
The densitybased solver solves the governing equations of continuity, momentum, and (where appropriate) energy and species transport simultaneously (i.e., coupled together). Governing equations for additional scalars will be solved afterward and sequentially (i.e., segregated from one another and from the coupled set) using the procedure described in Section 18.2. Because the governing equations are nonlinear (and coupled), several iterations of the solution loop must be performed before a converged solution is obtained. Each iteration consists of the steps illustrated in Figure 18.1.2 and outlined below:
These steps are continued until the convergence criteria are met.
In the densitybased solution method, you can solve the coupled system of equations (continuity, momentum, energy and species equations if available) using, either the coupledexplicit formulation or the coupledimplicit formulation. The main distinction between the densitybased explicit and implicit formulations is described next.
In the densitybased solution methods, the discrete, nonlinear governing equations are linearized to produce a system of equations for the dependent variables in every computational cell. The resultant linear system is then solved to yield an updated flowfield solution.
The manner in which the governing equations are linearized may take an "implicit'' or "explicit'' form with respect to the dependent variable (or set of variables) of interest. By implicit or explicit we mean the following:
In the densitybased solution method you have a choice of using either an implicit or explicit linearization of the governing equations. This choice applies only to the coupled set of governing equations. Transport equations for additional scalars are solved segregated from the coupled set (such as turbulence, radiation, etc.). The transport equations are linearized and solved implicitly using the method described in section Section 18.2. Regardless of whether you choose the implicit or explicit methods, the solution procedure shown in Figure 18.1.2 is followed.
If you choose the implicit option of the densitybased solver, each equation in the coupled set of governing equations is linearized implicitly with respect to all dependent variables in the set. This will result in a system of linear equations with equations for each cell in the domain, where is the number of coupled equations in the set. Because there are equations per cell, this is sometimes called a "block'' system of equations.
A point implicit linear equation solver (Incomplete Lower Upper (ILU) factorization scheme or a symmetric block GaussSeidel) is used in conjunction with an algebraic multigrid (AMG) method to solve the resultant block system of equations for all dependent variables in each cell. For example, linearization of the coupled continuity, , , momentum, and energy equation set will produce a system of equations in which , , , , and are the unknowns. Simultaneous solution of this equation system (using the block AMG solver) yields at once updated pressure, , , velocity, and temperature fields.
In summary, the coupled implicit approach solves for all variables ( , , , , ) in all cells at the same time.
If you choose the explicit option of the densitybased solver, each equation in the coupled set of governing equations is linearized explicitly. As in the implicit option, this too will result in a system of equations with equations for each cell in the domain and likewise, all dependent variables in the set will be updated at once. However, this system of equations is explicit in the unknown dependent variables. For example, the momentum equation is written such that the updated velocity is a function of existing values of the field variables. Because of this, a linear equation solver is not needed. Instead, the solution is updated using a multistage (RungeKutta) solver. Here you have the additional option of employing a full approximation storage (FAS) multigrid scheme to accelerate the multistage solver.
In summary, the densitybased explicit approach solves for all variables ( , , , , ) one cell at a time.
Note that the FAS multigrid is an optional component of the explicit approach, while the AMG is a required element in both the pressurebased and densitybased implicit approaches.