Equilibrium distribution functions

For Navier-Stokes equations

The Equilibrium Distribution Functions (EDF) depend on the physical problem and the mathematical model to be simulated. For example a fluid flow problem will require to simulate the Navier-Stokes (NS) equations. For low-Mach NS version, the EDF is the standard one defined in Overview of Lattice Boltzmann Methods by Eq. (353). For incompressible NS version different EDF are implemented. They are summarized in Equilibrium distribution functions. For single-phase with unique constant density Eq. (192) is sufficient. Two-phase flows with two bulk densities, the EDF is either Eq. (197) or Eq. (206) with a dimensionless pressure. The choice of one of them implies two different corrections. For the first one a source term is added Eq. (204). For the second one, two force terms are added Eq. (213) and Eq. (214).

For transport equations

For scalar EDP, such as transport or heat equation or phase-field equation, the equilibrium distribution functions are different. One method to derive them has been presented in Transport equations.

Summary: equilibrium distribution functions are defined in each Kernels

Because the EDF depends on the mathematical model (number of EDP, scalar or vector, etc.) we must refer to the description of each Kernels. If the documentation is not updated, you must see the LBMScheme.h file. For example for NSAC_Comp model, the EDF are implemented in file LBMScheme_NS_AC_Comp.h (folder src/kernels/NSAC_Comp).

Section author: Alain Cartalade