Two-Dimensional Potential Flows
4.1 STREAM FUNCTION
The velocity potential f was de¢ned in such a way that it automatically satis¢ed the condition of irrotationality. The continuity equation then showed that f had to be a solution of Laplace’s equation. A second function may be de¢ned by a complementary procedure for two-dimensional incom- pressible £uid £ows. That is, a function may be de¢ned in such a way that it automatically satis¢es the continuity equation, and the equation it must satisfy will be determined by the condition of irrotationality.
The continuity equation, in cartesian coordinates, for the £ow ¢eld under consideration is
@u
@xþ @v@y¼ 0
Now introduce a function c that is de¢ned as follows:
u¼ @c
@y ð4:1aÞ
v ¼ @c@x ð4:1bÞ
With this de¢nition, the continuity equation is satis¢ed identically for all functions c. The function c is called the stream function, and by virtue of its de¢nition it is valid for all two-dimensional £ows, both rotational and irrotational.
The equation that the stream function c must satisfy is obtained from the condition of irrotationality. Denoting the components of the vorticity vectorv by ðx; Z; zÞ, it is ¢rst observed that, in two dimensions, the only nonzero component of the vorticity vector is z, the component perpendicular to the plane of the £ow. Secondly, it is noted that z¼ @v=@x @u=@y. Thus, the condition of irrotationality is
@v @x @
u @y¼ 0
Substituting for u andv from Eqs. (4.1) shows that c must saisfy the following equation:
@2c
@x2þ @ 2c
That is, the stream function c, like the velocity potential f, must satisfy Laplace’s equation. The stream function c has some useful properties that will now be derived.
The £ow lines that correspond to c¼ constant are the streamlines of the £ow ¢eld. To show this, it is noted that c is a function of both x and y in general so that the total variation in c associated with a change in x and a change in y may be calculated from the expression
dc¼ @c @xdxþ @
c @ydy ¼ v dx þ u dy
where Eqs. (4.1) have been used. Then the equation of the line c¼ constant will be 0¼ v dx þ u dy or dy dx c ¼ v u
where the subscript denotes that this expression for dy=dx is valid for c held constant. But it was shown in Chap. 2 that this is precisely the equation of the streamlines in the xy plane. Hence the lines corresponding to c¼ constant are the streamlines, and each value of the constant de¢nes a di¡erent streamline. It is this property of the function c that justi¢es the name stream function.
Another property of the stream function c is that the di¡erence of its values between two streamlines gives the volume of £uid that is £owing between these two streamlines. To show this, consider two streamlines cor- responding to c¼ c1and c¼ c2as shown in Fig. 4.1. A control surface AB of arbitrary shape but positive slope is shown joining these two streamlines, and an element of this surface shows the positive volumetric £ow rates crossing it in the x and y directions per unit depth perpendicular to the £ow ¢eld. Then the total volume of £uid £owing between the streamlines per unit time per unit depth of £ow ¢eld will be
Q¼ Z B A u dy Z B A v dx
But it was observed earlier that dc¼ v dx þ u dy, so that, integrating this expression between the two points A and B, it follows that
c2 c1¼ Z B A v dx þ Z B A u dy
Comparing these two expressions con¢rms that c2 c1 ¼ Q.
Finally, it should be noted that the streamlines c¼ constant and the lines f¼ constant, which are called equipotential lines, are orthogonal to each other. This may be shown by noting that if f depends upon both x and y, the total change in f associated with changes in both x and y will be
df¼ @f @x dxþ @
f @y dy ¼ u dx þ v dy
where Eq. (II.4) has been used. Then the lines corresponding to f¼ constant will be de¢ned by
0¼ u dx þ v dy or dy dx f ¼ u v
FIGURE4.1 Two streamlines showing the components of the volumetric flow rate across an element of control surface joining the streamlines.
That is, dy dx f ¼ 1 ðdy=dxÞc
In words, the slope of the lines f¼ constant is the negative reciprocal of the slope of the lines c¼ constant, so that these sets of lines must be orthogonal. This property of the streamlines and the equipotential lines is the basis of a numerical procedure for solving two-dimensional potential-£ow problems. The method is referred to as the flow net.
4.2
COMPLEX POTENTIAL AND COMPLEX
VELOCITY
The velocity components u andv may be expressed in terms of either the velocity potential or the stream function. From Eqs. (II.4) and (4.1), these expressions are u¼ @f @x ¼ @ c @y v ¼@f @y ¼ @ c @x
That is, the functions f and c are related by the expressions @f @x¼ @ c @y @f @y ¼ @ c @x
But these will be recognized as the Cauchy-Riemann equations for the functions fðx; yÞ and cðx; yÞ. Then consider the complex potential FðzÞ, which is de¢ned as follows:
FðzÞ ¼ fðx; yÞ þ icðx yÞ ð4:3Þ
where z¼ x þ iy. Now if FðzÞ is an analytic function, it follows that f and c will automatically satisfy the Cauchy-Riemann equations. That is, for every analytic function FðzÞ the real part is automatically a valid velocity potential and the imaginary part is a valid stream function.
