Improved treatment of the open boundary in the method of lattice Boltzmann equation

Transcription

1 Improved treatment of the open boundary in the method of lattice Boltzmann equation Dazhi Yu, Renwei Mei and Wei Shyy Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL USA Abstract: The method of lattice Boltzmann equation (LBE) is a kineticbased approach for fluid flow computations. In LBE, the distribution functions on various boundaries are often derived approximately. In this paper, the pressure interaction between an inlet boundary and the interior of the flow field is analysed when the bounce-back condition is specified at the inlet. It is shown that this treatment reflects most of the pressure waves back into the flow field and results in a poor convergence towards the steady state or a noisy flow field. An improved open boundary condition is developed to reduce the inlet interaction. Test results show that the new treatment greatly reduces the interaction and improves the computational stability and the quality of the flow field. Keywords: lattice Boltzmann equation; boundary condition; inlet; bounce-back. Reference to this paper should be made as follows: Yu, D., Mei, R. and Shyy, W. (xxxx) Improved treatment of the open boundary in the method of lattice Boltzmann equation, Progress in Computational Fluid Design, Vol. x, No. x, pp.xxx xxx. Biographical notes: Dazhi Yu received his PhD in Aerospace from University of Florida. He is now a Postdoctoral Research Associate in Aerospace Department at Texas A&M University. His research interests include lattice Boltzmann method, parallel computation, and turbulence. Renwei Mei is Professor of the Department of Mechanical and Aerospace Engineering at University of Florida. His research areas include particle dispersion and collision in turbulence, boiling heat transfer and two-phase flow, lattice Boltzmann method, and desalination. He is interested in applying computational methods and applied mathematics to solve practical thermal fluid problem. Wei Shyy is currently Professor and Chairman of the Department of Mechanical and Aerospace Engineering at University of Florida (UF). He is the author or a coauthor of three books dealing with computational and modelling techniques involving fluid flow, interfacial dynamics, and moving boundary problems. He has written papers dealing with computational and modelling issues related to fluid dynamics, heat/mass transfer, combustion, material processing, parallel computing, biological flight and aerodynamics, 1 design optimisation, micro-scale and biofluid dynamics.

2 1 INTRODUCTION 1.1 General description of the method Since the last two decades or so, there has been a rapid progress in developing the method of the lattice Boltzmann equation (LBE) as an alternative, computational technique for solving a variety of complex fluid dynamic systems [1 16]. Adopting the macroscopic method for computational fluid dynamics (CFD), macroscopic variables of interest, such as velocity u and pressure p, are usually obtained by solving the Navier-Stokes (NS) equations (e.g., see [17,18]). In the LBE approach, one solves the kinetic equation for the particle velocity distribution function f(x,ξ,t), where ξ is the particle velocity vector, x is the spatial position vector, and t is the time. The macroscopic quantities (such as mass density ρ and momentum density ρu) can then be obtained by evaluating the hydrodynamic moments of the distribution function f. Lattice gas automata, the predecessor of LBE, were first proposed by Frisch et al. [19]. The theoretical foundation of LBE was established in the subsequent papers, notably in McNamara and Zanetti [20], Higuera and Jimenez [21], Koelman [22] and Qian et al. [23]. A popular kinetic model adopted in the literature is the single relaxation time approximation, the socalled BGK model [24]: f t + ξ f = 1 λ (f f (eq) ) (1) where f (eq) is the equilibrium distribution function (the Maxwell-Boltzmann distribution function), and λ is the relaxation time. To solve for f numerically, equation (1) is first discretised in the velocity space using a finite set of velocities {ξ α } without affecting the conservation laws [15, 24,25], f a t + ξ a f a = 1 λ (f a f a (eq) ) (2) In the above equation, f α (x,t) f(x,ξ α,t)isthe distribution function associated with the α-th discrete velocity ξ α and f α (eq) is the corresponding equi- Figure 1: A 2-D, 9-velocity lattice (Q9D2) model librium distribution function. The 9-bit square lattice model, which is often referred to as the D2Q9 model (Figure 1), has been successfully used for simulating 2-D flows. In the D2Q9 model, we use e α to denote the discrete velocity set and we have e 0 =0, e α = c(cos((a 1)p/4), sin((a 1)p/4)) forα =1, 3, 5, 7, e α = 2c(cos((a 1)p/4), sin((a 1)p/4)) forα =2, 4, 6, 8 (3) where c = δx/δt, δx and δt are the lattice constant and the time step size, respectively. The equilibrium distribution for D2Q9 model is of the following form [23] [ f a (eq) = w a ρ + 3 c 2 e α u + 9 2c 4 (e α u) 2 3 ] 2c 2 u u where w α is the weighting factor given by 4/9, α =0 w α = 1/9, α =1, 3, 5, 7 1/36, α =2, 4, 6, 8. (4) (5) With the discretised velocity space, the density and 2

