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potential theory is not valid there and this point has to be excluded when carrying out integrations in the ‡uid. The potential and stream functions of a circulation with tangential velocities in a clockwise direction are given by:¯¯¯¯ © = ¡ ¡ 2¼ ¢ µ ¯¯¯¯ (a) and ¯¯¯¯ ª = + ¡ 2¼ ¢ ln r ¯¯¯¯ (b) (3.72) or in Cartesian coordinates by: 3-16 CHAPTER 3. CONSTANT POTENTIAL FLOW PHENOMENA © = ¡ ¡ 2¼ ¢ arctan ³ y x ´ (a) and ª = + ¡ 2¼ ¢ ln p x2 + y2 (b) (3.73) Note that a circulation ‡ow is a counterpart of a source or sink ‡ow; © and ª have similar forms but they are exchanged. 3.5 Superposition of Basic Elements While each of the ‡ow patterns from the four potential ‡ow elements treated above can be quite easily visualized, they are not all that common, and a lot of interesting ‡ow patterns are much more complex as well. The ’power’ of potential theory lies in the fact that because of its linearity superposition (or ’adding up’) of ‡ow pattern components may be used to generate more complex ‡ow patterns. This section concentrates on producing some of these more complex and interesting ‡ow patterns. 3.5.1 Methodology Superposition may be carried out ’by hand’ using the following 5 steps; they are then most suited for simple cases. Superposition of only a few elements can be carried out by hand. A. Place each potential ‡ow element at its desired x; y location. B. Draw the ‡ow pattern from each element individually - without regard to the others. C. Assign appropriate numerical values to each (independent) stream line. D. The resulting stream function value at any point x; y can be found by adding the stream function values from each of the components at that point. (This is easiest at locations where streamlines from the various elements cross each other. Otherwise, a certain amount of interpolation may be needed to carry this out.) This yields a …eld of resulting stream function values, each associated with a particular location, x; y: E. Lines can now be sketched which connect locations having the same stream function values. This is much like drawing contour lines in map-making or a dot-to-dot drawing as in elementary school! Interpolation may again be necessary in this sketching process, however. These steps are illustrated in …gure 3.7 in which a uniform ‡ow (from left to right) is combined with source ‡ow. Parts A and B of the …gure show half-planes of the two independent potential ‡ow elements. These are superposed in part C of the …gure thus completing the …rst three steps above. The results of the addition work are shown in part D; part E of the …gure shows the resulting set of streamlines. Whenever a larger number of elements are to be combined, the interpolation work involved in carrying out the above summations becomes too cumbersome for ’handwork’. A slightly di¤erent approach is most appropriate for computer processing. Choose a single, X;Y coordinate system. 3.5. SUPERPOSITION OF BASIC ELEMENTS 3-17 Figure 3.7: Superposition Steps Carried out by Hand 3-18 CHAPTER 3. CONSTANT POTENTIAL FLOW PHENOMENA Place each of the potential ‡owelements at its desired X;Y location. Formulas are available to express each element’s stream and potential function in terms of its own (local) x; y coordinates. Transform the coordinates for each ‡ow element so that its contribution is expressed in terms of X and Y coordinates. These formulas will be used in step 5, below. Choose a rectangular grid of points in the X;Y coordinate system at which the functions are to be evaluated. The area covered by these points should be the area for which the stream and potential lines are to be plotted. A computer program - even a spreadsheet program will do - can be used to compute the relevant function value at each grid point. Even a modest, but modern PC can quickly generate values on a grid of 50 by 50 points. The resulting data table can then be transferred to a 3-D plotting program to generate contour lines of stream or potential values as a function of X and Y: These are the desired product. Obviously, this method is not really hindered by the number of ‡ow elements to be included; the formulas only become a bit longer in the program. Because the plotting program will interpolate anyway, it is no longer important that the computed grid point data have ’convenient’ values. The rest of this section details a number of simple and useful potential ‡ow element com- binations. 3.5.2 Sink (or Source) in Uniform Flow This …rst example - still a bit simple - could represent the ‡ow of oil to a well drilled into a reservoir through which the oil is ‡owing slowly. This situation has the opposite sense of the one used to illustrate the method, above; now, the ‡ow pattern is determined by adding a sink to a uniform ‡ow which goes from right to left. Using equations 3.52 and 3.66: ª = ¡ Q 2¼ ¢ arctan ³ y x ´ ¡ U1 ¢ y (3.74) Figure 3.8 shows the curve separating the ‡ow going to the sink from that passing (or ’escaping’) the sink. All ‡ow originating between the two asymptotes gets ’caught’ by the sink. Obviously, if the sink were to be replaced by a source, the ‡ow coming from the source would then remain within the asymptotes as well. 3.5.3 Separated Source and Sink Consider next a source and a sink with equal and opposite intensity, Q 2¼ , separated by a distance of 2s: Once again, superposition is used to yield - using equations 3.66: ªsource = Q 2¼ ¢ µ1 = Q2¼ ¢ arctan µ y x1 ¶ for the source (3.75) ªsink = ¡Q 2¼ ¢ µ2 = ¡Q 2¼ ¢ arctan µ y x2 ¶ for the sink (3.76) 3.5. SUPERPOSITION OF BASIC ELEMENTS 3-19 Figure 3.8: Sink in a Uniform Flow The resulting stream lines are then (in polar coordinates for a change) lines with: ª = Q 2¼ ¢ µ1 ¡ Q 2¼ ¢ µ2 = constant (3.77) These stream lines are a set of circles, all centered on the y axis, and all passing through source and sink; see …gure 3.9. Figure 3.9: Separated Source and Sink One can also write the resultant stream function in Cartesian coordinates as: ª = Q 2¼ ¢ arctan µ 2 y s x2+ y2 ¡ s2 ¶ (3.78) 3-20 CHAPTER 3. CONSTANT POTENTIAL FLOW PHENOMENA 3.5.4 Source and Sink in Uniform Flow The stream function is, in this case: ª = Q 2¼ ¢ arctan µ 2 y s x2 + y2 ¡ s2 ¶ + U1 y (3.79) The stream lines that one obtains now are shown in …gure 3.10. Notice that the ellipse which surrounds the source and the sink (drawn a bit heavier in the …gure) is a stream line; no ‡ow takes place through that ellipse. The ‡ow from source to sink stays inside; the constant current ‡ow stays outside and passes around the form. The physical interpretation of this is that one could obtain the same ‡ow by replacing that ellipse with an impermeable object in a uniform ‡ow. Figure 3.10: Source and Sink in Uniform Flow This approach can be extended even further as explained in the next section. 3.5.5 Rankine Ship Forms The Englishman W.J.M. Rankine extended the above-mentioned approach about 1870. He did this by including additional, matched pairs of sources and sinks along the x axis in the uniform ‡ow. They were always located symmetrically about the coordinate origin. By giving each matched source-sink pair its own spacing and strength (relative to the uniform ‡ow), he was able to generate a fatter or thinner, or more blunt or pointed shape; it was always symmetric with respect to both the x and y axes, however. Use of relatively weaker source-sink pairs near the ends of the shape make it more ’pointed’. Since these forms can be made to somewhat resemble a horizontal slice of a ship (neglecting the fact that the stern has a bit di¤erent shape from the bow), they came to be know as Rankine ship forms. Flow computations with such forms are quite simple to carry