Von Karman Integral Method BSL PRANDTL BOUNDARY LAYER

Von Karman Integral Method (BSL) PRANDTL BOUNDARY LAYER EQUATIONS for steady flow are 1 Continuity N-S (approx) 2 ¯ If we solve these, we can get Vx, (and hence ). ¯ Alternative: We can integrate this equation and obtain an equation in and shear stress t. 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 the shapes are very different IIT-Madras, Momentum Transfer: though July 2005 -Dec 2005

Von Karman Integral Method (BSL) Prandtl equations for steady flow are Continuity 1 N-S (approx) 2 Pressure gradient (approx) What is Vy? IIT-Madras, Momentum Transfer: July 2005 -Dec 2005 3 a 3 b

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

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

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

Von Karman Integral Method (BSL) Substituting in equation (5) 6 To write equation (6) in a more meaningful form: 1. To equation (6), add and subtract IIT-Madras, Momentum Transfer: July 2005 -Dec 2005

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

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

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

Von Karman Integral Method (BSL) Example Assume velocity profile It has to satisfy B. C. For zero pressure gradient For example, use 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= ? 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 No slip condition For zero pressure gradient IIT-Madras, Momentum Transfer: July 2005 -Dec 2005 OK OK

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

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

Calculation for comes out ok Calculation for Cf also comes out ok Even if velocity profile is not accurate, prediction is tolerable IIT-Madras, Momentum Transfer: July 2005 -Dec 2005