# Low-Speed Aerodynamics by Joseph Katz

By Joseph Katz

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Extra resources for Low-Speed Aerodynamics

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12 Velocity at point P due to a vortex distribution. 13 The velocity at point P induced by a vortex segment. 37 38 2 / Fundamentals of Inviscid, Incompressible Flow so that ∇× ζ dV = ∇ × |r0 − r1 | dl |r0 − r1 | Carrying out the curl operation while keeping r1 and dl fixed we get ∇× dl = |r0 − r1 | dl × (r0 − r1 ) |r0 − r1 |3 Substitution of this result back into Eq. 68a) A similar manipulation of Eq. 67a) The Velocity Induced by a Straight Vortex Segment In this section, the velocity induced by a straight vortex line segment is derived, based on the Biot–Savart law.

36) the velocity potential is not single valued if there is a nonzero circulation. 8 Uniqueness of the Solution The physical problem of finding the velocity field for the flow created, say, by the motion of an airfoil or wing has been reduced to the mathematical problem of solving Laplace’s equation for the velocity potential with suitable boundary conditions for the velocity on the body and at infinity. 37c) Since the body boundary condition is on the normal derivative of the potential and since the flow is in the region exterior to the body, the mathematical problem of Eqs.

6a) To evaluate the integral over the sphere, introduce a spherical coordinate system at P and since the vector n points inside the small sphere, n = −er , n · ∇ = −∂ /∂r , and ∇1/r = −(1/r 2 )er . 7) This formula gives the value of (P) at any point in the flow, within the region V , in terms of the values of and ∂ /∂n on the boundaries S. If, for example, the point P lies on the boundary S B then in order to exclude the point from V , the integration is carried out only around the surrounding hemisphere (submerged in V ) with radius , and Eq.