Derivation of the Linearized Supersonic Pressure Coefficient Formula

For the case of supersonic flow, let us write Equation (11.18) as

л^-4=о

Эх2 dy2

where Л = J— 1. A solution to this equation is the functional relation

Ф — f(x-ky) [12.2]

We can demonstrate this by substituting Equation (12.2) into Equation (12.1) as follows. The partial derivative of Equation (12.2) with respect to x can be written as

З ф. d(x — ky)

= / (x — ky)——- —

dx dx

In Equation (12.3), the prime denotes differentiation of / with respect to its argument, x — ky. Differentiating Equation (12.3) again with respect to x, we obtain

d-±= f" dx2 1

Similarly,

Substituting Equations (12.4) and (12.6) into (12.1), we obtain the identity

Xі f" – Xі f" = 0

Hence, Equation (12.2) is indeed a solution of Equation (12.1).

Examine Equation (12.2) closely. This solution is not very specific, because / can be any function of x — Xy. However, Equation (12.2) tells us something specific about the flow, namely, that ф is constant along lines of л – Xy = constant. The slope of these lines is obtained from

x — Xy — const

Hence, ± = ! =_______ ‘___

dx x !Mlc -1

From Equation (9.31) and the accompanying Figure 9.25, we know that

tan/r = , : [12.8]

where p. is the Mach angle. Therefore, comparing Equations (12.7) and (12.8), we see that a line along which ф is constant is a Mach line. This result is sketched in Figure 12.1, which shows supersonic flow over a surface with a small hump in the middle, where 9 is the angle of the surface relative to the horizontal. According to Equations (12.1) to (12.8), all disturbances created at the wall (represented by the perturbation potential ф) propagate unchanged away from the wall along Mach waves. All the Mach waves have the same slope, namely, dy/dx — (M^. — 1)~1/2. Note that the Mach waves slope downstream above the wall. Hence, any disturbance at the wall cannot propagate upstream; its effect is limited to the region of the flow downstream of the Mach wave emanating from the point of the disturbance. This is a further substantiation of the major difference between subsonic and supersonic flows mentioned in previous chapters, namely, disturbances propagate everywhere throughout a subsonic flow, whereas they cannot propagate upstream in a steady supersonic flow.

Keep in mind that the above results, as well as the picture in Figure 12.1, pertain to linearized supersonic flow [because Equation (12.1) is a linear equation]. Hence, these results assume small perturbations; that is, the hump in Figure 12.1 is small,

and thus в is small. Of course, we know from Chapter 9 that in reality a shock wave will be induced by the forward part of the hump, and an expansion wave will emanate from the rearward part of the hump. These are waves of finite strength and are not a part of linearized theory. Linearized theory is approximate; one of the consequences of this approximation is that waves of finite strength (shock and expansion waves) are not admitted.

The above results allow us to obtain a simple expression for the pressure coeffi­cient in supersonic flow, as follows. From Equation (12.3),

and from Equation (12.5),

~ 9<^ і f’

v = — = – A./
dy

Eliminating /’ from Equations (12.9) and (12.10), we obtain

Figure 1 2.2 Variation of the linearized pressure coefficient with Mach number (schematic).

portion. This is denoted by the (+) and (—) signs in front of and behind the hump shown in Figure 12.1. This is also somewhat consistent with our discussions in Chapter 9; in the real flow over the hump, a shock wave forms above the front portion where the flow is being turned into itself, and hence p > whereas an expansion wave occurs over the remainder of the hump, and the pressure decreases. Think about the picture shown in Figure 12.1; the pressure is higher on the front section of the hump, and lower on the rear section. As a result, a drag force exists on the hump. This drag is called wave drag and is a characteristic of supersonic flows. Wave drag was discussed in Section 9.7 in conjunction with shock-expansion theory applied to supersonic airfoils. It is interesting that linearized supersonic theory also predicts a finite wave drag, although shock waves themselves are not treated in such linearized theory.

Examining Equation (12.15), we note that Cp oc (Af£, — l)-l/2; hence, for su­personic flow, Cp decreases as M0c increases. This is in direct contrast with subsonic flow, where Equation (11.51) shows that Cp cx (1 — M^,)^1/2; hence, for subsonic flow, Cp increases as M^ increases. These trends are illustrated in Figure 12.2. Note that both results predict Cp —>■ oo as M —> 1 from either side. However, keep in mind that neither Equation (12.15) nor (11.51) is valid in the transonic range around Mach 1.