Issues associated with Galilean invariance on a moving solid boundary in the lattice Boltzmann method Cheng Peng, Nicholas Geneva Department of Mechanical Engineering, University of Delaware, Newark, DE, 19716-3140, USA Zhaoli Guo State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan, P.R. China Lian-Ping Wang Department of Mechanical Engineering, University of Delaware, Newark, DE, 19716-3140, USA and State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan, P.R. China In the lattice Boltzmann method (LBM), it is well known that the hydrodynamic force evaluation based on the mesoscopic momentum exchange method often fails to satisfy the Galilean invariance. While multiple methods have been proposed in the past to restore Galilean invariance of the evalu- ated hydrodynamic force acting on the solid boundary, the related Galilean-invariance property of the fluid particle bounce-back distributions near the solid boundary has so far been largely ignored or unnoticed. In this work, we first illustrate this problem through a simple physical analysis, then demonstrate through numerical simulations that this problem can severely contaminate LBM re- sults in some cases. Several LBM simulations are presented to show how this problem can render a simulation inaccurate when the entire flow is subjected to a moving reference frame. This reveals the incompleteness of previous modifications on the momentum exchange method. To address this issue, a new bounce-back scheme based on coordinate transformation is proposed. Numerical tests in both laminar and turbulent flows show that the new scheme can effectively eliminate the errors associated with the usual bounce-back implementation on a no-slip solid boundary and maintain an accurate momentum exchange calculation with minimal computational overhead. PACS numbers: 01.55.+b, 01.65.+g 2 I. INTRODUCTION In the past forty years, the lattice Boltzmann method (LBM) has been developed into an viable alternative for solving the Navier-Stokes (N-S) equations governing viscous fluid flows. An important reason for its popularity is its apparent simplicity for handling the no-slip boundary condition within complex geometries, which makes it extremely suitable in simulating particle-laden flows [1{4] and flows through porous media [5{7]. In such simulations, the no-slip boundary condition on moving or fixed (typically curved) solid surfaces is usually realized by a bounce-back scheme, which was first proposed in a simple direct bounce back form [1] and later modified to achieve at least a second-order accuracy on curved boundaries [5, 6, 8, 9]. Also inherited from the gas kinetic theory, the hydrodynamic force acting on a solid body in a LBM simulation can be evaluated by summing the net momentum change between the incident and reflected fluid particles at the boundary nodes during the bounce-back process. This mesoscopic treatment is known as the momentum exchange method (MEM) [1, 10], which is computationally more efficient than its macroscopic counterpart, namely, the stress integration method (SIM), as demonstrated in [10, 11]. While bounce-back schemes together with MEM allow different complex geometries to be implemented in LBM, the existence of potential violation to the Galilean invariance (VGI) in these methods has been exposed both theoretically and numerically [12{16]. By the VGI error, we mean that the physical result of the flow, such as the hydrodynamic force, could change if a constant translational velocity is added to the entire system. Although it is well known that the Galilean-invariant N-S equations can be derived from the LBM scheme via different methods, e.g., Chapman- Enskog analysis [17], asymptotic expansion [18, 19], and linear analysis [20], these derivations often do not consider the presence of a solid boundary. In special cases such as the laminar channel flow, the numerical errors due to a bounce-back scheme can be analyzed theoretically [6, 21]. In general it is very difficult to theoretically study the numerical errors due to a bounce-back scheme alone. Although numerical errors for laminar benchmark flows such as a channel flow or cavity flow are routinely gathered, these errors could contain both the boundary-implementation error and the inherent discretization error within LBM. In order to correct the VGI error associated with the treatment of a solid boundary, several different solutions have been proposed to modify the MEM to achieve Galilean invariance (GI) in the calculation of the hydrodynamic force acting on the solid boundary. To our knowledge, Caiazzo & Junk were among the first to realize the lack of Galilean invariance in the MEM and proposed a corrected MEM by subtracting the link-wise VGI error directly on each link [12]. Later, Clausen & Aidun (hereafter CA09) presented a much more straightforward correction that acts in a