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Finite difference gas-kinetic BGK scheme forcompressible inviscid flow computationAshraf A. Omar a , Ong J. Chit b , Lim J. Hsuh b & Waqar Asrar aa Department of Mechanical Engineering , Faculty of Engineering, International IslamicUniversity Malaysia , P. O. Box 10, 50728, Kuala Lumpur, Malaysiab Faculty of Mechanical Engineering, University Technology MARA Pulau Pinang , 13500,Permatang Pauh, Penang, MalaysiaPublished online: 24 Apr 2008.

To cite this article: Ashraf A. Omar , Ong J. Chit , Lim J. Hsuh & Waqar Asrar (2008) Finite difference gas-kinetic BGK schemefor compressible inviscid flow computation, International Journal of Computational Fluid Dynamics, 22:3, 183-192, DOI:10.1080/10618560701812824

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Finite difference gas-kinetic BGK scheme for compressible inviscid flow computation

Ashraf A. Omara*, Ong J. Chitb, Lim J. Hsuhb and Waqar Asrara

aDepartment of Mechanical Engineering, Faculty of Engineering, International Islamic University Malaysia, P. O. Box 10, 50728

Kuala Lumpur, Malaysia; bFaculty of Mechanical Engineering, University Technology MARA Pulau Pinang, 13500 Permatang Pauh,

Penang, Malaysia

(Received 5 July 2007; final version received 27 October 2007 )

In this paper, the gas-kinetic Bhatnagar–Gross–Krook (BGK) scheme is developed using the finite difference method

and applied to simulate 2D compressible inviscid flow. The numerical scheme employed in this study, namely the BGK

scheme, is based on the collisional Boltzmann model. The high-order resolution of this scheme is obtained by utilising

the monotone upstream-centred schemes for conservation law-type primitive variable reconstruction. As for the time

integration part of the scheme, a multistage Runge–Kutta method is employed. In order to assess the computational

characteristics of the numerical scheme, namely accuracy and robustness, three flow problems are selected in this study.

They are supersonic channel flow, supersonic wedge cascade and circular arc bump. The computed results of these test

cases are validated with available exact solutions and numerical results from the central difference scheme with total

variation diminishing (TVD) formulation. It shows that the BGK scheme is an accurate and robust scheme in comparison

to the central difference scheme with TVD when used to simulate 2D compressible inviscid flows.

Keywords: gas-kinetic BGK scheme; compressible inviscid flow; Boltzmann model; MUSCL; multistage

Runge–Kutta method

1. Introduction

Throughout the history of computational fluid dynamics

development, many numerical schemes have been created

to solve practical application of gas dynamics problems.

The key design criterion of any numerical schemes is to

maximise robustness and accuracy. This requirement is

particularly important in compressible flow solutions

involving high-speed flow where intense shock waves and

boundary layers may simultaneously exist. Among those

notable and successful are the Godunov-type and flux

vector splitting schemes. Besides these numerical

schemes that stem from the discretisation of the Euler

equations, the gas-kinetic schemes have attracted much

attention in recent years due to their high robustness and

accuracy in solving compressible flow problems.

Recent developments have seen the emergence of

another class of scheme known as the gas-kinetic schemes

that are developed based on the Boltzmann equation (Xu

1998a, Ong 2004). Mainly, there are two groups of gas-

kinetic schemes and the difference lies within the type of

Boltzmann equation used in the gas evolution stage. One

of them is the well-known kinetic flux vector splitting

(KFVS) scheme which is based on the collisionless

Boltzmann equation and the other is based on the

collisional Bhatnagar–Gross–Krook (BGK) model (Ong

et al. 2004a) where the BGK scheme is derived. Like any

other flux-vector-splitting (FVS) method, the KFVS

scheme is very diffusive and less accurate in comparison

with the Roe-type flux-difference-splitting (FDS)method.

The diffusivity of the FVS schemes is mainly due to the

particle or wave-free transport mechanism, which sets the

Courant-Friedrichs-Lewy (CFL) time step equal to

particle collision time (Xu 1998b). In order to reduce

diffusivity, particle collisions have to be modelled and

implemented into the gas evolution stage. One of the

distinct approaches to take particle collision into

consideration in gas evolution can be found in Xu

(1998a). In this method, the collision effect is considered

by the BGK model as an approximation of the collision

integral in theBoltzmannequation. It is found that this gas-

kinetic BGK scheme possesses accuracy that is superior to

the flux vector splitting schemes and avoids the anomalies

of FDS-type schemes (Chae et al. 2000, Ong et al. 2003,

Ong et al. 2004b, Abdusslam et al. 2006, Ong et al. 2006).

