Richardson Extrapolation


This homework could get a little ugly, but I've chosen an example that illustrates the process without too many side calculations.  If I ask you to study sensitivity to the spatial mesh in a flow you might start with a volume:

            -------------------------
            | x         |
            -------------------------
and split it in two.  
            -------------------------
            | x |     x   |
            -------------------------
 The problem you've got is that answers for  state variables like temperature and pressure from the fine mesh never align with those for the coarser mesh.  To apply what you learned about Richardson extarpolation, you need to interpolate results on the fine mesh to get values aligned with those on the coarser mesh.  If you don't chose an interpolation method with an order of accuracy higher than the original difference method, you'll get results from a three mesh Richardson that tell you about your interpolation and not about the underlying difference method.

To avoid this complication I want you to study the spatial accuracy of the wall conduction solution.  It evaluates temperatures at the volume edges, so cutting the mesh spacing in half always leaves a fine mesh node aligned with each coarser mesh node.   The simplest conduction mesh has a temperature node at each surface of the metal:

            -------------------------
            x         x
            -------------------------
to split the mesh spacing in two, one temperature node is added in the middle:  
            -------------------------
            x x       x
            -------------------------
The next halving gives:
            -------------------------
            x x x     x   x
            -------------------------

The Problem:

A one meter long section of pipe has an inside diameter 0.25m. The of 0.01m thick wall is Inconel 600, and should be assumed to be perfectly insulated on its outer surface. Liquid enters at 300K and 50 m/s and exits to a pressure of 2.0e5 Pa.  Assume no wall friction or irrecoverable losses. The initial metal temperature is 400K.  The initial water temperature is 300K.  Use a Semi-Implicit method for the fluid and only one fluid volume.  Study the sensitivity to the radial conduction mesh spacing of the temperature at the insulated surface of the metal after 20 seconds of transient time.  Start with just one temperature node at each metal surface (2 temperature nodes, one metal volume), and refine the mesh spacing by factors of 1/2 until you can demonstrate a discretization error of less than 0.05K.  Use the three mesh Richardson extrapolation to predict the spatial order of accuracy (p), and apply that value of p to the standard Richardson error formula to predict error on your finest mesh.  After the you've got 3 mesh refinements, you will apply this procedure to the 3 finest meshes to estimate error, until you get a mesh with discretization error below 0.05K.  I recommend that you use a spreadsheet for these calculations.

With your final spatial mesh use the same procedure with fixed spatial mesh and successively halved time steps until you demonstrate the order of accuracy in time, and a time discretization error less than 0.05K for the outer surface temperature at 20 seconds.




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