Part 23 Gas-Kinetic BGK Schemes

Transcription

1 Part 23 Gas-Kinetic BGK Schemes

2 A New High-Order Multidimensional Scheme Qibing Li, Kun Xu and Song Fu Abstract A third-order accurate multidimensional gas-kinetic BGK scheme is constructed through the high-order expansion of the distribution function and the highorder reconstruction of conservative variables. With several typical test cases the good performance of the new scheme is validated in both smooth flow and the flow with strong discontinuity. The theoretical validity for such an approach is due to the fact that the kinetic equation has no specific requirement on the smoothness of the initial data, as well as the simple particle transport mechanism and the inherent multidimensional characteristics on the microscopic level. The present study shows a new hierarchy to construct a high-order multidimensional method, and the Navier- Stokes flux function obtained from the present work can be adapted to many other high-order CFD methods. 1 Introduction The high-order numerical methods for the Navier-Stokes (NS) equations has attracted many researches due to its advantages in wide-range applications [1, 6]. The development of a solution under piecewise discontinuous high-order initial reconstruction for the NS equations directly is urgent, but difficult due to the mathematical inconsistency of the discontinuous initial data and the hyperbolic-parabolic Qibing Li Department of Engineering Mechanics, Tsinghua University, Beijing , China, lqb@tsinghua.edu.cn Kun Xu Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong e- mail: makxu@ust.hk Song Fu Department of Engineering Mechanics, Tsinghua University, Beijing , China, fsdem@tsinghua.edu.cn 1

3 2 Qibing Li, Kun Xu and Song Fu nature of the NS equations. Furthermore, due to the lack of a multidimensional Riemann solution, it is also a great challenge to construct a genuinely multidimensional scheme. However, based on the gas-kinetic theory, it is easy to develop a generalized NS flow solver, BGK scheme [7, 3], due to the fact that the Boltzmann equation has no specific requirement on the smoothness of the initial data and the the simple particle transport mechanism at the microscopic level, including the coupling of the particle free transport and collisions, and the inherent multidimensional characteristics. The extension of the method from 2nd-order to high-order has been investigated, such as the directional splitting method [4], or the multidimensional version for smooth flow [5]. This paper will further extend the method to construct a general high-order multidimensional gas-kinetic BGK method (HBGK-MD), suitable for not only smooth flow, but also flow with discontinuity. 2 A High-Order Multidimensional BGK Scheme 2.1 Fundamental of Gas-Kinetic BGK Scheme The construction of gas-kinetic BGK scheme [7] is briefly described as follows. First, the BGK-Boltzmann equation for two-dimensional (2-D) flow is written as f t u f i = g f x i τ, i = 1,2, (1) where τ = µ/p is the particle collision time. f = f (x,t,u,ξ ) is the gas distribution function, and g is the equilibrium state approached by f, assumed to be a Maxwellian distribution, g = ρ(2πrt ) (K2)/2 e λ( u U 2 ξ 2), (2) where ξ 2 = ξ1 2 ξ ξ K 2 represents the internal energy of particles with total number of degrees of freedom K = (4 2γ)/(γ 1). During the particle collisions, f and g satisfy the conservation constraint, (g f )ψdξ = 0, ψ = (1,u,( u 2 ξ 2 )/2) T, (3) at any point in space and time for the conservation of mass, momentum and energy. Here dξ = du 1 du 2 dξ 1 dξ 2...dξ K is the volume element in the phase space. If the distribution function f is known, the macroscopic conservative quantities Q and the flux F can be obtained through the integration over the phase space Q = (ρ,ρu,ρv,ρe) T = f ψdξ, F = u f ψdξ. (4)

