Parallel Implementation of Gas-Kinetic BGK Scheme

on Unstructured Hybrid Grids

Murat Ilgaz∗

Defense Industries Research and Development Institute, Ankara, 06261, Turkey

and

Ismail H. Tuncer†

Middle East Technical University, Ankara, 06531, Turkey

The gas-kinetic BGK scheme on unstructured hybrid grids is presented. In order toaccurately resolve the boundary layers in wall bounded viscous flow solutions, quadrilateralgrid cells are employed in the boundary layer regions normal to solid surfaces while therest of the domain is discretized by triangular cells. The computation time, which is asignificant deficiency of the gas-kinetic schemes, is improved by performing computationsin parallel. The parallel algorithm for hybrid grids is based on the domain decompositionusing METIS, a graph partitioning software. Several numerical test cases are presented toshow the accuracy and robustness of the proposed approach.

I. Introduction

The gas-kinetic methods have become popular for the simulation of compressible fluid flows in the lastdecade. The most promising ones are the Equilibrium Flux Method,1 the Kinetic Flux Vector Splitting2,3

and the Gas-Kinetic BGK method.4 Due to the inclusion of intermolecular collisions with BGK simplification,the gas-kinetic BGK scheme gives a more complete and realistic description of the flow and has been wellstudied.5–12 Although the first studies about the gas-kinetic BGK method were based on finite volumeformulation on structured grids, the obvious advantages of unstructured grids over structured counterpartsfor complex configurations have led researcher to adopt the scheme on unstructured meshes.13–15 Recently,in the previous work of the present authors, the gas-kinetic flux vector splitting and the gas-kinetic BGKschemes on unstructured grids were given.16 The solutions were obtained in parallel in order to reducethe computational inefficiency common in gas-kinetic schemes. Several numerical test cases were presentedand it was shown that the gas-kinetic schemes are efficient in parallel computations and do not experiencenumerical instabilities as their classical counterparts. However, in general, unstructured grids are not sosuitable for the simulation of viscous flows especially for the well resolution of the boundary layers.

In the present work, a high-order gas-kinetic scheme for the Navier-Stokes equations on unstructuredhybrid grids is given. The finite volume formulations are combined with explicit Runge-Kutta time-steppingscheme. The parallel algorithm and the domain decomposition for unstructured hybrid meshes are explainedand several numerical test cases are presented to show the accuracy and robustness of the present approach.

II. Gas-Kinetic Theory

In gas-kinetic theory, gases are comprised of small particles and each particle has a mass and velocity.Due to the existence of large number of particles in a small volume at standard conditions (e.g. at STP,∼ 1019 air molecules in 1 cm3 volume), it is very difficult to trace individual particle motion. Instead, aparticle distribution function is defined to describe the probability of particles to be located in a certainvelocity interval

∗Senior Research Engineer, Aerodynamics Division, AIAA Student Member.†Professor, Department of Aerospace Engineering, AIAA Member.

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f(xi, t, ui) (1)

Here xi = (x, y, z) is the position, t is the time and ui = (u, v, w) are the particle velocities. The macroscopicproperties of the gas can be obtained as the moments of the distribution function. For example, the gasdensity can be written as

ρ =∑

i

m · ni (2)

where m is the particle mass, ni is the number density. Since, by definition, distribution function is theparticle density in phase space, it is concluded that

m · ni = f(xi, t, ui), (3)

ρ =∫ ∫ ∫

f dudvdw. (4)

The time evolution of the distribution function is governed by the Boltzmann equation17

ft + ui · fxi + ai · fui = Q(f, f). (5)

Here ai shows the external force on the particle in the ith direction and Q(f, f) is the collision operator.When the collision operator is equal to zero, collisionless Boltzmann equation is obtained and the solutionof this equation gives the Maxwellian (equilibrium) distribution function

g = ρ · (λπ

)N+3

2 · exp −λ · [(ui − Ui)2 + ξi2] (6)

where ξi = (ξ1, ξ2, ...., ξN ) are the particle internal velocities, N is the internal degrees of freedom, Ui =(U, V,W ) are the macroscopic velocities of the gas and λ is a function of temperature given by

λ =1

2 ·R · T(7)

R being the gas constant.

