The lattice Boltzmann method for compressible flows at highMach number

Yao Deng1, Fue-Sang Lien1, and Eugene Yee2

1 Department of MME, The University of Waterloo, Waterloo, Canada2 Defence Research and Development Canada, Suffield Research Centre, Medicine Hat, Alberta, Canada

Email: y27deng@uwaterloo.ca

ABSTRACT

The lattice Boltzmann method (LBM) is a mesoscopic-level “particle-based” method representing a densitydistribution that can be used to solve certain types ofpartial differential equations (PDEs). More specif-ically, LBM provides a very simple numerical pro-cedure for simulating the Boltzmann equation at themicroscopic level, whose appropriate coarse grainingleads to the standard hydrodynamic equations (in thelong wavelength limit) that express the conservationof mass (continuity equation) and momentum (NavierStokes equation) at the continuum level. Owing tothe fact that LBM offers numerous advantages suchas simplicity, efficiency and accuracy over more tra-ditional methodologies, it has been increasingly usedby researchers in recent years for the simulation of tur-bulence and multi-phase and multi-component flows inporous media.

In the initial development of LBM, researchers fo-cused on the simulation of very low-Mach-number in-compressible and isothermal flows [2]. These method-ologies are not suitable (e.g., unstable and unreliable)for the simulation of fluid flows at larger Mach num-bers. Currently, there is no consensus on the “cor-rect” method for application of LBM to the simula-tion of compressible high-Mach-number flows (whichnecessarily requires the development of a model forLBM that leads to the correct Newtonian thermo-hydrodynamics in the long wavelength limit). Nev-ertheless, there has been some important recent de-velopments on models for LBM that can simulatehigh-Mach-number compressible flows, most notablythe Kataoka-Tsutahara (KT) model [4] and Qu’smodel [7].

In this paper, we present and compare results for anumber of benchmark test cases of compressible flowsobtained with the KT model and Qu’s model. To this

purpose, these thermal LBM models have been im-plemented using a number of different finite volumeadvection schemes such as the simple upstream dif-ferencing scheme (UDS) and various total variationdiminishing (TVD) schemes, such as the MonotonicUpstream-Centered Scheme for Conservation Laws(MUSCL) and the weighted essentially non-oscillatory(WENO) schemes [7, 10]. Typical benchmark testssuch as the shock tube problem and the double Machreflection problem have been implemented in order tovalidate the two compressible LBM models and theproposed advection schemes used with these models.These simulations demonstrate the capability of LBMin the simulation of compressible flows at high Machnumbers and will provide the guidance for future re-searchers who are interested in finding the most appro-priate LBM models and advection schemes for specificcompressible flow applications.

1 INTRODUCTION

The idea of LBM is developed from the lattice gascellular automata (LGCA) [12]. People researchinginto LGCA have developed a set of rules for the cellssimilar to a set of rules for traffic . These rules havebeen derived from some statistical results or equationsthat can describe the macroscopic world. The typ-ical LGCA models are HPP [3] and FHP [6]. TheHPP model, introduced by Hardy, de Pazzis, andPomeau was the first LGCA model. However thismodel has some intrinsic shortcomings [9], such aslack of Galilean invariants, high viscosity, noise, etc.These serious problems make it unsuitable for usein real world cases. In 1986, Frisch, Hasslacherand Pomeau established a better hexagonal symmetrymodel, named FHP. Although this model can recoverthe 2-D and 3-D incompressible Navier-Stokes equa-tions, it also encounters the same kinds of problems as

23rd Annual Conference of the Computational Fluid Dynamics Society of CanadaJune 7-10, 2015, Waterloo, Ontario, Canada

HPP. These early models are all based on the Fermi-Dirac distribution, which confers an exponential timecomplexity in the simulation of these models.To overcome the shortcomings of these previous mod-els, the single relaxation time (SRT) model was estab-lished by Chen et al. [2] in 1991. It used the Maxwell-Boltzmann distribution instead of the Fermi-Dirac dis-tribution. The Maxwell-Boltzmann distribution can beused as an approximation to the Fermi-Dirac distribu-tion in the classical regime. The main contribution ofSRT is the introduction of a linear collision operationto make the model explicit. It is based on the work byBhagtnagar, Gross, and Krook in 1954 [1]. As such,this approximation is also known as the BGK model .Furthermore, the model does not violate Galilean in-variance.Most of the models developed in the last century arelimited to low-Mach-number incompressible isotropicflows. Therefore, nowadays, researchers seek to over-come this limitation. Researchers such as Yan et al.[14], Shan et al. [8], Kataoka and Tsutahara (KT)[4, 5], Qu [7] and others have all formulated modelsfor compressible flows. Shan’s model, which is basedon the Taylor expansion of the Maxwell distributionfunction up to various higher-order terms, can onlysolve compressible flows with a Mach number below0.7. This is because as the Mach number increases,the truncation errors cannot be neglected. The higherthe Mach number is, the more important it is to in-clude ever higher-order terms in the Taylor expansionof the Maxwell distribution function. However, someother models such as the KT model and Qu’s modelcan be applied to supersonic flows. These two modelsare adopted in this work.In the incompressible flow models, the molecules aremoving from one cell to another. The magnitude of themolecular velocity is equal to the cell spacing. There-fore, one can easily apply the streaming process. How-ever, the compressible flow models have irregular ve-locities, so the streaming process is more difficult toapply. Therefore, applying different finite volume ad-vection schemes is an alternative to the streaming pro-cess. Also, with the finite volume schemes, the ac-curacy can be controlled. Of the many finite volumeschemes, the UDS, the TVD scheme with minmodlimiter, the MUSCL [7], and the WENO [11] schemesare the most typical finite volume methods and will beincorporated into the KT and Qu’s models.This paper is structured as follows. The two compress-ible models, namely, the KT model and Qu’s model aredescribed in Section 2. The finite volume advectionschemes are presented in Section 3. Section 4 showsthe results of some benchmark test cases; namely, theSod shock-tube problem, 2-D Riemann problem and

the double Mach reflection problem. The last sectionis a summary of the work presented.

2 COMPRESSIBLE MODELS

The lattice Boltzmann equation (LBE) with BGK ap-proximation is

∂ fi

∂t+ξi ·∇ fi =−

fi− f eqi

τ, (1)

where fi is the discrete distribution function, i =1, · · · ,N, N is the total number of discrete particle ve-locities, ξi is the discrete particle velocity and τ is therelaxation time.

2.1 KT model

Kataoka and Tsutahara developed the D2Q16 modelfor the Navier-Stokes equations [4] in 2004. Thespecific-heat ratio γ can be chosen freely in this model.The non-dimensional macroscopic variables for theKT model are

ρ =N

∑i=1

fi, (2a)

ρuα =N

∑i=1

fiξiα, (2b)

ρ(bT + u2α) =

N

∑i=1

fi(ξ2iα + η

2i ), (2c)

where ρ, uα, T are the non-dimensional hydrodynamicdensity, flow velocity and temperature, respectively(and α is the index for the spatial direction so α ≡ x,y in two dimensions). The specific-heat ratio is re-lated to the total number of degrees of freedom, b, of amolecule as

γ = (b+2)/b. (3)

The coefficient ηi is used to control the specific-heatratio in the microscopic field. The 16-velocity latticeof this model is

(ξix, ξiy) =

cyc : (±1,0) 1≤ i≤ 4,cyc : (±6,0) 5≤ i≤ 8,√

2(±1,±1) 9≤ i≤ 12,3√2(±1,±1) 13≤ i≤ 16.

(4)

The non-dimensional equilibrium distribution functionf eqi (a function of the non-dimensional variables ρ, uα,

T ) is given by

f eqi = ρ[a0i +a1iT +a2iT 2 +(a3i +a4iT )u2

α +a5iu2αu2

β

+(b0i +b1iT +b2iu2α)uβξiβ +(d0i +d1iT +d2iu2

α)

× uβξiβuχξiχ + eiuαξiαuβξiβuχξiχ] (5)

where i = 1, · · · ,16 and the indices α, β, χ correspondto the coordinate direction x or y. The coefficientsa0i, · · · ,ei in Equation (5) are given in Reference [4].

2.2 Qu’s model

Qu’s (Quk’s) model [7] uses a new method to derivethe equilibrium distribution function. It adopts a circu-lar distribution function in the place of the Maxwell-Boltzmann distribution function. The simple circularfunction is

Feq =

ρ

2πcif ‖ξ−u‖= c =

√D

pρ,

0 others ,(6)

where D is the spatial dimension, ξ is the particle ve-locity, and ρ and p are the density and pressure, re-spectively.In Qu’s D2Q13L2 model, the hydrodynamic momentscan be calculated using the following relations:

ρ =13

∑i=1

2

∑υ=1

fiυ, (7a)

ρuα =13

∑i=1

2

∑υ=1

fiυξiα, (7b)

ρ(bT + u2α) =

13

∑i=1

2

∑υ=1

fiυ

(ξ

2iα +2λ

2iυ

), (7c)

where fiυ is the non-dimensional distribution function,υ denotes the energy level and λiυ is the rest energy.The distribution functions are derived from the conser-vation laws and can be found in Reference [7].The non-dimensional discrete particle velocities of thismodel are

(ξix, ξiy) =

(0,0) i = 1,cyc : (±1,0) 2≤ i≤ 5,(±1,±1) 6≤ i≤ 9,cyc : (±2,0) 10≤ i≤ 13.

(8)

3 DISCRETIZATION

If the first-order time-stepping scheme is applied, fi attime step t +dt becomes

f t+dti = f t

i −dt(ξi ·∇ fi)−dtτ( f t

i − f eqi ). (9)

The second term on the RHS of Eq. (9), namely,ξi ·∇ fi, is the flux term. The UDS, TVD, MUSCLor WENO schemes can be applied to approximate thisterm. In 2D Cartesian coordinates, the flux term canbe written as

ξi ·∇ fi = ξix∂ fi

∂x+ξiy

∂ fi

∂y=

∂Fi

∂x+

∂Gi

∂y, (10)

where Fi = ξix fi, and Gi = ξiy fi are the flux terms in xand y directions, respectively.The semi-discretized form for the flux term is

ξi ·∇ fi = (uFi,I+1/2,J− uFi,I−1/2,J)/dx

+( uGi,I,J+1/2− uGi,I,J−1/2)/dy, (11)

where uFi,I±1/2,J and uGi,I,J±1/2 are the fluxes on the in-terface of the cell (I,J).According to the flux-splitting method, the flux can bedivided into two parts depending on the sign of the ve-locity. For instance, in the x-direction,

F+i,I,J =

12(ξix + |ξix|) fi,I,J , (12)

F−i,I,J =12(ξix−|ξix|) fi,I,J . (13)

Because of flux splitting, two fluxes (one from the leftside and the other from the right side) are at the sameinterface. The total flux at the interface is the summa-tion of the left and right fluxes. So,

uFi,I+1/2,J = (FL)i,I+1/2,J +(FR)i,I+1/2,J . (14)

3.1 First-order UDS

The upwind difference scheme is the simplest fluxmethod. The fluxes on the interface are related directlyto the variables at the centre of the adjacent cells. Theleft and right fluxes are{

(FL)i,I+1/2,J = F+i,I,J ,

(FR)i,I+1/2,J = F−i,I+1,J .(15)

3.2 TVD (minmod) and MUSCL schemes

The TVD scheme with the minmod limiter is a second-order scheme. The left and right fluxes at (I + 1/2,J)are

FL = F+i,I,J +

12

minmod(∆F+i,I+1/2,J ,∆F+

i,I−1/2,J),

FR = F−i,I+1,J−12

minmod(∆F−i,I+1/2,J ,∆F−i,I+3/2,J),

(16)

where∆F±i,I+1/2,J = F±i,I+1,J−F±i,I,J . (17)

The definition of the minmod function is

minmod(a,b) =

{0, if ab≤ 0,sgn(a) ·min(|a|, |b|) if ab > 0.

(18)The MUSCL scheme was developed as the higher-order extension of the TVD scheme. Using theMUSCL approach, the spatial accuracy can be im-proved up to the third order. The fluxes in the MUSCLscheme are

uFi,I+1/2,J =

{ξix fL if ξix ≥ 0,ξix fR if ξix < 0,

(19)

wherefL = fi,I,J +

{ s4[(1−κs)∆−+(1+κs)∆+)]

}I,

fR = fi,I+1,J−{ s

4[(1−κs)∆++(1+κs)∆−)]

}I+1

,

(20)κ = 1/3 and s is the Van Albada limiter defined as

s =2∆+∆−+ ε2

∆2++∆2

−+ ε. (21)

The ε in Eq. (21) has no physical meaning and is sim-ply a very small number that is included to avoid di-vision by zero. The value of ε can be set to 10−10

(depending on to the machine-precision). The othervariables ∆+ and ∆− are defined as follows:{

(∆+)I = fi,I+1,J− fi,I,J ,

(∆−)I = fi,I,J− fi,I−1,J .(22)

3.3 WENO scheme

The weighted essentially non-oscillatory (WENO)schemes refer to a class of nonlinear finite volumediscretization methods. The key idea of the WENOmethod is to automatically choose the smoothest localstencil, in order to obtain higher-order accuracy fluxesfrom the lower-order ones.In the WENO scheme the numerical flux is defined as

uFi,I+1/2,J = w1 uF1i,I+1/2,J +w2 uF2

i,I+1/2,J +w3 uF3i,I+1/2,J .

