. (2) is just the Poisson equation, which also can be effectively solved by the lattice Boltzmann method [49–53]. In thepresent study, the model adopted in our previous work [38] is employed because compared with others this model is moreefficient and more accurate to solve the Poisson equation. The evolution equation for Eq. (2) reads [38]

fkð~xþ c~ekDt; t þ DtÞ � fkð~x; tÞ ¼ Xk þX0k; ð12Þ

where Xk ¼ �s�1w ½fkð~x; tÞ � f ðeqÞ

k ð~x; tÞ�, X0k ¼ DtfkHD and D ¼ c2

2 ð0:5� swÞ. sw is the dimensionless relaxation time, which can bechosen arbitrarily except 0.5 [38]. f ðeqÞ

k is the equilibrium distribution function, and defined by

f ðeqÞk ¼

ðn0 � 1:0Þw : k ¼ 0;nkw : k ¼ 1;2;3;4;

�ð13Þ

where nk and fk are weight parameters given as n0 ¼ 0, nk ¼ 1=4ðk ¼ 1 . . . 4Þ, f0 ¼ 0, fk ¼ 1=4ðk ¼ 1 . . . 4Þ. w is obtained by

w ¼ 11� n0

XkP1

fk: ð14Þ

The detailed derivation of Eqs. (7) and (12) to Eqs. (1) and (2) can be found in our previous work [38].In the present model Eqs. (3) and (4) are solved by the central finite difference scheme.

4. Numerical results

To validate the present model, numerical experiments are implemented on flow in a two-dimensional cavity, which is abenchmark test in CFD. Iliescu et al. used two related types of LES models to simulate the two-dimensional square lid-drivencavity flows and compared their numerical performance [54]. Larchevêque et al. employed LES to simulate compressible flowpast a deep cavity [55]. Matos and his cooperators conducted the numerical experiment of turbulent flow over a two-dimen-sional cavity with temperature fluctuations with LES [56]. Hou et al. developed a large-eddy-based LB model to simulatetwo-dimensional turbulent cavity flows with Re up to 106 [57]. More recently, Rubio et al. investigated the flow-inducedoscillation modes in rectangular cavities using LES [58], to cite only a few. In this study, the flow in a square one-sidedlid-driven cavity is investigated with Re ranging from 400 to 50000. The grid resolution is 128� 128 for Re 6 20000 and200� 200 for Re ¼ 50000. Flow in a square one-sided lid-driven cavity is an interesting research problem because manyimportant flow phenomena such as corner vortices, longitudinal vortices, Taylor–Görtler vortices, transition and turbulenceall occur in the same closed geometry [7,38,59–61]. As Figs. 1–4 illustrate, when Re 6 10000, the flow is laminar and steady.For low Re 6 1000, only three vortices appear in the cavity, a primary one near the center and a pair of secondary ones in thelower corners of the cavity. At Re ¼ 5000, a third secondary vortex is seen in the upper left corner. And a fourth secondary

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Fig. 1. Contour lines of w of flow in one-sided square lid-driven cavity: Re ¼ 400.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Fig. 2. Contour lines of w of flow in one-sided square lid-driven cavity: Re ¼ 1000.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Fig. 3. Contour lines of w of flow in one-sided square lid-driven cavity: Re ¼ 5000.

594 S. Chen / Applied Mathematics and Computation 215 (2009) 591–598

vortex appears at the bottom right corner when Re ¼ 10000. We can also see that the center of the primary vortex movestoward the center of the cavity as Re increases. From Fig. 5, we can derive that the flow region near the wall is the first partchanged into turbulence, but different parts of the flow domain remain at different levels of turbulence activity and someparts may remain laminar even though unsteady when Re is up to 20000. However, when Re ¼ 50000, the flow in the cavitybecomes turbulent completely, even though the plot of streamline for Re ¼ 50000 shows a relatively stable primary vortex,however, the shape and the structure of flow change greatly with time, this can be validated by the plot of vortex (see Fig. 6).The results obtained by the present model agree well with previous studies [7,38,59–61]. To quantify the results, the loca-tions of the primary center vortex and the two secondary ones are listed in Table 1. From the table, we can see that the loca-tions of the vortices predicted by the present model agree well with those of previous work [7,59–61] for all the Reynoldsnumbers considered.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Fig. 4. Contour lines of w of flow in one-sided square lid-driven cavity: Re = 10,000.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Fig. 5. Contour lines of w of flow in one-sided square lid-driven cavity: Re = 20,000.

Table 1Locations of vortex of the one-sided lid-driven square cavity flow: ð�Þc primary vortex; ð�Þl lower left vortex; ð�Þr lower right vortex.

