Defining Equation for the Stream Function
Recall from Eq. 7.7 that the continuity equation for axisymmetric flow is:
which may be written as follows:
dux 1 d
Now, we multiply Eq. 7.18 through by r. Because x and r are independent variables, the coefficient r in the first term may be included within the x-derivative, and Eq.
d(^x) + d(™r) = 0
Eq. 7.19 is simply an alternate way of writing the continuity equation Eq. 7.18. Examination of Eq. 7.19 shows that it is satisfied by a scalar function (i. e., the stream function) such that:
л = rux, ^ = – rur, dr x dx r
which defines the stream function, y(x, r) for axisymmetric flow. As in the twodimensional, planar case, the derivative of the stream function yields a velocity component orthogonal to the derivative direction. Because y is related to the volume flow rate, a negative sign goes with the dy/dx expression in Eq. 7.20 so that the radial-velocity ur component has the correct sign to properly reflect continuity.
For an irrotational flow, the curl of the velocity vector (i. e., the vorticity) is zero. Applying the curl operator for axisymmetric flow, it follows that:
Substituting the definition of the stream function, Eq. 7.20, into the irrotationality condition, Eq. 7.21, yields:
T f1T1 Vf f1 ? V 0. (7-22)
dr f r dr ) dx f r dx )
Finally, expanding Eq. 7.22 and rearranging:
d2 у d2 у 1д^_ о dx2 dr2 r dr
Eq. 7.23 is the defining equation for the stream function. If solutions for this equation can be found, then the velocity components can be determined by using Eq. 7.23, and the Bernoulli Equation provides the corresponding pressure distribution.
As in the two-dimensional, planar case (see Chapter 4), Eq. 7.23 is a linear equation so that elementary solutions may be superposed to generate solutions for more complex flows. As before, it is not necessary to solve Eq. 7.23 directly because useful elementary solutions for the stream function may be constructed, as demonstrated later.
Notice that the defining equation for the stream-function equation, Eq. 7.23, is not the Laplace’s Equation in the case of axisymmetric, incompressible, irrotational flow. Compare Eq. 7.23 with Eq. 7.16; the sign before the third term is not the same. Contrast this with two-dimensional, planar, incompressible, irrota – tional flow, where the velocity potential and the stream function both satisfy the Laplace’s Equation.
Following the same procedure used in the planar case, elementary solutions for the stream function are constructed next and then superposed to generate more complex flow fields. The process begins with consideration of a three-dimensional point-source flow. A brief digression follows to examine the flow about a sphere in uniform flow. Then, the source flow and a uniform flow are superposed so as to construct the solution for a stream function that describes the flow around an axisymmetric body.