Small perturbation potential theory:
THE VELOCITY POTENTIAL EQUATION
The inviscid, compressible, subsonic flow over a body immersed in a uniform stream is irrotational; there is no mechanism in such a flow to start rotating the fluid elements (see Sec. 2.12). Thus, a velocity potential (see Sec. 2.15) can be defined. Since we are dealing with irrotational flow and the velocity potential, review Sec. 2.12 and 2.15 before progressing further.
Let us proceed to obtain an equation for c/J which represents a combination of the continuity, momentum, and energy equations. Such an equation would be very useful, because it would be simply one governing equation in terms of one unknown, namely f.
The continuity equation for steady, two-dimensional flow is obtained from Eq. (2.43) as
This equation holds for a steady, compressible, inviscid flow and relates p and V along a streamline. It can readily be shown that Eq. (3.12) holds in any direction throughout an irrotational flow, not just along a streamline (try it yourself). Therefore, from Eqs. (3.12) and (11.2a and b), we have
Since a0 is a known constant of the flow, Eq. (11.13) gives the speed of sound, a, as a function of f. Hence, substitution of Eq. (11.13) into (11.12) yields a single partial differential equation in terms of the unknown ¢. This equation represents a combination of the continuity, momentum, and energy equations. In principle, it can be solved to obtain 1> for the flow field around any two-dimensional shape, subject of course to the usual boundary conditions at infinity and along the body surface. These boundary conditions on f are detailed in Sec. 3.7, and are given by Eqs. (3.47a and b) and (3.48b).
Because Eq. (11.12) [along with Eq. (11.13)] is a single equation in terms of one dependent variable, f, the analysis of isentropic, irrotational, steady, compressible flow is greatly simplified-we only have to solve one equation instead of three or more. Once ¢ is known, all the other flow variables are directly
obtained as follows:
Although Eq. (11.12) has the advantage of being one equation with one unknown, it also has the distinct disadvantage of being a nonlinear partial differential equation. Such nonlinear equations are very difficult
to solve analytically, and in modern aerodynamics, solutions of Eq. (11.12) are usually sought by means of sophisticated finite-difference numerical techniques. Indeed, no general analytical solution of Eq. (11.12) has been found to this day. Contrast this situation with that for incompressible flow, which is governed by Laplace’s equation-a linear partial differential equation for which numerous analytical solutions are well known.
Given this situation, aerodynamicists over the years have made assumptions regarding the physical nature of the flow field which are designed to simplify Eq. (11.12). These assumptions limit our considerations to the flow over slender bodies at small angles of attack. For subsonic and supersonic flows, these assumptions lead to an approximate form of Eq. (11.12) which is linear, and hence can be solved analytically. These matters are the subject of the next section.
Keep in mind that, within the framework of steady, irrotational, isentropic flow, Eq. (11.12) is exact and holds for all Mach numbers, from subsonic to hypersonic, and for all two-dimensional body shapes, thin and thick.
LINEARIZED VELOCITY POTENTIAL EQUATION:
Consider the two-dimensional, irrotational, isentropic flow over the body shown in Fig. 11.2. The body is immersed in a uniform flow with velocity V¥ oriented in the x direction. At an arbitrary point P in the flow field, the velocity is V with the x and y components given by u and v, respectively. Let us now visualize the velocity V as the sum of the uniform flow velocity plus some extra increments in velocity. For example, the x component of velocity, u, in Fig. 11.2 can be visualized as V¥ plus an increment in velocity (positive or negative). Similarly, the y component of velocity, v, can be visualized as a simple increment itself, because the uniform flow has a zero component in the y direction. These increments are called perturbations, and
In contrast, Eq. (11.18) is not valid for thick bodies and for large angles of attack. Moreover, it cannot be used for transonic flow, where 0.8 < M < 1.2, or for hypersonic flow, where M > 5.
We are interested in solving Eq. (11.18) in order to obtain the pressure distribution along the surface of a slender body. Since we are now dealing with approximate equations, it is consistent to obtain a linearized expression for the pressure coefficient-an expression which is approximate to the same degree as Eq. (11.18), but which is extremely simple and convenient to use. The linearized pressure coefficient can be derived as follows.
First, recall the definition of the pressure coefficient Cp given in Sec. 1.5: