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MAE 5130: VISCOUS FLOWS

MAE 5130: VISCOUS FLOWS. Homework #3 Solutions Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk. PROBLEM 3.2. Consider axial Couette flow of Fig. 3-3 with both cylinders moving

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MAE 5130: VISCOUS FLOWS

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  1. MAE 5130: VISCOUS FLOWS Homework #3 Solutions Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

  2. 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

  3. PROBLEM 3.2: NON-DIMENSIONAL REPRESENTATION Radius non-dimensionalized by r0 u non-dimensionalized by U0

  4. PROBLEMS 3-14 and 3-22: OSEEN AND TAYLOR VORTICESG0=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

  5. 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

  6. PROBLEM 3-55 • 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 a b

  7. 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

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