10, September 2008 EMA521, Fall 2008, Assignment 2, Due 9/18/2008 1. The nozzle in the figure has circular cross section and its contour is described by R(z) = Ri z0 z0 + z
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in the cylindrical coordinate system shown. The inlet radius is Ri . The flow is steady, inviscid, incompressible, axisymmetric and swirl-free (i.e. V = 0 everywhere). The z-component of the velocity is uniform across any nozzle cross section normal to the z-axis. The z-component of the velocity at the inlet is Vi . a) Write at least one condition that the radial component of the velocity, Vr (r, z) must satisfy. NOTE: if you can't think of any conditions at first, they will probably become clear when you solve part d) below. b) Write all the conditions that the velocity field must satisfy. c) Determine the functional form of the axial component of the velocity, = Vz Vz (z). d) Determine the EXACT (NOT the simplest) functional form of the radial component of the velocity, Vr = Vr (r, z). e) Determine all the components of the acceleration vector a(r, z). f) Determine the functional form of the streamlines in a diametral plane (as that shown in the figure) rs = rs (z). g) Using a computer, plot a few streamlines.
r
Ri
Vi
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R(z) z
Figure 1: Contoured nozzle
2. Show that (V) = V + V
September 10, 2008 3. The radial component of the velocity of an ...