Lattice Boltzmann simulations of 2D laminar flows past two tandem cylinders

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Abstract

We apply the lattice Boltzmann equation (LBE) with multiple-relaxation-time (MRT) collision model to simulate laminar flows in two-dimensions (2D). In order to simulate flows in an unbounded domain with the LBE method, we need to address two issues: stretched non-uniform mesh and inflow and outflow boundary conditions. We use the interpolated grid stretching method to address the need of non-uniform mesh. We demonstrate that various inflow and outflow boundary conditions can be easily and consistently realized with the MRT-LBE. The MRT-LBE with non-uniform stretched grids is first validated with a number of test cases: the Poiseuille flow, the flow past a cylinder asymmetrically placed in a channel, and the flow past a cylinder in an unbounded domain. We use the LBE method to simulate the flow past two tandem cylinders in an unbounded domain with Re = 100. Our results agree well with existing ones. Through this work we demonstrate the effectiveness of the MRT-LBE method with grid stretching.

Introduction

In recent years the lattice Boltzmann equation (LBE) has become a viable means for computational fluid dynamics (CFD) (cf. [1] and references therein). As opposed to conventional CFD methods based on direct discretizations of the Navier–Stokes equations, the LBE method is derived from the Boltzmann equation and kinetic theory [2], [3]. The kinetic origin of the LBE method differentiates it from conventional CFD methods in several ways. In the LBE method, one deals with the discretized particle velocity distribution functions {fi} instead of the hydrodynamic variables. Therefore, one must also deal with the boundary conditions for the distribution functions {fi} instead of that for the hydrodynamic variables. This kinetic nature of boundary conditions in the LBE method has some times caused confusions and the boundary conditions in the LBE are still a topic of active research.

The most popular LBE implementation of the Dirichlet boundary conditions on the flow velocity u or pressure p is the bounce-back (BB) boundary conditions (BC). When the bounce-back boundary conditions are coupled with the simple lattice Bhatnagar–Gross–Krook (BGK) collision model with one single relaxation parameter τ, the exact location where the Dirichlet boundary conditions for u or p are satisfied depends on the viscosity ν (or the relaxation parameter τ) [4], [5], [6], [7], [8]. This problem in the lattice BGK (LBGK) equation with the bounce-back BCs has generate numerous papers (e.g. [9], [10], [11], [12], [13], [14], [15]). Unfortunately, none of these works provides a rigorous analysis of or a systematic remedy to the problem due to inaccurate BB-BCs with the LBGK model, in spite of the fact that this problem can be analyzed [4], [5], [6], [7] and removed [16], [7], [8] by using the lattice Boltzmann equation with the multiple-relaxation-time (MRT) models [17], [18], [19], [20] and improved boundary conditions [16], [7], [8]. It should also be noted that asymptotic analysis [21], [22] has been used to address this problem in LBE [23].

In addition to flow-solid boundary conditions, LBE implementations of various inflow and outflow boundary conditions have been considered. Various approaches have been proposed previously in the context of the LBGK equation [9], [10], [13], [14]. In flow simulations, boundary conditions only specify the values of hydrodynamic variables at boundaries. However, hydrodynamic boundary conditions are insufficient for the lattice Boltzmann equation, because the distribution functions {fi} have non-equilibrium moments which are not specified by the values of hydrodynamic variables. Similarly, hydrodynamic initial conditions are insufficient to completely specify the LBE initial conditions but can be solved by an iterative procedure [24]. In this work we will demonstrate the inflow and outflow boundary conditions of either Dirichlet or Neumann type can be easily and consistently realized with the MRT-LBE method.

The LBE method usually employs uniform Cartesian meshes in both two and three dimensions. To be computationally efficient, non-uniform and adaptive meshes must be used. To this end, two approaches have been used in the LBE: the grid refinement [15], [25], [26] and the interpolated grid stretching [27], [2], [28]. In grid refinement, Cartesian meshes are used, and grid spacing is divided by an integer, which is usually a power of 2, to the next refined grid level. Interpolations are used at the interface between two connected meshes of different grid spacings, and at solid boundaries [29], [8]. With the interpolated grid stretching method, one can use body-fitted meshes [30], [31]. Except at flow-solid boundaries, interpolations have to be used throughout the entire mesh where the discretized distribution functions {fi} cannot be propagated exactly from one grid to another in the advection process. The simple bounce-back boundary conditions can be used with body-fitted meshes. We will use the latter approach, i.e., the interpolated grid stretching, in the present work.

