1. Potential Flow and Computational Fluid Dynamics Numerical Analysis C8.3 Saleh David Ramezani BIEN 301 February 14, 2007

2. Problem Given: Consider plane inviscid flow through a symmetric diffuser as shown below. Only the upper half is shown. The flow is to expand from inlet half-width h to exit half width 2h. The expansion angle θ is 18.5˚. Given: Consider plane inviscid flow through a symmetric diffuser as shown below. Only the upper half is shown. The flow is to expand from inlet half-width h to exit half width 2h. The expansion angle θ is 18.5˚.

3. Problem Asked : Set up a non-square potential flow mesh for this problem, and calculate the plot (a) the velocity distribution and (b) the pressure coefficient along the centerline. Assume uniform inlet and exit flows. Asked : Set up a non-square potential flow mesh for this problem, and calculate the plot (a) the velocity distribution and (b) the pressure coefficient along the centerline. Assume uniform inlet and exit flows.

4. Assumptions Incompressible flow Incompressible flow Frictionless flow Frictionless flow Neglected gravity Neglected gravity Steady flow Steady flow

5. Free Body Diagram θ 2h L h V

6. Mesh Model We can make our mesh model with long and high rectangles. 5h 2h

7. Stream Function For a non-square mesh use equation 8.108 to find the stream function. For a non-square mesh use equation 8.108 to find the stream function.where

8. Stream Function Simplify to get:

9. Boundary Conditions Boundary values are not given. Boundary values are not given. Assume your own stream boundary values along the walls. Assume your own stream boundary values along the walls. Choose 100 m 2 /s along the top wall, and 0 along the lower wall. Choose 100 m 2 /s along the top wall, and 0 along the lower wall. Use Excel to iterate for stream function nodal values. Use Excel to iterate for stream function nodal values.

10. Stream Function Nodal Values

11. Velocity Velocity at any point in the flow can be computed from equation 8.107: Velocity at any point in the flow can be computed from equation 8.107: Using Excel and our previously computed Stream nodal values we can find the velocity nodal values. Using Excel and our previously computed Stream nodal values we can find the velocity nodal values.

12. Velocity Nodal Values

13. Pressure Coefficient Pressure coefficient can be computed from Bernoulli’s equation. Pressure coefficient can be computed from Bernoulli’s equation. The simplified form of this equation is found in example 8.5 of the textbook. The simplified form of this equation is found in example 8.5 of the textbook.

14. Pressure Coefficient V was previously computed from stream function. V was previously computed from stream function. V 1 is the velocity at the entrances found from the stream function (here 200 m 2 /s) V 1 is the velocity at the entrances found from the stream function (here 200 m 2 /s)

15. Pressure Coefficient Nodal Values

16. Stream Distribution

17. Velocity Distribution

18. Pressure Coefficient Distribution

19. Biomedical Application Not all blood vessels or passage ways in our body have a uniform thickness or shape. In fact most of them are characterized by complicated geometries. The most obvious biomedical application of this problem is the numerical analysis of velocities or flow rates through those more complex shaped passage ways. Not all blood vessels or passage ways in our body have a uniform thickness or shape. In fact most of them are characterized by complicated geometries. The most obvious biomedical application of this problem is the numerical analysis of velocities or flow rates through those more complex shaped passage ways.