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 3.10P
Prove that the velocity potential and the stream function for a uniform flow, Equations
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Q1
(a)
If a flow field is compressible, what can you say about the material
derivative of density? What about if the flow field is incompressible?
Explain your answer.
(a) State the equation describing the condition at streamline boundary for a two-dimensional flow with velocity V and displacement ds
Using the equation, derive the equation of a streamline given by Eq. (4.1): ) (See attached images)
(b) A steady, incompressible, two-dimensional velocity field is given by Eq. (4.2) : (see attached images)
Where: the x- and y-coordinates are in metres and the magnitude of the velocity is in m/s. Calculate the material acceleration at the points of (x = 2, y= 3)
Outer pipe wall
Consider the steady, incompressible, laminar flow of a Newtonian fluid in
an infinitely long round pipe annulus of inner radius R, and outer radius
Ro. Assume that the pressure is constant everywhere there is no forced
pressure gradient driving the flow, Pi = P2. However, let the inner
cylinder be moving at steady velocity V to the right, essentially a piston.
The outer cylinder is stationary. This makes an axisymmetric Couette
flow. Use cylindrical coordinates and the equations of motion to generate
an expression for the x-component of velocity u as a function of r. Ignore
the effects of gravity.
Fluid: p, H
iP
R; R,
ƏP_ P2- P1
ax x2-X1
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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Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.Similar questions
- 1. Consider a three-dimensional steady incompressible flow with components: u =-x(y+z), v= y, w=÷(z²-2yz), (a) Is the flow locally rotating? Justify your answer. (b) Determine the equation of vortex lines. (c) Are the vortex lines perpendicular to the streamlines?arrow_forwardThe stream function of a flow field is y = Ax3 – Bxy², where A = 1 m1s1 and B = 3 m-1s1. (a) Derive the velocity vector (b) Prove that the flow is irrotational (c) Derive the velocity potentialarrow_forwardConsider a uniform stream of magnitude V inclined at angle ?. Assuming incompressible planar irrotational flow, find the velocity potential function and the stream function. Show all your work.arrow_forward
- One of the corner flow patterns of Fig. 8.18 is given by thecartesian stream function ψ = A(3yx2 - y3). Which one?Can the correspondence be proved ?arrow_forwardAn incompressible stream function is given byψ = aθ +br sin θ. ( a ) Does this flow have a velocitypotential? ( b ) If so, find it.arrow_forwardFor the flow field described by u = 2, v = yz2t, w = −z3t/3. (a) Is this flow one-, two-, or three-dimensional? (b) Is this flow steady? (c) Is this flow incompressible? (d) Find the z-component of the acceleration vector.arrow_forward
- 6. Find the stream function associated with the two-dimensional incompressible, possible flow with velocity components given by a2 Ur = b|1- cos e and ug = -b1+- sin 0 r2 Where a, and b are known constants.arrow_forwardExample 6.5. For a three-dimensional flow field described by V = (y? + 22) i + (x² +z?) j + (x²+y²) k find at (1, 2, 3) (i) the components of acceleration, (ii) the components of rotation.arrow_forward(b) Two velocity components of a steady, incompressible flow field are given as follows; u = 2ax + bxy + cy? v = axz – byz? where a, b and c are constants. Determine an expression for w as a function of x, y, and z.arrow_forward
- A two-dimensional, incompressible flow has the velocitycomponents u = 4 y and v = 2 x . ( a ) Find the accelerationcomponents. ( b ) Is the vector acceleration radial? ( c ) Sketcha few streamlines in the first quadrant and determine if anyare straight lines.arrow_forwardA fluid flows along a flat surface parallel to the x-direction. The velocity u varies linearly with y, the distance from the wall, so that u = ky. (a) Find the stream function for this flow. (b) Is this flow irrotational?arrow_forward. Find the stream function associated with the two-dimensional incompressible, possible flow with velocity components given by a2 cos 0 and u, = -b 1+ --(1+)sino a2 b1 r2 r2 Where a, and b are known constants.arrow_forward
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