# An Accelerated Approach to Steady State for Confined Flow

It is well known that one of ** FLOW-3D**'s strengths is its accuracy in computing transient solutions to complex flow problems. However, there may be occasions where users may only need a steady state solution. Computing the transient result is time consuming and unnecessary in these situations. Fortunately, under certain circumstances, techniques can be applied which drastically accelerate the solution to a steady state. These techniques can only be applied to simulations of incompressible, confined flows. The steady state solution will be the same result as if the transient had been computed but far less computation will be required.

## Three settings are required to accelerate a solution to steady state

- SOR Pressure solver
- Multiplier for Dynamically Adjusted Convergence Criterion
- Relaxation Factor

### SOR Pressure Solver

First select the SOR pressure solver on the Model Setup, Numerics tab. Only the SOR pressure solver is guaranteed to reach a steady state solution.

### Multiplier for Dynamically Adjusted Convergence Criterion

Figure 1: Convergence controls dialog for Pressure solver options

Next, select the Convergence Controls button on the Numerics tab (see Figure 1) and set the Multiplier for Dynamically Adjusted Convergence Criterion to a large number (~106). The result will be that only one pressure iteration will be required each time step. This causes the solution to proceed through time quickly but has the result of making the transient solution meaningless.

### Relaxation Factor

Finally, set the Relaxation Factor to 1.0 on the Convergence Controls dialog (see Figure 1). This setting also allows the solution to proceed through time quickly without making unnecessary adjustments to the pressure.

## 2-D fluid flow between horizontal parallel plates 2.4 inches apart

An example problem will be studied next to demonstrate the techniques described above. The example problem consists of incompressible, confined flow between two plates. Two cases will be examined. In Case A, the full transient solution will be computed until a steady state is reached. In Case B, the steady state acceleration techniques described above will be applied.

The specifications of the simulation are:

Fluid: density = 1.9 slugs/ft3, kinetic viscosity = 0.002 ft3/s;

Mesh: cell size = 0.02 ft, length = 200ft, height = 0.2 ft;

Boundary: volume flow rate 1.2 slugs/s at left boundary, outflow at right boundary, walls at top and bottom boundaries;

## Case A (EPSADJ=1.0, OMEGA=1.0) Results:

In Case A, the Multiplier for Dynamic Adjustment of Convergence Criterion is set to 1.0 (default) which results in a typically small convergence criteria and a fairly large number of pressure iterations each time step (See Figure 2a). The simulation was run until the Mentor indicated that a steady state solution had reached after an Elapsed CPU time of 6.6e4 seconds.

Figure 2a. The simulation message when EPSADJ=1.0 and OMEGA=1.0.

Figure 2b. The pressure contour between x=90 ft and x=190 ft when EPSADJ=1.0 and OMEGA=1.0.

## Case B (EPSADJ=1.e6, OMEGA=1.0) Results

In Case B, the Multiplier for Dynamically Adjusted Convergence Criterion is set to 106 and the Relaxation Factor is set to 1.0.b. This results in a large convergence criterion and only 1 pressure iteration per time step. Note the column labeled CPU in Figure 3a below and compare it to the values in the CPU column in Case A. The solution of Case B proceeds very quickly through time, relative to Case A. The Mentor indicated that Steady State had been achieved after 8.8e3 seconds.

Figure 3a. The simulation message when EPSADJ=1.e6 and OMEGA=1.0.

Figure 3b. The pressure contour between x=90 ft and x=190 ft when EPSADJ=1.e6 and OMEGA=1.0

## Comparison of Computational Time

Figure 4: The computational time with EPSADJ=1.e6 (blue) and EPSADJ=1.0 (black).

## Conclusion

The pressure field results in Figures 2b and 3b indicate the pressure drop in steady state at 100 ft along the plates (midpoint of domain) is identical for both cases, 670 lbf/ft2, which is within 2% of the theoretical value 684lbf/ft2 (Frank M. White, Fluid Mechanics). Also, the Elapsed Time required to compute the steady state solution was reduced by a factor of 7.5.