Navier-Stokes Equations in a Bend of 180 Degrees.

The Problem.
The Navier-Stokes equations are given by

where
  • U1 is the velocity in the x-direction,
  • U2 is the velocity in the y-direction,
  • U3 is the vorticity.
The domain is a bend with the following 4 boundaries:
These conditions are assigned to the boundary pieces:

Use PDELab to ...

Why use PDELab?
 In this case study, we use the CADSOL Package (Copyright University of Karlsruhe 1992) to solve a system of Navier-Stokes equations on a bend of 180 degrees. CADSOL has been designed for the numerical solution of non-linear systems of 2D elliptic or parabolic equations, with non-linear boundary conditions on an arbitrary domain overlaid by a rectangular index grid.
The original CADSOL Package requires users to write fortran functions to define the systems of PDEs and boundary conditions. The Jacobians of the PDEs and boundary conditions must be computed by the user and defined via additional fortran functions. Users must also write a main driver program which allocates the necessary memory and calls the appropriate solver routines. A body-oriented grid must be defined via a fortran function, and solution output is the responsibility of the user.
PDELab simplifies the process of specifying a problem to CADSOL by providing The purpose of this case study is to demonstrate the ease of using the CADSOL solver within the PDELab environment to solve interesting problems.

Define the Problem
Step 1:When PDELab top level window appears, select 2-D and FDM and click on New File. The PDELab 2-D Finite Difference Method Session appears. The Session toolkit is to the right of the Session window.

Step 2: Click on the Framework Editor in the Session toolkit.

Step 3: Choose CADSOL from the Framework menu. The Framework Editor allows users to select the framework for defining the problem. In this case, selecting CADSOL tells the symbolic processor that Jacobians must be computed for the PDE System and for the boundary conditions which are entered by the user. The framework selection also signals the fortran code generator that CADSOL format functions must be generated.
Enter 3 for the Number of Equations and click on PDE System. Enter the Navier-Stokes equation in the PDE System editor as shown here. Click on Done when finished.

Step 4: Since CADSOL requires that the general domain specified by the user must be mapped to a rectangle, the number of boundary conditions for a CADSOL problem is 4. Therefore, the Number of Boundary Pieces field cannot be modified when the CADSOL framework is selected. Click on Boundary Conditions to enter the four sets of boundary conditions. Three equations are entered for each of the four boundary pieces in the mapped domain. Click on OK when finished.
Click on the Non-Linear Info button so that the specialized parameters for non-linear problems can be entered. These parameters are Tolerance, Maximum Number of Iterations, and the Initial Guess for the solution components U1, U2 and U3. We use the default values for all parameters, and use zero as the initial guess for all solution components. Click on OK when finished.

Step 5:The information for the PDE system and boundary conditions has now been entered. Click on Generate Framework. The Maxima symbolic processor is accessed for the Jacobian computations, and the GenCray fortran code generator generates the appropriate fortran functions for the CADSOL problem specification. Any input errors are tagged by Maxima and printed out in the Maxima window of the Framework Editor.

When the input has been successfully processed, select Quit from the File menu. The PDELab language program appears in the Session window. Click here to see a portion of this generated program. The entire program, including all symbolic computations and all CADSOL definition fortan routines, is generated automatically by PDELab. Look at the functions that define the PDE and boundary conditions (cdsu01, cdsu02), the functions that define the Jacobian (cdsu03, cdsu04), and the "guess" function. The call to the CADSOL solver appears as a PDELab triple. The default values for order (2), linear system solver method (10), and tolerance (10-4) appear as parameters to the CADSOL triple. Note that one of the CADSOL parameters requests PDELab format output . Additional output in the neutral format (for the mesh) and component format (for the solution values) will be generated automatically, unless otherwise specified.

