Fundamentals of Aerodynamics
6th Edition
ISBN: 9781259129919
Author: John D. Anderson Jr.
Publisher: McGraw-Hill Education
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Chapter 9, Problem 9.19P
Repeat Problem 9.18, except with
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By using the expression for the shear stress derived in class (and in BSL), show that the shear force on asphere spinning at a constant angular velocity in a Stokes’ flow, is zero.This means that a neutrally buoyant sphere (weight equal buoyancy force) that is made to spin in aStokes’ flow, will neither rise nor fall, nor translate in any preferential direction in the (x-y) plane.
expressions for velocity are:
v_r (r,θ)= U_∞ [1-3R/2r+R^3/(2r^3 )] cosθ
v_θ (r,θ)= -U_∞ [1-3R/4r-R^3/(4r^3 )] sinθ
Where v_r and v_θ are the radial and angle velocity, U_∞ is the velocity of fluid coming to sphere which very faar away from the sphere. And R is the radius of sphere.
Flow over a sphere is given by the superposition of uniform flow and 3D doublet flow. In
cylindrical coordinates the flow resembles that pictured below. See Lecture Pack 3 for more
information on vector operations in polar and cylindrical coordinates. Uniform flow in the
z direction is o = Uz. Doublet flow is
U„R³z
+ z2
where R is the radius of the sphere.
1. Calculate ur, uo, and uz as functions of r and z for superposed uniform and doublet
flow.
2. Show thatr 0, z ±R are stagnation points.
Problem 1: Write the boundary conditions for the following flows:
a) Flow between parallel plates (Fig. 1) without a pressure gradient. The
upper plate is moving with velocity V.
y = +h
y = -h
Fixed
Fig. 1
V
u(y)
Chapter 9 Solutions
Fundamentals of Aerodynamics
Ch. 9 - A slender missile is flying at Mach 1.5 at low...Ch. 9 - Consider an oblique shock wave with a wave angle...Ch. 9 - Equation (8.80) does not hold for an oblique shock...Ch. 9 - Consider an oblique shock wave with a wave angle...Ch. 9 - Consider the flow over a 22.2 half-angle wedge. If...Ch. 9 - Consider a flat plate at an angle of attack a to a...Ch. 9 - A 30.2 half-angle wedge is inserted into a...Ch. 9 - Consider a Mach 4 airflow at a pressure of 1 atm....Ch. 9 - Consider an oblique shock generated at a...Ch. 9 - Consider the supersonic flow over an expansion...
Ch. 9 - A supersonic flow at M1=1.58 and p1=1atm expands...Ch. 9 - A supersonic flow at M1=3,T1=285K, and p1=1atm is...Ch. 9 - Consider an infinitely thin flat plate at an angle...Ch. 9 - Consider a diamond-wedge airfoil such as shown in...Ch. 9 - Consider sonic flow. Calculate the maximum...Ch. 9 - Consider a circular cylinder (oriented with its...Ch. 9 - Consider the supersonic flow over a flat plate at...Ch. 9 - (The purpose of this problem is to calculate a...Ch. 9 - Repeat Problem 9.18, except with =30. Again, we...Ch. 9 - Consider a Mach 3 flow at 1 atm pressure initially...Ch. 9 - The purpose of this problem is to explain what...
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- Plot the streamlines and potential lines of the flow due to aline source of strength 3m at (a, 0) plus a sink -m at(-a, 0). What is the pattern viewed from afar?arrow_forwardTwo free vortices of equal strength, but opposite direction of rotation, are superimposed with a uniform flow as shown in Fig. 4 Ų = -[±T(2#)] In r. (a) Develop an equation for the x-component of velocity, u, at point P(x,y) in terms of Cartesian coordinates x and y. (b) Compute the x-component of velocity at point A and show that it depends on the ratio I'/H. 4- 1. The stream functions for these two vorticies are Sketch and describe the flow then determine the stagnation points. Plx, y) U H Figure (4)arrow_forwardTwo free vortices of equal strength. but opposite direction of rotation, are superimposed with a uniform flow as shown in Fig. 4 1. The stream functions for these two vorticies are V = -[±T/2#)] In r. (a) Develop an equation for the x-component of velocity, u, at point P(x,y) in terms of Cartesian coordinates x and y. (b) Compute the x-component of velocity at point A and show that it depends on the ratio I'/H. 4- Sketch and describe the flow then determine the stagnation points. •Plx, y) Figure (4)arrow_forward
- Problem 1 Given a steady flow, where the velocity is described by: u = 3 cos(x) + 2ry v = 3 sin(y) + 2?y !! !! a) Find the stream function if it exists. b) Find the potential function if it exists. c) For a square with opposite diagonal corners at (0,0) and (47, 27), evaluate the circu- lation I = - f V.ds where c is a closed path around the square. d) Calculate the substantial derivative of velocity at the center of the same box.arrow_forwardPlot the streamlines and potential lines of the fl ow due to aline source of strength m at (a, 0) plus a source 3m at(2a, 0). What is the fl ow pattern viewed from afar?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_forward
- Please indicate the given, assumption and illustration. A source with strength 0.25 m2/s and a vortex with strength 1 m2/s (counter-clockwise) are located at the origin. After working out the equations for the stream function and velocity potential components, determine the following velocity components at a point P(1, 0.5): A) The Radial Velocity component in meters/second. B) The Tangential Velocity Component in meters/second.arrow_forwardThe streamlines in a particular two-dimensional flow field are all concentric circles, as shown in Fig. 5 the equation v, = wr where w is the angular velocity of the rotating mass of fluid. Determine the circulation around the path ABCD. The velocity is given by B Figure 5arrow_forwardProblem 1: Consider a uniform flow (U. ) past a 2D cylinder (1) Derive analytical expression for the the drag and lift forces exerted on a 2D cylinder in an inviscid flow, using either the stream function w or velocity potential ø provided below: y =U_r 1-: sin0, ø =l 1+ cos0. (2) Plot the streamlines around the cylinder. Discuss its differences in the vortex structures comparing with the other cases at finite Reynolds numbers. (a) Re << 1 (b) Re= 10 (c) Re 60 (d) Re 1000arrow_forward
- Problem 4: Vorticity of flow A two-dimensional velocity is given by the equation: v = 6yi +6xj Is the flow rotational or irrotational? Show the details of your work for full credit.arrow_forward6.6 Determine an expression for the vorticity of the flow field described by V = = -xy³i + y*f Is the flow irrotational?arrow_forwardthis is not simple couette flow! the walls on the side have an effect on the velocity profile, but im not sure how. 1. Consider a two-dimensional rectangular region which is very long and narrow, and filled with a fluid. Along one of the long walls is a conveyor belt, which moves with constant velocity U. Near the center of the region, far from the ends, the flow may be assumed to be parallel flow. Find U(y).arrow_forward
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