A laminar boundary layer velocity profile is approximated by /U=2(y/8) - 2(y/8)³ + (y/6)* 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).
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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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a4d63fc-14a6-490f-8088-b0678285a71d%2F55ebb7e8-6c21-41f4-ba09-5599699a47c5%2Fxqhy2y7_processed.jpeg&w=3840&q=75)
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- Q4) 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.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.(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.
- (b) In two dimensional boundary layer, shear stress was changed linearly from the solid surface toward y-axis until it reach the value of zero at y = 6. Based on Table 2 and setting given to you; () Derive the equation of displacement thickness and momentum thickness using Von Karman Approximation Method ; and (ii) Determine the accuracy of this method in determining the value of displacement thickness and momentum thickness. Table 2 : Equation of Velocity Profile Setting Equation wU = 3(y/8)/2 – (y/8j?/2a. 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 ThicknessViscous fluid occupies the region 0 < z3. Viscous fluid occupies the region 0Current Attempt in Progress Incorrect. Consider the steady potential flow around the circular cylinder shown in the figure below. At what minimum integer distance, y/a (along the y axis), is the velocity within 4% of the free-stream velocity? 4=0 2U y/a = i !Consider an artificial boundary layer flow past a flat plate. The profile of the velocity field assumes =()¹², 0QI/ If the velocity profile of the boundary layer is 4 =-÷O find the thickness of boundary %3D layer, the shear stress at trailing edge and the drag force on one side of plate 2 m long , i if it is Kg immersed in water flowing with velocity of 0.4 m/s (p = 998 , 0= 1.007*10-6 m2/s m3(b) In two-dimensional boundary layer, shear stress was changed linearly from the solid surface toward y-axis until it reaches the value of zero at y = 8. Based on Table 2 and setting given to you; (i) Derive the equation of displacement thickness and momentum thickness using Von Karman Approximation Method; and (ii) Determine the accuracy of this method in determining the value of displacement thickness and momentum thickness. Table 2: Equation of Velocity Profile Equation u/U = 3(y/8)/2 – (y/8)³/2 Setting 2Q4:- 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 SideSEE MORE QUESTIONSRecommended textbooks for youElements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill EducationControl Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEYElements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill EducationControl Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY