Superposition of Elementary Solutions
Although a superposition can be carried out using either the elementary velocity potentials or stream functions, the superpositions discussed here are in terms of the stream function. The advantage of using the stream function is that the streamlines of the complicated flow then are generated automatically; recall that lines of у = constant are streamlines. In principle, for any flow generated by superposition, we can choose different constant values for the stream function; find pairs of values of (x, y) or (r,0), which make the superposed stream function equal to this constant; plot these coordinate pairs; and then find the streamlines by “joining the dots” (e. g., with a contour-plotting software package). In some cases, the streamline pattern may be found analytically from the superposition expression.
Uniform Flow Plus Source at Origin
This superposition uses the two solutions, Eqs. 4.29 and 4.32. Namely:
У = У UF + ¥s = V y+2Пtan 1 xy. (4.39)
For an illustrative flow problem, V and Л are constants that may be chosen arbitrarily. The streamlines generated by joining the dots (as previously explained) can be examined by running the software Program PSI.
The following comments are pertinent to running Program PSI:
1. The flow is symmetrical about the x-axis. One streamline with a value у = —
passes through a stagnation point located on the x-axis. This may be seen by differentiating Eq. 4.39 to find u, v, and then setting these two velocity components equal to zero (because at a stagnation point, V = 0) and solving for (x, y). The result is:
(The student should verify this result.) The value of this x, y pair, when substituted into Eq. 4.39, yields the value of у on that stagnation streamline.
2. Recall that any streamline can be “cross-hatched” mentally to represent the surface of a solid body. The streamline passing through the stagnation point thus may be thought of as the surface of an open-ended (i. e., semi-infinite) body opening to the right. Vary the source strength, Л, and the freestream velocity values in the program to verify that the body shape changes as anticipated. Note that the location of the stagnation point along the x-axis changes as these parameters are varied. This is to be anticipated because the stagnation point occurs when the oncoming freestream flow is just balanced by the opposing source-flow streamline directed upstream along the x-axis.
3. Notice that the point source is located at the origin and, hence, inside the open-ended body. Thus, the fact that the velocity is infinite at the source is of no concern because the source is not within the external flow field around the body.
4. One of the other streamlines approaching the body in the second or third quadrants may be “cross-hatched.” This streamline describes the flow along a plain and over a hill of continually increasing elevation.
5. Note that although the body shape (or hill) may be varied by varying the flow parameters, as in comment (2), the body shape comes out of the solution. Thus, we cannot easily specify a certain body shape in advance—“You get what you get.”