4 A New High-Order Multidimensional Scheme 3 From Eqs. (1) and (3), the finite volume formulation of the BGK scheme is formed as (Q ) n1 lm = (Q ) n lm 1 t n t F dtds (5) S lm t n where the computational cell is indexed by l and m with the area S lm and boundary s. The superscript represents the variable in the global coordinates. The flux F is calculated through the coordinate transformation from that in the local coordinates F. For convenience, the calculation of F is presented through an example at a cell interface x s = (x l1/2,y m ) T with x l1/2 = 0 and y/2 y m y/2. Now the question is how to solve the BGK equation (1) to obtain the gas distribution function f. To avoid the great difficulty of direct solving method, the gas-kinetic BGK scheme adopts a most ingenious method based on the Chapmann-Enskog expansion and the integral solution of the BGK equation, f (x,t,u,ξ ) = 1 τ t 0 g(x,t,u,ξ )e (t t )/τ dt e t/τ f 0 (x ut,u,ξ ) (6) where x = x u(t t ) is the trajectory of a particle motion and f 0 is the initial gas distribution function at the beginning of each time step (t = 0). The Chapman- Enskog expansion is used to construct f 0 and g around the cell interface (l 1/2,m). Thus the time dependent distribution function f can be easily deduced and then the fluxes across the cell interface can be calculated with Eq. (4) and finally the conservative variables at the next time step can be calculated via the finite volume formulation (5). Details can be found in the corresponding reference. It should be noted that in the above-mensioned BGK method, f 0 and g can be constructed according to different purpose, such as that to approach high-order macro equation, i.e. BGK-Burnett [8], or to approximate NS equations with higher order accuracy [4, 5]. 2.2 High-Order Multidimensional Scheme In order to develop a high-order accurate gas-kinetic BGK scheme, we can construct the high-order accurate initial distribution function f 0 and the equilibrium distribution g through the expansion to third-order in both spatial and temporal directions. The scheme for 1-D flow and 2-D flow with directional splitting method has been developed in our previous study [5]. Here, the genuinely multidimensional scheme can be constructed with the following f 0 and g, including both the normal and tangential slopes, [ f 0 (x,0,u,ξ ) = 1 a l ix i τ(a l iu i A l ) τ(a l ia l Ci l )x i ] (a l ia l j b l i j)( τu i x j x i x j /2) (1 H(x 1 ))g l [1 a r i x i τ(a r i u i A r ) τ(a r i A r C r i )x i

5 4 Qibing Li, Kun Xu and Song Fu (a r i a r j b r i j)( τu i x j x i x j /2) ] H(x 1 )g r. (7) g(x,t,u,ξ ) = g 0 [1 a i x i At (a i a j b i j )x i x j /2 (a i A C i )x i t (A 2 B )t 2 /2 ] (8) where g 0 is the initial local Maxwellians and H is the Heaviside function. The local terms a i,b i j,c i,b and A are from the Taylor expansion of a Maxwellian and take the form, a = a (α) ψ α, α = 1,2,3,4, where all coefficients, a (α),...,a (α), are local constants from the first and second derivatives of g. These coefficients, as well as g 0 are related to the reconstructed conservative variables Q and their slopes, which can be evaluated through the condition on Chapman-Enskog expansion, same as that in BGK-Burnett method [8, 5]. Then the distribution function at the cell interface can be deduced, f (x s,t,u,ξ ) = (1 e t/τ )g 0 ( τ (τ t)e t/τ )a i u i g 0 (t τ τe t/τ )Ag 0 (1 e t/τ )(a 2 2 b 22 )(x 2 2/2)g 0 (τ 2 (t 2 /2 τt τ 2 )e t/τ )(a i a j b i j )u i u j g 0 (t 2 /2 τt τ 2 τ 2 e t/τ )(A 2 B )g 0 (2τ 2 τt (2τ 2 τt)e t/τ )(C i a i A)u i g 0 ( e t/τ 1 (t τ)a l iu i τa l τt(ci l a l ia l )u i ((a l ) 2 b l 22)x2/2 2 ) (τt t 2 /2)(a l ia l j b l i j)u i u j H(u 1 )g l e t/τ ( 1 (t τ)a r i u i τa r τt(c r i a r i A r )u i ((a r ) 2 b r 22)x 2 2/2 (τt t 2 /2)(a r i a r j b r i j)u i u j ) (1 H(u1 ))g r. (9) In the above equation, the variation of f along the tangential direction of the cell interface x 2 is represented through the tangential slopes, such as a 2,b 12,b 22,C 2. However, the terms explicitly in proportion to x 2 are omitted, as the integration is zero. The terms containing x2 2 are retained, which is necessary for the scheme to achieve the third-order accuracy with only ONE integral point (the center of the cell interface). Furthermore, the above solution, or (6) allows the movement of particles in any direction. That is, the present high-order-accurate scheme simulates a multidimensional transport process across a cell interface. Thus it is a truely multidimensional scheme. For smooth flow field, the present method goes back to the previous simple scheme [4] (the term τtb is unnecessary at the level of NS equations), and the computational cost can be decreased remarkably. It should be noted that it is difficult to achieve high-order reconstruction of macro conservative variables for multidimensional flow. In the present study, the least-square method is adopted and the coefficients can be calculated in advance for only one time to decrease the computational cost. The 2nd-order PFGM limiter [9] (p = 2) is used mostly for the direct reconstruction of conservative variables when the flow contains discontinuities.

