Issues associated with Galilean invariance on a moving solid boundary in the lattice Boltzmann method

Abstract

In lattice Boltzmann simulations involving moving solid boundaries, the momentum exchange between the solid and fluid phases was recently found to be not fully consistent with the principle of local Galilean invariance (GI) when the bounce-back schemes (BBS) and the momentum exchange method (MEM) are used. In the past, this inconsistency was resolved by introducing modified MEM schemes so that the overall moving-boundary algorithm could be more consistent with GI. However, in this paper we argue that the true origin of this violation of Galilean invariance (VGI) in the presence of a moving solid-fluid interface is due to the BBS itself, as the VGI error not only exists in the hydrodynamic force acting on the solid phase, but also in the boundary force exerted on the fluid phase, according to Newton's Third Law. The latter, however, has so far gone unnoticed in previously proposed modified MEM schemes. Based on this argument, we conclude that the previous modifications to the momentum exchange method are incomplete solutions to the VGI error in the lattice Boltzmann method (LBM). An implicit remedy to the VGI error in the LBM and its limitation is then revealed. To address the VGI error for a case when this implicit remedy does not exist, a bounce-back scheme based on coordinate transformation is proposed. Numerical tests in both laminar and turbulent flows show that the proposed scheme can effectively eliminate the errors associated with the usual bounce-back implementations on a no-slip solid boundary, and it can maintain an accurate momentum exchange calculation with minimal computational overhead.

A comparative study of immersed boundary method and interpolated bounce-back scheme for no-slip boundary treatment in the lattice Boltzmann method: Part I, laminar flows

“…With the use of bounce-back schemes, the natural way to calculate the hydrodynamic force and torque acting on a solid surface is the momentum exchange method [12,36,37]. Although the combinations of bounce-back schemes and momentum exchange method do not ensure the instantaneous Galilean invariance [38], their accuracy has been proven to be sufficient in most simulations [39,40]. In particular, the Galilean invariant momentum exchange method (GIMEM) proposed by Wen et al [37],…”

Unified directional parabolic-accurate lattice Boltzmann boundary schemes for grid-rotated narrow gaps and curved walls in creeping and inertial fluid flows

“…The violation of the Galilean invariance (VGI) of the LBM velocity solution is attributed either to (1) third-order momentum truncation [100]; (2) incorrect viscous stress [101,102] due to absence of its cubic-velocity correction by the quadratic NSE term E (u) q in Eq. (3b); (3) inaccuracy of the boundary scheme on a static solid surface subject to the tangential motion [103]; and (4) deficient reconstruction on the moving solid-fluid interface due to (i) mass-flux [8,79,83], (ii) "refill" of the new-born nodes [82,83,104], and (iii) momentumexchange algorithm [105], and also a combination of all these effects, where it has been understood [59] that the drag measured on the cylinder surface shows a much better agreement between the static and moving frames when the multireflection-based "refill" combines with the same accuracy-order modified momentum-exchange [59], which improves the standard force computation [2,4,55] for the stress approximation from the middle of the cut link towards the solid surface.…”

“…We then examine in Sec. VI E the GI property of the boundary schemes on the static walls following [103], but in the context of an exact, Galilean invariant grid-rotated bulk solution (16c). This study will show that the closure relation independence of the NSE term gradient ∂ q E (u) q is crucial for the mass-leakage reduction and the solution equivalence in the moving and static frames, when the Reynolds number grows with the tangential wall velocity.…”

“…We prescribe now a tangential velocity u x (y = ±h/2) = u w on the two channel walls and examine the Galilean invariance (GI) property for different boundary schemes. Following [103], the center-line velocity u c = u x (y = 0) is fixed, the steady-state velocity profile u x (y , u w ) and its mean value…”

“…A boundary scheme is considered to better satisfy the GI property when its accuracy in the wall-moving reference frame is less dependent on the wall velocity, i.e., when the velocity error-estimate takes nearly the same value in the static and moving coordinate system x -u w . Unlike in work [103], we then measure the error estimate in the moving frame for u x ( r)-u w with respect to the same no-slip solution u x ,0 (y ) = u x (y , u w = 0):…”