III. Numerical Implementation

The proposed numerical method is based on the cell-centered finite volume formulation. The numericalfluxes are calculated at the edges of the cells and the conservative variables at the cell centers are updatedusing a third-order explicit Runge-Kutta time-stepping scheme. In the following, the numerical implemen-tation of the proposed approach is presented in terms of the initial reconstruction of the conservative flowvariables and gas evolution stages.

A. Initial Reconstruction

The initial reconstruction of conservative variables is based on the method proposed by Frink.18 In thismethod, a universal expression for the Taylor series expansion within a triangular cell is derived, whichrequires the knowledge of the state at the nodes.

The Taylor series expansion of the conservative variables in the neighborhood of each cell center (xcc, ycc)is

Q(x, y) = Q(xcc, ycc) +∇Qcc ·∆r +O(∆r2) (8)

where

Q =

ρ

ρU

ρV

ρE

. (9)

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Figure 1. Arbitrary triangular cell.

This formulation requires the gradients of the flow variables at the cell centers. The evaluation of the gradientvectors is based on the Green’s theorem:∫ ∫

Ω

∇Q dA =∮

∂Ω

Q n dl (10)

The average gradient vector at the cell center, ∇Qcc, is then given by the closed boundary integral over thecell

∇Qcc =1AΩ

∮∂Ω

Q n dl (11)

where the unit normal n is assumed to point outward from the domain. Finally, the flow variables at themiddle of the cell edges, Qci, (Fig. 1) is given by Frink18

Qci = Qcc +13[12(Q2 + Q3)−Q1] (12)

where the nodal values are calculated from the surrounding cell center values. Mitchell19 stated that thisexpression is an average of one central difference term and two upwind terms and recovers a family of schemescommon with structured grid schemes. Rewriting Eq. (12)

Qci =13[12(Q2 + Q3) + 2(

32Qcc −

12Q1)], (13)

Qci = w[C.D.] + (1− w)[Upwind] (14)

where w = 1 is fully central differencing, w = 0 is fully upwinding. Note that w = 1/3 recovers Eq. (12)which is incidently suggested by Mitchell as the optimum value for a variety of test functions and meshdistributions.19

The reconstruction algorithm given for triangular cells may similarly be applied to quadrilateral cells.For an arbitrary quadrilateral cell shown in Fig. 2, the triangular cell in red is used for the reconstructionof the flow variables on cell edge 2-3:

Qci = Qcc′ +13[12(Q2 + Q3)−Qcc] (15)

For the remaining edges, the corresponding triangular cells within the quadrilateral cell are used for recon-struction. It should be noted that since the reconstruction procedure involves the simple averaging of theflow variables, the formal accuracy of the method may be less than second-order.

B. Gas Evolution

The gas evolution is based on the gas-kinetic BGK scheme. The gas-kinetic BGK scheme is based onthe Boltzmann BGK equation where the collision operator (see Eq. (5)) is replaced by the Bhatnagar-Gross-Krook model.20 The Boltzmann BGK equation in two-dimensions can be written as

ft + u · fx + v · fy =g − f

τ(16)

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Figure 2. Arbitrary quadrilateral cell.

where f is the particle distribution function, g is the equilibrium state approached by f over particle collisiontime τ , u and v are the particle velocities in x- and y-directions, respectively. The equilibrium state is usuallyassumed to be a Maxwellian

g = ρ · (λπ

)K+2

2 · exp −λ · [(u− U)2 + (v − V )2 + ξ2] (17)

where ρ is the density, U and V are the macroscopic velocities in x- and y-directions and K is the dimensionof the internal velocities. The general solution of the particle distribution function f at the cell edge ci andtime t is

f(sci, t, u, v, ξ) =1τ

∫ t

0

g(s′, t′, u, v, ξ) · exp [−(t− t′)/τ ] · dt′ + exp (−t/τ) · f0(sci − ut− vt). (18)

Here s′ = sci−u ·(t−t′)−v ·(t−t′) is the particle trajectory of a particle and f0 is the initial gas distributionfunction at the beginning of each time step. The relation between mass ρ, momentum ρU , ρV and totalenergy ρE densities with the distribution function is defined based on Eq. (4)