6 A New High-Order Multidimensional Scheme 5 3 Numerical Results Here we present some results computed by the newly developed HBGK-MD scheme. The first one is the inviscid isentropic vortex advection problem [10]. The computational domain is set to [ 5,5] 2 divided by unform cells. The limiter is not used in this case. Figure 1 shows the grid convergence of computed density, from which the third-order accuracy of the present method can be clearly observed L 1 error L error rd order Fig. 1 Errors in density vs. cell size at t = 10 for the isentropic vortex advection. The initial mean flow and the perturbation value are given by u = v = 1, p = 1, T = 1, and (δu,δv) = (5/2π)e (1 r2 )/2 ( y,x), δt = 25(γ 1)/(8γπ)e 1 r2. Error x Figure 2 shows the velocity profiles at different streamwise locations for a boundary layer flow with Mach number 0.15 and Reynolds number Re = U L/ν = 10 5, where L = 100 is the length of a plane plate. The computational domain is chosen as [ 40,100] [0,50] and grid cells are adopted with cells locate ahead of the plate. The minimal cell sizes are x m = 0.1 and y m = One can see that the velocity distributions, not only for the streamwise component, but also for the transverse one, can be accurately predicted with only four cells, which shows the good performance of the present scheme in viscous flow. 1 U/(νU /x) 1/ x/l= x/l= x/l= Blasius V/(νU /x) 1/ x/l= x/l= x/l= Blasius y/(νx/u ) 1/ y/(νx/u ) 1/2 Fig. 2 Velocity profiles at different streamwise locations in a boundary layer.

7 6 Qibing Li, Kun Xu and Song Fu Fig. 3 Density contours for viscous shock tube problem with Re = 1000 and Pr = 0.73 at time t = 1 (left: cell number , right: ). The two-dimensional shock tube problem [2] is calculated with the non-slip boundary at both two side walls and two end walls. Only half of the shock tube is considered divided by uniform cells, due to the symmetry of the flow in the vertical direction. The initial flow field is stationary, with sound speed a = 1 and a strong density jump at x = 0.5: ρ = 120 on the left and 1.2 on the right. As shown in figure 3, the complicated flow structures, such as the shock, boundary layer, vortex and their interactions, are well captured by the present scheme. Good grid convergence is achieved in the present study. 4 Conclusion In the present study, a high-order multidimensional gas-kinetic scheme is developed and validated with typical numerical tests. The constructed valuable NS flux function can be implemented to many high-order computational fluid dynamic methods. The comparison between the directional splitting scheme and a mutidimensional method, and the effect of the limiter for reconstruction require further study. Acknowledgements This work was supported by National Natural Science Foundation of China (Project No ). References 1. Cockburn, B., Karniadakis, G.E., Shu, C.W. (eds.): The development of discontinuous Galerkin methods. Springer, Berlin (2000) 2. Daru, V., Tenaud, C.: Numerical simulation of the viscous shock tube problem by using a high resolution monotonicity-preserving scheme. Comput. Fluids 38, (2009) 3. Li, Q.B., Fu, S.: On the multidimensional gas-kinetic BGK scheme. J. Comput. Phys. 220, (2006) 4. Li, Q.B., Fu, S.: A high-order accurate gas-kinetic BGK scheme. In: H. Choi, H.G. Choi, J.Y. Yoo (eds.) Computational Fluid Dynamics Springer (2009)

