Air flows in a cylindrical duct of diameter D = 6 in. At section ①, the turbulent boundary layer is of thickness δ1 = 0.4 in. and the velocity in the inviscid central core is U1 = 80 ft/s. Further downstream, at section ②, the boundary layer is of thickness δ2 = 1.2 in. The velocity profile in the boundary layer is approximated well by the
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- 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?arrow_forwardA pipe bore diameter D and length L has fully developed laminar flow throughout the entire length with a centre line velocity u0. Given that the drag coefficient is given as CD = 16/Re where Re = pu,D/µ. Derive the expression of the drag force on the inside of the pipe to be FD =8TTHU,L c. and hence show the pressure loss due to skin friction is pL = 32µu,L/D²arrow_forwardFlow 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.arrow_forward
- From the laminar boundary layer the velocity distributions given below, find the momentum thickness θ, boundary layer thickness δ, wall shear stress τw, skin friction coefficient Cf , and displacement thickness δ*1. A linear profile, u(x, y) = a + by 2. von K ́arm ́an’s second-order, parabolic profile,u(x, y) = a + by + cy2 3. A third-order, cubic function,u(x, y) = a + by + cy2+ dy3 4. Pohlhausen’s fourth-order, quartic profile,u(x, y) = a + by + cy2+ dy3+ ey4 5. A sinusoidal profile,u = U sin (π/2*y/δ)arrow_forward(a) Use the y-momentum equation to show that the pressure gradient across the boundary layer is approximately zero i.e. = 0 . Assume the boundary layer to be a two-dimensional ду steady and incompressible flow. Neglect gravitational forces. State clearly all assumptions made. Use the Bernoulli's equation to prove that the pressure difference is given by (b) P2-P1 = -4pU? for a fluid with constant density p flowing from point 1 to point 2 where pi, U1, A1 are the pressure, velocity and flow cross-section area at point 1 and p2, U2, A2 are the pressure, velocity and flow cross-section area at point 2 respectively. The ratio of the cross-section area A1 = 3.arrow_forwardA vertical air stream flowing at a velocity of 100 m/s supports a ball of 60 mm in diameter. Taking the density of air as 1.2 kg/m³ and kinematic viscosity as 1.6 stokes, the weight of the ball that is supported is (if coefficient of drag C= 0.8)arrow_forward
- Fluid enters a square duct as shown in the figure below. Considering boundary layer is laminar, please use the "displacement thickness" to estimate (a) the velocity Ucore and (b) the pressure Pcore in the core of the flow at the position x. U。。 Square Duct Boundary layers x Ucorearrow_forwardAir flows through the test section of a small wind tunnel at speed V = 7.5 ft/s. The temperature of the air is 80°F, and the length of the wind tunnel test section is 1.5 ft. Assume that the boundary layer thickness is negligible prior to the start of the test section. Is the boundary layer along the test section wall laminar or turbulent or transitional?From above problem.assume the flow remains laminar, and estimate the boundary layer thickness, the displacement thickness, and the momentum thickness of the boundary layer at the end of the test section. Give your answers in inches, compare the three results, and discuss. .arrow_forward3. Consider the laminar flow of an incompressible fluid over a flat plate at y = 0. (a) Assume the velocity profile of the boundary layer u is a sinusoidal function. Solve = fm), where U is the freestream velocity and 7 = y/5. (b) From momentum integral equation, express Cf, Tw in terms of Rex.arrow_forward
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