Fox and McDonald's Introduction to Fluid Mechanics
9th Edition
ISBN: 9781118912652
Author: Philip J. Pritchard, John W. Mitchell
Publisher: WILEY
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Textbook Question
Chapter 9, Problem 11P
Evaluate the displacement thickness δ* and the momentum thickness θ for a velocity profile given by
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The sketch shows a flat plate aligned parallel to an incompressible airstream. Assume a velocity profile, and use the momentum integral to find the boundary layer thickness δ as a function of x, ν, and U. (ν=μ/ρ).
Solve it
(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.
Chapter 9 Solutions
Fox and McDonald's Introduction to Fluid Mechanics
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A simple but effective anemometer to measure wind...Ch. 9 - The Willis Tower (formerly the Sears Tower) in...Ch. 9 - It is proposed to build a pyramidal building with...Ch. 9 - Calculate the drag forces on a 1/200 scale model...Ch. 9 - A circular disk is hung in an air stream from a...Ch. 9 - A vehicle is built to try for the land-speed...Ch. 9 - An F-4 aircraft is slowed after landing by dual...Ch. 9 - A tractor-trailer rig has frontal area A = 102 ft2...Ch. 9 - A 180hp sports car of frontal area 1.72 m2, with a...Ch. 9 - An object falls in air down a long vertical chute....Ch. 9 - Prob. 99PCh. 9 - A light plane tows an advertising banner over a...Ch. 9 - The antenna on a car is 10 mm in diameter and 1.8...Ch. 9 - Consider small oil droplets (SG = 0.85) rising in...Ch. 9 - Standard air is drawn into a low-speed wind...Ch. 9 - A small sphere with D = 6 mm is observed to fall...Ch. 9 - A tennis ball with a mass of 57 g and diameter of...Ch. 9 - A water tower consists of a 12-m-diameter sphere...Ch. 9 - 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A light airplane, with mass M = 1000 kg, has a...Ch. 9 - A light airplane has 35-ft effective wingspan and...Ch. 9 - Assume the Boeing 727 aircraft has wings with NACA...Ch. 9 - Jim Halls Chaparral 2F sports-racing cars in the...Ch. 9 - Some cars come with a spoiler, a wing section...Ch. 9 - Roadside signs tend to oscillate in a twisting...Ch. 9 - Air moving over an automobile is accelerated to...Ch. 9 - A class demonstration showed that lift is present...Ch. 9 - Rotating cylinders were proposed as a means of...Ch. 9 - A baseball pitcher throws a ball at 80 mph. 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- 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_forwardhelparrow_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
- The 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/m3.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_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
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- Water flows at a velocity of 1.2 m/s over a flat plate 1.2 m long. Assume 1/7thpower law and determine the boundary layer thickness and displacementthickness. Compare the values with values calculated using laminar flowcorrelations. υ = 1 × 10−6m2/s.arrow_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_forwardQ4:- If the velocity profile for the laminar boundary layer is given by :- 24 -²)-¹()*+()* ² ( 2 ) - ² ( ² ) ² + ( ²3/ = 2 00 Starting from the integral momentum equation, find an expressions in terms of the Reynolds number to evaluate 1- Boundary Layer Thickness 2-Wall Shear Stress 3- Average Skin Friction Coefficient 4- Drag Force exerted on One Sidearrow_forward
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