Potential Flow

Potential flow is based on the concept that the flow field can be represented by a potential function ф such that:

(2.35)

This linear partial differential equation is popularly known as Laplace equation. Derivatives of ф with respect to the space coordinates x, y and z give the velocity components Vx, Vy and Vz, respectively, along x-, y – and z – directions. Unlike the stream function ф, the potential function can exist only if the flow is irrotational, that is, when viscous effects are absent. All inviscid flows must satisfy the irrotationality condition:

(2.36)

For two-dimensional potential flows, by Equation (2.30), we have the vorticity Z as:

dV dV

Zz = — —x = 0.

dx dy

Using Equation (2.33), we get the vorticity as:

This shows that the flow is irrotational. For two-dimensional incompressible flows, the continuity equation is:

V V =

dx dy

In terms of the potential function ф, this becomes:

d2 ф d2 ф

—— – I—— = 0

dx2 dy2

that is:

V2ф = 0.

This linear equation is the governing equation for potential flows. For potential flows, the Navier-Stokes equations (2.23) reduce to:

Equation (2.37) is known as Euler’s equation.

At this stage, it is natural to have the following doubts about the streamline and potential function, because we defined the streamline as an imaginary line in a flow field and potential function as a mathe­matical function, which exists only for inviscid flows. The answers to these vital doubts are the following:

• Among the graphical representation concepts, namely the pathline, streakline and streamline, only the first two are physical, and the concept of streamline is only hypothetical. But even though imaginary, the streamline is the only useful concept, because it gives a mathematical representation for the flow field in terms of stream function ф, with its derivatives giving the velocity components. Once the velocity components are known, the resultant velocity, its orientation, the pressure and temperature associated with the flow can be determined. Thus, streamline plays a dominant role in the analysis of fluid flow.

• Knowing pretty well that no fluid is inviscid or potential, we introduce the concept of potential flow, because this gives rise to the definition of potential function. The derivatives of potential function with the spatial coordinates give the velocity components in the direction of the respective coordinates and the substitution of these velocity components in the continuity equation results in Laplace equation. Even though this equation is the governing equation for an impractical or imaginary flow (inviscid flow), the fundamental solutions of Laplace equation form the basis for both experimental and computational flow physics. The basic solutions for the Laplace equation are the uniform flow, source, sink and free or potential vortex. These solutions being potential can be superposed to get the mathematical functions representing any practical geometry of interest. For example, superposition of a doublet (source and a sink of equal strength in proximity) and uniform flow would represent flow past a circular cylinder. In the same manner, suitable distribution of source and sink along the camberline and superposition of uniform flow over this distribution will mathematically represent flow past an aerofoil. Thus, any practical geometry can be modeled mathematically, using the basic solutions of the Laplace equation.