3 momentum fluxes can be evaluated as and ρ = ρu = f α = k=0 e α f α = k=1 k=0 k=1 f (eq) α (6) e α f (eq) α (7) The speed of sound in this model is c s = c/ 3 and the equation of state is that of an ideal gas, p = ρc 2 s (8) Equation (2) can be further discretised in space and time. The completely discretised form of equation (1), with the time step δt and space step e α δt, is: f α (x i + e α δt, t + δt) f α (x i,t) = 1 τ [f α(x i,t) f (eq) α (x i,t)] (9) where τ = λ/δt, and x i is a point in the discretised physical space. The above equation is the discrete lattice Boltzmann equation [20,21,26] with BGK approximation [24]. Equation (9) is usually solved in the following two steps where represents the postcollision state. The viscosity in the NS equation derived from equation (9) is ν =(τ 1/2)c 2 sδt (10) This modification of the viscosity formally makes the LBGK scheme a second-order method for solving incompressible flows [25,27]. The positivity of the viscosity requires that τ> 1 / 2. It is noted that the collision step is completely local and the streaming step takes very little computational effort. Equation (9) is explicit, easy to implement, and straightforward to parallelise. 1.2 Issues in the boundary condition treatment Like in any other fluid flow computations, the numerical boundary condition is a very important issue in the LBE method. Two classes of boundaries are often encountered in computational fluid dynamics: open boundaries and the solid wall. The open boundaries typically include lines (or planes) of symmetry, periodic cross-sections, infinity, and inlet and outlet. At these boundaries, velocity or pressure is usually specified in the macroscopic description of fluid flow. Unlike solving the NS equations where the macroscopic variables, their derivatives, or a well established constraint (such as the mass continuity for pressure distributions) can often be explicitly specified at boundary [28], in the LBE method, these conditions need to be converted into distribution function f. 2 COMPUTATIONAL ASSESSMENT In the computational cases to be presented and discussed, the following questions are addressed: Will the interpolation-based superposition scheme give the same accuracy as those of bounce back if both conditions lead to converged results? How does the inlet/free-stream boundary condition impact the convergence of the solution and the quality of the flow field? At the symmetric and periodic boundaries, the conditions for f α s can be given exactly. At the outlet of a computational domain, f α s can usually be approximated by simple extrapolation. In general, the boundary condition for f α s on a solid wall with a known velocity can only be given approximately. A popular approach is to employ the bounce-back scheme [29 31]. It is intuitively derived from lattice gas automata and has been extensively applied in lattice Boltzamnn equation simulations. The same idea for the solid wall has been directly extended to the inlet boundary with known velocity to provide the required inlet condition for f α s [31 34]. Let us consider a horizontal flow at an inlet (Figure 2). The first vertical grid line is at x A. The inlet is arbitrarily x I which is between lattice node A at x A and located at node B at x B. The node B at x B is the first fluid node next to the inlet. Using the bounce 3

4 based inlet condition results in slower convergence towards steady or dynamically steady state and causes stronger interaction between the inlet and interior field. Such interaction may affect the longterm behaviour of the solution, especially at higher Re. The interpolation-based superposition scheme can reduce the impact of the boundary-interior interaction on the unsteady development of the flow field, improves the convergence rate, improves the computational stability, and improves the quality of the overall solution. ACKNOWLEDGEMENTS Figure 2: Schematic of lattice nodes at inlet back on the link scheme [29], one typically requires that the known velocity profile u(x I ) be specified at x A e 1 dt with Ω = 0.5, where Ω is the fraction of a link in the fluid region intersected by inlet boundary, Ω= x B x I x B x A 0 Ω 1 (11) The standard bounce-back scheme for fᾱ(x A )atthe inlet x A gives: fᾱ(x A ) = f a (x A + eᾱδt) +2w α ρ(x A e ādt) 3 c 2 e ᾱ u(x(12) i ) for ᾱ = 1, 2, 8 and α = 5, 6, 4. We note that f ᾱx ( A ) in equation (12) depends on f αx ( A + eᾱδt) and ρ(x A + 1/2eᾱδt). Thus, it creates a mechanism for the pressure waves to propagate along the e α direction through f αx ( A + eᾱδt)and to reflect back to the interior of computational domain along the eᾱ direction through fᾱ(x A ). 3 CONCLUSIONS Computations based on the bounce-back based inlet condition and interpolation-based superposition condition are performed for various cases. Bounce-back The work reported in this paper has been partially supported by NASA Langley Research Center, with David Rudy as the project monitor. The authors thank Dr. Li-Shi Luo for many helpful discussions. The work reported in this paper has been partially supported by NASA Langley Research Center, with David Rudy as the project monitor. The authors thank Dr. Li-Shi Luo for many helpful discussions. REFERENCES 1 Alves, A.S. (Ed.) (1991) Discrete Models of Fluid Dynamics, Figueira da Foz, Portugal, September 19 22, 1990, World Scientific, Singapore. 2 Verheggen, T.M.M. (Ed.) (1992) Numerical methods for the simulation of multi-phase and complex flow, Proceedings of a Workshop held at Koninklijke/Shell-Laboratorium, Amsterdam, The Netherlands, May 30th June 1, 1990, Springer, Berlin. 3 Lebowitz, J.L., Orszag, S.A. and Qian, Y.H. (Eds.) (1995) J. Stat. Phys., Vol. 81, Nos. 1/2, pp.1 538, Proceedings of the 5th International Conference on Discrete Models for Fluid Mechanics, Princeton, New Jersey, June 27 29, Boghosian, B.M., Alexander, F.J. and Coveney, P.V. (Eds.) (1997) Int. J. Mod. Phys., Vol. 8, No. 4, pp , Proceedings of 1996 Conference on Discrete Models for Fluid Mechanics, Boston, Massachusetts, August 26 28, Boghosian, B.M. (Ed.) (1998) Int. J. Mod. Phys. C, Vol. 9, No. 8, pp , Proceedings of the 7th 4

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