2. Numerical methods

The 2D normalised compressible Euler equations can be

written in the strong conservation form as

›W

›tþ

›F

›xþ

›G

›y¼ 0; ð1Þ

ISSN 1061-8562 print/ISSN 1029-0257 online

q 2008 Taylor & Francis

DOI: 10.1080/10618560701812824

http://www.informaworld.com

*Corresponding author. Email: aao@iiu.edu.my

International Journal of Computational Fluid Dynamics

Vol. 22, No. 3, March 2008, 183–192

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where

W ¼

r

rU

rV

r1

26666664

37777775; F ¼

rU

rU 2 þ p

rUV

ðr1þ pÞU

26666664

37777775;

G ¼

rV

rUV

rV 2 þ p

ðr1þ pÞV

26666664

37777775;

where r, U, V, p and 1 are the macroscopic density,

x-component of velocity, the y-component of velocity,

the pressure and total energy, respectively. Subsequently,

Equation (1) is transformed into curvilinear coordinates

(j, h). Hoffmann and Chiang (1993) provide a detailed

description about this matter.

A standard BGK scheme is based on the collisional

Boltzmann equation and it is written in two dimensions

as (Xu 1998a)

›f

›tþ u

›f

›xþ v

›f

›y¼

ðg2 f Þ

t; ð2Þ

where f is the real particle distribution function and g is the

equilibrium state approached by f within a collision time

scale t. Both f and g are functions of space x, y; time t;

particle velocity u, v; and internal degrees of freedom 6.

The equilibrium state g in the 2D BGK model for the

Euler equations is the Maxwell–Boltzmann distribution

function and it has the following form

g ¼ rl

p

� �ðKþ2Þ=2

e2l½ðu2UÞ2þðv2VÞ2þ6 2�; ð3Þ

where l is a function of density and pressure, l ¼ r/2p. 6

is a K dimensional vector which accounts for the internal

degrees of freedom such as molecular rotation,

translation and vibration. The dimensional vector, K is

related to the specific heat ratios and the space dimension

by the relation K ¼ (4 2 2g)/(g 2 1), where for a

diatomic gas g ¼ 1.4.

The relations between the densities of mass r,

momentum (rU, rV), and total energy 1 with the

distribution function f are derived from the following

moment relation

r

rU

rV

1

0BBBBB@

1CCCCCA ¼

ðfC dJ; ð4Þ

where dJ ¼ dudvd6 is the volume element in the phase

space while C is the vector of moments given as

C ¼

1

u

v

12ðu2 þ v2 þ 62Þ

0BBBBB@

1CCCCCA; ð5Þ

with the moment relation defined in Equation (4), a

similar approach could be adopted in obtaining the

numerical fluxes across cell interfaces and they are given

as

Fx ¼

ðufC dJ Gy ¼

ðvfC dJ; ð6Þ

where Fx and Gy are the physical flux in the x- and y-

direction, respectively.

A general solution for f of Equation (3) at the cell

interface (xiþ1/2, yj) in two-dimensions is obtained as Ong

et al. (2004b)

f ð0; 0; t; u; v; 6Þ ¼ ð12 wÞgo þ wf oð2ut;2vtÞ; ð7Þ

where w ¼ e 2t/t is an adaptive parameter. For a first-

order scheme w can be fixed in the numerical

calculations. When the BGK scheme is extended to

high-order, the value of w should depend on the real flow

situations.

Finally, the gas-kinetic BGK numerical flux across

the cell interface in the x-direction can be computed as

Fx ¼

ðuf ð0; 0; t; u; v; 6ÞC dJ

Fx ¼ ð12 wÞFex þ wFf

x;

ð8Þ

where Fex is the equilibrium flux function and Ff

x is the

non-equilibrium or free stream flux function. Hence, the

numerical flux for the BGK scheme at the cell interface

in the x-direction are obtained from Equation (8) as

Fiþ1=2; j ¼ ð12 wÞFeiþ1=2; j þ wF

f

iþ1=2; j ð9Þ

while the numerical flux at the cell interface in the

y-direction is obtained in a similar manner and the

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resulting relation is presented as