Step 6: We will now specify the domain. We could draw the domain using the 2D BoundaryTool, but decide instead to write our own parameterization of the boundary of the domain directly in the Session window. Click on the bar at the top of the session window : Click here to edit session. Change the boundary segment from 

boundary. 
cadsol  on x = 0 
        on x = 1 
        on y = 0 
        on y = 1
to 
boundary.
cadsol  on x=cos(t),y=sin(t) for t= 0. to 3.1416
cadsol  on x=t, y=0 for t= -1.0 to -2.0
cadsol  on x=2.*cos(t),y=2.*sin(t) for t=3.1416 to 0.
cadsol  on x=t, y=0 for t= 2.0 to 1.0

Step 7: Click on the 2D BoundaryTool. The parameterized boundary we just defined is dynamically loaded into PDELab (assuming you had no typing errors in the boundary definition). If you click on Set Conditions, you will see that the conditions are set to "cadsol". This is correct! You have already specified the boundary conditions in the Framework Editor, and the functions describing those conditions are in the subprograms segment. Click on Cancel to dismiss the Boundary Conditions Editor.

WARNING! In the CADSOL framework, it is the user's responsibility to make sure that the numbering of the boundary conditions given in the Framework Editor corresponds to the numbering of the boundary pieces in the domain definition. Click on Show Piece # and check that the piece numbering does indeed correspond to the order in which the boundary conditions were given in the Framework Editor.

Specify the Solution
Step 8:We will now generate the body-oriented grid. Do not close the BoundaryTool; it must be up when the Grid Generator is invoked. Click on the GridGenerator. The domain boundary appears in the window with the piece numbers identified.

Click on Define Mapping. The default mapping of a four piece boundary to a rectangle is the Identity Map. Note that when your domain has more than four pieces, you must enter a correct mapping of those pieces to a rectangle. Click on Cancel to dismiss the Define Mapping editor, and click on Set Mesh Points. Pieces 1 and 3 will have 60 grid points, and pieces 1 and 3 will have 10 points (the default). Enter 60 for pieces 1 and 3 and then click on Continue.

We can now generate the body-oriented grid. Click on Generate Mesh. The resulting mesh is shown to the left. Click on Save to save the grid to a PDELab format file. A File Name dialog appears, and you must specify a name for the body-oriented grid file. Then click on OK to dismiss the dialog. A mesh segment will appear in the Session window. It is placed immediately after the boundary segment and indicates that a file containing the body-oriented grid now exists and will be used in the problem solution process. 

mesh.
   read structured from file &
        (filename='navier.grid', &
        i1mtyp=5, nx=11, ny=61, meshtype=5)
You can now close the Grid Generator by clicking on Quit. Then close the BoundaryTool by clicking on Quit.

Step 9: you can view the CADSOL solver parameters by bringing up the `triple' menu. Any CADSOL parameter that may be set by the user is listed here. You can view and/or modify the values for any of the parameters. Dismiss the Module Specification Editor by clicking on Cancel.

Step 10: You can save the PDELab language program to a `.e' file by clicking on the SaveAs button in the Session toolkit. A File Name dialog requests a name for the .e file.

Execute the Problem
Step 11: We are now ready to enter the ExecuteTool environment, where the PDELab language program will be processed. A fortran main program will be generated from the .e file. It will be compiled and linked with the PDELab module libraries and the standard PDELab I/O libraries. Click on the ExecuteTool button in the Session toolkit.

Step 12: The PDELab .e file is loaded automatically. Click on the Run button The trace of the process for compiling and linking the PDELab executable appears in the trace window. When the compiling and linking process is complete, the executable program is run and the solution and trance files are generated and placed in your user directory. Click on Quit to exit the ExecuteTool.

Visualize the Solution
Step 13: The solution output has been generated during execution. You should find the output file containing the PDELab format data in the location specified by the save segment. If no path was specified, this file is located in the execution directory. The file contains the solution and its derivatives. Click on the OutputTool in the Session toolkit. Select PDELab Solution in the File menu. Select the output file that was generated by your program when the OutputTool requests the file name. The Select Solution Box is displayed after the two files are loaded. The selection box contains the solutions for U1, U2 and U3.


Velocity in the X-direction.

Velocity in the Y-direction.

The Vorticity.

Step 14: Select U1 from the Select Solution Box. Then choose the Contour2D visualization package and press Run Tool. When the visualizer is displayed, select Render Contour from the View menu. The solution u is shown. Select File-Quit-OK to exit the tool. You can return to the OutputTool and from the Solutions Selection Panel under Solutions in the OutputTool menu bar, de-select U1 and select U2. In the same manner, use Contour2D to visualize U2. Select File-Quit-OK to exit the tool. Similary, view U3 with Contour2D.