The Equivalent Differential Equation of the Gas-Kinetic BGK Scheme

The Equivalent Differential Equation of the Gas-Kinetic BGK Scheme Wei Gao Quanhua Sun Kun Xu Abstract The equivalent partial differential equation is derived for the gas-kinetic BGK scheme (GKS) [Xu J. Comput. Phys. vol. 7 pp. 89-335 ()] following its detailed numerical procedure. The GKS scheme is a one-step scheme where the flux reconstruction employs the accurate time-dependent solution of the BGK equation. The derived equation confirms that the GKS scheme solves the Navier-Stokes equations with a predetermined bulk viscosity. It shows that the directional splitting method fails to include the contributions from velocity gradients in tangential direction to the viscous flow flux which can be resolved using a multi-dimensional scheme. The GKS scheme is second-order accurate in both space and time at the level of Euler equations. However it is first-order accurate in time for viscous flow simulations. Keywords Equivalent differential equation Gas kinetic BGK scheme Navier- Stokes equations Compressible flow Mathematics Subject Classification () 35Q3 65M8 76M5 76N5 8B W. Gao Q. Sun (corresponding author) State Key Laboratory of High-temperature Gas Dynamics Institute of Mechanics Chinese Academy of Sciences No. 5 Beisihuan Xi Rd Beijing 9 China e-mail: qsun@imech.ac.cn Tel: +86 () 853 K. Xu Mathematics Department Hong Kong University of Science and Technology Clear Water Bay Kowloon Hong Kong

Introduction In the past years a gas-kinetic BGK scheme (GKS) has been developed for compressible flow simulations [-3]. The GKS scheme evaluates the interface flux by integrating the velocity distribution function of gas molecules which is different from conventional finite volume method where macroscopic properties are employed directly for flux evaluation. In the GKS scheme the gas distribution function follows the accurate time-dependent solution of the BGK equation thus the free transport and collision mechanisms are physically coupled. It has been shown that the GKS scheme is very successful for many engineering applications [5] and is more robust than generalized Riemann problem (GRP) scheme for compressible flow simulations [6]. The GKS scheme is designed using the distribution function governed by the BGK equation under the conditions where the Navier-Stokes (NS) equations are valid. The scheme does not solve directly the gas distribution function using the finite volume method. Instead the distribution function is used to construct the evolution process for interface flux. It is more like a physical model than a numerical scheme. The validity of the scheme relies heavily on the construction of the distribution function on the interface. In an early scheme [] the initial gas distribution is reconstructed with a slope term having the spatial derivation of a Maxwellian distribution. Later Xu [] added a nonequilibrium term to account for the nonequilibrium states obtained from the Chapman-Enskog expansion. Ohwada [7] analyzed the kinetic schemes using a railroad method and showed that solutions in the GKS scheme were nd -order accurate in both space and time. Recently the GKS scheme has been extended to multi-dimensional schemes [8] high-order schemes [9] and unified schemes for rarefied gas flows []. Particularly the gradients of flow variables in both normal and tangential directions are explicitly included in the distribution function at cell interface in multi-dimensional schemes. or high-order schemes it is important to have the initial distribution and its evolution reconstructed using high-order processes. The validity of GKS schemes has been demonstrated by various test problems in the literature. The underlying equations however are not as clear as those schemes directly solving macroscopic equations such as the Navier-Stokes equations. Although

it may be more suitable to employ physical means to resolve problems such as the flow in discontinuous regions it is beneficial to know the governing equations of the GKS scheme in smooth flow regions. If the equivalent differential equation can be obtained the numerical error will be obvious and may indicate research directions for further improvement of the scheme. The objective of this study is to find the equivalent partial differential equation (EDE) of the GKS scheme when it is applied to a smooth flow region and show its connection to the Navier-Stokes equations. or this purpose the GKS scheme is introduced in Sect. along with its flux expressions. The EDE is then derived in Sect. 3. The connection between the derived equation and the NS equations is discussed in Sect.. Some concluding remarks are given in the last section. The Gas-Kinetic BGK Scheme and its lux Evaluation Several gas-kinetic schemes have been reported in the literature. The most important one seems to be the one published in [] and is termed as the GKS scheme. Thus the GKS scheme is analyzed in this paper. As the scheme has been detailed in [] only related expressions are presented here. The GKS scheme evaluates the interface flux with the help of the velocity distribution function that follows the BGK equation f f f f g f u v w t x y z () where f is the gas distribution function and g is the equilibrium state approached by f. Both f and g are functions of space (x y z) time t molecular velocities (u v w) and internal variable ξ. The internal variable has K degrees of freedom where K is equal to 5 3 and γ is the ratio of specific heats. The relaxation time τ is related to the gas viscosity and pressure ( p ). The equivalent state g is a Maxwellian distribution K 3 u U v V w W g e

