Question

130 points] The one meter wide duct shown below has the linear velocity profile shown. A screen downstream causes the flow to have the uniform velocity profile downstream shown. Assume the density ρ is known. 1. a. [10 points] Assuming the linear velocity profile is V()-my+b, show that the velocity profile is r(y)-r_ (1-% ) assuming the velocity at the wall is zero and the velocity at the centerline is F-. b. [10 points] Show that the mass flowrate for this linear velocity profile is pl [5 points] Show that the mass flow rate for the uniform velocity profile is m-2 ·gh-w. c. d. [5 points] Show that the average velocity for the uniform profile is V,-'m%

Answer 1

V(y) = my + b therefore delta* = integral^y_1_0 (1 - u_x, y/u_infinity du) delta* = delta integral^1_0 (1 - u(n)/u_infinity) dn du = delta dn therefore 1/2 delta (x)^2 = 140/13 v(n)/u_infinity u(n) = c/(12e)^1/2 (x/c)^1/2 times 0.69 u du/du + v du/du = v d^2 u/du^2 delta x lessthanorequalto 1/2 u_i, j (delta y)^2/v \r\n mass flow rate for linear velocity profile: therefore v_avg = 1/2 v_lin m = e v_avg h.w therefore velocity profile for the uniform flow is given as: v/u = y/s v = uy/delta delta = integral^infinity_0 (1 - u/u) dy = integral^integral_0 (1 - u/u) dy V_avg = v_max/2 \r\n Linear velocity profile (uniform) delta_liner = integral^delta_0 (1 - y/s) dy delta_linear = integral^1_0 (1 - n) delta dn = delta [n - 1/2 n^2] = 1/2 delta m^dot = 2 e v_avg middot hw