Gi; jþ1=2 ¼ ð12 wÞGei; jþ1=2 þ wG

f

i; jþ1=2: ð10Þ

In extending the numerical scheme to high-order

spatial accuracy, the monotone upstream-centred

schemes for conservation law approach (Hirsch 1990)

is adopted together with the minmod limiter. Hence, the

left and right states of the primitive variables r, U, V, p at

a cell interface could be obtained through the non-linear

reconstruction of the respective variables and are given as

Ql ¼ Qi; j þ1

2f

DQiþ1=2; j

DQi21=2; j

� �DQi21=2; j

Qr ¼ Qiþ1; j 21

2f

DQiþ3=2; j

DQiþ1=2; j

� �DQiþ1=2; j;

ð11Þ

where Q is a primitive variable and the subscripts l and r

correspond to the left and right sides of a considered cell

interface. In addition, DQiþ1/2, j ¼ Qiþ1, j 2 Qi, j.

The minmod limiter used in the reconstruction of flow

variables in Equation (11) is given as

fðVÞ ¼ minmodð1;VÞ ¼ max½0;minð1;VÞ�: ð12Þ

For the time integration of steady state problems, an

explicit formulation is chosen for the current solver

which utilises a fourth-order Runge–Kutta method.

Applying this method to the generalised 2D Euler

equations provides the following results

�Wð1Þi; j ¼ �W

ni; j

�Wð2Þi; j ¼ �W

ni; j 2

Dt

4

› �F

›j

� �ð1Þ

i; j

þ› �G

›h

� �ð1Þ

i; j

" #

�Wð3Þi; j ¼ �W

ni; j 2

Dt

3

› �F

›j

� �ð2Þ

i; j

þ› �G

›h

� �ð2Þ

i; j

" #

�Wð4Þi; j ¼ �W

ni; j 2

Dt

2

› �F

›j

� �ð3Þ

i; j

þ› �G

›h

� �ð3Þ

i; j

" #

�Wnþ1i; j ¼ �W

ni; j 2 Dt

› �F

›j

� �ð4Þ

i; j

þ› �G

›h

� �ð4Þ

i; j

" #:

ð13Þ

3. Results and discussions

In this paper, the numerical algorithm of the BGK

scheme will be tested with three typical test cases for

computing 2D inviscid compressible flow, namely

supersonic channel flow, supersonic wedge cascade and

circular arc bump. All the simulations executed for the

test cases are done using second-order accuracy.

The computed numerical solutions of the BGK scheme

are compared with available exact solutions and results

from central difference scheme with TVD for compara-

tive and validation purposes. A grid independence study

is implemented to determine the most feasible mesh size

to be used for the test cases. Only a brief discussion about

this aspect is included, which is located in the first test

case, i.e. supersonic channel flow. The same procedure is

adopted for the remaining test cases without any further

elaboration.

3.1 Supersonic channel flow

This particular flow problem is taken from Hoffmann and

Chiang (1993) where a channel which includes both

compression and expansion corners is used to illustrate

the formation of oblique shock and expansion waves and

their reflection and interaction. The rationalisation

behind this selection is simply because analytical results

can be calculated and used to validate the numerical

results computed from the BGK scheme and central

difference scheme with TVD.

The physical dimensions of the domain is given by

the grid and shown in Figure 1. The compression/expan-

sion angle for the channel is 108. The number of grid

points for the domain is 241 £ 131. The following free

stream conditions are used to initialise the flow:

p ¼ 100 kPa T ¼ 300K M ¼ 2:0;

where p, T and M are pressure, temperature and Mach

number, respectively. The boundary conditions used for

this problem consist of a supersonic inlet and outlet at the

left and right of the physical domain while an inviscid

wall condition is applied at the lower and upper wall.

The pressure and density contours showing the

formation of the oblique shock, expansion wave and

their reflection and interaction are shown in Figures 2 and 3

for the BGK scheme, and in Figures 4 and 5 for the

central scheme with TVD. The pressure and density

contours for both of the schemes seem to agree with each

other very well. In order to obtain a better understanding

Figure 1. Physical mesh for the supersonic channel.

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Figure 2. Pressure contours of the BGK scheme.

Figure 3. Density contours of the BGK scheme.

Figure 4. Pressure contours of the central scheme.

Figure 5. Density contours of the central scheme.

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of the computational characteristics of both schemes, the

computed pressure and density distributions along the

lower surface are compared to the analytical solutions

calculated from Hoffmann and Chiang (1993).