where ρ is the density (U V W) is the macroscopic velocity in the (x y z) directions and λ is equal to m kt. Here m is the molecular mass k is the Boltzmann constant and T is the temperature. The general solution of f in () at a cell interface x j and time t is t tt t f xj tuvw gx t uvw e dt e fxj ut () where x x ut t j is the molecular trajectory and f is the initial state of f at the beginning of each time step (t=). unctions g and f are unknown and are reconstructed as l r g g H x a xh x a x At l l l l r r r r f H x g a x au A H x g a x a u A (3) () x l r where Hx g g g are the corresponding Maxwellian x distributions constructed at the interface the left hand side and the right hand side of the interface. The variables a and A are the spatial and temporal derivatives of related distributions respectively. Substituting (3) and () into () function f can be expressed as t t j t t l r e te a Hua Huug t l l r r e A Hug A Hug f x t u v w e g t e Ag e u t a H u g u t a H u g t l l r r Expression (5) is rather complex and can be greatly simplified in smooth flow l r l r l r l r regions where g g g a a a a A A A. The function f then becomes as f g au t A. (6) With the above expression the flux through an interface can be easily integrated. Particularly the mass momentum and energy fluxes in the x direction are calculated as (7) ug au t A d (5)

where d dudvdwd dk a a a ua va wa u v w uvw u v w 3 5 A A Au Av A w A u v w 3 5. The derivatives a and A are determined using the slopes of macroscopic quantities. Namely ga d b (8) gad gaud (9) j j where b x ju j ju j b x jvj jvj b3 x jwj jwj b x b 5 j E j je j x 3 and K E U V W. The subscript refers to the reconstructed value at the interface j+/. Notice that slopes b are approximated using the neighboring cell values and are nd-order accurate. With the values of a and A from (8) and (9) the fluxed in (7) are derived as t U K U V WbK UbVb3Wbb 5 K 3 (a) u K U b Ub K 3 K 5 K 3U 3V 3W Ub t K 3 K 5 3 6 UV Vbb K U b UVb3 Wb b5 3 v K 5U V W Vb t K 3 K 3U b V K U b V b W b b UW Wbb 3 3 5 w (b) (c) (d) t K 5U V W Wb K 3 WK Ub Vb3 Wb b5k 3U b

K 5 E Pr U U V W K K 3K 5 3 U V W b K 3 K Ub Vb3 Wb K 5b 5 U V W K U V W K 5K 7U V W b. (e) K U K V W t K 3 K 3K 5 Ub K 5 5U V W Vb3 Wb b5 K 3U b 5 It is well known that the Prandtl number in the BGK equation is fixed at. In order to extend the GKS scheme to flows with arbitrary Prandtl number a post-processing step is used in [] and the energy flux is updated as q () Pr E EPr q uut uut vvt wwt fd which is calculated as where EPr t 5 t t t t u t v t w q U Q U V W Q U V W () T and Q Q Q3 Q Q5 f d Q Ut Q3 Vt Q Wt. The expressions of Q Q Q Q are derived as follows Q tb (3a) t Q U K U V W bk Ub Vb3 Wb b 5 K 3 (3b) 3 3 Q V t UVb Vb Ub (3c) Q W t U W b W b U b (3d) K 3 Q5 U V W K K 5 K K 5 U V W U b U V W b K 3 3 t K 5 UVb3Wb b5 K 3 K (3e)

It should be emphasized that () and (3) use values of flow variables at the interface. Thus the interface values are reconstructed as the averages of the neighboring cell values. or instance j j. Numerical error of this approximation can be identified using Taylor expansion at the interface as x O x 8 x A summary of related expressions for interface values is listed in Appendix (A.). () Notice that Ot ( ) is introduced when U t V t W are approximated as shown in t (A-A8). With these expressions the fluxes are derived with cell size x and time t as follows U p t U t O x t x x 3 K U U K 5 U p u t U p t p U O x K 3 x x K 3 x x 3 t t t V U V p v t UV t V O x x x x t t W U W p w t UW t W O x x x x t t (5a) (5b) (5c) (5d) K K 5 x K 3 Pr U U V W K K 5 U p x K 3 x t p K 8U V W K 5RT Ot x x PrK U U p V V W W U Pr K 3 x x x x x x x 5 E t U U V W K Up U V W RT t t (5e) Expressions (5) are the fluxes of the GKS scheme in smooth regions. These fluxes are evaluated at time t using values of macroscopic flow variables at the beginning of the time step. It should mention that the present derivation of fluxes is based on the first order expansion of the velocity distribution function as in (3) and (). or higher-order gas-kinetic BGK schemes the flux expressions may be slightly different. With the flux expressions conservative flow variables Q in the cell are updated

