Supersonic Flow Past a Wave-Shaped Wall

Suppose that the unperturbed flow in the problem considered in the previous section is supersonic, so that ${\rm Ma}_\infty^{\,2}-1\equiv \lambda^{\,2} > 0$ . The perturbed flow is governed by Equation (15.127), which is now of the hyperbolic type

$$\frac{\partial^{\,2}\phi}{\partial x^{\,2}} -\frac{1}{\lambda^{\,2}}\,\frac{\partial^{\,2}\phi}{\partial y^{\,2}}\simeq 0$$ (15.151)

(Arfken 1985). As is easily verified, the previous equation has the general solution

$$\phi(x,y)= f(x-\lambda\,y) + g(x+\lambda\,y),$$ (15.152)

where $f$ and $g$ are arbitrary functions (Fitzpatrick 2013). It is clear that $f$ is constant along lines $x-\lambda\,y={\rm const.}$ , whereas $g$ is constant along lines $x+\lambda\,y={\rm const.}$ . These lines are inclined at the Mach angle, $\mu=\cot^{-1}[({\rm Ma}_\infty^{\,2}-1)^{\,1/2}]$ , to the undisturbed flow. They are, in fact, the Mach lines, or characteristics, of the unperturbed flow. (See Section 15.3.) The former characteristics are inclined downstream. In other words, they originate at the wall. The latter characteristics are inclined upstream. In other words, they originate at infinity. Because there are no sources at infinity, the latter characteristics carry no perturbation, which implies that $g=0$ .

The boundary condition at the wall, (15.129), yields

$$-\lambda\,f'(x)=-U\,\epsilon\,\alpha\,\cos(\alpha\,x),$$ (15.153)

where $'$ denotes derivative with respect to argument. It follows that

$$f(x) = \frac{U\,\epsilon}{\lambda}\,\sin(\alpha\,x),$$ (15.154)

and, hence, that

$$\phi(x, y) = f(x-\lambda\,y) = \frac{U\,\epsilon}{\lambda}\,\sin[\alpha\,(x-\lambda\,y)],$$ (15.155)

$$u' =-\frac{\partial\phi}{\partial x} = -\frac{U\,\epsilon\,\alpha}{\lambda}\,\cos[\alpha\,(x-\lambda\,y)],$$ (15.156)

$$v' =- \frac{\partial\phi}{\partial y}=U\,\epsilon\,\alpha\,\cos[\alpha\,(x-\lambda\,y)].$$ (15.157)

Note that, unlike the case of subsonic flow, the amplitude of the perturbed velocity does not decrease with increasing distance from the wall.

The pressure coefficient at the wall is

$$C_p \simeq \frac{2}{U}\left.\frac{\partial\phi}{\partial x}\right... ... = \frac{2\,\epsilon\,\alpha}{\sqrt{{\rm Ma}_\infty^{\,2}-1}}\,\cos(\alpha\,x).$$ (15.158)

It can be seen, by comparison with Equation (15.145), that the maxima and minima of the pressure are now phase-shifted by $\pi/2$ radians with respect to the corresponding maxima and minima in the subsonic case. Hence, the pressure distribution at the wall is anti-symmetric with respect to the crests and troughs of the wall. Consequently, a net drag force is exerted on the wall. The mean coefficient of drag per wavelength is [see Equation (15.56)]

$$C_D=\frac{1}{l}\int_0^lC_p\,\frac{dy}{dx}\,dx,$$ (15.159)

which is obtained by replacing the sine of the slope of the wall by the tangent, $dy/dx$ . This approximation is valid within the context of the small-perturbation theory. However, according to Equations (15.130) and (15.158), $C_p$ can be written

$$C_p =\frac{2}{\sqrt{{\rm Ma}_\infty^{\,2}-1}}\,\frac{dy}{dx},$$ (15.160)

which implies that

$$C_D = \frac{2}{\sqrt{{\rm Ma}_\infty^{\,2}-1}}\,\overline{\left(\frac{dy}{dx}\right)^2}.$$ (15.161)

Here, the bar denotes a period average of the form

$$\overline{\left(\frac{dy}{dx}\right)^2}=\frac{1}{l}\int_0^l\left(\frac{dy}{dx}\right)^2 dx.$$ (15.162)

Equations (15.160) and (15.161) are valid for any small-amplitude, periodic modulation of the wall. For the particular sinusoidal modulation under consideration in this section, we find that

$$C_D = \frac{(\epsilon\,\alpha)^{\,2}}{\sqrt{{\rm Ma}_\infty^{\,2}-1}}.$$ (15.163)

The discussion of the range of validity of the approximations used in deriving the previous results follows the same lines as in the subsonic case.