LINEARIZED COMPRESSIBLE POTENTIAL FLOW

The foregoing treatment based on Prandtl-Meyer and oblique shock relationships is somewhat tedious to apply. Also, the general behavior of supersonic airfoils is not disclosed by this approach. Therefore we will now consider a linearized solution that holds for slender profiles and for Mach numbers that are not too close to unity or not too high.

Assuming that the free-stream velocity is only perturbed by the presence of a slender body at a small angle of attack, the x and у components of the local velocity can be written as

Vx=V~+u Vy = v

д2фс _ 1 д2фі дХ2 ~P дХ2

Substituting this into the left side of Equation 5.56 gives

Since the terms within the parentheses are equal to zero, it follows that Equation 5.58 is a solution of Equation 5.56.

Now consider a body contour У(х). At any point along the contour, the following boundary condition must hold.

dY v dx Vo, + и

_ v

Vac

Equation 5.59 holds to the first order in the perturbation velocities.

Relating v to the incompressible perturbation velocity potential leads to

(5.60)

In the compressible case,

(5.61)

дф^ду can be expanded in a Maclaurin series to give

дфі(х, у) дфі(х, 0) /f £2)

dy ~ ду 1 ‘ ’

Thus, by comparing Equations 5.60 and 5.61, it follows that the body contour for which фс holds is the same (to a first order) as that for </>,.

We are now in a position to determine the pressure distribution for a given slender body shape as a function of Mach number. Along a streamline the resultant velocity, U, in terms of the perturbation velocities, can be written as

U = [(V„+uf+v2im = V» + и (to a first order)

Euler’s equation along a streamline was derived earlier in differential form. Expressed in finite difference form, it can be written as

UAU + ^- = 0

Using Equation 5.63, this becomes

Finally,

p (ll2)pVj

(t/2)PVj

(5.64)

Since и = дфідх, it follows from Equations 5.64 and 5.58 that the pressure distribution over a slender body at a finite subsonic Mach number is related to the pressure distribution over the same body at M = 0 by

(5.65)

This was assumed earlier in this chapter as Equation 5.2.

Thus, to predict the lift and moment on a two-dimensional shape such as an airfoil, one simply calculates these quantities in coefficient form for the incompressible case and then multiplies the results by the factor 1//3.

The three-dimensional case is somewhat more complicated, but not much. Here,

<t>c = – jp4>i(x, f3y, f3z) (5.66)

Hence, to find the compressible flow past a three-dimensional body with coordinates of x, y, and z, one solves for the incompressible flow around a body having the coordinates x, fiy, and jQz. The pressure coefficients are then related by

(5.67)