Fundamentals of Aerodynamics
Fundamentals of Aerodynamics
6th Edition
ISBN: 9781259129919
Author: John D. Anderson Jr.
Publisher: McGraw-Hill Education
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Chapter 4, Problem 4.9P

Starting with Equations ( 4 . 35 ) and ( 4 . 43 ) , derive Equation ( 4 . 62 ) .

Expert Solution & Answer
Check Mark
To determine

To derive the equation cm,le=π2(A0+A1A22) using the given equations.

Answer to Problem 4.9P

The moment coefficient about the leading edge cm,le=π2(A0+A1A22)

Explanation of Solution

Given equation:

  γ(θ)=2V(A0( 1+cosθ)sinθ+n=1Ansin nθ) and,

The total moment about the leading edge, MLE'=ρV0cξγ(ξ)dξ

Calculation:

The moment coefficient about the leading edge can be written as,

  cm,le=M LE'qc2or,qc2×cm,le=MLE'

Substitute the value of MLE' in the above equation,

  qc2×cm,le=ρV0cξγ(ξ)dξor,ρV2c22×cm,le=ρV0cξγ(ξ)dξVc22×cm,le=0cξγ(ξ)dξ______(1)

Here, ξ =c2(1-cosθ),dξ =c2sinθ.dθand,

  γ(θ)=2V(A0( 1+cosθ)sinθ+n=1Ansin nθ)

Substituting all these values in equation (1),

  Vc22×cm,le=0πc2( 1-cosθ) .2V( A 0 ( 1+cosθ ) sinθ + n=1 A n sin nθ )c2sinθ.dθVc22×cm,le=c2V20π[A0 ( 1- cos 2 θ ) sinθ+ n=1 A n ( 1-cosθ )sin nθ] sinθ.dθcm,le=0πA0( 1- cos2θ)dθ +0π n=1 A n ( 1-cosθ )sin nθ sinθ.dθ

By integrating the above equation, the equation obtained is as follows:

  cm,le=π2A0π2A1+π4A2cm,le=π2(A0 +A1 A 2 2)

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