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MAE 5130: VISCOUS FLOWS Homework #3 Solutions Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk 1 PROBLEM 3.2 • Consider axial Couette flow of Fig. 3-3 with both cylinders moving – Find velocity distribution u(r) and plot for (a) U1=U0, (b) U1=-U0, and (c ) U1=-2U0 – 0: inner cylinder coordinate, 1: outer cylinder coordinate • • Sum of separate solutions for moving inner or outer cylinder Superposition is possible because N/S equations are linear for this flow 2 PROBLEM 3.2: NON-DIMENSIONAL REPRESENTATION Radius non-dimensionalized by r0 u non-dimensionalized by U0 3 PROBLEMS 3-14 and 3-22: OSEEN AND TAYLOR VORTICES G0=C=1.0, n=1x10-5 • • • Taylor profiles are flatter than Oseen Taylor profiles are less peaked Decay by dropping quickly near the axis with a reversal in vorticity away from the axis 4 PROBLEM 3-32: JEFFERY-HAMEL WEDGE FLOW, Re=0, a≠0 • Case where a = 0º is the Poiseuille parabola for channel flow • Case where a = 90º is the separation point • For a > 90º, separation or backflow must occur in a diverging flow even at zero Reynolds number 5 PROBLEM 3-55 a b • Both circumscribed forces are greater than the actual forces • Both inscribed forces are less than the actual forces • The ratios are significantly different from unity and vary by a factor of 2 to 3 6 EXAMPLE: DRAG CALCULATION FROM A WAKE • A uniform stream flows past an immersed object as shown above, producing a broad, low-velocity wake, which is idealized as a V-shape. • Pressures at p1 and p2 are essentially equal • The flow is 2D and incompressible, with width b into page • Derive a formula for drag force on the object and obtain expression for drag coefficient, CD 7