Practice Problems on Boundary Layers. Answer(s): D = 107 N D = 152 N. C. Wassgren, Purdue University Page 1 of 17 Last Updated: 2010 Nov 22

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1 BL_01 A thin flat plate 55 by 110 cm is immersed in a 6 m/s stream of SAE 10 oil at 20 C. Compute the total skin friction drag if the stream is parallel to (a) the long side and (b) the short side. D = 107 N D = 152 N C. Wassgren, Purdue niversity Page 1 of 17 Last pdated: 2010 Nov 22

2 BL_02 sing the momentum integral theorem, determine the friction coefficient, c f, dimensionless boundary layer momentum thickness, M /, and the dimensionless boundary layer displacement thickness, D /, for laminar flat plate flow with no pressure gradient assuming a sinusoidal velocity profile: u y sin 2 Compare your answers with the Blasius eact laminar boundary layer solution Re 2 M Re 1 2 D Re C f Re 1 2 C. Wassgren, Purdue niversity Page 2 of 17 Last pdated: 2010 Nov 22

3 BL_03 One method proposed to decrease drag and avoid boundary layer separation on aircraft is to use suction to remove the low momentum fluid near the aircraft surface. By removing the low momentum fluid near the surface, the boundary layer remains more stable and transition to a turbulent boundary layer is delayed. Only recently has this method been attempted in practice (Aviation Week & Space Technology, Oct. 12, 1998, pg. 42). Airbus is currently testing a micro-perforated titanium skin on an Airbus A320 aircraft fin. The ultimate goal of Airbus tests is to reduce wing drag by 10-16% and empenage/nacelles drag by nearly 5%. Fuel consumption is epected to be decreased by as much as 13%. To analyze this flow, consider a laminar boundary layer on a porous flat plate. Fluid is removed through the plate at a uniform velocity, V. The thickness of the boundary layer is denoted by and the velocity outside the boundary layer is a constant,. Assuming that the velocity profile, u, is given by a power law epression (n is a positive constant describing the shape of the profile and y is the vertical distance from the surface of the plate): u y 1 n Determine: 1. the momentum thickness of the boundary layer in terms of 2. the drag acting on the plate over a length L if the plate has a depth b into the page (epress your answer in terms of M.) y fluid velocity through porous plate, V porous plate with depth, b, into page L M 2 D b n n1n2 n V L n n 1 2 or 2 V D b M L C. Wassgren, Purdue niversity Page 3 of 17 Last pdated: 2010 Nov 22

4 BL_04 A wind tunnel has a test section 1 m square by 6 m long with air at 20C moving at an average velocity of 30 m/s. To account for the growing boundary layer, the walls are slanted slightly outward. At what angle should the walls be slanted between =2 m and =4 m to keep the test-section velocity constant? 0.1 C. Wassgren, Purdue niversity Page 4 of 17 Last pdated: 2010 Nov 22

5 BL_05 A thin smooth sign is attached to the side of a truck as shown. Estimate the skin friction drag on the sign when the truck speed is 55 mph. 5 ft 20 ft 3 ft GO BOILERS!! 4 ft D 1.57 lb sign f C. Wassgren, Purdue niversity Page 5 of 17 Last pdated: 2010 Nov 22

6 BL_06 Flow straighteners are arrays of narrow ducts placed in wind tunnels to remove swirl and other in-plane secondary velocities. They can be idealized as square boes constructed by vertical and horizontal plates as shown in the figure. The cross-section of the bo is a by a and the bo length is L. Assuming laminar flat plate flow and an array of N by N boes, derive a formula for: a. the total drag on the bundle of boes. b. the effective pressure drop across the bundle. N boes L N boes a flow direction a D NN cells N Dcell Re p N a D N N cells L 2 2 La N C. Wassgren, Purdue niversity Page 6 of 17 Last pdated: 2010 Nov 22

7 BL_10 A laminar boundary layer subjected to a favorable pressure gradient is to be approimated by a profile of the form: 2 3 y y y y u y 1 1 a. sing the Kármán Momentum Integral Equation, determine the differential equation which must be satisfied by () and (). b. Show that if ()=c 1/9, the solution to this equation is of the form ()=A 4/9. c. Find A in terms of c and the kinematic viscosity,. Answers are currently unavailable. C. Wassgren, Purdue niversity Page 7 of 17 Last pdated: 2010 Nov 22

