A MACINTOSH BASED CFD PROGRAM FOR ENGINEERING STUDENTS
DEVELOPED USING THINK C CLASS LIBRARIES
ABSTRACT
A two-dimensional computational fluid dynamics package called QUICK 'n SIMPLE was developed for use in undergraduate and graduate fluid mechanics courses. It combines both a graphical user interface for specification of the problem and an iterative control-volume finite-difference solution algorithm. The program was written for the Macintosh computer using Symantec THINK C with the object-oriented Think Class Libraries (TCL). The program solves the general problem of two-dimensional convection with diffusion of a passive scalar. It uses either the SIMPLE or SIMPLER coupling algorithms, and allows the user to select from three popular upwinding schemesΓÇö pure upwinding, power-law differencing scheme (PLDS), and the quadratic upstream interpolation for convective kinematics (QUICK) scheme.
INTRODUCTION
Modern computational fluid dynamics (CFD) tools have dramatically changed the way engineers in industry solve fluid flow and heat transfer problems. Unfortunately, many engineering students have not had any hands-on exposure to CFD. This is primarily because the complexity and training time required to use a full-featured CFD package is excessive for a typical engineering course. For this reason, we have developed a CFD program called QUICK 'n SIMPLE (QnS) which is extremely easy to use, yet powerful enough to give the student exposure to real fluid flow simulation.
There are two major challenges in developing such a program: 1) the creation of the user interface, and 2) the actual solution of the problem. The program was developed for the Macintosh because of its consistent and powerful graphical user interface (GUI) tools. The entire program is written with Symantec THINK C. The GUI portion was written using the object-oriented approach and taking advantage of the THINK Class Libraries (TCL), which handle the standard user interface items. The CFD solution algorithm was written for ANSI-C compatibility. In addition to incorporating the CFD solver in the QUICK 'n SIMPLE application, we also have a stand-alone version which runs on a UNIX workstation or supercomputer. This stand-alone version reads the problem specification file created by QUICK 'n SIMPLE and allows for speedy solution of large problems. The stand-alone version is optional, and the only limitations on solving problems directly on the Macintosh are dictated by the amount of installed memory on your computer, and the amount of time you are willing to wait for a solution.
In an effort to reduce the complexity of the user interface, the program has been confined to the solution of a single class CFD problems, namely two-dimensional convection with diffusion of a passive scalar. This constraint, along with the intuitive nature of the GUI, allows the user to learn QnS with a minimum amount of time investment. In fact, students with some prior experience with Macintosh computers can obtain simple results within the first hour of use. In the future, we plan to extend the program to solve more complex physical problems, such as those involving heat transfer and turbulent flow.
USING QUICK 'n SIMPLE
In defining the problem to be solved, there are two primary levels of operation: (1) problem specification mode, and (2) solution grid manipulation mode. Initially, the problem specification mode is used to define the geometry of the domain and the boundary conditions before the computational grid is generated. In this way the definition of the physical problem is separated from the definition of the grid. The grid, after all, is an artifact of the solution technique, and has nothing to do with the physics of the problem you are trying to analyze. If accuracy is important, then the solution results should be compared for a variety of different solution grids to evaluate the effect of the grid. Ideally the grid is refined to the point that further changes in the grid have no effect on the solution. When this point has been reached we say that we have "grid independence". Any further refinement of the grid merely requires additional computational resources with no net gain in accuracy.
The user specifies the problem by "drawing" it with tools similar to those found in modern computer-aided drafting software. Figure 1 shows an example of the Macintosh screen midway through the specification of a problem. The tools palette, located in the upper-left corner, determines how the cursor functions. There are three main sections: 1) arrow/zoom tool (top), 2) interior block/monitor /concentration source tool (left), and 3) wall boundary inflow/outflow/symmetry tool (right).
With the interior block tool selected, internal obstacles can be added by pressing the mouse button, dragging the mouse, and releasing the button. The object will be constrained to the closest reference grid line if the "Snap-to-Grid" menu item is selected. The blocks can be moved with the mouse, or their precise position can be specified by double-clicking the object, or by selecting the object and choosing the "Get Info" menu item.
Inflows and outflows are placed in the walls in a similar fashion, with the specific type of boundary determined by the currently selected wall boundary tool. QUICK 'n SIMPLE supports three types of inflow/outflow constraints ( constant, linear, and parabolic ) as well as a general unconstrained outflow condition. A symmetry boundary conditions for the flow field can also be specified by using the symmetry line tool.
The physical properties dialog box, shown in Figure 1, allows the user to specify the density and absolute viscosity or use the standard properties of air or water at a specified temperature. The system of units, also shown in this dialog box, may be selected, but it must be the same as those used in the specifications of the dimensions of the problem. The coefficient of diffusion for the passive scalar can also be specified. The default value is for the diffusion of CO2 gas in air at 20┬░C.
Once all of the objects and wall boundary conditions have been specified, the solution grid can be placed. This is done by selecting the "Generate Grid" menu item in the "Grid Pane" menu. Once the grid has been generated, the program enters the grid manipulation mode, and objects and boundary conditions can no longer be moved. If the user wishes to add an new object or change the position of an existing object, then the solution grid must be deleted, which reverts the program back to the problem specification mode.
The control-volume finite difference technique used in the solution of the problem requires that grid lines be located at the edges of all obstacles and inflow/outflows. This is
 