The pressure distributions for both of the schemes

compared against the analytical solution is shown in

Figure 6. It can be seen from this plot that the computed

results for the BGK and central scheme with TVD agree

very well to the analytical solution, except the BGK

scheme is able to give a higher pressure value after the

wave reflection and interaction region. As for the density

distributions, Figure 7 illustrates a comparison made

between the BGK scheme and central scheme with TVD

against the analytical solutions. This figure shows that

the BGK scheme is able to provide a better resolution of

the inviscid compressible flow in comparison with the

central scheme especially in regions after the oblique

shock and expansion wave.

A quantitative understanding about the numerical

results presented for this test case can be obtained by

calculating the percentage errors between the numerical

results with the analytical solutions. The following relation

is used to calculate the percentage error:

% error ¼Numerical2 Analyticalj j

Analytical£ 100%:

Figures 8 and 9 contain the data calculated for the pressure

and density percentage errors along the lower surface of the

channel, respectively. The percentage errors in the figures

are calculated based on the available analytical data which

starts from location x ¼ 0 to 0.52m only. Overall, the

percentage errors for the BGK scheme are lower than the

central scheme. This deduction is made through a close

examination of the data points contained in Figures 8 and 9.

Figure 6. Comparison of pressure distributions along thelower surface.

Figure 7. Comparison of density distributions along the lowersurface.

Figure 8. Percentage error distributions for the pressure alongthe lower surface.

Figure 9. Percentage error distributions for the density alongthe lower surface.

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These outcomes can be translated to state that the BGK

scheme is more capable of resolving the flow field than the

central scheme.

The choice for the mesh size is determined by

conducting a grid independence study. This is done

by opting for a coarser grid size to start with, followed by

finer grid sizes until a satisfactory numerical result is

achieved. For this test case, the chosen grid is about

31,571 (241 £ 131) points. Two additional mesh sizes

are needed in order to implement the grid independence

study. Hence, a coarser mesh size of 7865 (121 £ 65)

points and a finer mesh size of 70,756 (361 £ 196) points

are selected. The former accounts for about 75% of size

reduction and the latter is about 124% of size increment in

comparison to the current grid size being used. These two

mesh sizes are tested with the numerical solver for this

flow problem and the results produced are shown in

Figure 10. As can be seen from this figure, the pressure

distribution generated by the mesh size of 7865 is not as

accurate as the two other mesh sizes. This implies that the

solution produced by the mesh size of 7865 is still

dependable on the grid resolution. However, when

observing the pressure distributions for both the 31,571

and the 70,756 mesh sizes, they are in good agreement

with the analytical data and to each other. This shows that

the grid has reached independence with respect to the flow

solution. Thus, the mesh with 241 £ 131 grid points is

used in this test problem in order to avoid unnecessary

extra computing time in using a finer mesh.

3.2 Supersonic wedge cascade

Another test case of internal flow is selected to further

evaluate the computational characteristics of the BGK

scheme. This particular flow problem is taken from Roe

(1982) and Fottner (1990) where a cascade of wedges

is used to illustrate the shock capturing ability of the

method for oblique shocks. The rationalisation behind

this selection is simply because analytical results can be

calculated from characteristic theory and oblique shock

relations (Fottner 1990) to validate the numerical results

computed from the BGK scheme and compare with the

central scheme.

Figure 10. Pressure distributions along the lower surface forgrid independence study.

Figure 11. Physical mesh of the wedge cascade.

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The physical domain for this wedge cascade is shown

in Figure 11. The compression/expansion angle for the

wedge is 5.738. The number of grid points for the domain

is 150 £ 50. The following flow conditions are used

to initialise the flow:

M ¼ 1:6 a ¼ 608 P2=P01 ¼ 0:3536;

where M and a are the inlet Mach number and angle of

attack, respectively. Since the flow at the inlet is

supersonic, all the flow variables are fixed to free stream.

At the exit, as required by the theory of characteristics,

for subsonic flow, static pressure is fixed while total

pressure, total temperature and flow angle are extrapo-

lated from the interior. However, if the exit flow is

supersonic, all four variables are extrapolated from the

interior. The lower and upper boundaries consist of both

periodic and solid boundaries. An inviscid wall condition

is applied on the wedge surface while periodic condition

is applied to the upstream of the leading edge and

downstream of the trailing edge.

