Fundamentals of Aerodynamics
Fundamentals of Aerodynamics
6th Edition
ISBN: 9781259129919
Author: John D. Anderson Jr.
Publisher: McGraw-Hill Education
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Chapter 2, Problem 2.11P

Assuming the velocity field given in Problem 2.6 pertains to an incompressible flow, calculate the stream function and velocity potential. Using your results, show that lines of constant ϕ are perpendicular to lines of constant ψ .

Expert Solution & Answer
Check Mark
To determine

To find:

The stream function and velocity potential for the given velocity functions and to prove it’s perpendicularity.

Answer to Problem 2.11P

Equation of the velocity potential is ϕ=c2(x2y2).

Equation of the stream function is ψ=cxy+d.

Explanation of Solution

Given:

The horizontal and vertical velocity component of velocity is given as below ( problem 2.6).

u=cx....................(1)v=cy.................(2)

Stream function and velocity potential has to be found and prove perpendicularity of the stream function and velocity potential.

Stream function:

The equation of the stream function is given as follows:

u=ψyandv=ψxPut the value of the u:ψy=cxψ=cxyIntegrating both sides,ψ=cxyψ=cxy+f(x)...............................(3)Where f(x) is the integration constant.Differentiate equation (3) with respect to x.ψ=cxy+f(x)x(ψ)=x(cxy)+f'(x)ψx=cy+f'(x)Put the vaue of the ψx=cy.cy=cy+f'(x)f'(x)=0f(x)=dPut the value of the f(x) in equation 1.ψ=cxy+0ψ=cxy+d.........................(4)answerWhere d is constant.

Equation of the stream function is ψ=cxy+d

Velocity potential:

The equation of the velocity potential is given as follows:

u=ϕxandv=ϕyPut the value of the u.ϕx=cxϕ=cxxIntegrating both sides,ϕ=cxxϕ=cx22+f(y)...............................(5)Where f(y) is the integration constant.Differentiate equation (1) 1 with respect to y.ϕ=cx22+f(y)y(ϕ)=y(cx22)+f'(y)ϕy=0+f'(y)Put the vaue of the ϕy=cy.f'(y)=cyf(y)=cy22Put the value of the f(y) in equation 2.ϕ=cx22cy22ϕ=c2(x2y2).........................(6)answer

Equation of the stream function is ϕ=c2(x2y2).

Proof of the perpendicularity:

Differentiate equation (2) and (3) with respect to x keeping φ and ψ constant.

ψ=cxyddx(ψ)=c(xdydx+y)0=xdydx+ydydx=yx..............................(7)

Similarly,

dϕdx=c2(2x2ydydx)0=c2(2x2ydydx)2x2ydydx=0dydx=xy..............................(8)

Compare equations 7 and 9.

(dydx)at constant ϕ=1(dydx)at constant ψ

Hence,the stream function and velocity potential are perpendicular to each other.

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Students have asked these similar questions
Problem 3: Obtaining the stream function from velocity components The steady state, incompressible flow field for two-dimensional flow is given by the following velocity components: V =16y-x and v, =16x + y %3D Determine the equation for the stream function and make sure continuity is satisfied.
2. Consider a stream function given by = (²+x²). (a) Does this flow satisfy conservation of mass? Show your work. (b) Plot the streamlines for this flow. Let K= 2. Be sure to indicate the direction of the flow. (c) Is this flow irrotational? If so, find the velocity potential for this flow. If not, show that a velocity potential does not exist. (d) Describe the flow represented by this stream function.
A two-dimensional flow field has an x-component of velocity given in Cartesian coordinates by u = 2x − 3y. (a) Find v, the y-component of velocity, if the flow is incompressible and v = 0 when x = 0. (b) If the flow follows the Bernoulli equation, find an expression for the pressure distribution as a function of x and y, given that the pressure is p0 at the stagnation point.  
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