In this paper we intend to use the LBE method to simulate laminar flows past two tandem cylinders in an unbounded domain in two-dimensions (2D). We shall restrict ourselves to the athermal (or isothermal) LBE models, in which the internal energy is not a conserved quantity. The main objective of this work is to investigate the effectiveness of the MRT-LBE with non-uniformly stretched grids for flow simulations. Implementations of inflow and outflow boundary conditions with the MRT-LBE method will also be validated through a number of test cases.

The remainder of this paper is organized as follows. We discuss the LBE method in Section 2, including descriptions of the MRT-LBE method, the interpolated bounce-back boundary conditions for arbitrary curved boundaries, inflow and outflow boundary conditions, force evaluation at flow-solid boundaries, and non-uniform grid stretching method. To validate the grid stretching method and inflow and outflow boundary conditions, we provide a number of test cases in Section 3. These test cases include: the Poiseuille flow, the flow past a cylinder asymmetrically placed in a channel at the Reynolds number Re = 20 and 100 [32], and the flow past a cylinder in an unbounded domain. In Section 4 we present the LBE results for the flow past two tandem cylinders in an unbounded domain with Re = 100 [33], [34], [35], [36], [37], [38], [39], [40]. We will compare our results with existing ones obtained with conventional Navier–Stokes solvers [33], [37], [38]. Finally we conclude the paper with Section 5.

Section snippets

Multiple-relaxation-time lattice Boltzmann method

We will use the lattice Boltzmann equation with the multiple-relaxation-time (MRT) collision model [17], [18], [20],f(xj+cδt,tn+δt)=f(xj,tn)-M-1·Sˆ·[m-m(eq)(ρ,u)](xj,tn),where ρ and u are the macroscopic density and velocity, respectively, the boldface symbols such as f denote Q-tuple vectors, and Q is the number of discrete velocities:f(f0,f1,,fQ-1)T,f(xj+cδt)(f0(xj),f1(xj+c1δt),,fQ-1(xj+cQ-1δt))T,m(m0,m1,,mQ-1)T,m(eq)(m0(eq),m1(eq),,mQ-1(eq))T,where T denotes the transpose operator.

We

Validation of the numerical method

In this section we validate our proposed approach to realize Dirichlet boundary conditions and non-uniform mesh with stretched grids. All the validations are carried out in two-dimensional flows. We first compare our proposed boundary conditions with the non-equilibrium bounce-back scheme for the Poiseuille flow and the results are presented in Section 3.1. Our second validation test for the boundary conditions is the flow past a cylinder asymmetrically placed in a channel with the Reynolds

Flows past two tandem cylinders at Re = 100

Our code is written in C++ with a open-source version of the Message Passing Interface library (MPICH). The numerical simulations presented in this work were carried out on cluster computers available to us at the Department of Computer Science, Old Dominion University (ODU) and Politecnico di Torino.

Conclusions

In this paper we use the lattice Boltzmann equation with multiple-relaxation-time collision model to simulate laminar flows in two dimensions. For the no-slip boundary conditions at flow-solid boundaries, we apply the interpolated bounce-back boundary conditions [29], [8]. For the inflow and outflow boundary conditions, we use a general bounce-back boundary conditions which can be easily, naturally and consistently realized with the MRT-LBE in particular. To enhance the computational efficiency

Acknowledgements

The authors are grateful to Dr. Meelan M. Choudhari of NASA Langley Research Center for suggesting this problem to them and for his insightful comments of this work, Prof. Manfred Krafczyk for bringing our attention the effects of the Mach-number on the results obtained by the LBE method, and Dr. Th. Zeiser for his careful reading of our paper and his critical comments. A.M. would like to thank Politecnico di Torino for financial support, and the Department of Mathematics & Statistics and

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