8 BL_12 The working section of a water tunnel consists of a duct with a rectangular cross-section. The width of the crosssection, b (perpendicular to the sketch), is constant but the height, h(), may vary with longitudinal distance,, measured along the centerline of the duct: 1 / 2 h() 1 / 2 h() Laminar boundary layers form on the upper and lower surfaces of the working section and would cause an acceleration of the flow outside the layers if the height h were constant (A similar effect would be caused by the front and back surfaces but we ignore this for the purposes of this problem and assume that there are no boundary layers on the front and back surfaces.) A water tunnel designer wishes to select the function h() in order to ensure that the pressure and velocity outside the boundary layer (say, on the centerline) vary with distance,, in a specified way. The designer decides to use functions of the form: k h h0 H where h 0, H, and k are constants and the boundary layers begin at =0. Find the value of k which produces zero longitudinal pressure gradient in the tunnel. Also find the epression for H in terms of b, the kinematic viscosity,, and the velocity of the flow at the centerline,. k 1 2 H 2* C. Wassgren, Purdue niversity Page 8 of 17 Last pdated: 2010 Nov 22

9 BL_15 Air, with a density of 1.23 kg/m 3 and a kinematic viscosity of 2.5*10-6 m 2 /s, enters a long horizontal ventilation duct of circular cross-section (radius of 0.25 m) with a velocity of 1.0 m/s. At the entrance it is assumed that this velocity is uniform over the entire cross-section. However, as the flow proceeds down the duct a thin laminar boundary develops on the inside wall of the duct. If we first assume that this is like the boundary layer on a flat plate and that the velocity away from the boundary layer remains at 1.0 m/s, find the displacement thickness in meters at a distance (in meters) from the entrance. Having calculated this displacement thickness we recognize that the velocity outside the boundary layer cannot remain precisely constant at 1 m/s. sing the above calculated displacement thickness, find the uniform velocity outside the boundary layer at a point 200 m from the entrance. What is the pressure difference between the entrance and this point 200 m from the entrance? Describe in words how you might now proceed to a more accurate boundary layer calculation which takes this pressure gradient into account. D = (2.7*10-3 m 1/2 ) 1/2 =200 m = 1.4 m/s p =200 m = N/m 2 C. Wassgren, Purdue niversity Page 9 of 17 Last pdated: 2010 Nov 22

10 BL_16 A measured dimensionless laminar boundary layer profile for flow past a flat plate is given in the table below. se the momentum integral equation to determine the 99% boundary layer thickness. Compare your result with the eact (Blasius) result. y/ u/ Re 2 C. Wassgren, Purdue niversity Page 10 of 17 Last pdated: 2010 Nov 22

11 BL_18 A four-bladed Apache helicopter rotor rotates at 200 rpm in air (with a density of 1.2 kg/m 3 and kinematic viscosity 1.5*10-5 m 2 /s). Each blade has a chord length of 53 cm and etends a distance of 7.3 m from the center of the rotor hub. To greatly simplify the problem, assume that the blades can be modeled as very thin flat plates at a zero angle of attack (no lift is generated). blade tip 7.3 m 200 rpm 53 cm a. At what radial distance from the hub center is the flow at the blade trailing edge turbulent? b. What is the (99%) boundary layer thickness at the blade tip trailing edge? c. Assuming that the flow over the entire length of the four blades is turbulent, estimate the power required to drive the helicopter rotor (neglecting all other effects besides aerodynamic drag). r crit 0.68 m P 4-blades = 58.3 kw 3 9.1*10 m 9.1 mm C. Wassgren, Purdue niversity Page 11 of 17 Last pdated: 2010 Nov 22

