Livro do Journee - Offshore Hydromechanics

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