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Fig. 6. Treatment of boundary velocities. Although conceptually elegant, the staggered grid formulation can be an indexing nightmare for the programmer. This is where the spreadsheet solution becomes pedagogically useful. Students can set up the staggered grid in a visibly clear fashion on the spreadsheet, and use cell formatting o distinguish between u, v, and p grid points. Once he finite difference equations have been entered for one u, v, and p grid point, they can be simply copied to the remaining interior points.

Figure 6 Treatment of boundary velocities. Although conceptually elegant, the staggered grid formulation can be an indexing nightmare for the programmer. This is where the spreadsheet solution becomes pedagogically useful. Students can set up the staggered grid in a visibly clear fashion on the spreadsheet, and use cell formatting o distinguish between u, v, and p grid points. Once he finite difference equations have been entered for one u, v, and p grid point, they can be simply copied to the remaining interior points.

Related Figures (12)

‘ig. 1. Wiring harness bracket geometry and boundary conditions. The computational grid is superimposed on the geometry. For this grid, Ax = Ay = 1.25cm. If the circular reference solution capability of Excel is used, the finite difference equations must be solved explicitly for the nodal temperatures T; ; before entering them as cell formulas. This extra manipulation is not necessary when using Solver. In order to use Solver, two regions of the Discretization proceeds from either the govern- ing two-dimensional differential equation or by applying an energy balance on the region surrounding each node (a good description of these techniques is found in Incropera and DeWitt [8]). Using the energy balance approach,
‘ig. 1. Wiring harness bracket geometry and boundary conditions. The computational grid is superimposed on the geometry. For this grid, Ax = Ay = 1.25cm. If the circular reference solution capability of Excel is used, the finite difference equations must be solved explicitly for the nodal temperatures T; ; before entering them as cell formulas. This extra manipulation is not necessary when using Solver. In order to use Solver, two regions of the Discretization proceeds from either the govern- ing two-dimensional differential equation or by applying an energy balance on the region surrounding each node (a good description of these techniques is found in Incropera and DeWitt [8]). Using the energy balance approach,
Fig. 2. The two regions on the spreadsheet. The first region contains the temperature values Solver iterated on to satisfy the finite difference equations in the second region.
Fig. 2. The two regions on the spreadsheet. The first region contains the temperature values Solver iterated on to satisfy the finite difference equations in the second region.
Fig. 3. The Solver tool menu. The use of Solver for conduction problems provides students with an intuitive method for obtaining finite difference solutions. Its advantages over the circular reference approach become even more apparent when the staggered grid formula- tion for the Navier-Stokes equations is utilitzed, as described in the next section. Solver is then chosen from the ‘Tools’ menu, and
Fig. 3. The Solver tool menu. The use of Solver for conduction problems provides students with an intuitive method for obtaining finite difference solutions. Its advantages over the circular reference approach become even more apparent when the staggered grid formula- tion for the Navier-Stokes equations is utilitzed, as described in the next section. Solver is then chosen from the ‘Tools’ menu, and
Fig. 4. The driven cavity. A condition for pressure must also be specified to result in a determined system; it is sufficient to specify pressure at a point, for instance. For iterative solutions of the system of equations resulting from discretization, the specification of pressure at a point is implicitly done through initial guesses for variables at all grid points.
Fig. 4. The driven cavity. A condition for pressure must also be specified to result in a determined system; it is sufficient to specify pressure at a point, for instance. For iterative solutions of the system of equations resulting from discretization, the specification of pressure at a point is implicitly done through initial guesses for variables at all grid points.
Fig. 5. The staggered grid.
Fig. 5. The staggered grid.
Fig. 7. The two spreadsheet regions for the driven cavity problem. Cell formatting has been used to distinguish pressure and velocity cells. In the region on the left, cells outside the bold borders contain boundary conditions. interior grid points in the left grid subject to the constraints that the cells containing formulas in the right grid are equal to zero. When Solver has completed its iterations, the grid on the left contains the solution and the grid on the right is filled with zeroes. Consideration of Equations (6) through (8) will indicate why the spreadsheet circular reference function will not work for the discretization given above. For the circular reference function to work, an equation for the variable at each node must exist. If one were to solve Equation (7) for Uj+1/2,, and Equation (8) for v; j+1/2, that would leave Equation (6) for the node corresponding to pi; Equation (6), however, does not contain pressure! This quandary brings out the essence of the pressure problem for solution of the incom pressible flow equations. Matrix inversion could be used, but that requires additional manipulation of the finite difference equations in order to fill the operator matrix. The Solver function is an excel- lent tool that allows direct solution for the unknowns without muddying the waters with complicated preprocessing.
