Ansys Forte employs implicit methods in solving the algebraic finite-volume equations that result from the differencing methods described in Discretization of the Governing Equations . The advantages of implicit methods are:

Specifically, Ansys Forte uses a modified version of the SIMPLE method [68] , which is a two-step iterative procedure used to solve for the flow field variables. Velocities at each time step in the numerical simulation need to be computed from time-advanced pressure gradients. This requires an iterative solution procedure, because the time-advanced pressures depend on accelerations and the velocities computed from the pressures. The SIMPLE method extrapolates the pressure, iteratively solves for velocities, then temperature, and finally the pressure.

Within the Ansys Forte 3-stage time-step advance procedure, Stage 2 involves selecting a predicted value of the pressure p _b . The pressure field is then frozen during solution of other flow quantities where diffusion terms are handled fully implicitly. Then the resulting values of the diffusion terms are frozen and we solve for the corrected pressure field using equations that difference the pressure terms implicitly. In the second step of this iteration Ansys Forte simultaneously solves the cell-face velocity equations, the volume change equations, and a linearized form of the equation of state. By algebraically eliminating the volumes and cell-face velocities from these equations in favor of the pressures, we are essentially solving a Poisson equation for the pressure in this step. Next, the predicted and corrected pressures are compared. If they agree to within a specified convergence tolerance, the equations have been solved, and we proceed to Stage 3. If the difference between the pressure fields exceeds the convergence tolerance, the corrected pressure field becomes the new predicted pressure field, and we return to the first step of the iteration and repeat the process. Each pass through the two steps is referred to as an outer iteration.

The mass fractions are used in the calculation of the Stage 2 pressure, but the values of the Stage 2 pressures and velocities do not influence the solution of the mass fractions. Thus the species diffusion terms are solved to update the species mass fractions before beginning the outer iteration This results in a considerable computational time savings over other schemes, such as those that calculate implicit convection during the SIMPLE iteration and which include the mass fraction equations in the outer iteration. We often have many chemical species (10s to 100s) in our combustion applications, such that solving species and mass equations in the outer iteration would greatly increase computational times.

For the Stage 2 calculation of k and ε , the flow field influences their values through the turbulence production terms after completion of the outer iteration. The finite-difference equations have been derived in such a way as to assure this one-way coupling. Mathematically, the values of k and ε influence the flow through the turbulent diffusivity and the turbulent pressure . This coupling could be accounted for by using Stage 2 values of k and ε to evaluate the turbulent diffusivity and but this would greatly increase computational times and is not necessary for stability. Furthermore, it is usually not necessary for accuracy when time steps are used that satisfy the stability constraints.

For the Ansys Forte implementation of the SIMPLE method, then, the only equations in the outer iteration are the momentum equation, internal energy equation, and the pressure equation. Because the equations for the species mass fractions, k and ε, are weakly coupled to the flow-field solution, these equations are not included in the outer iteration.