Fundamentals of Aerodynamics
6th Edition
ISBN: 9781259129919
Author: John D. Anderson Jr.
Publisher: McGraw-Hill Education
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Chapter 2, Problem 2.3P
Consider a velocity field where the x and y components of velocity are given by
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Consider
the velocity field represented by
V = K (yĩ + xk)
Rotation about z-axis is
velocity field is given by:
A two-dimensional
V = (x - 2y) i- (2x + y)Ĵj
a. Show that the flow is incompressible and irrotational.
b. Derive the expression for the velocity potential, (x,y).
c. Derive the expression for the stream function, 4(x,y).
1. For a flow in the xy-plane, the y-component of velocity is given by v = y2 −2x+ 2y. Find a possible x-component for steady, incompressible flow. Is it also valid for unsteady, incompressible flow? Why? 2. The x-component of velocity in a steady, incompressible flow field in the xy-plane is u = A/x. Find the simplest y-component of velocity for this flow field.
Chapter 2 Solutions
Fundamentals of Aerodynamics
Ch. 2 - Consider a body of arbitrary shape. If the...Ch. 2 - Consider an airfoil in a wind tunnel (i.e., a wing...Ch. 2 - Consider a velocity field where the x and y...Ch. 2 - Consider a velocity field where the x and y...Ch. 2 - Consider a velocity field where the radial and...Ch. 2 - Consider a velocity field where the x and y...Ch. 2 - The velocity field given in Problem 2.3 is called...Ch. 2 - The velocity field given in Problem 2.4 is called...Ch. 2 - Is the flow field given in Problem 2.5...Ch. 2 - Consider a flow field in polar coordinates, where...
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- An unsteady velocity field V = xy'ti + zxj – t°k exists at the 3D plane along a streamline that passes through the point (3,-1,2) at t = 0. Find the equation representing this streamline.arrow_forwardConsider a velocity field where the radial and tangential components of velocity are Vr = 0 and Vθ = cr, respectively, where c is a constant. Obtain the equations of the streamlines.arrow_forwardAn unsteady velocity field V = xy^2ti + zxj −t^3k exists at the 3D plane along a streamline that passes through the point (3,-1,2) at t = 0. Find the equation representing this streamline.arrow_forward
- At t = 0, an unstable velocity field V = xy²ti + zxj – t³k, emerges in the 3D plane as a result of a streamline passing through the points (3, -1, 2). Determine the equation that can act for the streamline.arrow_forwardConsider the velocity field given by u = y/(x2 + y2) and v = −x/(x2 + y2). Calculate the equation of the streamline passing through the point (0, 5).arrow_forwardIn three-dimensional fluid flow, the velocity component an u = * + y z, v = - (xy + yz + zx). Determine the %3D satisfy the continuity equation.arrow_forward
- Consider a velocity field where the x and y components of velocity are given by u = cx and v = -cy, where c is a constant. Obtain the equations of the streamlines.arrow_forwardIn a certain steady, two-dimensional flow field the fluid den- sity varies linearly with respect to the coordinate x; that is, p = Ax where A is a constant. If the x component of velocity u is given by the equation u = y, determine an expression for v.arrow_forwardConsider a velocity field where the x and y components of velocity aregiven by u = cx and v = −cy, where c is a constant. Assuming the velocity field given is pertains to an incompressible flow, calculate the stream function and velocity potential.Using your results, show that lines of constant φ are perpendicular to linesof constant ψ.arrow_forward
- A fluid has a velocity field defined by u = x + 2y and v = 4 -y. In the domain where x and y vary from -10 to 10, where is there a stagnation point? Units for u and v are in meters/second, and x and y are in meters. Ox = 2 m. y = 1 m x = 2 m, y = 0 No stagnation point exists x = -8 m, y = 4 m Ox = 1 m, y = -1 m QUESTION 6 A one-dimensional flow through a nozzle has a velocity field of u = 3x + 2. What is the acceleration of a fluid particle through the nozzle? Assume u, x and the acceleration are all in consistent units. O 3 du/dt 9x + 6 1.5 x2 + 2x O Oarrow_forward1. For a velocity field described by V = 2x2i − zyk, is the flow two- or threedimensional? Incompressible? 2. For an Eulerian flow field described by u = 2xyt, v = y3x/3, w = 0, find the slope of the streamline passing through the point [2, 4] at t = 2. 3. Find the angle the streamline makes with the x-axis at the point [-1, 0.5] for the velocity field described by V = −xyi + 2y2jarrow_forward1. Let V = Vx ((x+ yz)i) be the velocity field for a fluid flow. (a) Verify that this fluid flow is two-dimensional and incompressible. (b) Describe the streamlines for this flow.arrow_forward
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