13 - Unsteady Incompressible Potential Flow

Summary

We have seen in the previous chapters that in an incompressible, irrotational fluid the velocity field can be obtained by solving the continuity equation. However, the incompressible continuity equation does not directly include time-dependent terms, and the time dependency is introduced through the boundary conditions. Therefore, the first objective is to demonstrate that the methods of solution that were developed for steady flows can be used with only small modifications. These modifications will include the treatment of the “zero normal flow on a solid surface” boundary conditions and the use of the unsteady Bernoulli equation. Furthermore, as a result of the nonuniform motion, the wake becomes more complex than in the corresponding steady flow case and it should be properly accounted for. Consequently, this chapter is divided into three parts, as follows:

1. a. Formulation of the problem and of the proposed modifications for converting steady-state flow methods to treat unsteady flows (Sections 13.1–13.6).

2. b. Examples of converting analytical models to treat time-dependent flows (e.g., thin lifting airfoil and slender wing in Sections 13.8–13.9).

3. c. Examples of converting numerical models to treat time-dependent flows (Sections 13.10–13.13).

For the numerical examples only the simplest models are presented; however, application of the approach to any of the other methods of Chapter 11 is strongly recommended (e.g., can be given as a student project).

In the general case of the arbitrary motion of a solid body submerged in a fluid (e.g., a maneuvering wing or aircraft) the motion path is determined by the combined dynamic and fluid dynamic equations.