Evaluate the displacement thickness δ* and the momentum thickness θ for a power law velocity profile given by
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- A laminar boundary layer velocity profile is approximated by u/U=2(y/8)-2(y/8)³ + (y/8)* for y ≤ 8 and u = U for y> 8. (a) Show that this profile satisfies the appropriate boundary conditions. (b) Use the momentum integral equation to determine the boundary layer thickness, 8 = 8(x).arrow_forwardThe velocity distribution in the boundary layer is given by :u/U=(y/δ)^0.23 where u is the velocity at a distance y from the plate and u=U at y=δ , δ being boundary layer thickness. Find: a-The displacement thickness .b- the momentum thickness .c- the energy thickness. d- Boundary layer shape factor .e- energy loss due boundary layer if a particular section. The boundary layer thickness is 28mm and the free stream velocity is 19 m/s. if the discharge through the boundary layer region is 8m3/s per meter width, express this energy loss in term of meters of head. Take ρ=1.2 kg/m3arrow_forwardConsider the boundary layer over a flat plate at 45° angle as shown. The exact flow field in this configuration is described by the Falkner-Skan similarity solution with n = 1/3 (see Figure 10.8 of the textbook, the Falkner-Skan profile chart). The objective is to find the approximate solution to this problem using the Thwaites method and calculate its error. Ve 11/4 Assume that for this approximate solution the free stream velocity is U₂(x) = ax" where a is an unknown constants and n = 1/3. Use the Thwaites method to find the momentum 0/x and 8*/x displacement thicknesses as well as the friction coefficient cf = 0.5, as functions of Re₂ = Uer/v, where is the shear stress at the wall. (No need to interpolate the Thwaites method table values; you can pick the nearest numbers.) Using the Falkner-Skan profile chart approximate the friction coefficient c; (by estimating the slope of the corresponding velocity profile at the wall). How does this value compare with your prediction in part…arrow_forward
- An equation for the velocity for a 2D planar converging nozzle is Uy u =U1+ w=0 L Where U is the speed of the flow entering into the nozzle, and L is the length. Determine if these satisfy the continuity equation. Write the Navier-Stokes equations in x and y directions, simplify them appropriately, and integrate to determine the pressure distribution P(x.y) in the nozzle. Assume that at x = 0, y = 0, the pressure is a known value, P.arrow_forward(c) If the velocity profile in a boundary layer was approximated by the quadratic distribution u = Ay² + By + C, what are the boundary condition(s) that would be required to satisfy the following profile: U U 2 • (-/-) – ( - ) ² = 2 Prove that the expression for wall shear stress using Von Karman Momentum Integral equation based on the above velocity profile approximation to be as: = PU² √Rex To 0.365-arrow_forwardIBL, Flat Plate. Apply the integral boundary layer analysis to a flat plate turbulent flow as follows. Assume the turbulent profile u/U = (y/δ)1/6 to compute the momentum flux term on the RHS of IBL, but on the LHS of IBL, use the empirical wall shear stress, adapted from pipe flow: ?w = 0.0233 ⍴U2 (v/Uδ)1/4 where the kinematic viscosity ν = μ/⍴. It is necessary to use this empirical wall shear relation because the turbulent power law velocity profile blows up at the wall and cannot be used to evaluate the wall shear stress. Compute (a) (δ/x) as a function of Rex; (b) total drag coefficient, CD, L as a function of ReL; (c) If ReL = 6 x 107 compare values for this IBL CD,L and those empirical ones given in Table 9.1 for both smooth plate and transitional at Rex = 5 x 105 cases. Note: You must show all the algebra in evaluating the IBL to get full credit. Ans OM: (a) (δ/x) ~ 10-1/(Rex)1/5; (b) CD,L ~ 10-2/(ReL)1/5; (c) CD,IBL ~ 10-3; CD,Smooth ~ 10-3; CD,Trans ~ 10-3arrow_forward
- a. If the velocity distribution for the laminar boundary layer over a flat plate is given by :- ** (²) ² - 21 + A₂ x U A₁ + A₂ x Determine the form of the velocity profile by using the necessary boundary conditions. After that by using the Von-Karman integral momentum equation find an expression in terms of the Reynolds number to evaluate 1- Boundary Layer Thickness Force 2-Wall Shear Stress 3-Drag 5- Displacement Thickness + A₁ x 4- Local and Average Skin Friction Coefficients 6- Momentum Thickness 7- Energy Thicknessarrow_forwardQ4) Define the boundary layer, then find the thickness of boundary layer, shear stress, drag force and coefficiet of drag in terms of Re for the velocity profile of laminar boundary layer u/U=4(y/8)-6(y/8)5. Calcate the boundary layer thickness and drag force if the air flows over a sharp edged flat plate 1m long and 0.5m wide at a velocity 0.9m/s, the air density 1.23 kg/m³ and the kinematic viscosity is 1.46*10-5 m/s².arrow_forwardAn airplane has a rectangular wing with a chord length of 8 ft. If the airplane is flying at 200knots TAS and at 20,000 ft density altitude, find the Reynolds number at the trailing edge of the wing.arrow_forward
- B) for the velocity profiles given below, state whether the boundary layer has separated or on the verge of separation or will remain attached ( i) u/U=2(y/8) -(y/8)² ii) u/U=-2(y/8) +0.5(y/8)³ iii) u/U-1.5(y/8) +0.5(y/8)³ Q3: Find the displacement, momentum thickness and energy thickness for the velocity distribution in the boundary layer: F u/U=0.5(y/8) + 1.5(y/8)³ Q4: A) Find the velocity distribution and expression of the maximum velocity and shear stress for a flow between two stationary plates. C B) A laminar flow of oil between two horizontal fixed parallel plates with a maximum velocity 2 m/s and 100mm apart. Canulate the pressure gradient and shear stress. Take µ-2.4525 s/m².arrow_forwardA tornado is simulated by a line sink m = -1000 m2/s plus aline vortex K = 1600 m2/s. Find the angle between anystreamline and a radial line, and show that it is independent ofboth r and θ. If this tornado forms in sea-level standard air, atwhat radius will the local pressure be equivalent to 29 inHg?arrow_forwardLocal boundary layer effects, such as shear stress and heattransfer, are best correlated with local variables, rather usingdistance x from the leading edge. The momentum thicknessθ is often used as a length scale. Use the analysis of turbulentflat-plate flow to write local wall shear stress τw in terms ofdimensionless θ and compare with the formula recommendedby Schlichting: Cf ≈ 0.033 Reθ -0.268.arrow_forward
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