Fig. 7. The two spreadsheet regions for the driven cavity problem. Cell formatting has been used to distinguish pressure and velocity cells. In the region on the left, cells outside the bold borders contain boundary conditions. interior grid points in the left grid subject to the constraints that the cells containing formulas in the right grid are equal to zero. When Solver has completed its iterations, the grid on the left contains the solution and the grid on the right is filled with zeroes. Consideration of Equations (6) through (8) will indicate why the spreadsheet circular reference function will not work for the discretization given above. For the circular reference function to work, an equation for the variable at each node must exist. If one were to solve Equation (7) for Uj+1/2,, and Equation (8) for v; j+1/2, that would leave Equation (6) for the node corresponding to pi; Equation (6), however, does not contain pressure! This quandary brings out the essence of the pressure problem for solution of the incom pressible flow equations. Matrix inversion could be used, but that requires additional manipulation of the finite difference equations in order to fill the operator matrix. The Solver function is an excel- lent tool that allows direct solution for the unknowns without muddying the waters with complicated preprocessing.
Fig. 9. Comparison of spreadsheet results and results from reference [11] for Re=0. The figure shows wu along the cavity vertical centerline. A drawback in the Solver function is a relatively low limit on the number of constraints and chan- ging cells. The Solver function in the version of Excel used to calculate results presented here could not handle more than 100 unknowns. For the Because the pressure and stream function regions refer back to the solution, the contour
Fig. 9. Comparison of spreadsheet results and results from reference [11] for Re=0. The figure shows wu along the cavity vertical centerline. A drawback in the Solver function is a relatively low limit on the number of constraints and chan- ging cells. The Solver function in the version of Excel used to calculate results presented here could not handle more than 100 unknowns. For the Because the pressure and stream function regions refer back to the solution, the contour
Fig. 8. Grid showing stream function values at pressure grid points and relevant velocity values.
Fig. 8. Grid showing stream function values at pressure grid points and relevant velocity values.
Fig. 10. Streamlines for Re=0. Results for Re=100 are compared to those of Burggraf as reported in Goda [12] in Fig. 11. As expected with such a coarse grid, the results are not very close to those from Burggraf’s finer grid, but are qualitatively reasonable. The streamline plot in Fig. 12 shows the expected shift of the central vortex to the right as described, for example, in Ghia et al. [13]. The Solver tool in Excel provides an intuitive way for students to learn the fundamentals of computational fluid dynamics and heat transfer. A spreadsheet solution allows students to
Fig. 10. Streamlines for Re=0. Results for Re=100 are compared to those of Burggraf as reported in Goda [12] in Fig. 11. As expected with such a coarse grid, the results are not very close to those from Burggraf’s finer grid, but are qualitatively reasonable. The streamline plot in Fig. 12 shows the expected shift of the central vortex to the right as described, for example, in Ghia et al. [13]. The Solver tool in Excel provides an intuitive way for students to learn the fundamentals of computational fluid dynamics and heat transfer. A spreadsheet solution allows students to
Fig. 11. Comparison of spreadsheet results and results from reference [12] for Re= 100. The figure shows u along the cavity vertical centerline.
Fig. 11. Comparison of spreadsheet results and results from reference [12] for Re= 100. The figure shows u along the cavity vertical centerline.
Fig. 12. Streamlines for Re = 100. clearly conceptualize the interrelationships among the nodal variables in the discretization without the burden of having to write and debug code. The spreadsheet links between plots and cell values allows instant visualization of para- metric changes.
Fig. 12. Streamlines for Re = 100. clearly conceptualize the interrelationships among the nodal variables in the discretization without the burden of having to write and debug code. The spreadsheet links between plots and cell values allows instant visualization of para- metric changes.
Fig. 13. Student assessment of using Solver to solve the driven cavity problem. Possible scores were 5=strongly agree, 4= agree, 3=neutral, 2 = disagree, | = strongly disagree.
Fig. 13. Student assessment of using Solver to solve the driven cavity problem. Possible scores were 5=strongly agree, 4= agree, 3=neutral, 2 = disagree, | = strongly disagree.
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