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25.1.1 Pressure-Based Solver

The pressure-based solver employs an algorithm which belongs to a general class of methods called the projection method [ 58]. In the projection method, wherein the constraint of mass conservation (continuity) of the velocity field is achieved by solving a pressure (or pressure correction) equation. The pressure equation is derived from the continuity and the momentum equations in such a way that the velocity field, corrected by the pressure, satisfies the continuity. Since the governing equations are nonlinear and coupled to one another , the solution process involves iterations wherein the entire set of governing equations is solved repeatedly until the solution converges.

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 where the governing equations are solved sequentially (i.e., segregated from one another). Because 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 for the solution variables (e.g., $u$, $v$, $w$, $p$, $T$, $k$, $\epsilon$, etc.) are solved one after another . Each governing equation, while being solved, is "decoupled" or "segregated" from other equations, hence its name. The segregated algorithm is memory-efficient, since the discretized equations need only be stored in the memory one at a time. However, the solution convergence is relatively slow, inasmuch as the equations are solved in a decoupled manner.

With the segregated algorithm, each iteration consists of the steps illustrated in Figure  25.1.1 and outlined below:

1.   Update fluid properties (e,g, density, viscosity, specific heat) including turbulent viscosity (diffusivity) based on the current solution.

2.   Solve the momentum equations, one after another, using the recently updated values of 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., source term for the carrier phase due to discrete particles).

7.   Check for the convergence of the equations.

These steps are continued until the convergence criteria are met.

Figure 25.1.1: Overview of the Pressure-Based Solution Methods
figure



The Pressure-Based Coupled Algorithm


Unlike the segregated algorithm described above, the pressure-based coupled algorithm solves a coupled system of equations comprising 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.

Since the momentum and continuity equations are solved in a closely coupled manner, the rate of solution convergence significantly improves when compared to the segregated algorithm. However, the memory requirement increases by 1.5 - 2 times that of the segregated algorithm since the discrete system of all momentum and pressure-based continuity equations needs to be stored in the memory when solving for the velocity and pressure fields (rather than just a single equation, as is the case with the segregated algorithm).


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