(23)Under the condition ξix ≥ 0,

uF1i,I+1/2,J =

13

Fi,I−2,J−76

Fi,I−1,J +116

Fi,I,J , (24a)

uF2i,I+1/2,J =−

16

Fi,I−1,J +56

Fi,I,J +13

Fi,I+1,J , (24b)

uF3i,I+1/2,J =

13

Fi,I,J +56

Fi,I+1,J−16

Fi,I+2,J , (24c)

where Fi,I,J = ξix fi,I,J . The weighting factors wq inEq. (23) are given by

wq =wq

w1 + w2 + w3, wq =

δq

(10−6 +σq)2 , (25)

with δ1 = 1/10, δ2 = 3/5 and δ3 = 3/10.

4 SIMULATION RESULTS

4.1 Sod shock-tube problem

The Sod shock-tube problem is used to verify the nu-merical accuracy and efficacy of a proposed method-ology for the solution of compressible flow problems.It simulates a tube whose cross section is circular orrectangular. The tube is separated into two parts bya diaphram. One side, called the driver section, con-tains a high-pressure gas. The other side, called theworking (or driven) section, is maintained at a lowerpressure. With the bursting of the diaphragm, a shockwave is produced. After a short time of propagation,the whole flow grows into several sections separatedby an expansion fan, contact discontinuity, and shock.This test case consists of the following parameters:ρL/ρ0

pL/p0uL/u0

=

1.01.00.0

, 0 < x/L≤ 0.5,

ρR/ρ0pR/p0uR/u0

=

0.1250.10.0

, 0.5 < x/L≤ 1, (26)

where ρ0 = 1.165 kg m−3, R = 287 J kg−1 K−1, T0 =303 K, Pr = 0.71, and the specific ratio γ = 1.4.For the KT model, the numerical settings of the Sodshock-tube case are:

• Model: the KT D2Q16 model;

• Advection scheme: UDS, TVD (minmod),MUSCL, WENO;

• Boundary: equilibrium conditions before arrivalof the shock wave;

• Grid: 100×10 nodes;

• Final time: t = 0.01 s = 0.1474t0;

• Reference temperature: Tr = T0.

Extra rules exists for the KT model. The time step dtshould be smaller than the relaxation time τ0. Usually,dt = τ/10, where τ0 = µ/(ρRT0). After normalization,

dt = τ0/(10t0). Figure 1 shows the profiles of ρ, u, T ,and p obtained with the KT model with the advectionterm discretized with different flux methods.From Figures 1a to 1d, it is observed firstly that the

x/L

0 0.5 1

ρ/ρ

0

0

0.2

0.4

0.6

0.8

1

1.2UDS

MUSCL

TVD

WENO

Exact

(a) ρ

x/L

0 0.5 1u

/u0

0

0.2

0.4

0.6

0.8

1

1.2

(b) u

x/L

0 0.5 1

T/T

0

0.6

0.7

0.8

0.9

1

1.1

1.2

(c) T

x/L0 0.5 1

p/p

0

0

0.2

0.4

0.6

0.8

1

1.2

(d) p

Figure 1: Density, velocity, temperature and pressureprofiles of Sod shock-tube problem with a grid of100×10 nodes.

KT model can be used to solve compressible CFDproblems. The small difference between the numer-ical results and the exact solution is due to the useof a coarse grid. Secondly, some oscillations occurat the diaphragm location when the UDS and TVDschemes are employed. These oscillations may causedivergence of the code in 2-D cases or higher pressure-ratio cases. Thirdly, the third-order accurate MUSCLscheme performs very well, achieving an equivalentaccuracy and a greater smoothness in comparison tothe fifth-order WENO scheme.The results of Qu’s model with four kinds of advectionschemes for the shock tube problem are very similar tothose of the KT model. MUSCL is selected for com-parison between these two models. Figure 2 showslittle difference between these two models. The resultobtained with the KT model agrees better with the ex-act solution. The results of our similations suggest thatit is the choice of the flux methods that have a majoreffect on the solution accuracy.