Figure 1 Tool Palettes and Physical Properties Dialog Box
automatically taken care of by the grid generation feature of QUICK 'n SIMPLE. The grid lines along the edges of objects and boundary inflow/outflows are fixed and represent constraints. The number and proportional spacing of the other grid lines can be interactively modified by the user. Grid manipulation is accomplished by defining grid "panes" which span the region between fixed grid lines. These grid panes may contain one or more control volumes. The number and spacing of the control volumes within any grid pane is adjusted by clicking the vertical or horizontal grid panes until it is highlighted. The grid lines within a grid pane are shown in the small scrolling windows to the left of the main window, as shown in Figure 2.
 
Figure 2 Grid Pane Selection If the user wishes to change the number or location of the grid lines within the selected grid pane, the user clicks the "Modify" button above the appropriate vertical "X-Grid" or horizontal "Y-Grid" listing.
HOW QUICK 'n SIMPLE SOLVES THE PROBLEM
Once the problem has been specified and the grid has been placed, the user selects "Save and Solve" from the "File" menu. This initiates a dialog box, shown in Figure 3, in which the user specifies the options for the numerical solution of the problem. In order to make intelligent choices with regard to these options, the user must understand how QnS arrives at a solution to the problem.
 
Figure 3 Setting the Solution Parameters
QUICK 'n SIMPLE uses an iterative control-volume finite-difference (CVFD) method to obtain a numerical solution to the equations governing steady, laminar, two-dimensional flow of an incompressible fluid and the conservation equation for a passive scalar. These equations are the conservation of mass and momentum of the fluidΓÇöthe continuity and Navier-Stokes equationsΓÇöand the generic advection-diffusion equation for the scalar. The scalar is passive in the sense that it does not affect the flow field. The flow field, however, will have a large effect on the scalar field. Therefore, we will always solve for the flow field first, and then solve for the scalar field. The solution of the scalar filed will normally be much quicker than the solution of the flow field.
QnS solves the flow equations in either Cartesian or axisymmetric coordinates. The governing equations in Cartesian coordinates are:
 
(1)
 
(2)
 
