Essentially Entropic Lattice Boltzmann Model

The entropic lattice Boltzmann model (ELBM), a discrete space-time kinetic theory for hydrodynamics, ensures nonlinear stability via the discrete time version of the second law of thermodynamics (the $H$ theorem). Compliance with the $H$ theorem is numerically enforced in this methodology and involves a search for the maximal discrete path length corresponding to the zero dissipation state by iteratively solving a nonlinear equation. We demonstrate that an exact solution for the path length can be obtained by assuming a natural criterion of negative entropy change, thereby reducing the problem to solving an inequality. This inequality is solved by creating a new framework for construction of Padé approximants via quadrature on appropriate convex function. This exact solution also resolves the issue of indeterminacy in case of nonexistence of the entropic involution step. Since our formulation is devoid of complex mathematical library functions, the computational cost is drastically reduced. To illustrate this, we have simulated a model setup of flow over the NACA-0012 airfoil at a Reynolds number of $2.88\times {10}^6$.

Figure 1

The entropic involution step: $H_i$ are different entropy levels ($H_1>H_2>H_3$). Triangle denotes the polytope of positivity. Note that the precollisional state $f$ and the mirror state $f^{\text{mirror}}$ are at the same entropy level. The postcollisional state $f^{*}$ is at a lower entropy level.

Figure 2

Residue related to the approximations of $\log (1+y)$ evaluated with the trapezoid rule, $\frac{1}{3}$ Simpson’s rule, Boole rule, i.e., the first-, second-, and third-order Newton-Cotes quadratures, respectively. Here $P_n$ are the integrating polynomials. The residues are positive and their magnitude progressively decreases with increase in the order of quadrature.

Figure 3

Density, velocity, and entropy plots from the BGK model, i.e., $\alpha =2$ (left column), Eq. (10) (right column) at time $t=500$ for viscosity $\nu =1.0\times {10}^{-5}$. At $t=0$, domain was initialized with step density as $\rho (x<0)=1.5$ in the left half of domain and $\rho (x>0)=0.75$.

Figure 4

Snapshot of $\alpha$ at the compressive shock front from Eq. (10) for the shock tube simulation.

Figure 5

Flow over the NACA-0012 airfoil.