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Advanced inviscid uid ow Egme 508 { Fall 2016 Homework No. 03, Quiz on October 5, 2016 Problem 1: Consider the one-dimensional Cartesian velocity eld: u =x=ti where is a constant. Find a spatially uniform, time-dependent density eld, =(t), such that = at t =t . o o Problem 2: Consider the situation depicted below. Wind strikes the side of a simple residential structure and is de ected up over the top of the structure. Assume the following: two-dimensional steady inviscid constant-density ow, uniform upstream velocity prole, linear gradient in the downstream velocity prole (velocity U at the upper boundary and zero velocity at the lower boundary as shown), no ow through the upper boundary of the control volume, and constant pressure on the upper boundary of the control volume. Using the control volume shown: a) Determine h in terms of U and h . 2 1 b) Determine the direction and magnitude of the horizontal force on the house per unit depth into the page in terms of the uid density , the upstream velocity U, and the height of the house h . 1 c) Evaluate the magnitude of the force for a house that is 10 m tall and 20 m long in wind of 22 m/sec (approximately 50 miles per hour). Problem 3: A Rankine vortex is modeled with the following velocity eld, u ;u = 0 and u (r) such that r z 8 < !r : rR 2 u = !R : : r>R r a) Determine whether this ow pattern is irrotational in either the inner or outer region. b) Use the r-momentum equation to determine the pressure distribution p(r) in the vortex, assuming p =p as r!1. 1 2 2 2 @u @u u @u @u u 1@p 1 @ @u 1 @ u @ u u 2 @u r r r r r r r r +u + +u = + r + + +g r z r 2 2 2 2 2 @t @r r @ @z r @r r@r @r r @ @z r r @ 1Problem 4: Start with the equations of motion in the rotating steadily coordinates, and prove Kelvins circulation theorem D = 0 Dt Assume the ow is inviscid and barotropic and that the body forces are conservative. Explain the result physically. Problem 5: An ideal...
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