12 BL_20 A small bug rests on the outside of a car side window as shown in the figure below. The surrounding air has a density of 1.2 kg/m 3 and kinematic viscosity of 1.5*10-5 m 2 /s. To first order, we can approimate the flow as flat plate flow with no pressure gradient and the start of the boundary layer begins at the leading edge of the window. window bug 35 cm 70 cm 40 cm 100 cm a. Determine the minimum speed at which the bug will be sheared off of the car window if the bug can resist a shear stress of up to 1 N/m 2. b. What is the total skin friction drag acting on the window at a speed of = 20 m/s? c. Ignoring the presence of the bug, at what streamwise location will the boundary layer separation point occur on the window? Justify your answer. Hence, the minimum required speed to shear off the bug is 20 m/s Boundary layer separation will not occur since there is no adverse pressure gradient in the flow (zero pressure gradient was assumed). C. Wassgren, Purdue niversity Page 12 of 17 Last pdated: 2010 Nov 22

13 BL_23 The flat plate formulas for turbulent flow over a flat plate assume that turbulent flow begins at the leading edge ( = 0). In reality there is an initial region of laminar flow as shown in the figure. y laminar flow turbulent flow 1. Derive an epression for the 99% boundary layer thickness in the turbulent region by accounting for the laminar part of the flow. 2. Plot the dimensionless boundary layer thickness, /, as a function of Reynolds number (10 4 Re 10 8, use a log scale for the Re ais) for your derived relation and for the turbulent relation that does not consider the laminar part. Assume a 1/7 th power law velocity profile for the turbulent boundary layer and an eperimental friction coefficient correlation of C f Re *10 Re 500, Re Re C. Wassgren, Purdue niversity Page 13 of 17 Last pdated: 2010 Nov 22

14 BL_26 A thin equilateral triangle plate is immersed parallel to a 1 m/s stream of air at standard conditions. Estimate the skin friction drag on this plate. 1 m/s 2 m D = 1.1*10-2 N C. Wassgren, Purdue niversity Page 14 of 17 Last pdated: 2010 Nov 22

15 BL_28 Air flows between two parallel flat plates as shown in the figure below. The upper plate is porous from point B to point C and additional air is injected through this surface. As a result, the free stream speed, (), varies as: 0 where 0 is the air speed entering the channel (at point A), is a constant, and is the distance downstream of the point B. A boundary layer develops along the lower surface. Assuming a linear velocity distribution in the boundary layer, estimate the rate of boundary layer growth, d/d, in terms of,, 0,, and the air properties. A B C y 0 d d 0 0 C. Wassgren, Purdue niversity Page 15 of 17 Last pdated: 2010 Nov 22

16 BL_29 Consider a thin disk of density, D, diameter, d D, and height, h D, resting on a submerged flat plate as shown in the figure below. Flowing over the plate is a fluid of density, F, and dynamic viscosity, F, with a free stream velocity,. There are no pressure gradients in the flow. g fluid with free stream velocity,, density, F, and dynamic viscosity, F disk of diameter, d D, height, h D, and density, D Assume the flow upstream of the plate is uniform, but then results in a boundary layer when the fluid contacts the plate. The effective static friction coefficient between the disk and the plate is (for simplicity assume that the static and dynamic friction coefficients are equal).for the following questions, assume the following: d D = 2 mm h D = 0.5 mm D = 2500 kg/m 3 F = 1000 kg/m 3 F = 1.0e-3 kg/(ms) = 1.5 m/s = 0.3 g = 9.81 m/s 2 a. Determine the effective friction force acting to hold the disk in place. b. If the disk is released at the leading edge of the plate, at what distance from the leading edge will the disk come to rest? (Neglect the inertia of the disk, i.e. treat the disk movement in a quasi-static manner). c. Neglecting the flow over the disk, at what distance from the leading edge will the boundary layer separate? Justify your answer. F F = 6.9e-6 N = m boundary layer will not separate C. Wassgren, Purdue niversity Page 16 of 17 Last pdated: 2010 Nov 22

17 BL_32 A flat plate of length c is placed inside a duct. By curving the walls of the duct, the pressure distribution on the flat plate can be set. Assume the walls of the duct are contoured in such a way that the outer flow over the plate gives the following velocity on the surface of the flat plate: ue 8 1 c c 1/5 1. Write an epression for the streamwise pressure gradient as a function of /c. 2. Determine which portions of the plate have a favorable pressure gradient and which portions have an adverse pressure gradient. Answers currently unavailable. C. Wassgren, Purdue niversity Page 17 of 17 Last pdated: 2010 Nov 22

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