Advanced lattice Boltzmann scheme for high-Reynolds-number magneto-hydrodynamic flows

Abstract

To cite this version:Alessandro de Rosis, Emmanuel Lévêque, Robert Chahine. Advanced lattice Boltzmann scheme for high-Reynolds-number magneto-hydrodynamic flows.ABSTRACT 11 Is the lattice Boltzmann method suitable to investigate numerically 12 high-Reynolds-number magneto-hydrodynamic (MHD) flows? It is shown 13 that a standard approach based on the Bhatnagar-Gross-Krook (BGK) collision 14 operator rapidly yields unstable simulations as the Reynolds number increases. In 15 order to circumvent this limitation, it is here suggested to address the collision 16 procedure in the space of central moments for the fluid dynamics. Therefore, an 17 hybrid LB scheme is introduced, which couples a central-moment scheme for the 18 velocity with a BGK scheme for the space-and-time evolution of the magnetic field. 19This method outperforms the standard approach in terms of stability, allowing us 20 to simulate high-Reynolds-number MHD flows with non-unitary Prandtl number 21 while maintaining accuracy and physical consistency. 22 KEYWORDS 23 Lattice Boltzmann method, magnetohydrodynamics, turbulence 24The use of the lattice Boltzmann (LB) method has become ubiquitous in many 25 areas of computational fluid dynamics, and now represents a consolidate alternative 26 to classical approaches based on the discretization of the incompressible Navier-Stokes 27 equations [1][2][3][4][5][6][7][8]. In short, the flow is inferred from the motion of distributions 28 of fictitious particles streaming and colliding along the links of a regular lattice. 29The LB method has practical advantages with respect to a continuum-based 30 formulation. In particular, LB dynamics is governed by a first-order partial differential 31 equation in which non-localities and non-linearities are well separated [5]. Conversely, 32 the integration of the Navier-Stokes equations requires the evaluation of first 33 and second-order derivatives, and possibly the application of a non-local Poisson 34 solver to obtain the pressure field. Moreover, the computational complexity of the 35 continuum-based approach becomes rapidly prominent and evident when the fluid 36 dynamics encompasses additional physical features such as magnetic effects. In that 37 case, the particulate nature of the LB approach offers some tangible advantages, as 38 will be demonstrated in this article. 39 The incompressible Navier-Stokes equations for magnetohydrodynamics (MHD) 40 drive the evolution of an electrically conductive fluid of kinematic viscosity ν and 41 CONTACT A. De Rosis. Email: derosis.alessandro@icloud.com magnetic diffusivity η in the form 42where ρ and u are the mass density and velocity of the fluid, respectively. The vector 47 field b denotes the magnetic field and j = ∇ × b is the electric current. The fluid 48 pressure p stems from the incompressibility constraint ∇ · u = 0. In comparison with 49 the non-magnetic case, here it is mandatory to integrate a coupled set of non-linear 50 partial differential equations for the velocity and magnetic fields, thus leading to heavy 5...

Role of higher-order Hermite polynomials in the central-moments-based lattice Boltzmann framework

“…As a consequence, we proved that it is possible to derive central-moments-based schemes for a broad range of governing equations, as preconditioning [37] and shallow water equations [38]. A compelling proof of the generality of the algorithm is provided in [39], where a central-moments-based scheme able to recover the solution of the incompressible Navier-Stokes equations for magnetohydrodynamics has been derived. If one compares the algorithm outlined in [39] with the nonconductive case [33], it is possible to appreciate that the structure of the algorithm is identical.…”

“…A compelling proof of the generality of the algorithm is provided in [39], where a central-moments-based scheme able to recover the solution of the incompressible Navier-Stokes equations for magnetohydrodynamics has been derived. If one compares the algorithm outlined in [39] with the nonconductive case [33], it is possible to appreciate that the structure of the algorithm is identical. The only difference lies in the equilibrium central moments, that are enriched by terms accounting for the magnetic field.…”

Universal formulation of central-moments-based lattice Boltzmann method with external forcing for the simulation of multiphysics phenomena

“…Its simulation represents a hard task since smaller and smaller structures appear in the domain as the Reynolds number grows. This test case is then a perfect candidate to assess (1) the accuracy of the present method to capture the existence of fine flow features, as well as ( 2) its ability to deal with high local gradients that arise during the simulation 66 .…”

“…Consequently, the proposed approach allows the derivation of CM-LBMs for any lattice of discrete velocities in a straightforward manner. This was thoroughly demonstrated by successfully recovering different sets of governing equations with this approach, hence allowing the simulation of a rich variety of physics problems such as shallow waters 65 , magnetohydrodynamic 66 , and multicomponent flows 67 among others.…”

Multiphysics flow simulations using D3Q19 lattice Boltzmann methods based on central moments

“…More recently, it was demonstrated that it is possible to adopt a CMs-based procedure where moments relax to whatever discrete equilibrium state and for whatever lattice discretization [22][23][24][25][26] . It is possible to switch from populations to CMs (and vice versa) by simply multiplying (or dividing) by a transformation matrix that depends on the adopted lattice and the local fluid velocity.…”