The leading edge of the wedge leads to an oblique

shock wave inside the channel which can be seen from

the Mach number contours plot of Figure 12. This

shock is then reflected on the suction side and cancelled

at the upstream corner giving uniform flow between the

two parallel surfaces. At the downstream corner, an

expansion shock wave is seen to be present on the

contours plot. Figures 13 and 14 show the surface Mach

number distributions computed by the BGK scheme

and central difference scheme, respectively, together

with the analytical solutions. From both of these

figures, it can be seen that the numerical results

produced by both schemes are in agreement with the

analytical result. Better flow resolution can be seen for

the BGK scheme in comparison with the central

scheme, especially at the region of the first corner and

Figure 12. Mach number contours by the BGK scheme for thesupersonic wedge cascade flow.

Figure 13. Comparison of surface Mach number distributionsfor the BGK scheme and analytical solutions.

Figure 14. Comparison of surface Mach number distributionsfor the central difference scheme and analytical solutions.

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parallel surfaces of the wedge cascade. The description

of the flow at the expansion corner is rather

unsatisfactory. Nonetheless, the numerical results

illustrated here are by far much better than the one

presented in Roe (1982). Another interesting aspect that

can be observed from Figures 13 and 14 is the shock

resolution capability of the BGK scheme where it is

better than the central difference scheme with TVD.

It can be seen that the BGK scheme is able to resolve

the shock location much better than the central

difference scheme with TVD.

For a more detailed analysis of the numerical results

presented above, the percentage errors for the Mach

number distributions on the suction and pressure sides of

the cascade are calculated for both the BGK and central

schemes. These are shown in Figures 15 and 16 for the

suction and pressure sides, respectively, where only a

few selected data points of interest are shown. Browsing

the percentage errors shown in these figures and

comparing the one from the BGK scheme with the

central scheme, one may observe that the percentage

error for the BGK scheme in general is less than the

central scheme.

3.3 Circular arc bump

The transonic flow over a circular arc bump is a typical

test case that is often used to assess the accuracy of the

numerical schemes. The descriptions of this test case

are taken from Ferziger and Peric (1996). The geometry

of the problem is shown in Figure 17 with a mesh size

of 181 £ 51. The thickness to chord ratio of the

circular arc is 10% with the chord length selected as

one unit. The length of the channel upstream and

downstream of the bump is chosen to be one unit as

well. The flow is initialised with a flow Mach number

of 0.675 at zero angle of attack. As for the boundaries,

the following conditions are applied. At the inlet, the

boundary condition is set according to the subsonic

flow. As for the outlet, the boundary condition is set

according to the theory of characteristics where for the

subsonic flow condition, static pressure is fixed and

three other flow variables are extrapolated from interior

points and for the supersonic flow condition, all flow

variables are extrapolated. The inviscid wall condition is

set at the lower and upper walls.

The contours for the Mach number for the BGK and

central schemes are shown in Figures 18 and 19,

respectively. As shown in the plots, there is a shock

occurring at the lower wall for a transonic flow

condition. Both of the schemes are able to predict the

location of this shock reasonably. The Mach number

distributions along the lower surface of the problem are

shown in Figure 20 for both the BGK scheme and

central scheme with TVD which are compared with the

numerical solution from Ferziger and Peric (1996). This

figure shows that both of the schemes are able to

resolve the shock and predict its location accurately.

However, the BGK scheme for inviscid flows is better

Figure 15. Percentage error distributions for the surface Machnumber on the suction side.

Figure 16. Percentage error distributions for the surface Machnumber on the pressure side. Figure 17. Physical mesh for the circular arc bump.

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than the central scheme in describing the flow after the

shock, especially in regions close to the outlet when

compared to the solution provided by Ferziger and

Peric (1996).

4. Conclusion

A numerical solver based on the collisional BGK

model of the Boltzmann equation has been successfully

developed to simulate compressible inviscid flow.

Three test cases of this flow realm are selected to

assess the computational characteristics of the devel-

oped numerical solver. The computed results of the test

problems clearly demonstrate that the BGK scheme is

able to provide a very good resolution of the flow when

compared against the central difference scheme with

TVD and available exact solutions. In brief, this paper

concludes that the BGK scheme formulated via the

finite difference method is an accurate and robust

numerical scheme for computing 2D compressible

inviscid flows.

Acknowledgements

The authors would like to acknowledge the support of the

Institute of Research, Development and Commercialisation

(IRDC) under the project No. 600-IRDC/ST 5/3/887.

Figure 18. Mach number contours of the BGK scheme.

Figure 19. Mach number contours of the central scheme.

Figure 20. Comparison of Mach number distributions alongthe lower surface.

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