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Derivation of Von Karman Integral Momentum Equation u v 0 x y Multiply this by edge velocity ue(x). Integrate by parts. We start with continuity: u u uu e u e x x x v vue ue y y uu e u vue Result : u e 0 x x y ue (1) Subtract from this the u-momentum equation: u 2 uv p xy x y x y (2) Use also the Euler equation: VdV+dp=0 which gives at boundary layer edge u p e ue e 0 x x (3) Result after some minor rearranging: u u u 2 vue vu e u e e 1 u e u e 1 x x eue u e e u e y xy y (4) Integrate equation (4) from y=0 to y=. u u u 2 u dy vue vu 0 e u e e e e 1 x u e e u e x 0 u 0 1 e u e dy xy dy y 0 Use definitions of displacement thickness and momentum thickness. Second term goes to zero, since v=- at wall, and u=ue at the edge. We get: u e u e2 e u e e * wall x x (5) Equation (5) works for laminar and turbulent, compressible and incompressible flows. Expand first term. Simplify this for incompressible flows, by assuming e is constant. We use ordinary derivatives from now on, since there are no derivatives with respect to y left. We get: u u d e u e2 2 e u e e H e u e e wall dx x x Divide thru’ by e u e2 We get Cf d 1 du e (2 H ) dx u e dx 2 (6)