1. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 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. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 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. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 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. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Substitute (3a) and (3b) in (2) Von Karman Integral Method (BSL) 4 Integrate (4) with respect to y, from 0 to infinity 5

5. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Integration by Parts. Let Von Karman Integral Method (BSL) Eqn. 5: On the RHS Eqn 5: On the LHS, for the marked part

6. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Von Karman Integral Method (BSL) This is for the marked region in LHS of Eqn 5

7. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 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. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 2. Note 7 Von Karman Integral Method (BSL) 3. Also... and multiply both sides by -1

9. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Combining the above two Von Karman Integral Method (BSL)

10. IIT-Madras, Momentum Transfer: July 2005-Dec 2005. 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. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Example Assume velocity profile It has to satisfy B.C. For zero pressure gradient For example, use Von Karman Integral Method (BSL)

12. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Von Karman Integral Method (BSL) Or for example, use What condition should we impose on a and b? What is the velocity gradient at y=  ?

13. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 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. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Von Karman equation gives Now, to substitute in the von Karman Eqn, find shear stress Also

15. IIT-Madras, Momentum Transfer: July 2005-Dec 2005

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

17. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Now numerical method are more common Conservation of mass Von Karman Method (3W&R)

18. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Conservation of mass Von Karman Method

19. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Substitute, rearrange and divide by  x Outside B.L.

20. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 If is const If we assume

21. IIT-Madras, Momentum Transfer: July 2005-Dec 2005