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 Section 2.12). Thus, a velocity potential (see Section 2.15) can be defined. Since we are dealing with irrotational flow and the velocity potential, review Sections 2.12 and 2.15 before progressing further.

Consider two-dimensional, steady, irrotational, isentropic flow. A velocity po­tential, ф = <p(x, y), can be defined such that [from Equation (2.154)]

V = V0 [11.1]

or in terms of the cartesian velocity components,

Эф r,

u = —— [11.2a]

dx

_ 9 ф

V dy

Let us proceed to obtain an equation for ф 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 the velocity potential ф.

The continuity equation for steady, two-dimensional flow is obtained from Equa­tion (2.52) as

We are attempting to obtain an equation completely in terms of </>; hence, we need to eliminate p from Equation (11.5). To do this, consider the momentum equation in terms of Euler’s equation:

dp = —pV dV

This equation holds for a steady, compressible, inviscid flow and relates p and V along a streamline. It can readily be shown that Equation (3.12) holds in any direction throughout an irrotational flow, not just along a streamline (try it yourself). Therefore, from Equations (3.12) and (11.2a and b), we have

dp = – pVdV = ~^d(V2) = ~^d(u2 + v2)

Recall that we are also considering the flow to be isentropic. Hence, any change in pressure dp in the flow is automatically accompanied by a corresponding isentropic change in density dp. Thus, by definition

dp dp)s

The right-hand side of Equation (11.7) is simply the square of the speed of sound. Thus, Equation (11.7) yields

[1 1.8]

Substituting Equation (11.8) for the left side of Equation (11.6), we have

Considering changes in the x direction, Equation (11.9) directly yields

Similarly, for changes in the у direction, Equation (11.9) gives

Эр = p /дф д2ф дф д2ф пі ці

Эу а2 Эт Эх ду ду ду2 )

Substituting Equations (11.10) and (11.11) into (11.5), canceling the p which appears in each term, and factoring out the second derivatives of ф, we obtain

[1 1.12]

which is called the velocity potential equation. It is almost completely in terms of ф only the speed of sound appears in addition to ф. However, a can be readily expressed in terms of ф as follows. From Equation (8.33), we have

Since ao is a known constant of the flow, Equation (11.13) gives the speed of sound a as a function of ф. Hence, substitution of Equation (11.13) into (11.12) yields a single partial differential equation in terms of the unknown ф. This equation represents a combination of the continuity, momentum, and energy equations. In principle, it can be solved to obtain ф 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 ф are detailed in Section 3.7, and are given by Equations (3.47a and b) and (3.48Z?).

Because Equation (11.12) Lalong with Equation (11.13)] is a single equation in terms of one dependent variable ф, the analysis of isentropic, irrotational, steady, compressible flow is greatly simplified—we only have to solve one equation instead of three or more. Once ф is known, all the other flow variables are directly obtained as follows:

1. Calculate и and v from Equations (11.2a and b).

2. Calculate a from Equation (11.13).

3. Calculate M = V/a — u2 + v2/a.

4. Calculate T, p, and p from Equations (8.40), (8.42), and (8.43), respectively. In these equations, the total conditions Tq, po, and po are known quantities; they are constant throughout the flow field and hence are obtained from the given freestream conditions.

Although Equation (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 modem aerodynamics, solutions of Equation (11.12) are usually sought by means of sophisticated finite-difference numerical techniques. Indeed, no general analytical solution of Equation (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 Equation

(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 Equation (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, Equation (11.12) is exact and holds for all Mach numbers, from subsonic to hyper­sonic, and for all two-dimensional body shapes, thin and thick.