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Von Karman Integral Method (BSL)

Von Karman Integral Method (BSL). 1. 2. PRANDTL BOUNDARY LAYER EQUATIONS for steady flow are. Continuity. N-S (approx). If we solve these, we can get V x , (and hence d) . Alternative: We can integrate this equation and obtain an equation in d and shear stress t.

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Von Karman Integral Method (BSL)

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  1. Von Karman Integral Method (BSL) 1 2 PRANDTL BOUNDARY LAYER EQUATIONS for steady flow are Continuity N-S (approx) • If we solve these, we can get Vx, (and hence d). • Alternative: We can integrate this equation and obtain an equation in d and shear stress t.

  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) Pressure gradient (approx) 1 2 3a 3b Prandtl equations for steady flow are Continuity N-S (approx) What is Vy?

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

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

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

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

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

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

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

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

  12. 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=d ?

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

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

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

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

  17. Von Karman Method Conservation of mass

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

  19. If is const If we assume

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