ρ

ρU

ρV

ρE

=∫f · ψ · dΞ (19)

where ψ represents the vector of moments for the distribution function

ψ =

1u

v12 (u2 + v2 + ξ2)

(20)

and dΞ is the volume element in phase space. Since mass, momentum and energy are conserved duringparticle collisions, f and g must satisfy the conservation constraint of∫

(f − g) · ψ · dΞ = 0, (21)

at all (x, y) and t. The high-order gas-kinetic BGK scheme on unstructured hybrid meshes is implementedas follows:

Consider the control volumes (triangular or quadrilateral), their neighbors and local coordinates givenin Fig. 3 where dots represent cell centers, the triangles or quadrilaterals in red show the control volumes,L and R the left and right states, xn and xt the local coordinate system normal and tangent to the celledge ci, respectively, x and y the global coordinate system. In cell-centered, finite volume gas-kinetic BGKscheme on unstructured hybrid meshes, the velocity components Un and Ut in local (normal and tangential)coordinate system are found from global coordinate system counterparts U and V for both mesh types.

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Figure 3. Control volumes, their neighbors and local coordinate system.

In the present work, the initial gas distribution function f0 and the equilibrium state g are assumed tobe

f0 =

gL · [1 + aL · (xn − xnci

)], if xn ≤ xnci

gR · [1 + aR · (xn − xnci)], if xn ≥ xnci

(22)

and

g =

g0 · [1 + aL · (xn − xnci

) +A · t], if xn ≤ xnci

g0 · [1 + aR · (xn − xnci) +A · t], if xn ≥ xnci

(23)

where gL, gR and g0 are the equilibrium distribution functions to the left, right and in the middle of the celledge ci, respectively, of the form

g∗ = ρ · (λ∗

π)

K+22 · exp −λ∗ · [(un − U∗

n)2 + (ut − U∗t )2 + ξ2], (24)

and aL, aR, aL, and aR are the spatial slopes and A is the time slope, given by

aL = aL1 + aL

2 · un + aL3 · ut + 1

2 · aL4 · (u2

n + u2t + ξ2)

aR = aR1 + aR

2 · un + aR3 · ut + 1

2 · aR4 · (u2

n + u2t + ξ2)

aL = aL1 + aL

2 · un + aL3 · ut + 1

2 · aL4 · (u2

n + u2t + ξ2)

aR = aR1 + aR

2 · un + aR3 · ut + 1

2 · aR4 · (u2

n + u2t + ξ2)

A = A1 +A2 · un +A3 · ut + 12 ·A4 · (u2

n + u2t + ξ2)

(25)

Here un and ut are the particle velocities normal and tangent to the cell edge, respectively. With thereconstructed values obtained in § III.A, the equilibrium distribution functions to the left and right of thecell edge as well as their slopes can be determined

∫gL · ψci · dΞ =

ρL

ci

ρULnci

ρULtci

ρELci

,

∫gR · ψci · dΞ =

ρR

ci

ρURnci

ρURtci

ρERci

(26)

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∫gL · aL ·ψci · dΞ =

(ρL

ci − ρLcc)/∆s

(ρULnci

− ρULncc

)/∆s(ρUL

tci− ρUL

tcc)/∆s

(ρELci − ρEL

cc)/∆s

,

∫gR · aR ·ψci · dΞ =

(ρR

cc − ρRci)/∆s

(ρURncc

− ρURnci

)/∆s(ρUR

tcc− ρUR

tci)/∆s

(ρERcc − ρER

ci)/∆s

(27)

where the subscript ci corresponds to the reconstructed values at the cell edge, the subscript cc refers to thecell center values, ∆s is the distance from the cell edge to the cell center and ψci stands for the vector ofmoments at the cell edge

ψci =

1un

ut12 (u2

n + u2t + ξ2)

(28)

Considering Eq. (24), all the parameters in gL and gR can be uniquely determined in Eq. (26). Once gL andgR are obtained, the slopes aL and aR can be computed from Eq. (27).

