About Equation Controls

Flow Property Equations

Equation controls temper the computational processes used to solve the discretized equations associated with the various flow properties. These equations are derived from the Navier-Stokes advection-diffusion equation, which takes the form:

\frac{\partial}{\partial t} (e ϕ) + \frac{\partial}{\partial x_{j}} (e u_{j} ϕ) = \frac{\partial}{\partial x_{j}} (k \frac{\partial}{\partial x_{j}} ϕ) + f_{i},

where

$ϕ$: is the flow property quantity (for example, pressure),
$e$: is a flow property coefficient,
$t$: is the time after each increment,
$x_{j}$: is the displacement of the quantity,
$u_{j}$: is the velocity at which the quantity is moving,
$k$: is the diffusivity of particle motion or heat transport, and
$f_{i}$: is the explicit body force.

Each term in the Navier-Stokes advection-diffusion equation has a physical meaning: transience, advection, diffusion, and body force. When you define your simulation information, you make choices that further define these terms to create a discretized form of the Navier-Stokes advection-diffusion equation. For example, if your particular simulation does not evaluate the development or degradation of flow over time (using a transient flow step), the transience term is removed from all flow property equations. If your simulation does not include a source or sink of a flow property quantity $ϕ$ , the explicit body force is removed from the corresponding flow property equation. Depending on the type of flow property you evaluate, the advection and diffusion terms may not be applicable to the flow property's equation. The table below details which terms are present for each of the discretized flow property equations.


Flow Property Equation	Advection	Diffusion
Coupled	—	—
Pressure	No	Yes
Momentum	Yes	Yes
Turbulence	Yes	Yes
Energy	Yes	Yes
Species	Yes	Yes
VOF	Yes	No
Radiation	—	—
Electromagnetics	No	Yes
Mesh Motion	—	—

Effect of Multispecies Diffusion on the Energy Equation

If your simulation includes multispecies sections, the app also computes the following diffusivity term as part of the energy equation:

- \nabla\cdot [T \sum_{j = 0}^{N} ρ_{0} C_{p j} D_{j} \nabla Y_{j}],

where

$j$: indicates one of the species in the multispecies section,
$0$: indicates the background species,
$N$: is the total number of species,
$Y_{i}$: is the mass fraction of the $j^{t h}$ species, and
$T$: is the temperature of the background fluid and the other species.

The discrete form of this equation, where $f$ indicates faces, is

- \sum_{f a c e s}^{n} T^{f} [\sum_{j = 0}^{N} ρ_{0}^{f} C_{p j}^{f} D_{j}^{f} \nabla Y_{j}^{f}] .

Linear Solver Controls

There are multiple types of linear solvers used to solve the various, applicable flow property equations, including:

Conjugate gradient (CG) solvers

Algebraic multi-grid conjugate gradient (AMG-CG)

Diagonally scaled conjugate gradient (DS-CG)

Symmetric successive over-relaxation conjugate gradient (SSOR-CG)
Flexible generalized minimum residual (FGMRES) solvers

Diagonally scaled flexible generalized minimum residual (DS-FGMRES)

Incomplete lower-upper factorization flexible generalized minimum residual (ILU-FGMRES)
Bi-conjugate gradient stabilized method (BiCGStab) solvers

Algebraic multi-grid bi-conjugate gradient stabilized method (AMG-BiCGStab)

Incomplete lower-upper factorization bi-conjugate gradient stabilized method (ILU-BiCGStab)

The CG solvers are the most effective solvers for resolving the flow property equations in which the matrices are symmetric-positive definite (for example, the pressure equation). The CG solver integrates the equation's diffusion term with respect to particle position. It then converts the flow property quantity $ϕ$ into matrix form for each cell in the mesh. Of the CG solvers available, AMG-CG is considered superior from a numerical linear algebra point of view.

The FGMRES solvers are effective for solving challenging problems with a large Krylov subspace. The FGMRES solvers may be more robust (that is, have a higher likelihood that the solver will converge on a solution) than other solver types, but they can be more computationally expensive. The FGMRES solver integrates the equation's advection and diffusion terms with respect to particle position. It then converts the flow property quantity $ϕ$ into matrix form for each cell in the mesh.

The BiCGStab solvers are superior to the FGMRES-type solvers in a number of ways. Like the FGMRES solvers, the BiCGStab solver integrates the equation's advection and diffusion terms with respect to particle position. It then converts the flow property quantity $ϕ$ into matrix form for each cell in the mesh.

There are additional linear solver controls to relax, filter, and smooth the convergence to minimize the time required to obtain accurate simulation results.

Under-Relaxation Factors

The under-relaxation factor, in general, has the largest impact on controlling the convergence rate of a steady-state simulation. You can manually specify the under-relaxation factor for each applicable flow property equation, or you can enable the app to calculate the under-relaxation factor for all applicable flow property equations. For more information, see About Under-Relaxation Factors.

Solid Energy Subcycling

Solid energy subcycling is a reduction in how often the linear solver computes the energy of the solids with respect to the energy of the fluids. By default, both the solid energy and the fluid energy are solved every iteration or increment using the energy equation. However, some CFD scenarios do not require repeated calculation of the solid energy, and doing so would prolong the simulation execution time. Enabling subcycling can, therefore, reduce the execution time by reducing the number of calculations, particularly for simulations with at least one of the following qualities:

The solid material's heat conductivity is orders of magnitude larger than the fluid material's heat conductivity. In this case, the solid material distributes the heat within a few iterations, while the fluid material requires many more iterations before there is a noticeable change in the heat distribution. Since the solid energy does not require repeated calculation past the first few iterations, enabling subcycling can significantly reduce the number of times the linear solver calculates the solid energy with respect to the fluid energy.
The number of solid section elements is of the same order of magnitude as the number of fluid section elements. In this case, the linear solver calculates the solid energy about as many times as the fluid energy. You are likely interested only in the results of the fluid section, so gratuitous calculations of the solid energy prolong the simulation time. Enabling subcycling allows the linear solver to skip these unnecessary solid energy calculations.