Recall from Section 2.12 that an irrotational flow is defined as a flow where the vorticity is zero at every point. From Equation (2.129), for an irrotational flow,
§ = V x V = 0 [2.152]
Consider the following vector identity: if ф is a scalar function, then
V x (V0) = 0 [2.153]
i. e., the curl of the gradient of a scalar function is identically zero. Comparing Equations (2.152) and (2.153), we see that
Equation (2.154) states that for an irrotational flow, there exists a scalar function ф such that the velocity is given by the gradient of ф. We denote ф as the velocity potential. ф is a function of the spatial coordinates; i. e., ф = ф(х, у, z), or ф = ф(г, в, z), or ф = ф(г, в, Ф). From the definition of the gradient in cartesian coordinates given by Equation (2.16), we have, from Equation (2.154),
дф дф дф
мі + uj + шк = -^-i + тр-j + тг“к [2.155]
dx dy dz
The coefficients of like unit vectors must be the same on both sides of Equation
(2.155) . Thus, in cartesian coordinates,
In a similar fashion, from the definition of the gradient in cylindrical and spherical coordinates given by Equations (2.17) and (2.18), we have, in cylindrical coordinates,
and in spherical coordinates,
dф _ 1 dф _ 1 dф
dr 6 г дв Ф r sin# ЗФ
The velocity potential is analogous to the stream function in the sense that derivatives of ф yield the flow-field velocities. However, there are distinct differences between ф and ф (or ф):
1. The flow-field velocities are obtained by differentiating ф in the same direction as the velocities [see Equations (2.156) to (2.158)], whereas ф (or ф) is differentiated normal to the velocity direction [see Equations (2.147) and (2.148), or Equations (2.150) and (2.151)].
2. The velocity potential is defined for irrotational flow only. In contrast, the stream function can be used in either rotational or irrotational flows.
3. The velocity potential applies to three-dimensional flows, whereas the stream function is defined for two-dimensional flows only.2
When a flow field is irrotational, hence allowing a velocity potential to be defined, there is a tremendous simplification. Instead of dealing with the velocity components (say, u, v, and w) as unknowns, hence requiring three equations for these three unknowns, we can deal with the velocity potential as one unknown, therefore requiring the solution of only one equation for the flow field. Once іjr is known for a given problem, the velocities are obtained directly from Equations (2.156) to (2.158). This is why, in theoretical aerodynamics, we make a distinction between irrotational and rotational flows and why the analysis of irrotational flows is simpler than that of rotational flows.
Because irrotational flows can be described by the velocity potential ф, such flows are called potential flows.
In this section, we have not yet discussed how ф can be obtained in the first place; we are assuming that it is known. The actual determination of ф for various problems is discussed in Chapters 3, 6, 11, and 12.