(3)
The conservation equation for the passive scalar is
 
(4)
Equations (1) through (4) are converted to a system of nonlinear, coupled, algebraic equations using the control-volume finite-difference method ( see, e.g., Patankar, 1980). To implement the above method, two decisions must be made : 1) how will the control volume interface values be interpolated from the values at the control volume centers, and 2) what method will be used to couple the continuity/momentum/ pressure equations.
The interface values are necessary in order to calculate the convective flux of the dependent variables between control volumes. Interface values must be computed from the nodal values at the center of each control volume. The type of interpolation affects both the accuracy and stability of the calculations. It is well known that upwind interpolation for the convective terms (left hand sides) in equations (2) through (4) eliminates spurious oscillations in the computed solution. Also, upwinding adds stability to solution if an implicit formulation is used and the discrete equations are solved iteratively. On the other hand, the types of upwinding that are most easily incorporated into finite-difference codes are also those with poor (first order in the mesh size) formal accuracy. To allow users to experiment with the affect that the upwind interpolation has on the accuracy of the finite-difference solution, QnS provides three different upwinding schemes: 1) pure upwinding, 2) the Power-Law Difference Scheme (PLDS), and 3) the Quadratic Upstream Interpolation for Convective Kinematics (QUICK) scheme (Leonard, 1979). The QUICK scheme is implemented with the deferred correction approach ( Zhu and Rodi 1991; Huang et al. 1985 ). This provides the necessary stability during the iterative solution of the governing equations.
QnS provides two means for coupling the continuity/ momentum/pressure equations: 1) the Semi-Implicit Method for Pressure Linked Equations ( SIMPLE ) algorithm, and 2) the SIMPLE-Revised ( SIMPLER ) algorithm (Patankar, 1980). Both SIMPLE and SIMPLER are iterative algorithms for resolving the coupling between equations (1) through (3). In both algorithms the u and u equations are artificially decoupled and linearized so these equations can be solved sequentially rather than simultaneously. An equation for the pressure is derived from the continuity equation. SIMPLE and SIMPLER differ in how the pressure field is iteratively updated, but both algorithms involve a correction step in which the u and u fields are adjusted to satisfy the continuity equation at the end of each global iteration. Several global iterations are required before the momentum and continuity equations are satisfied. The sequence in which the equations are solved, and the continuity-satisfying velocity correction, result in a solution to the coupled equations.
The SIMPLER algorithm requires more memory and more computations per iterations, but it generally converges in fewer iterations for an overall saving in execution time. Unlike the choice of upwinding described above, the accuracy of the converged solutions is unaffected by the u-u-p coupling algorithm chosen. Both algorithms converge to the same solution, and only the rate, and possibly the stability of the convergence, will be affected.
The user can control the convergence of the solution procedure by adjusting underrelaxation coefficients for the u, u and p equations. QnS provides default underrelaxation coefficients that should work for most problems. The dialog box shown in Figure3 allows the user to specify several parameters, including the total number of iterations, that control how the problem is solved.
VISUALIZING THE OUTPUT
QUICK 'n SIMPLE provides output in two forms : 1) a text file containing all of the velocity, pressure and scalar field data, and 2) separate data files for each field variable in a format compatible with Spyglass® Transform. Transform is a general purpose post-processing visualization program for two-dimensional arrays of data. It is available for Macintosh computers and UNIX workstations. ( Spyglass, Inc., 1800 Woodfield Dr., Savoy, IL 61874 (217) 355-6000).
EXAMPLE PROBLEM
Laminar flow over a backward-facing step was chosen as an example to demonstrate the program. This flow is interesting because even though the geometry is very simple, the flow structure is surprisingly complex. In addition, the characteristics of the resulting flow field are very sensitive to the Reynolds number. Figure 4 shows the streamlines predicted by QUICK 'n SIMPLE using the power-law differencing scheme for a inflow Reynolds number of 1000. This flow has been extensively studied, both numerically and experimentally, and there are a number of published papers detailing the accuracy of various computational methods for solution of this problem. We provide this result as a qualitative example of the kinds of problems which QUICK 'n SIMPLE is capable of solving.
 
Figure 4 Streamlines for flow over a backward-facing step : Re = 1000 - Upwinding : PLDS ( not to scale )
CONCLUSION
Our undergraduate curriculum spends a great deal of time on analytical rigor associated with fluid mechanics and the related discipline of heat transfer. Unfortunately, the student often leaves with a head full of equations, but very little insight or feeling for the nature of typical fluid flows. This lacking can be somewhat ameliorated by giving the student the computational tools to investigate some simple fluid flows.
Commercially available codes are difficult to integrate into the classroom because they typically cost several thousand dollars and require a significant investment in time spent learning to use the code. Furthermore, most commercial codes run only on UNIX workstations with which undergraduate students may be unfamiliar.
QUICK 'n SIMPLE is a two-dimensional CFD package which is appropriate for engineering educationΓÇöwhether the "student" be an undergraduate, graduate, or a practicing engineer. With its user-friendly graphical interface, one can easily study a variety of problems in laminar flow and mass diffusion.
In addition, QnS provides the user with a tool for comparing three popular upwinding interpolation schemesΓÇöpure, PLDS, and QUICK. Though it is widely accepted that the QUICK scheme is more accurate, the PLDS scheme is still widely used in practice. QnS provides a convenient way to study how the choice of interpolation scheme affects the computed flow field.
Version 1.0 of QUICK 'n SIMPLE is currently available as freeware. Please contact either of the authors (preferably by email) to request a copy. We expect to continue development of the product, and future versions will include heat transfer, turbulence modeling, direct linking with remote computers, and real time graphical display of the convergence of the solution.
REFERENCES
Hayase, T., J.A.C. Humphrey, and R. Grief, 1992. ΓÇ£A Consistently Formulated QUICK Scheme for Fast and Stable Convergence Using Finite-Volume Iterative Calculation Procedures,ΓÇ¥ J. Comput. Phys., vol. 98, pp. 108ΓÇô118.
Leonard, B.P. 1979. "A Stable and Accurate Convective Modelling Procedure Based on Quadratic Upstream Interpolation" Comp. Meth. in App. Mech. vol. 19, pp. 59ΓÇô98.
Patankar, S.V., 1980, Numerical Heat Transfer and Fluid Flow , Hemisphere Press, NY.
Zhu, J., and W. Rodi. 1991. ΓÇ£Zonal Finite-Volume Computations of Incompressible Flow,ΓÇ¥ Comp. in Fluids, vol. 20, no. 4, pp. 411ΓÇô420.
