FLUENT numerical solvers

FLUENT allows you to choose one of two numerical methods: [33]

• pressure-based solver

• density-based solver

In both methods the velocity field is obtained from the momentum equations. In the density-based approach, the continuity equation is used to obtain the density field, while the pressure

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field is determined from the equation of state. On the other hand, in the pressure-based approach, the pressure field is extracted by solving a pressure or pressure correction equation which is obtained by manipulating the continuity and momentum equations.

Using either method, FLUENT will solve the governing integral equations for the conservation of mass and momentum, and, when appropriate, for energy and other scalars such as turbulence and chemical species. In both cases a control-volume-based technique is used.

Pressure-based solver [33]

Two pressure-based solver algorithms are available in FLUENT. A segregated algorithm, and a coupled algorithm. These two approaches are discussed in the sections below.

The Pressure-based segregated algorithm

The pressure-based solver uses a solution algorithm in which the governing equations are solved sequentially (i.e., segregated from one another). Since the governing equations are non-linear and coupled, the solution loop must be carried out iteratively in order to obtain a converged numerical solution.

In the segregated algorithm, the individual governing equations are solved one after another for the solution variables. During the solution, each governing equation is decoupled from the other equations. The segregated algorithm is memory-efficient, since the individual equations need only be stored in the memory one at a time. However, the solution convergence is relatively slow, due to so many equations being solved in a decoupled manner.

In the segregated algorithm, each iteration consists of the steps illustrated in Fig. 3.3. and outlined below:

1. Update the fluid properties (e,g, density, viscosity, specific heat) using the turbulent viscosity (diffusivity) of the current solution.

2. Solve the momentum equations, one after another, using the recently updated values for pressure and face mass fluxes.

3. Solve the pressure correction equation using the recently obtained velocity field and the mass-flux.

4. Correct face mass fluxes, pressure, and the velocity field using the pressure correction obtained from Step 3.

5. Solve the equations for additional scalars, if any, such as turbulent quantities, energy, species, and radiation intensity using the current values of the solution variables.

6. Update the source terms arising from the interactions among different phases (e.g., a source term for the carrier phase due to discrete particles).

53 7. Check for the convergence of the equations.

These steps are continued until the convergence criteria are met.

Figure 3.3: Overview of the pressure-based solution methods The Pressure-based coupled algorithm

Unlike the segregated algorithm described above, the pressure-based coupled algorithm solves a coupled system of equations comprised of the momentum equations and the pressure-based continuity equation. Thus, in the coupled algorithm, Steps 2 and 3 in the segregated solution algorithm are replaced by a single step in which the coupled system of equations are solved. The remaining equations are solved in a decoupled fashion as in the segregated algorithm.

Density-based solver [33]

The density-based solver solves the governing equations of continuity, momentum, and, where appropriate, energy and species transport simultaneously (i.e., coupled together).

Update properties

Solve sequentially:

Uvel Vvel Wvel

Solve pressure-correction (continuity) equation

Update mass flux, pressure, and velocity

Solve energy, species, turbulence, and other scalar equations

Update properties

Update mass flux

Solve energy, species, turbulence, and other scalar equations

Solve simultaneously:

system of momentum and pressure-based continuity equations

Pressure-based segregated algorithm Pressure-based coupled algorithm

Converged Yes Stop

No No Converged Yes

Stop

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Governing equations for additional scalars will be solved afterward and sequentially, that is, segregated from one another and from the coupled set. Since the governing equations are non-linear 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 Fig. 3.4 and outlined below:

1. Update the fluid properties based on the current solution. (If the calculation has just begun, the fluid properties will be based on the initialized solution.)

2. Solve the continuity, momentum, and (where appropriate) energy and species equations simultaneously.

3. Where appropriate, solve equations for scalars such as turbulence and radiation using the previously updated values of the other variables.

4. When interphase coupling is to be included, update the source terms in the appropriate continuous phase equations with a discrete phase trajectory calculation.

5. Check for the convergence of the equation set.

These steps are continued until the convergence criteria are met.

Figure 3.4: Overview of the density-based solution method

In the density-based solution method, the coupled system of equations (continuity, momentum, energy and species equations if available) can be solved by using either the coupled-explicit formulation or the coupled-implicit formulation.

The manner in which the governing equations are made linear may take an "implicit'' or

"explicit'' form with respect to the dependent variable (or set of variables) of interest.

Update properties

Solve continuity, momentum, energy and species equations simultaneously

Solve turbulence and other scalar equations

Converged Yes Stop

No

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Implicit: For a given variable, the unknown value in each cell is computed using a relation that includes both existing and unknown values from neighboring cells. Therefore, each unknown will appear in more than one equation in the system, and these equations must be solved simultaneously to give the unknown quantities.

Explicit: For a given variable, the unknown value in each cell is computed using a relation that includes only existing values. Therefore, each unknown will appear in only one equation in the system, and the equations for the unknown value in each cell can be solved one at a time to give the unknown quantities.