Introduction

Recall that for incompressible, irrotational flow, excellent results could be obtained using the Laplace equation

Where F is defined such that

i.e.

High-speed flows can also be irrotational, so that it should be possible to define a potential, and to describe such flows using a potential equation, and to get valuable insight and calculation techniques from this description.

Objectives

1. Derive the potential equation for compressible flow.
2. Reduce to linearized form for small-perturbation analysis.
3. Apply results to thin airfoils and slender axisymmetric bodies.

Generally, flight vehicles are designed to create as little "perturbation" as possible, because large disturbances cause large drag.
 

We will derive the potential equation for 2-D flows. At the end, it will be obvious how to extend it to 3-D flow, so we will just write down the 3-D equation.

Assume:

1. Isentropic flow.
2. Steady flow.

Momentum Conservation:

u - component

v - component

 

Speed of sound

i.e.

 

Substitute (3a) in (2),

 

For homework,

1. Multiply (4a) by u, and (4b) by v and add. Call this eqn (5)
2. Expand eqn (1) and substitute in eqn (5)

This equation contains derivatives of u and v. If we could define a potential, we could reduce the number of variables.
 

Assume Irrotational Flow

If this is true, then a potential f can be defined such that

i.e.

Substituting in eqn. (6) (as shown in homework)

or using the notation

What is 'a', the local speed of sound? To see this, go to the energy equation for steady adiabatic flow.

or

Thus,

or, 
 
 

Extension to 3-D flow

By inspection, (and by verification by looking in the text), equations (8) and (11) can be written, for 3-D potential flow, as:

where,

Note:

1. This is still an exact equation. We have not made any approximations. In other words, if the flow satisfies our assumptions (steady, irrotational, isentropic), the solution of this will still give exact results. Note the very important distinctions between this and the approximate equations that we will presently derive based on "engineering estimates".
2. We have not made the assumption that disturbances are small; although we have implicitly assumed that the shocks are absent (or quite weak). Thus, this equation can be, and is used to calculate transonic flows over very complex configuration, such as fighter aircraft and rotor blades.
3. This equation is highly non-linear in f .
4. Superposition of solutions will not work.
5. Equally valid for subsonic and supersonic flows.
6. We have used , and