Acceleration Potential

Another useful potential function which is used in aerodynamics is the acceleration potential. If we recall the momentum equation for barotropic flows:

Dq

Dt

As seen in the left hand side of the equation, the material derivative of the velocity vector is obtained from the gradient of a function of pressure and density only. Hence, we can define the acceleration potential as follows

Dq

Dt

As a result of last line the momentum equation reads as,

vw + v Z — = 0 J p

The integral form of the last equation becomes

W = -/ ^ + F(t)

p

The pressure term integrated at the right hand side of the equation from free stream to the point under consideration gives,

w = Pi—p p

Because of the direct relation between the pressure and the acceleration potential, this potential is also called the pressure integral. Let us rewrite the Kelvin’s equation in gradient form

We can now find the relation between the velocity potential and the accelera­tion potential as follows

The integral of the last equation

00/+qr – w=F(t)

Once again if we choose F(t) = U2/2 we can satisfy the flow conditions at infinity. Hence, the acceleration potential becomes,

, 0/ q2 U2

w = а7 + Т – IT

With small perturbation approach, the linear form of the last line reads

If the linear operator

operates on Eq. 2.24b to give

Interchanging the operators and utilizing Eq. 2.25 gives us the final form of the equation for the acceleration potential

(2.26)