Entropic Lattice Boltzmann Method (ELBM)

Lattice Boltzmann methods provide access to high Reynolds numbers through keeping low the Mach number and the Knudsen number as the two independent parameters of the simulation

Lattice Boltzmann methods(LBM) were introduced in the late 80’s – early 90’s as a new approach to CFD, and begun to find wide acceptance during the past decade. In LBM, one does not attempt a direct discretization of the governing fluid dynamics equations for mass, momentum and energy; instead, a kinetic equation of the Boltzmann type for a controlled number of discrete velocities is solved numerically on a regular grid. The entropic LBM is an advancement of LBM which satisfies the Second Law of thermodynamics (entropy of the system always increases).

The simplest Entropic lattice Boltzmann equation for the incompressible flow simulations can be understood with this example. Let {v}_i, i=1,\dots,Q be a set of discrete velocities representing links of a regular -dimensional lattice, and  f_i(x,t) be the populations of the velocities at the node  x at the discrete time t. Using the notion of the entropy H(f)=\sum_i f_i\ln(f_i/W_i) (weights W_i depend on the choice of the lattice), the equilibrium f_i^{\rm eq}(\rho,u) (analog of the Maxwell distribution) is derived as the minimizer of  H under fixed local (nodal) density \rho=\sum_i f_i and momentum \rho u=\sum_i f_i v_i. The entropic lattice Bhatnagar-Gross-Krook equation (ELBGK) describes the dynamics of the populations due to the free streaming of particles along the direction of the lattice links and the local relaxation to the equilibrium at the nodes:

f_i(x+v_i,t+1)-f_i(x,t)=\alpha\beta(f_i^{\rm eq}-f_i)

where  \beta is a parameter relates to the kinematic viscosity while function  \alpha maintains the entropy balance in the relaxation step at every grid node and is found as the non-trivial root of the equation (termed the entropy estimate)

\begin{equation}H(f+\alpha(f^{\rm eq}-f))=H(f) \end{equation}
Entropy estimate tells us that the entropy value should stay constant at the vanishing viscosity (\beta=1). This condition defines   \alpha as the maximal step of the over-relaxation without violating the Second Law (H-function decrease in the relaxation). Entropy estimate results in a confinement of the populations within the entropy contour during the relaxation, and leads to the unconditional stability of ELBGK. Observe that the entire nonlinearity (collision) in the ELBGK equation is on the right hand side, and is completely local in space, while the propagation in space (left hand side) is linear and exact. Furthermore, if the simulation is fully resolved, the entropy estimate leads to \alpha=2, and the ELBGK equation becomes its predecessor, the LBGK equation

f_i(x+v_i,t+1)-f_i(x,t)=2\beta(f_i^{\rm eq}-f_i)

With this stunningly simple formulation, the ELBM overcomes the stability problems of regular lattice Boltzmann method, while still retaining its locality, efficiency and flexibility.

With the proper choice of the lattice, the LBGK (or resolved ELBGK) recovers the Navier-Stokes equation with the kinematic viscosity, \nu=c_{\rm s}^2\left(\frac{1}{2\beta}-\frac{1}{2}\right), where  c_{\rm s} is the lattice speed of sound (a constant depending on the choice of the lattice), so that the relaxation parameter   \beta can be matched to the desired value. Note that the kinematic viscosity in the LBM formulation is independent of the time step which is one of the major findings of the method enabling it to reach low viscosity leading to high Reynolds numbers flow regimes.

While the LBGK cannot reach this limit due to disruptive numerical instabilities at the sub-grid scale, the ELBGK is unconditionally stable by respecting the Second Law of thermodynamics.