After determining f0, the equilibrium state g0 can be found using the compatibility condition∫g0 · ψci · dΞ =

∫u>0

gL · ψci · dΞ +∫

u<0

gR · ψci · dΞ (29)

from which the values of ρ0, Un0 , Ut0 and λ0 in g0 are determined. Then the slopes aL and aR can beobtained through the relations

∫g0 · aL · ψci · dΞ =

(ρ0 − ρL

cc)/∆s(ρUn0 − ρUL

ncc)/∆s

(ρUt0 − ρULtcc

)/∆s(ρE0 − ρEL

cc)/∆s

,

∫g0 · aR · ψci · dΞ =

(ρR

cc − ρ0)/∆s(ρUR

ncc− ρUn0)/∆s

(ρURtcc

− ρUt0)/∆s(ρER

cc − ρE0)/∆s

(30)

Up to this point, the equilibrium states to the left, to the right and in the middle of the cell edge aswell as the corresponding spatial slopes are determined. The only unknown term is the time slope term A.Since both f and g contain the time slope A, the conservation constraint (Eq. (21)) at the cell edge ci canbe applied and integrated over the time step ∆t∫ ∆t

0

∫(f − g) · ψci · dΞ · dt = 0, (31)

from which A can be calculated.Substituting Eqs. (22) and (23) into Eq. (18), the final gas distribution function at the cell edge ci is

expressed as

f(xnci , t, u, v, ξ) = [1− exp (−t/τ)] · g0+τ · [−1 + exp (−t/τ)] + t · exp (−t/τ) · [aL ·H(u) + aR · (1−H(u))] · u · g0+τ · [t/τ − 1 + exp (−t/τ)] ·A · g0+exp (−t/τ) · [(1− u · t · aL) ·H(u) · gL + (1− u · t · aR) · (1−H(u)) · gL]

(32)

where H(u) is the Heaviside function

H(u) =

0, if u < 01, if u > 0

. (33)

The local numerical fluxes for the mass, momentum and total energy across the cell edge ci can then becomputed as

Fρ

FρUn

FρUt

FρE

=∫ ∆t

0

∫un · f |xnci

· ψci · dΞ · dt (34)

The fluxes at the cell faces are first computed and the total flux for both triangular and quadrilateral cellsis then obtained by adding the flux contributions.

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IV. Parallel Processing

Parallel processing is based on domain decomposition. The unstructured hybrid grid is partitioned usingMETIS software package. METIS needs the graph file for the unstructured hybrid mesh, which is actuallythe neighbor connectivity of the cells. A sample unstructured hybrid mesh and the corresponding graph isgiven in Fig. 4. The first line in the graph represents the number of cells, number of nodes, weighting optionand number of weights, respectively. The remaining lines show the number of faces and the indices of theneighbors of the corresponding cell. Partitioning of the graph is performed using kmetis program. Duringthe partitioning, each cell is weighted by its number of edges so that each partition has about the samenumber of total edges to improve the load balancing in parallel computations.

Figure 4. Sample computational mesh and the corresponding graph.

Parallel Virtual Machine (PVM) message-passing library routines are employed in a master-worker algo-rithm. The master process performs all the input-output, starts up pvm, spawns worker processes and sendsthe initial data to the workers. The worker processes first receive the initial data, apply the interface and theflow boundary conditions, and solve the flow field within the partition. The flow variables at the interfaceboundaries are exchanged among the neighboring partitions at each time step for the implementation ofinter-partition boundary conditions.

V. Results and Discussions

In order to show the accuracy and robustness of the present approach, the gas-kinetic BGK scheme isapplied to a laminar flow over a flat plate and a transonic viscous flow over an airfoil. The results arecompared to the analytical solutions and available experimental data.

A. Case 1: Laminar Flow over a Flat Plate

A flat plate at zero angle of attack is selected as a first test case for which the analytical Blasius solutionis available. The freestream conditions and the computational mesh for this case are given in Table 1 andFig. 5, respectively.

Table 1. Freestream conditions for Case 1.

Mach Number Angle of Attack, deg Pressure, psia Temperature, R0.1 0.0 6.0 700

The above conditions correspond to a Reynolds number of 200000. The flat plate is placed between 0 and1 and the boundary layer region is meshed with quadrilateral cells while triangular cells are used outsidethe boundary layer. Fig. 6 shows the velocity vectors inside the boundary layer on the flat plate and the

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x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

Figure 5. Computational mesh for Case 1 (zoomed).

x

y

0.296 0.298 0.3 0.302 0.3040

0.002

0.004

0.006 Mach

0.090.080.070.060.050.040.030.020.01

Figure 6. Mach number contours and velocity vectors (zoomed).

