Fundamentals of Aerodynamics
Fundamentals of Aerodynamics
6th Edition
ISBN: 9781259129919
Author: John D. Anderson Jr.
Publisher: McGraw-Hill Education
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Chapter 3, Problem 3.1P

For an irrotational flow. show that Bernoulli’s equation holds between any points in the flow, not just along a streamline.

Expert Solution & Answer
Check Mark
To determine

To Show: Bernoulli’s equation for irrotational flow.

Answer to Problem 3.1P

 p+12ρV2=constant

Explanation of Solution

Given Information:

Flow type is irrotational.

Calculation:

Considering steady and inviscid flow:

xmomentum: ρuux+ρvuy+ρwuz=px(1)ymomentum:ρuvx+ρvvy+ρwvz=py(2)zmomentum:ρuwx+ρvwy+ρwwz=pz(3) ss

Multiplying (1), (2) and (3) by dx,dyanddzrespectively

 uuxdx+vuydx+wuzdx=1ρpxdx(4)uvxdy+vvydy+wvzdy=1ρpydy(5)uwxdz+vwydz+wwzdz=1ρpzdz(6)

Now, adding equation 4,5 and 6

u(uxdx+vxdy+wxdz)+v(uydx+vydy+vxdy)+w(uzdx+vzdy+wxdz)=1ρ(pxdx+pydy+pzdz)(7)

We know that for irrotational flow ×V=0

hencewy=vz;uz=wx;vx=uy(8)

Substituting Equation (8) into (7) we get

udu+vdv+wdw=1ρdp

12d(u2+v2+w2)=12d(V2)=VdV=1ρdρdp=ρVdVwhichintegratesto  p+12ρV2=constant

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