8 A New High-Order Multidimensional Scheme 7 5. Li, Q.B., Xu, K., Fu, S.: A high-order gas-kinetic Navier-Stokes flow solver. J. Comput. Phys. 229, (2010) 6. Toro, E.F., Millington, R.C., Nejad, L.A.M.: Towards very high order Godunov schemes. In: E.F. Toro (ed.) Godunov Methods: Theory and Applications, pp Kluwer/Plenum Academic Publishers (2001) 7. Xu, K.: A gas-kinetic BGK scheme for the Navier-Stokes equations, and its connection with artificial dissipation and Godunov method. J. Comput. Phys. 171, (2001) 8. Xu, K., Li, Z.: Microchannel flows in slip flow regime: BGK-Burnett solutions. J. Fluid Mech. 513, (2004) 9. Yang, M., Wang, Z.J.: A parameter-free generalized moment limiter for high-order methods on unstructured grids. Adv. Appl. Math. Mech. 4, (2009) 10. Yoon, S.H., Kim, C., Kim, K.H.: Multi-dimensional limiting process for three-dimensional flow physics analyses. J. Comput. Phys. 227, (2008)

9 A Direct Boltzmann-BGK Equation Solver for Arbitrary Statistics Using the Conservation Element/Solution Element and Discrete Ordinate Method Bagus Putra Muljadi and Jaw-Yen Yang Abstract This work presents a computational algorithm using discrete ordinate method with conservation element and solution element (CE/SE) scheme for solving the semiclassical Boltzmann equation with relaxation time approximation of Bhatnagar, Gross and Krook. The method is implemented on gases that obey arbitrary statistics distributions. 1 Introduction In the area of classical gas flows, the implementation of discrete ordinate method in the rarefied flow computations has been developed [Yang and Huang(1995)] and has been able to cover wide Knudsen number flow regimes. Under the same motivation, the present work is built using discrete ordinate method to describe the hydrodynamic properties of rarefied gases of all the three statistics. First, the discrete ordinate method is used to discretize the velocity space in the semiclassical Boltzmann-BGK equation into a set of equations in physical space with source terms. Second, the resulting equations can be treated as scalar hyperbolic conservation laws with stiff source terms whose evolution in space and time is modeled by an explicit, second-order CE/SE scheme developed by Chang [Chang and To(1991)]. Bagus Putra Muljadi Institute of Applied Mechanics, National Taiwan University, Taipei 10764,TAIWAN d @ntu.edu.tw Jaw-Yen Yang Institute of Applied Mechanics, National Taiwan University, Taipei 10764,TAIWAN yangjy@spring.iam.ntu.edu.tw 1