Re xc yc xl yl xr yr

400 a 0.5547 0.6055 0.0508 0.0469 0.8906 0.1250b 0.5608 0.6078 0.0549 0.0510 0.8902 0.1255c – – – – – –d – – – – – –e 0.5500 0.6125 0.0625 0.0501 0.8820 0.1250

1000 a 0.5313 0.5625 0.0859 0.0781 0.8594 0.1094b 0.5333 0.5647 0.0902 0.0784 0.8667 0.1137c 0.5300 0.5650 0.0833 0.0783 0.8633 0.1117d – – – – – –e 0.5310 0.5700 0.0901 0.0800 0.8501 0.1100

5000 a 0.5117 0.5352 0.0703 0.1367 0.8086 0.0742b 0.5176 0.5373 0.0784 0.1373 0.8078 0.0745c 0.5150 0.5350 0.0733 0.1367 0.8050 0.0733d 0.5156 0.5352 0.0742 0.1328 0.8086 0.0742e 0.5040 0.5001 0.0950 0.1100 0.8285 0.0745

10000 a – - – – – –b – – – – – –c 0.5117 0.5300 0.0583 0.1633 0.7767 0.0600d 0.5117 0.5313 0.0469 0.1593 0.7773 0.0586e 0.5117 0.5313 0.0585 0.1655 0.7813 0.0625

20000 a – – – - – –b – – – – – –c 0.5100 0.5267 – – – –d 0.5078 0.6094 – – – –e 0.5078 0.5313 – – – –

Note. (a) Ghia et al. [59]; (b) Hou et al. [60]; (c) Erturk et al. [61]; (d) Chai et al. [7]; (e) present work.

S. Chen / Applied Mathematics and Computation 215 (2009) 591–598 595

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Fig. 6. Contour lines of w of flow in one-sided square lid-driven cavity: Re = 50,000.

Table 2Comparison of the computational efficiency between different models for the one-sided lid-driven square cavity flow.

Re Ref. [37] Present

400 Iterations 90,200 4400Time 1094 583

1000 Iterations 94,000 6200Time 1141 825

5000 Iterations 326,900 18,200Time 4074 2408

596 S. Chen / Applied Mathematics and Computation 215 (2009) 591–598

Table 2 shows the computational performance, number of iterative step and computational time(Unit second), of thepresent model, comparing with that of the latest primitive-variables large-eddy-based LB model designed in Refs. [37].The grid size 128� 128 is used. All simulations were performed on a Pentium 4 (3.0G CPU). It is clear that in all casesthe present model is faster than the primitive-variables large-eddy-based LB model [37].

5. Conclusion

In this study we propose a novel and simple large-eddy-based LB model for turbulence simulation. Unlike existing prim-itive-variables large-eddy-based LB models, the target macroscopic equations of the present model are vorticity-streamfunc-tion equations. Therefore the present model can be employed straightforwardly for geophysical turbulence simulation.Moreover, thanks to the intrinsic features of vorticity-streamfunction equations, the calculation of se of the present modelis much simpler and more efficient than that of primitive-variables large-eddy-based LB models.

The present model is validated by two-dimensional one-sided square lid-driven cavity flow. The numerical results ob-tained by the present model agree well with previous studies besides higher computational efficiency.

Though the present model is designed for two-dimensional turbulent flows, its extension to three-dimensional problems,will be considered in future studies.

Acknowledgements

This work was partially supported by the Alexander von Humboldt Foundation, Germany. The present author gratefullyacknowledges Prof. B.C. Shi and Dr. H.J. Liu, Huazhong University of Science and Technology, China, for sharing their originalturbulent LB computer source code developed in Ref. [37] during this work. The present author also would like to thank thereferees for their valuable advice and comments.

Appendix A

For two-dimensional problems, a; b ¼ x; y, so

2SabSab ¼ 2ouox

� �2

þ 2ovoy

� �2

þ ouoy

� �2

þ 2ouoy

� �ovox

� �þ ov

ox

� �2

ðA:1Þ

¼ 2ouoxþ ov

oy

� �2" #

þ ouoy� ov

ox

� �2

þ 4ouoy

ovox� ou

oxovoy

� �:

S. Chen / Applied Mathematics and Computation 215 (2009) 591–598 597

With the aid of the continuum equation

ouoxþ ov

oy¼ 0; ðA:2Þ

and the definition of vorticity

x ¼ ouoy� ov

ox; ðA:3Þ

Eq. (A.1) is reduced as

2SabSab ¼ x2 þ 4ouoy

ovox� ou

oxovoy

� �: ðA:4Þ

For incompressible flow, there exists [42]

r2p ¼ 2ouoy

ovox� ou

oxovoy

� �¼ Oðe2Þ; ðA:5Þ

where e is a small quantity of Chapman–Enskog expansion [46]. With the aid of Eq. (A.5), (A.4) is further reduced as

2SabSab ¼ x2 þ Oðe2Þ: ðA:6Þ

Consequently,

jSj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2SabSab

q¼ jxj; ðA:7Þ

with second-order accuracy, consistent with the numerical accuracy of the LB method.

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