using the finite volume method. Namely where Q U V W E t n n Q Q d dt Vol S (6) is the flux along the surface and Vol is the cell volume. To derive the governing equation for Q (6) is expressed in the integral form t Q dvol dt (7) t As (7) is applied to every cell and any time step its differential form can be obtained as Q G H (8) t x y z where G and H are component fluxes in x y and z directions. It should emphasize that Q G and H are evaluated using variables with instantaneous values at local position. 3 The Equivalent Differential Equation In order to derive the equivalent differential equation of GKS the fluxes in (8) are to be evaluated locally at time t. Thus the variables at t= in (5) are converted to values at time t using Taylor expansions. Taking the mass flux (5a) as example we have and U U U t O t t t t t tt (9) then or U p U p O t x x x x t U U p t U t t O t x tt t tt x x tt tt () U U p U t t Ot x t x x Similarly the other fluxes are derived as follows. (a)

K U K U K 3 x t K 3 x u U p t U p 3 U K 5 U p t p 3 U Ot x x K 3 x x V V U V p v UV t UV t V Ot x x t x x x W W U W p w UW t UW t W Ot x x t x x x x K 3 K 5 K K 5 U U V W Up U RT E 5 W K 5 K V W K 5 t UU V W Up U RT t x K 3 Pr U U V W K K U p x K 3 x t p K 8U V W K 5RT Ot x x K U U p V V W W U Pr K 3 x x x x x x x V Pr (b) (c) (d) (e) All variables in () are now in time t. Then () presents the instantaneous numerical flux where the difference between the exact one and the numerical one is on the order of Ot x. or a finite volume conservative GKS which is the same for any other finite volume method the error in the instantaneous flux is the only source for the error in the differential equations of (8). Therefore the differential equation for the GKS must have the following form Q G H Ot x y z () t x y z where U K U U p K 3 x V UV x W UW x K V W K 5 E pu U RT x K 3 Pr

V U UV y K V V p G K3 y W VW y K U W K 5 E pv V RT y K 3 Pr W U UW z V. VW H z K W W p K3 z K U V K 5 E pw W RT z K 3 Pr G and H are obtained be rotating variables and coordinates in. rom () the time derivative terms in () are derived as follows t x U U p y z Ot x y z (3a) K U U p K 5 U U p U p Ot x y z t K3 x x x K3 x 3 3 uy uz u (3b) V VU p UV V vy vz v Ot x y z t x x x W WU p UW W wy wz w Ot x y z t x x x Ey Ez E 5 K V W T E pu U t x K 3 x K K U K 7 p U U V W p U ERT x K 3 x x O t x y z (3c) (3d) (3e) where the expressions of y z u y u z u v y v z v w y w z are given in Appendix (A.) and K 5 w E y E z E transfer coefficient. Substituting (3) to () the final expressions for fluxes are as follows y z R is the heat Pr U t O t x y z (a) K U U p t O t x y z K 3 x V v UV tv y v z v Ot x y z x W w UW t w y w z w Ot x y z x u u y u z u (b) (c) (d)

E E pu U x K 3 x K V W T Ey Ez E t O t x y z (e) where KU pu V V W W E E U. Pr K 3 x x x x x x x With () the governing equation (8) is updated as follows where t Q G H t Gt Ht t Ot x y z t x y z x y z x those in () and y (5) z are the time step and cell size G H are the same as y z uy uz u t vy vz v wy wz w Ey Ez E G G G G G G G G G G G G G G G G z x uz ux u t vz vx v wz wx w Ez Ex E H H H H H H H H H H H H H H H H x y ux uy u t vx vy v wx wy w Ex Ey E. The detailed expressions of G t and H t are not given here but can be easily obtained by rotating the coordinate and velocity components in t. Clearly Eq. (5) is the equivalent partial differential equation for the GKS scheme. It shows that the GKS scheme is st -order accurate in time and nd -order accurate in space. The st -order time error comes from the reconstruction of the interface flux. Detailed analysis is given in the next session. Connection with the Navier-Stokes Equations The GKS scheme is based on the Maxwell distribution and its gradients which means that the scheme is targeted at solving the continuum equations. The Navier-Stokes equations are the well-known equations for fluid flow at this level. Thus the equivalent differential equation of the GKS scheme is compared with the NS equations. In the NS equations the fluxes in the x direction can be written as