10 2 Bagus Putra Muljadi and Jaw-Yen Yang 2 Semiclassical Boltzmann-BGK Equation and Hydrodynamic Properties We first adopt the relaxation time concept of Bhatnagar, Gross and Krook, thus the semiclassical Boltzmann-BGK equation reads ( f t p ) ( ) δ f m x U(x,t) p f (p,x,t) = = f f (0) δt coll. τ see [Bhatnagar et al(1954)bhatnagar, Gross, and Krook]. Here, U is the externally applied potential, m is the particle mass, p is particle momentum, τ is the relaxation time. The equilibrium distribution function for general statistics is expressed as (1) f (0) 1 (p,x,t) = { } z 1 exp [p mu(x,t)] 2 /2mk B T (x,t) θ (2) where m is the particle, u(x,t) is the mean velocity, T(x,t) is temperature, k B is the Boltzmann constant and z(x,t) = exp(µ(x,t)/k B T (x,t)) is the fugacity, where µ is the chemical potential. In (2), θ = 1, denotes the Fermi-Dirac statistics, θ = 1, the Bose-Einstein statistics and θ = 0 denotes the Maxwell-Boltzmann statistics. The conservation laws of macroscopic properties in terms of macroscopic quantities i.e., number density n(x, t), momentum density nu(x, t), and energy density ε(x,t) are given by n(x,t) x j(x,t) = 0 (3) t mj(x,t) dp x t h 3 p p m f (p,x,t) = n(x,t) xu(x,t) (4) ε (x,t) dp p p 2 x t h 3 m 2m f (p,x,t) = j(x,t) xu(x,t) (5) In one spatial dimension, the general distribution is given by 1 f (p x,x, t) = { } z 1 exp [p x mu x ] 2 /2mk B T (x,t) θ (6) whereas n(x,t), j(x,t) and ε(x,t) are given by d px n(x,t) = h f (0) (p x,x,t) = Q 1/2(z) (7) λ d px p x j(x,t) = h m f (0) (p x,x,t) = n(x,t)u x (x,t) (8) d px p 2 x ε(x,t) = h 2m f (0) (p x,x,t) = Q 3/2(z) 2βλ 1 2 mnu2 x (9)

11 Title Suppressed Due to Excessive Length 3 βh Here, λ = 2 2πm is the thermal wavelength and β = 1/k BT (x,t). Quantum functions Q υ (z) of order υ are defined for Fermi-Dirac and Bose-Einstein statistics as x υ 1 F υ (z) 1 dx Γ (υ) 0 z 1 e x 1 l 1 zl ( 1) l=1 l υ (10) B υ (z) 1 x υ 1 dx Γ (υ) 0 z 1 e x 1 z l=1 l υ (11) Here, F υ (z) applies for Fermi-Dirac integral and B υ (z) for Bose-Einstein s, whereas Γ (υ) is gamma function. The normalized semiclassical Boltzmann-BGK equation is given by ˆf ( ˆυ x, ˆx, ˆt) ˆt ˆυ x ˆf ( ˆυ x, ˆx, ˆt) ˆx = ˆf ˆf (0) ˆτ From this part, our formulations are all considered normalized and we shall omit the hat sign for simplicity. (12) 3 Application of Discrete Ordinate Method The application of the discrete ordinate method to Eq. (12) results in f σ (x,t) t υ σ f σ (x,t) x = f σ f σ (0) τ with f σ and υ σ represent the values of respectively f and υ x evaluated at the discrete velocity points σ. Gauss-Hermite quadrature rule reads, exp( υ x 2 )[exp(υ x 2 ) f (υ x )]dυ x N σ= N where the discrete points υ σ and weight W σ can be found through W σ = (13) W σ exp(υ σ 2 ) f (υ σ ) (14) 2n 1 n! π n 2 [H n 1 (υ σ )] 2 (15) with n = 2N and υ σ are the roots of the Hermite polynomial H n (υ). The repeated Simpson s rule is used at high temperatures instead. The macroscopic quantities can be described accordingly: e.g., normal density is given by n(x,t) = [ f (υ x,x,t)e υ x 2 ]e υ x 2 dυ x

