The subgrid scale model: allows to take into account small-scale physical processes (energy transfer from large scales to small ones through dissipation) that occur at length- scales that cannot possibly be resolved by the grid cell size. The SISM, presented here- after, is a modified version of the Smagorinsky subgrid scale model which belongs to the family of eddy viscosity models [159]. For the latters, underresolved turbulence is taken into account through the eddy viscosity νt(Boussinesq hypothesis) which is simply added
112 Appendix A : Industrial Solver - LaBS
such a way that it tends to zero at walls:
νtSISM = (Cs∆)2|S − S| S=S
−→
wall 0, (A.2.1)
where Sαβ = (∂αuβ + ∂βuα)/2 and S is computed here through a temporal moving
average of S i.e S = 1
N PN −1
n=0 S(t − n∆t).
The selective spatial filtering: consists in a spatial average of quantities of interest, and leads to a modified kinematic viscosity. The latter is then wavelength-dependent which allows to dissipate spurious small spatial scales (high wave numbers) keeping the largest ones (small wave numbers) unaffected. As an example, the more robust (and less costly) approach is based on the spatial filtering of macroscopic quantities [126]:
hρi= ρ(x) − σ D X α=1 N X n=−N dnρ(xα− nxα) hui = u(x) − σ D X α=1 N X n=−N dnu(xα− nxα) , (A.2.2)
where N is the number of points of the damping stencil (here N = 3 which leads to a 7-point stencil in each spatial direction), 0 ≤ σ ≤ 1 is the strength of the filter and dn = d−n are the damping coefficients (d0 = 5/16, d1 = −15/64, d2 = 3/32 and
d3 = −1/64 [126]).
The finite-difference boundary conditions: allows to reconstruct missing popula- tions using information from the macroscopic quantities [99]. Indeed, at the Navier-Stokes level of physics
f = feq+ fneq ≈ f(0)(ρ, u, T ) + f(1)(ρ, u, T, ∇(u), ∇(T )). (A.2.3)
Thus, macroscopic quantities (and their gradients) can be used to build distribution functions compliant (in the continuum limit 1) with the Navier-Stokes set of equations.
The inverse distance weighting method: is a spatial interpolation that estimates the unknown quantities of interest with a weighted average of the values available at neighbor points:
A(x) =X
k
γk(x)A(xk) with A= ρ, u, or T (A.2.4)
and γk(x) = d(x, xk)−p X k d(x, xk)−p (A.2.5) where d(x, xk) = q
(x − xk)2+ (y − yk)2+ (z − zk)2 is the euclidean distance between
A.3 3D extension of high-order RR-LBMs 113
allows to extend the above boundary conditions to curved ones.
Let us now continue with recursive regularized (RR) LBMs that have been studied as possible candidates for the simulation of fully compressible flows.
A.3 3D extension of high-order RR-LBMs
Thanks to the Hermite polynomial expansion framework, the 3D extension of the new recursive regularized (RR) LBM is straightforward. Hence the focus was put on the investigation of 3D third- and fourth-order LBMs, which were constructed following the preservation of the orthogonality properties of Hermite polynomials. Furthermore, it was chosen to only consider lattices with a minimal number of velocities, and as compact as possible, to not deteriorate too much HPC efficiency of the ‘Collide & Stream’ algorithm. Flowing from the work of Shan [120], two possible candidates were found for the simulation of isothermal flows without Mach number restrictions (third-order LBMs). They are both based on a 39-velocity discretization of the Boltzmann equation. Regarding the simulation of fully compressible flows, Shan also proposed to use the D3Q103 [120]. They are illustrated in Fig. A.1, and their characteristics are summarized in Tab.D.3.
In addition, the use of DDF models was adopted to further reduce the number of equations that need to be solved at each time step and for each grid point (see App. C
for theoretical details about DDF approaches). The coupling between the two high-order LBMs was also investigated. The Boussinesq coupling was successfully implemented and validated. Nevertheless, the choice regarding the ideal gas coupling remains an open question. Indeed, it can either be done implicitly through the equilibrium VDF [64], or explicitly changing the definition of the equation of state using a forcing term [8,163]. The last approach is the most stable one, but it introduces error terms that need to be corrected. This is even worse when second-order LBMs (D3Q19 or D3Q27 lattices) are considered for the DDF model.
The last step to further improve the efficiency of compressible LBMs is to replace the second (thermal) LBM by an energy equation discretized using standard numerical schemes. This is differed to future investigations.
Regarding boundary conditions, several layers of boundary conditions are required to properly reconstruct missing populations. Current models are based on regularized boundary conditions coupled with the inverse distance weighted interpolation.
114 Appendix A : Industrial Solver - LaBS • First Layer D3Q15 Second Layer D3Q26 + (0,±1, ±2)F S Third Layer D3Q14 + (±1, ±1, ±3)F S • First Layer D3Q15 Second Layer D3Q18 Third Layer D3Q6 • First Layer D3Q19 Second Layer D3Q14 Third Layer D3Q6
Figure A.1 – Illustration of 3D high-order LBMs of interest: D3Q103, D3Q39a/b lattices (from top to
bottom). They are all composed of three layers of velocities. If not otherwise stated, each component of the velocities belongs to ±1, ±2 and ±3 for the first, second and third layer respectively. D3Q6, D3Q14 and D3Q18 lattices correspond to D3Q7, D3Q15 and D3Q19 lattices where the velocity (0, 0, 0) has been discarded. Data are compiled from [120].
Appendix B
Background on multivariate Hermite
tensors
This appendix aims at providing the main definitions and properties of multivariate Hermite tensors (or polynomials), and their coefficients. They are first given in the continuous case related to the continuous velocity space, whereas the last part of this appendix highlights the main differences with the discrete case used after both the velocity space and the numerical discretizations. If not otherwise specified, all properties below are given for any vector in RD.