4.2 2-D Riemann problem

The 2-D Riemann problem is a very challenging testcase. It consists of four domains with different initial

x/L0 0.5 1

ρ/ρ

0

0

0.2

0.4

0.6

0.8

1ExactKTQu

Figure 2: Density profile of the Sod shock-tube prob-lem with Qu’s model and the KT model on a grid of100×10 nodes.

states. The non-dimensional parameters are shown inthe Table 1. The KT model failed in this test case.

ρ2 = 0.5323u2 = 1.206v2 = 0p2 = 0.3

ρ1 = 1.5u1 = 0v1 = 0p1 = 1.5

ρ3 = 0.138u3 = 1.206v3 = 1.206p3 = 0.029

ρ4 = 0.5323u4 = 0v4 = 1.206p4 = 0.3

Table 1: The non-dimensional initial conditions for the2-D Riemann problem.

Only Qu’s model with the MUSCL scheme was ableto obtain a solution. This indicates that the Qu’s modelis more stable for high-Mach number flows.The numerical settings for the 2-D Riemann problemare as follows:

• Model: Qu’s D2Q13L2 model;

• Advection scheme: MUSCL;

• Boundary: use the 1-D exact solution;

• Grid: 100×100 nodes;

• Final time: t = 0.5t0;

• Reference temperature: Tr = 2.5T0;

• Non-dimensional relaxation time: τ = 10−4;

• Time step: dt = τ/4.

Figure 3 shows that Qu’s model can recover the detailsof the 2-D Riemann problem.

Figure 3: Density profiles of the 2-D Riemann problemwith Qu’s model and the MUSCL scheme on a grid of100×100 nodes.

4.3 Double Mach reflection problem

The double Mach reflection problem occurs when astrong shock moving horizontally encounters an in-clined wedge. It was first described by Woodwordet al. in 1984 [13]. In order to simplify this test casefor numerical calculation, we use a flat slip wall witha shock injected at an angle.

x

y

Ma = 10ρ = 8ρ0

p = 116.5p0

u = 8.25sin(π/3)

v =−8.25cos(π/3)

ρ = 1.4ρ0

p = p0

u = 0

v = 0

A

B C

D

E

F

Figure 4: Numerical geometry of the double Mach re-flection case.

In Figure 4, AB is the inlet and CD is the outlet. Thewhole domain is divided into two parts by the line EF .The left side of EF is the high-pressure inlet flow withMa = 10; the right side is the low-pressure area withzero initial velocity. FD is the reflection wall. The to-tal simulation time for the development of the flow ist = 0.2t0.The numerical settings for this double Mach reflectioncase are:

• Model: Qu’s D2Q13L2 model;

• Advection scheme: UDS, MUSCL, WENO;

• Reference temperature: Tr = 75T0;

• Grid: 500×125 nodes.

(a) UDS

(b) MUSCL

(c) WENO

Figure 5: Density contours of the double Mach re-flection problem with the UDS, MUSCL and WENOschemes from top to bottom utilizing a grid of 500×125 nodes.

All three flux methods yield good results on the doubleMach reflection problem. The TVD minmod schemeis not used because it diverges after thousands of timesteps. The following observations are drawn from acomparison of the various results (Figures 5a to 5c)obtained with the three advection schemes:

• The UDS method yields the smoothest results.The lines dividing the domain are very clear. Lit-tle noise is present, except at the stagnation point.

• The fifth-order WENO scheme results in muchmore noise than the other two methods, but pro-duces more details inside the two triangular areas.

• The MUSCL scheme is the best among the threeschemes examined here. It generates little noiseand yet captures the details of the vortex in thetwo triangular areas. Additionally, it is only athird-order scheme. The CPU time needed forthis method is less than that for the fifth-orderWENO scheme.

5 CONCLUSION

Both the KT model and Qu’s model can deal withhigh-Mach-number compressible flows. In the presentstudy, four kinds of finite volume advection schemesare introduced for the discretization of the flux terms.They are the UDS, TVD (minmod), MUSCL, andWENO schemes. Of course, higher-order schemes areslower but more accurate. Among them, the MUSCLscheme achieves a good compromise between effi-ciency and accuracy. Benchmark test cases are stud-ied for various canonical compressible flows. The KTand Qu’s models perform well for the shock-tube prob-lem. However, Qu’s model is more stable than the KTmodel in the more complicated 2-D cases such as inthe 2-D Riemann and the double Mach reflection prob-lems. These results demonstrate a great potential ofLBM when applied to solve compressible flows at highMach numbers.

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