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Numerical Solution

Let's now investigate how we can achieve a solution in FLUENT. One must keep in mind that the governing equations we are attempting to find an approximate solution to are non-linear. This means that in order for a CFD program, such as FLUENT to solve it, it must go through an iterative process. This process is briefly described in the flow-chart below (click for a clearer image).
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figure 5.1 - solution process
 
Before we can start the solution process, we must provide FLUENT with some initial information. Firstly, we will be describing the solution method we want FLUENT to follow. The options can be found with:

Solution > Solution Methods

In this case, we are using the first order solvers. (If time permits, try using second order solvers and determine what kind of difference it makes to the convergence time and the final solution).

Before we begin, we must also set the convergence limit. The convergence limit tells FLUENT how close the number must be to 0 before we accept the result as a solution to the governing equations. Note there is a trade-off between computational time and the convergence criteria. Also, if the grid size of the mesh is large, having a very small convergence limit would not make any difference to the solution. Most of the inaccuracies would be generated by the large mesh. Keep this in mind when optimizing results by refining the mesh (if the mesh size is reduced, often the convergence limit also needs to be reduced to achieve a closer approximation to the exact solution).

In FLUENT we can adjust the convergence limit by:

Solution > Monitors > Residuals - Print, Plot > Edit...

For this example, we will use the default convergence criteria is of 0.001. Also make sure Plot box is checked. Click OK when the value has been inputted.

Next, we need to set an initial guess at the solution. Note the better or closer the guess to the converged solution, the less time it takes to compute. The initial guess / condition can be entered by:
Solution > Solution Initialization

get FLUENT to solve our nonlinear BVP. It'll introduce discretization and linearization errors in the process, as discussed in the Pre-Analysis step. We'll check the level of numerical errors later in the Verification & Validation step. There are lots of knobs in the Solution menu that you can twiddle to improve your numerical solution to the BVP. We'll not mess with most of these since the default settings yield an adequate numerical solution for our problem. We could get a slight improvement in accuracy by fiddling various knobs which we'll refrain from doing here.

Solution > Methods

The FLUENT solver converts our BVP to a set of algebraic equations through a process called discretization. We'll use second-order discretization for which the error is of the order of the square of the mesh spacing. This is more accurate (albeit less stable) than first-order discretization where the error is of the order of the mesh spacing.  Choose Second-Order Upwind for all equations as shown below. Set Pressure-Velocity Coupling to SIMPLE if it is not by default.

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To set the convergence criterion identified in the flowchart above , select:

Solution > Monitors > Residual

We see that we need to provide a convergence criterion for each PDE that is being solved. The solver will stop iterating when mass, momentum, energy, k and epsilon imbalances (called residuals) fall below the convergence tolerance. We'll use a residual tolerance of 10^-6 for all six PDE's being solved. FLUENT will consider the iterations have converged when all six residuals have fallen below this tolerance. Set the residuals tolerance as shown in the figure below. Make sure to scroll down and set the tolerance for k and epsilon equations also. 

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Also make sure Plot box is checked as shown above. This will help you monitor how/whether the solution is proceeding to convergence as the iterations are carried out. Click OK.

Next, we set the initial guess indicated in the flowchart. The initial guess can be entered using:

Solution > Initialization

We need to provide FLUENT with an initial guess for the flow variables (velocity, pressure etc.) to start the iterations. For this example, we know the conditions at the inlet of the pipe . Therefore, we can set the inlet conditions (except for pressure which is set to zero gauge by default). Initialize the entire flowfield to the specified values at the inlet: First, select Standard Initialization, then under Compute from, select Inlet and click Initialize.

This essentially assumes that the initial guess for the entire pipe area to be the inlet conditions. From here we can now use FLUENT to solve the problem.

Please refer back to (figure 5.1 - solution process). FLUENT solves this non-linear problem by taking an initial guess at the solution, solving the equations with the initial guess, comparing the convergence of the solution and if the criteria is not met, it will repeat the process over many iterations until the criteria is met. However, there are cases when the solution does not converge (this is usually the result of a mistake in any of the previous sections). To prevent the computer from iterating indefinitely, we need to set an iterations limit.

This can be done by:
Solution > Run Calculation 
Enter 500 for Number of Iterations and click Calculate. Note that 500 is the maximum number of iterations we want the solver to carry out. You will see a window message saying Calculating the solution... Wait for FLUENT to finish the calculation. Our solution converges in about 350 - 400  iterations. So the solver will stop the iterations before it reaches the maximum number of iterations specified (500). You should see a residual plot on screen as the computation is being performed. It should look something like this:

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below. At this point, we have a reasonable solution to the set of algebraic equations generated from the BVP using discretization. 

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Save project and exit FLUENT:

File > Save Project 

File > Close FluentNow that the computation is completed, we can go check out the results!

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