EBK FUNDAMENTALS OF AERODYNAMICS
EBK FUNDAMENTALS OF AERODYNAMICS
6th Edition
ISBN: 9781259681486
Author: Anderson
Publisher: MCGRAW HILL BOOK COMPANY
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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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