node-wise manner for an arbitrarily orientated surface [13]. The link-wise correction intends to remove the VGI error on each link; it is a brute-force treatment that ensures the error be eliminated on all boundary links. The main drawback of the link-wise correction is that it is debatable whether such a correction is over constrained, since our goal is to achieve GI on each node, not on each link. The node-wise correction on the other hand is designed to remove the overall VGI error on each boundary node. Compared to the link-wise treatment, it appears to be more physical, but the major issue for such node-wise correction is that its performance may be affected by the boundary link configuration. Another problem for both correction methods is that they require the errors be formulated and calculated beforehand. The specification of the VGI error involves certain assumptions, so these corrections may not be accurate. Chen et al. [14] proposed a corrected MEM (hereafter Chen13) to incorporate the impulsive force [2] when a solid node becomes a fluid node, or vice versa; Chen13 essentially formulated a continuous correction term to replace an impulsive force (see Section 2.3 for further details). Their treatment has later been interpreted as a link-wise correction that essentially achieves the similar node-wise correction in certain ideal cases [22]. More recently, Wen et al. designed a Galilean-invariant momentum exchange method (GIMEM, hereafter Wen14), where the lattice fluid velocity relative to the wall was used to realize node-wise Galilean invariance [15]. Due to the incompleteness of this scheme later discussed in Section II, this method may not work well in all cases. However, since this formulation does not require explicit computation of the VGI error, which is usually obtained based on the assumption that the system is fully relaxed, this method could potentially work better under insufficient grid resolutions. A similar correction was proposed by Krithivasan et al. around the same time [16]. Numerical simulations of both laminar and turbulent particle-laden flows indicate that the above corrected force evaluation schemes of CA09, Chen13, and Wen14 lead to physically correct particle motions [15, 22, 23]. Further details of these methods are discussed in Wen et al.'s review article [24]. The above modified momentum exchange schemes provide reasonable corrections to the VGI error when the hy- drodynamic force acting on the solid boundary or a suspended solid particle is of concern. However, they should not be regarded as a complete fix to the VGI error. The incompleteness is easily seen by simply considering the Newton's Third law. Namely, if there exists a VGI error in the hydrodynamic force acting on the solid phase, there must be a corresponding VGI error in terms of interaction force experienced by the fluid phase, with the latter being equal and opposite to the former. This rather obvious and critical problem remains essentially untouched by all the proposed corrections listed above. As we will discuss in detail later in this paper, for some cases, the VGI error on the fluid may be corrected impulsively through the momentum gain/loss when a given node switches phase, namely, a fluid node 3 becomes solid or vise versa. However, in many cases in which the impulsive fix does not occur, e.g., the case where the wall moves along a fixed line (in 2D) or surface (in 3D) relative to the lattice nodes, the VGI error on the fluid phase is left uncorrected. Furthermore, the one-way correction on the hydrodynamic force can lead to the violation to the Newton's Third Law. We will discuss this issue in detail in Sec. II. The rest of this paper is arranged as follows. In Sec. II, we shall analyze the VGI error on both the solid and fluid phases and compare the corresponding physical mechanisms on the corrections. For the solid phase, three representative modified momentum exchange schemes proposed by CA09, Chen13 and Wen14, will be further analyzed in detail. The VGI error for the fluid phase and the potential remedies that currently exist will then be thoroughly discussed. We will also point out situations in which these remedies could fail, leading to inaccurate physical results. Then in Sec. III, a new boundary scheme is presented to maintain GI for both the fluid and solid phases. The proposed scheme is validated in Sec. IV, using a Poiseuille flow between two inclined walls and a turbulent channel flow. Finally, the contents of this work are highlighted in Sec. V with a summary of key conclusions. II. ANALYSIS OF DEVIATIONS FROM THE GALILEAN INVARIANCE IN LBM In LBM, the mesoscopic fluid-particle distribution function fi (x; t) is governed by fi (x + ei; t + δt) − fi (x; t) = Ωi; (1) where i refers to a discrete particle velocity ei, x and t are discrete node and time, respectively, δt is the time step size.