U U V W U V W U p B 3 x y z x y z V U UV NS x y W U UW x z U V W U V W V U W U T EpU U BU V W 3 x y z x y z x y x z x (6) where B is the bulk viscosity. In order to compare with the GKS flux (6) is separated into two terms NS U U V W U p B B 3 x 3 y z V U UV x y W U UW x z U V W T E pu V W U U B U V W B U V W 3 x x x x 3 y z y z The fluxes in the EDE of the GKS scheme can be written as U K U U p 3 3 9 y z K x V uy uz u UV GKS x t vy vz v Ot x y z W wy wz w UW x Ey Ez E K U V W T E pu U V W 3 3K 9 x x x x Comparing (7) and (8) the connection between EDE and the NS equations can be clearly identified. irst under the limitation of Δt Δx Δy Δz and quasi-d assumption (eg. y z for fluxes in the x-direction) the EDE of GKS equals to the NS K equations only that the EDE fixes the bulk viscosity at. In the NS equations 3K 9 the bulk viscosity is a free parameter which is usually neglected for many computational fluid dynamics applications. In the GKS it is zero for monatomic gases (K=) which is physically true. or other gases it accounts for the difference between the equilibrium pressure and kinetic pressure. The BGK model predicts the value of bulk viscosity as /5 of the dynamic viscosity when K= (or γ=7/5). However the (7) (8)

accurate value for the bulk viscosity is generally unknown. In [3] it is reported that the bulk viscosity of nitrogen at moderate temperatures is around.8 times of the dynamic viscosity when the absorption of sound waves was measured at frequencies below the relaxation frequency of rotational energy. Second the EDE lacks the gradients of flow variables in other directions which is the outcome of directional splitting used in the initial reconstruction. This is the main reason that multi-dimensional schemes [8] are desired for viscous flows. The EDE of multi-dimensional schemes can be derived in a similar procedure as in this paper. or instance if the reconstructed flux includes the flow gradients in other directions using the multi-dimensional scheme [] it can be shown in (9) that the y-derivatives and z-derivatives will be recovered as in the NS equations. U U V W U p B B 3 x 3 y z V U UV x y u GKS W U UW t v Ot x y z x z w U V W E p U B U V W 3 x x x E T V W U U B U V W x 3 y z y z (9) Third the EDE is nd -order accurate in space and st -order accurate in time. The first-order terms in (8) are related to the y-derivative z-derivative and gas viscosity. These y-derivatives and z-derivatives however can be removed if multi-dimensional splitting schemes are used as shown in (9). The viscous related terms can be traced back to (5) where they will show up when the inviscid fluxes are converted from t= to time t. This is the same as the way to evaluate A in (6) using (9) which is equivalent to the Euler equations. In other words the viscous terms in the NS equations have no contribution to the time evolution of the flow variables at the interface. If these terms are to be cancelled the reconstructed distribution function (5) should contain a nonlinear term t. In other words a high-order expansion beyond the traditional Chapman-Enskog for the NS needs to be used. This has been done in [9]. Therefore the EDE is nd -order accurate in both space and time at the level of

