By Maurice Holt
Simple advancements in Fluid Dynamics
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15. The wetted surface FIG. 15. Arbitrary body profile; fixed and smooth detachments. 1a) where s is the arc length measured from A along the body contour T, and S is the total arc length of the wetted surface. These functions and their first derivatives are assumed to be Holder continuous in s for 0 ^ s ^ S. 1b) which also satisfies the Holder condition: For any two points s1? 1c) where A and \i are positive constants (A is called the Holder constant, and \i the Holder index). The solution of this flow problem can be conveniently derived by a limiting process applied to the solution of an N-sided polygon, letting the number N increase beyond all bounds; the face lengths lk all tend to zero, and the turning angles sk all become vanishingly small, except possibly at a finite number of isolated points at which the obstacle has sharp corners.
It is obvious that the above w satisfies 25 INVISCID CAVITY A N D WAKE FLOWS the conditions on arg w over the solid boundary. 9) maps the circle \t\ = 1 onto a circular arc |w| = 1 by virtue of the well-known property of the bilinear trans formation T = (t - z)/(xt - 1) with - 1 < T < 1. 9). It is noted that the above w(t) is in general a multivalued function of t; it has zeros at the stagnation point and the concave corners, and has logarithmic singularities at the convex corners. The physical z plane can be determined from z (L5 10) \TtdL t wit) at The solution given by Eqs.
0 = (}(— 1). 3a) where Cl(t) = - I 71 J-! 3b) 40 T. YAO-TSU WU Clearly CD = log(l/w) and Q are related by w(t)= -\ogt + Q(f). 3c) The exact solution is here expressed parametrically as / = f(t), w = w(r), with/(r) given by Eq. 3). The present result of w(t) reduces to Eq. 9) for polygonal bodies when iV_1 dB = - I - (skn)S(t dx fct ! / Tfc), b(t — zk) being the Dirac delta function. 2) depends on the body curvature. 4) where s(— 1) = 0 and Kb is positive (or negative) when the body surface is concave (or convex) to the flow.
Basic Developments in Fluid Dynamics by Maurice Holt