A flow of 68°F air develops in a flat horizontal duct following a well-rounded entrance section. The duct height is H = 1 ft. Turbulent boundary layers grow on the duct walls, but the flow is not yet fully developed. Assume that the velocity profile in each boundary layer is u / U = ( y / δ ) 1/7 . The inlet flow is uniform at V = 40 ft/s at section ①. At section ②, the boundary-layer thickness on each wall of the channel is δ 2 = 4 in. Show that, for this flow, δ * = δ /8. Evaluate the static gage pressure at section ①. Find the average wall shear stress between the entrance and section ②, located at L = 20 ft.
A flow of 68°F air develops in a flat horizontal duct following a well-rounded entrance section. The duct height is H = 1 ft. Turbulent boundary layers grow on the duct walls, but the flow is not yet fully developed. Assume that the velocity profile in each boundary layer is u / U = ( y / δ ) 1/7 . The inlet flow is uniform at V = 40 ft/s at section ①. At section ②, the boundary-layer thickness on each wall of the channel is δ 2 = 4 in. Show that, for this flow, δ * = δ /8. Evaluate the static gage pressure at section ①. Find the average wall shear stress between the entrance and section ②, located at L = 20 ft.
A flow of 68°F air develops in a flat horizontal duct following a well-rounded entrance section. The duct height is H = 1 ft. Turbulent boundary layers grow on the duct walls, but the flow is not yet fully developed. Assume that the velocity profile in each boundary layer is u/U = (y/δ)1/7. The inlet flow is uniform at V = 40 ft/s at section ①. At section ②, the boundary-layer thickness on each wall of the channel is δ2 = 4 in. Show that, for this flow, δ* = δ/8. Evaluate the static gage pressure at section ①. Find the average wall shear stress between the entrance and section ②, located at L = 20 ft.
Flow straighteners consist of arrays of narrow ducts placed in a flow to remove swirl and other transverse
(secondary) velocities. One element can be idealised as a square box with thin sides as shown below.
Calculate the pressure drop across a box with L=22 cm and a= 2.7 cm, if air with free-stream velocity of
Uo = 11 m/s flows though the straightener. Use laminar flat-plate theory and take u = 1.85 x 10-5 Pa.s
and p = 1.177kg/m³ .
%3D
%3D
a
Uo
Figure 1: Flow across straighteners.
A two-dimensional diverging duct is being designed to diffuse the high-speed air exiting a wind tunnel. The x-axis is the centerline of the duct (it is symmetric about the x-axis), and the top and bottom walls are to be curved in such a way that the axial wind speed u decreases approximately linearly from u1 = 300 m/s at section 1 to u2 = 100 m/s at section 2 . Meanwhile, the air density ? is to increase approximately linearly from ?1 = 0.85 kg/m3 at section 1 to ?2 = 1.2 kg/m3 at section 2. The diverging duct is 2.0 m long and is 1.60 m high at section 1 (only the upper half is sketched in Fig. P9–36; the halfheight at section 1 is 0.80 m). (a) Predict the y-component of velocity, ?(x, y), in the duct. (b) Plot the approximate shape of the duct, ignoring friction on the walls. (c) What should be the half-height of the duct at section 2?
PART A
-5
For Air Assume: v=1.46×10 m²/s and p =1.225 kg/m³
Question A1
A flat plate with unit width is placed in a uniform steady two-dimensional flow of air with
negligible pressure gradient. The velocity distribution of the boundary layer forming on
either side of this plate has been given as:
U /Us = 2.0(y/ 8) - (y/8)2.0, where U = 10 m/s.
1- Find and expression for the boundary layer thickness as a function of Rex.
2- At x=1.0 m calculate the shear stress corresponding to the points y=0.0,3.0 and
10.0 mm.
3- Calculate the boundary layer displacement and momentum thicknesses at
x=1.0m. By using the calculated momentum thickness determine the drag force
acting on one side of the plate between its leading edge and x = 1.0m.
4- At x=1.0m, calculate the mass flow rate through the boundary layer (mkg/s per
unit width of the plate). What is the corresponding mass flow rate if the flow was
inviscid (m; kg/s per unit width of the plate)?
5- By using appropriate equation (s) explain how the…
Chapter 9 Solutions
Fox And Mcdonald's Introduction To Fluid Mechanics
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