Euler equations but is st -order accurate in time when the flow is viscous. In practical applications however the time step is usually much larger than the relaxation time. Thus the st -order time error that originated from the nonlinear term t may be included in the nd -order time error. The numerical time error can still be nd -order for the GKS scheme when simulating high-reynolds viscous flows. It should mention that Ohwada [7] found that the solution of the GKS scheme was nd -order accurate in space and time which is consistent with the present result since the error in conservative variables Q can be easily obtained by integrating (5) over the time step. The expressions of fluxes given in [7] however show difference in the time error term. This is because Ohwada s analysis is mainly for illustration purpose where specific expressions of the error in terms of flow variables have not been presented. 5 Conclusion In this paper we used a simplified version of the evolution solution of the gas distribution function at the interface formulated the fluxes with instantaneous values of macroscopic flow variables using Taylor expansions and derived the equivalent partial differential equation for the gas-kinetic BGK scheme. It is confirmed that the GKS scheme solves the Navier-Stokes equations with a predetermined bulk viscosity. The derived equation shows that multi-dimensional splitting schemes are desired to eliminate flux error from the directional splitting method for viscous flow simulations. The GKS scheme is nd -order accurate in both space and time at the level of Euler equation. However there is st -order error in time for viscous flows. To our knowledge numerical schemes using the finite volume method are seldom studied to obtain the equivalent partial differential equation. The analysis presented in this paper can then be used to study similar schemes. or instance with the same GKS scheme (multi-dimensional one) it is unclear how the order of scheme will be improved by using a high-order initial reconstruction for the slopes in space such as the use of high-order WENO methods. What is the equivalent differential equation of the high-order GKS schemes in [9]? What will happen if the fluxes are solved

using the Riemann solver such as the exact one? Answering these questions may be useful to the CD community. Acknowledgments This work has been supported by the National Natural Science oundation of China through grants No. 963 and No. 3735. Appendix: Some Auxiliary Results A. Taylor expansions at the interface for some variables 3 x b O 3 x x x 3 U x U b O 3 x x x 3 V x V b3 O 3 x x x 3 W x W b O 3 x x x 3 E x E b5 O 3 x x x (A) (A) (A3) (A) (A5) x O x 8 x x O x 8 x x U U U U O x 8 x x x V V V V O x 8 x x x W W W W O x 8 x x x U U V V W W T T RT R O x 8 K 3 x x x x x x x x x (A6) (A7) (A8) (A9) (A) (A)

x U U V V W W T T O x 8 T K3 R x x x x x x x x x (A) Ox x U U V V W W T T T K3R x x x x x x x x x x Q t U O O t (A3) x ( ) (A) x U p Ut U tu O x O t x x V Vt V tu O x O t x W Wt W tu O x O t x (A5) (A6) (A7) K5 T KU U p V V W W q R t U U Ox Ot (A8) x K3 x x x x x x x A. Some terms contributed to fluxes y z UV (A9) y UW (A) z K U U U (A) K 3 x x y y z z u UV p K5 V uy V p y y K 3 y (A) UW p K5 W uz W p z z K 3 z (A3) T K 8 U K U U V V W W U K 3x x K 3 x x K 3 K 3 x x x x x x T U U U K V V W W U K 3y y y y K 3 y y K 3 y y y y T U U U V V K W W U K 3 z z z z K 3 z z z z K 3 z z K U K 3 t x (A)

K U U U U T U U K 3 x x y y z z t x K 3 T x x x U U U p U V W x y z x K 3 T T T U V W RU V W p x y z x y z K U U V V W W V V (A5) U K 3 x x y y z z x x K 3R T x W W U U W W U U V V x x y y y y z z z z T T T x x y y z z Ot x y z vy UV p U (A6) y y vz UVW z (A7) K U V U K V v V U V U K 3 x x x x y y K 3 y y (A8) U V V V U z z z z t x V K V V V x x K 3 y y z z t x x V V V p U V W x y z y K 3 T T T U V W RU V W p x y z x y z K U U V V W W V V V K 3 x x y y z z x x (A9) K 3R T x W W U U W W U U V V x x y y y y z z z z T T T x x y y z z Ot x y z wz wy UVW (A3) y UW p U (A3) z z

K U W U W w W U W U K 3 x x x x y y y y (A3) U K W W W U z z K 3 z z t x W W K W W x x y y K 3 z z t x x W W W p U V W x y z z K 3 T T T U V W RU V W p x y z x y z K U U V V W W V V W K 3 x x y y z z x x (A33) K 3R T x W W U U W W U U V V x x y y y y z z z z T T T x x y y z z Ot x y z p K 5pUV K 5 V (A3) y y y K 3 y Ey UV U V W UV pu p K 5pUW K 5 W z z z K 3 z (A35) Ez UW U V W UW pu K U K U V W E ERT U U V W K 3 x x K 3 x x x x x x K5 KU U V V W W T U K3 K 3 x x x x x x x x U K5 U U KV V W W ERT U y y K3 y y K3 y y y y U K V W K5 T UU V W y y K3 y y y y K3y y (A36) U K5 U U V V KW W ERT U z z K3 z z z z K3 z z U V K W K5 T UU V W z z z z K 3 z z K 3 z z K V W K5 U RT t x K3 Pr

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