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u/U

η

0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

4

5

6

7

8

Exactx = 0.25x = 0.40x = 0.55

Figure 7. u-velocity profile in the boundary layer.

comparison of u-velocity profile at three different locations with the Blasius solution is given in Fig. 7.The results compare quite well with the exact Blasius solution. It should be noted that the gas-kineticBGK method produces Navier-Stokes solution in smooth flow regions as expected. Moreover, the use ofquadrilateral cells near the wall region leads to the better resolution of the boundary layer profile while theuse of triangular cells outside reduce the overall number of cells. The laminar flow solution given aboveverifies the accuracy and robustness of the present approach.

B. Case 2: Transonic Flow over an Airfoil

RAE 2822 transonic airfoil, which has available experimental data, is selected as a second test case.The freestream conditions for this case are given in Table 2. The computational mesh and the partitionsgenerated using the METIS software is presented in Fig. 8.

Table 2. Freestream condition for Case 2.

Mach Number Angle of Attack, deg Pressure, psia Temperature, R0.729 2.31 15.81 460

Fig. 9 shows the Mach number contours obtained from the gas-kinetic BGK scheme on the unstructuredhybrid grid. As shown, the flow solution is smooth and the shock is captured well. The comparison of thesurface pressure distribution with the experimental data is given in Fig. 10. As seen, the shock location andthe strength compare quite well and the overall pressure distribution is in excellent agreement. In Fig. 11,the present viscous solution is compared to the inviscid Euler solution. The viscous effects are clearly seenin the prediction of shock location and the pressure distribution downstream of the shock.

The parallel computations are performed on an Itanium cluster running on Linux. Dual Itanium2processors operate at 1.3GHz with 3MB L2 cache and 2GB of memory for each. The parallel efficiency of

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Figure 8. Computational mesh and partitions for Case 2 (zoomed).

x

y

-1 -0.5 0 0.5 1

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Mach

1.501.391.281.181.070.960.850.750.640.530.420.320.210.10

Figure 9. Mach number contours.

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x

Pres

sure

Coe

ffic

ient

-0.5 -0.25 0 0.25 0.5

-1

-0.5

0

0.5

1

ExperimentGas-Kinetic BGK

Figure 10. Pressure distribution on the airfoil surface (every other three data point is displayed).

x

Pres

sure

Coe

ffic

ient

-0.5 -0.25 0 0.25 0.5

-1

-0.5

0

0.5

1

Inviscid SolutionViscous Solution

Figure 11. Viscous versus inviscid solution (every other three data point is displayed).

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Table 3. Parallel efficiency of computations.

Number of Nodes Computational Efficiency, sec/iter1 0.09502 0.04824 0.02448 0.012116 0.0063

the computations is given in Table 3 and Fig. 12. It is observed that the high parallel efficiency of the gas-kinetic BGK scheme is maintained as the number of processors is increased, which is attributed to the highcomputing to communication ratio. Although the gas-kinetic BGK schemes are computationally expensiveand require longer computational time than classical schemes in serial computations,16 it may not be anissue in parallel computations.

Number of Processors

Spee

dU

p

2 4 6 8 10 12 14 16

2

4

6

8

10

12

14

16

IdealGas-Kinetic BGK

Figure 12. Computational speed up.

VI. Conclusion

In this paper, the gas-kinetic BGK scheme on unstructured hybrid grids are presented. The high-orderfinite volume formulations are given. The solutions are obtained in parallel and the results are comparedto the available analytical/experimental data. The unstructured hybrid grids used in the present studyremoves the difficulties faced in viscous flow computations with triangular grids. The viscous flow solutionswith the gas-kinetic BGK scheme on unstructured hybrid grids agree well with analytical/experimental data.In addition, the parallel computations improve the computational efficiency of the gas-kinetic BGK schemessignificantly.

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Acknowledgments

The authors acknowledge the support of the Defense Industries Research and Development Institute(TUBITAK-SAGE) under project SAM.

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