Governing Equation for Irrotational, Incompressible Flow: Laplace’s Equation
We have seen in Section 3.6 that the principle of mass conservation for an incompressible flow can take the form of Equation (3.39):
V • V = 0 [3.39]
In addition, for an irrotational flow we have seen in Section 2.15 that a velocity potential ф can be defined such that [from Equation (2.154)]
Therefore, for a flow that is both incompressible and irrotational, Equations (3.39) and (2.154) can be combined to yield
V • (V</>) = 0
Equation (3.40) is Laplace’s equation—one of the most famous and extensively studied equations in mathematical physics. Solutions of Laplace’s equation are called harmonic functions, for which there is a huge bulk of existing literature. Therefore, it is most fortuitous that incompressible, irrotational flow is described by Laplace’s equation, for which numerous solutions exist and are well understood.
For convenience, Laplace’s equation is written below in terms of the three common orthogonal coordinate systems employed in Section 2.2:
From Equations (3.40) and (3.46), we make the following obvious and important conclusions:
1. Any irrotational, incompressible flow has a velocity potential and stream function (for two-dimensional flow) that both satisfy Laplace’s equation.
2. Conversely, any solution of Laplace’s equation represents the velocity potential or stream function (two-dimensional) for an irrotational, incompressible flow.
Note that Laplace’s equation is a second-order linear partial differential equation. The fact that it is linear is particularly important, because the sum of any particular solutions of a linear differential equation is also a solution of the equation. For example, if ф], <p2, 03, …, фп represent n separate solutions of Equation (3.40), then the sum
Ф — Ф + Фі + • • • + Фп
is also a solution of Equation (3.40). Since irrotational, incompressible flow is governed by Laplace’s equation and Laplace’s equation is linear, we conclude that a complicated flow pattern for an irrotational, incompressible flow can be synthesized by adding together a number of elementary flows that are also irrotational and incompressible. Indeed, this establishes the grand strategy for the remainder of our discussions on inviscid, incompressible flow. We develop flow-field solutions for several different elementary flows, which by themselves may not seem to be practical flows in real life. However, we then proceed to add (i. e., superimpose) these elementary flows in different ways such that the resulting flow fields do pertain to practical problems.
Before proceeding further, consider the irrotational, incompressible flow fields over different aerodynamic shapes, such as a sphere, cone, or airplane wing. Clearly, each flow is going to be distinctly different; the streamlines and pressure distribution over a sphere are quite different from those over a cone. However, these different flows are all governed by the same equation, namely, V2</> = 0. How, then, do we obtain different flows for the different bodies? The answer is found in the boundary conditions. Although the governing equation for the different flows is the same, the boundary conditions for the equation must conform to the different geometric shapes, and hence yield different flow-field solutions. Boundary conditions are therefore of vital concern in aerodynamic analysis. Let us examine the nature of boundary conditions further.
Consider the external aerodynamic flow over a stationary body, such as the airfoil sketched in Figure 3.18. The flow is bounded by (1) the freestream flow that occurs (theoretically) an infinite distance away from the body and (2) the surface of the body itself. Therefore, two sets of boundary conditions apply as follows.