Fundamentals of Aerodynamics
6th Edition
ISBN: 9781259129919
Author: John D. Anderson Jr.
Publisher: McGraw-Hill Education
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Textbook Question
Chapter 3, Problem 3.14P
Derive the velocity potential for a doublet; that is, derive Equation
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Problem 1
Given a steady flow, where the velocity is described by:
u = 3 cos(x) + 2ry
v = 3 sin(y) + 2?y
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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.
Chapter 3 Solutions
Fundamentals of Aerodynamics
Ch. 3 - For an irrotational flow. show that Bernoullis...Ch. 3 - Consider a venturi with a throat-to-inlet area...Ch. 3 - Consider a venturi with a small hole drilled in...Ch. 3 - Consider a low-speed open-circuit subsonic wind...Ch. 3 - Assume that a Pitot tube is inserted into the...Ch. 3 - A Pilot tube on an airplane flying at standard sea...Ch. 3 - At a given point on the surface of the wing of the...Ch. 3 - Consider a uniform flow with velocity V. Show that...Ch. 3 - Show that a source flow is a physically possible...Ch. 3 - Prove that the velocity potential and the stream...
Ch. 3 - Prove that the velocity potential and the stream...Ch. 3 - Consider the flow over a semi-infinite body as...Ch. 3 - Derive Equation (3.81). Hint: Make use of the...Ch. 3 - Derive the velocity potential for a doublet; that...Ch. 3 - Consider the nonlifting flow over a circular...Ch. 3 - Consider the nonlifting flow over a circular...Ch. 3 - Consider the lifting flow over a circular cylinder...Ch. 3 - The lift on a spinning circular cylinder in a...Ch. 3 - A typical World War I biplane fighter (such as the...Ch. 3 - The Kutta-Joukowski theorem, Equation (3.140), was...Ch. 3 - Consider the streamlines over a circular cylinder...Ch. 3 - Consider the flow field over a circular cylinder...Ch. 3 - Prove that the flow field specified in Example 2.1...
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- 55. Derive the relation for angular velocity in terms of the velocity components for fluid rotation in a two-dimensional flow field. [Hint: Use the schematic for ro- tation in Figure IIa.3.5 and find the angular velocity for line oa as @a = doddt. Substitute for da= dl,/dx and for dl, from dl, = (JV,/dx)dxdt. Do the same for line ob to find @p. The z-component of rotation vector is the average of @a and @p. Do the same for x- and y- components].arrow_forwardPlease 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_forwardPlease check this step-by-step and correct if wrong:arrow_forward
- Q4: The velocity components for a two dimensional incompressible flow are u = - 6xy, v=-3x2 + 3y2 1) Is the flow satisfied the continuity equation?. 2) Obtain an expression of stream function 3) Obtain an expression for the velocity potential if it is exsits.arrow_forwardFluid dynamicsarrow_forwarda. Derive an equation for the material acceleration vector.b. Obtain the vorticity vector for the velocity field.c. Is the flow rotational or irrotational? Show through your derivation.d. Is the flow incompressible or compressible? Show through your derivation.arrow_forward
- Provide complete and readable solution, use values from Perry's handbook if necessary.arrow_forwardConsider the incompressible, irrotational, 2D flow. where the stream function is given by: 4 = 424 1) determine the Velocity field and prove the flow is Physically Possible and irrotutional ₂) Calculate and graph (do not sketch) the streamline pattern. 3) find the velocity potential for this flow and graph the lines of constant potential on the same graph as the streamlines.arrow_forwardQ.2 A flow is described by the stream function v = 25xv, The coordinates of the point at which velocity vector has a magnitude of 4 units and makes an angle 150 ° with the X-axis is A x=1.0, y=0.5774 B X=0.5774, Y=1.0 WRONG C X=1, Y=-0.5774 D X=-1, Y=0.5774arrow_forward
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