12 4 Bagus Putra Muljadi and Jaw-Yen Yang = N W ξ [ f σ e υ σ 2 ] (16) σ= N 4 Numerical Method To solve the set of equations (13), a space-time CE/SE c τ method with wiggle suppressing term is introduced. The two marching variables are described as n f σ j t 2 ( f σ n 1/2 j (0) f n 1/2 σ j ) = τ 1/2[(1 c) f σ n 1/2 j 1/2 (1 c) f σ n 1/2 j1/2 (1 c 2 )[( f σ x ) n 1/2 j 1/2 ( f σ x) n 1/2 j1/2 ]] (17) where c = υ σ ( t/ x) is the Courant number and ( f σ x ) n j = ( x/4)( f σ x) n j is the normalized form of ( f σ x ) n j. After f σ n j is known, another marching variable ( f σ x) n j is determined by where W are functions given by W (x,x ;α) = ( f σ x ) n j = (w ) n j ( ˆf σ x ) n j (w) n j ( ˆf σ x ) n j (18) (w±) n j = W ± (( ˆf σ x ) n j,( ˆf σ x ) n j;α) (19) x α x α x α, W x α (x,x ;α) = x α x α, (20) for real variables x, x and α 0. The arguments in the W functions are specified as ( ˆf σ x ) n j = (( f σ ) n j [ f σ (2c 1 τ c ) f σ x ] n 1/2 j1/2 )/(1 τ c) ( ˆf σ x ) n j = ([ f σ (2c 1 τ c ) f σ x ] n 1/2 j1/2 ( f σ ) n j)/(1 τ c ) (21) t has to be less than τ. It also shall not violate the CFL stability condition, t S = c ( x/(υ σ )max). Thus, t = min( t C, t S ) (22) Numerical root finding methods are used to solve z(x,t), which is the root of Ψ 1 (z) = ε Q 3/2 4π ( n ) 3 1 Q 1/2 2 nu2 x (23)

13 Title Suppressed Due to Excessive Length 5 5 Computational Results Q 3/2 (z) 8nT (1 χ) Q 5/2 (z) The results are shown in terms of 1-D shock tube problems. CE/SE is used with α = 4. i. Mesh refinement test. The condition applied to Fermion gas is (n l,u l,t l ) = (0.557,0,1) and (n r,u r,t r ) = (0.341,0,0.6). These correspond to z l = 0.4 and z r = 0.3. Three Uniform grid systems are used with 100, 200 and 400 cells. CFL = 0.5 and τ = The quadrature used is Gauss-Hermite with 20 abscissas. ii. Varying relaxation times.the initial condition applied to Fermion gas at both sides of the tube are (n l,u l,t l ) = (0.724,0,4.38) and (n r,u r,t r ) = (0.589,0,8.97) which correspond to z l = and z r = τ ranges from 0.1 to iii. Varying Knudsen numbers. The relaxation time will vary with Kn number according to τ = 5 π Kn. Initial condition of case ii are used with Knudsen number ranging from 0.1 to iv. Test on high temperatures. In this test, the temperatures at both sides are set as: T l = and T r = whereas fugacity and velocity are kept fixed. The results can be seen in Fig. 1. In case i the density profiles converge to the highest grid points. The same behavior is observed in Case ii and iii where those with the highest Kn and τ converge to Euler limit. Case iv illustrates how the three statistics recover to the classical limit at high temperatures. 6 Concluding Remarks A direct algorithm that applies discrete ordinate method and CE/SE for solving semiclassical Boltzmann-BGK transport equation for particles of all statistics is constructed. Different aspects of the numerical method are tested. The feasibility of this algorithm have been illustrated without much major constraints. 7 Acknowledgements This work is done under the auspices of National Science Council, TAIWAN through grants NSC I References [Bhatnagar et al(1954)bhatnagar, Gross, and Krook] Bhatnagar PL, Gross EP, Krook M (1954) A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems. Phys Rev 94(3): , DOI /PhysRev [Chang and To(1991)] Chang SC, To WM (1991) A new numerical framework for solving conservation laws: The method of space-time conservation element and solution element. NASA TM (104495), DOI /PhysRevE

14 6 Bagus Putra Muljadi and Jaw-Yen Yang [Yang and Huang(1995)] Yang JY, Huang JC (1995) Rarefied flow computations using nonlinear model boltzmann equations. Journal of Computational Physics 120(2): , DOI DOI: /jcph , URL 45NJJFW-1J/2/275dde267bf1a59e6a3cf435fa6e68ce (a) case i (b) case ii (c) case iii (d) case iv Fig. 1: case i ( : 100 grids; : 200 grids; solid line : 400 grids); case ii ( : Euler; : τ = ; : τ = 0.001; : τ = 0.01; : τ = 0.1.); case iii ( : Euler; : Kn = ; : Kn = 0.001; : Kn = 0.01; : Kn = 0.1); case iv ( : MB; : FD; : BE)