1. Von Karman Integral Method (BSL) PRANDTL BOUNDARY LAYER EQUATIONS for steady flow are Continuity N-S (approx) 12  If we solve these, we can get V x, (and hence .  Alternative: We can integrate this equation and obtain an equation in  and shear stress 

2. Von Karman Integral Method (BSL) ¯ If we assume a rough velocity profile (for the boundary layer), we can get a fairly accurate relationship ¯ Integration is ‘tolerant’ of changes in shape ¯ For all the above 3 curves, the integration (area under the curve) will provide the same result (more or less), even though the shapes are very different

3. Von Karman Integral Method (BSL) Prandtl equations for steady flow are Continuity N-S (approx) What is V y ? 12 Pressure gradient (approx) 3a 3b

4. Substitute (3a) and (3b) in (2) Von Karman Integral Method (BSL) 4 Integrate (4) with respect to y, from 0 to infinity 5

5. Integration by Parts. Let Von Karman Integral Method (BSL) Eqn. 5: On the RHS Eqn 5: On the LHS, for the marked part

6. Von Karman Integral Method (BSL) This is for the marked region in LHS of Eqn 5

7. Von Karman Integral Method (BSL) 1. To equation (6), add and subtract Substituting in equation (5) 6 To write equation (6) in a more meaningful form:

8. 2. Note 7 Von Karman Integral Method (BSL) 3. Also... and multiply both sides by -1

9. Combining the above two Von Karman Integral Method (BSL)

10. . First term is momentum thickness. Second term is displacement thickness. (Note: The density term is ‘extra’ here) Von Karman Integral Method (BSL) Equation (7) becomes. Note: Integral method is not only applied to Boundary Layer. It can be applied for other problems also.

11. Example Assume velocity profile It has to satisfy B.C. For zero pressure gradient For example, use Von Karman Integral Method (BSL)

12. Or for example, use What condition should we impose on a and b? What is the velocity gradient at y=  ?

13. Von Karman Integral Method (BSL) What is the velocity at y=  ? Check for other two Boundary Conditions For zero pressure gradient OK No slip condition

14. Von Karman equation gives Now, to substitute in the von Karman Eqn, find shear stress Also

16. Calculation for  comes out ok Calculation for Cf also comes out ok Even if velocity profile is not accurate,  prediction is tolerable

17. Now numerical method are more common Conservation of mass Von Karman Method (3W&R)

18. Conservation of mass Von Karman Method

19. Substitute, rearrange and divide by  x Outside B.L.

20. If is const If we assume