The foregoing result suggests a very simple way of establishing solutions to the equations of two-dimensional potential £ows. By equating
the real part of a given analytic function to f and the imaginary part to c, the theory of complex variables guarantees that H2f¼ 0 and H2c¼ 0 as required. The £ow ¢eld corresponding to that analytic function may be determined by studying the streamlines c¼ constant. The corresponding velocity components may be calculated from Eqs. (II.4) or (4.1), and the pressure may be obtained using Eq. (II.6). This approach has the dis- advantage of being inverse in the sense that a problem is ¢rst solved and then examined to see what the physical problem was in the ¢rst place. However, for teaching purposes this is of no consequence. Another disadvantage is that the method cannot be generalized to three-dimensional potential £ows. On the other hand, this approach avails itself of the powerful results of com- plex variable theory and avoids the di⁄culties of solving partial di¡erential equations. For these reasons the complex-potential approach will be used in this chapter.
Another quantity of prime interest, apart from the complex potential FðzÞ, is the derivative of FðzÞ with respect to z. Since FðzÞ is supposed to be analytic, dF=dz will be a point function whose value is independent of the direction in which it is calculated. Then, denoting this derivative by W, its value will be given by
WðzÞ ¼dF dz ¼ @ F @x ¼ @f @xþ i @ c @x that is, WðzÞ ¼dF dz ¼ u iv ð4:4Þ
where use has been made of Eqs. (4.3), (II.4), and (4.1b). In view of this result the quantity WðzÞ is called the complex velocity, although its imaginary part isiv. Equation (4.4) o¡ers a convenient alternative to Eqs. (II.4) and (4.1) for ¢nding the velocity components corresponding to a given complex potential.
A useful property of the complex velocity is that,when multiplied by its own complex conjugate, it gives the scalar product of the velocity vector with itself. To show this, consider WðzÞ and its complex conjugate WðzÞ.Then
W W ¼ ðu ivÞðu þ ivÞ
The signi¢cance of this result is that the quantity uu ¼ =f=f ¼ u2þ v2appears in the Bernoulli equation.
Frequently it is advantageous to work in cylindrical coordinates rather than cartesian coordinates. An expression for the complex velocity may be readily obtained in cylindrical coordinates by converting the cartesian components of the velocity vectorðu; vÞ to cylindrical components ðuR; uyÞ.
Figure 4.2 shows a velocity vector OP decomposed into its cartesian com- ponents (shown solid) and also its cylindrical components (shown dotted). From this ¢gure each of the cartesian velocity components may be expressed in terms of the two cylindrical components as follows:
u¼ uRcos yþ uycos p 2 y ¼ uRcos y uysin y v ¼ uRsin yþ uysin p 2 y ¼ uRsin yþ uycos y
Substituting these expressions into Eq. (4.4) gives the expression for the complex velocity W in terms of uRand uy.
FIGURE4.2 Decomposition of a velocity vector OP into its cartesian components ðu; vÞ and its cylindrical components ðuR; uyÞ.
W ¼ ðuRcos y uysin yÞ iðuRsin yþ uycos yÞ
¼ uRðcos y i sin yÞ iuyðcos y i sin yÞ
that is,
W ¼ ðuR iuyÞeiy ð4:6Þ
The foregoing results [Eqs. (4.3) to (4.6)] are su⁄cient to establish the £ow ¢elds, which are represented by simple analytic functions.
4.3
UNIFORM FLOWS
The simplest analytic function of z is proportional to z itself, and the corre- sponding £ow ¢elds are uniform £ows.
First, consider FðzÞ to be proportional to z where the constant of pro- portionality is real. That is,
FðzÞ ¼ cz where c is real. Then, from Eq. (4.4),
WðzÞ ¼ u iv ¼ c
Then, by equating real and imaginary parts of this equation, the velocity components corresponding to this complex potential are
u¼ c v ¼ 0
But this is just the velocity ¢eld for a uniform rectilinear £ow as shown in Fig. 4.3a. Thus the complex potential for such a £ow whose velocity mag- nitude is U in the positive x direction will be
FðzÞ ¼ Uz ð4:7aÞ
Next consider the complex potential to be proportional to z with an imaginary constant of proportionality. Then
FðzÞ ¼ icz
where c is real. The minus sign has been included to make the velocity com- ponent positive when c is positive. For this complex potential
so that the velocity components are u¼ 0 v ¼ c
This is a uniform vertical £ow as shown in Fig. 4.3b. Then the complex potential for such a £ow whose velocity magnitude is V in the positive y direction will be
FðzÞ ¼ iVz ð4:7bÞ
Finally, consider a complex constant of proportionality so that FðzÞ ¼ ceiaz
where c and a are real. For this complex potential WðzÞ ¼ u iv ¼ c cos a ic sin a Hence the velocity components of the £ow ¢eld are
u¼ c cos a v ¼ c sin a
This corresponds to a uniform £ow inclined at an angle a to the x axis as shown in Fig. 4.3c. Hence the complex potential for such a £ow whose velo- city magnitude is V will be
FðzÞ ¼ Veiaz ð4:7cÞ
FIGURE4.3 Uniform flow in (a) the x direction, (b) the y direction, and (c) an angle a to the x direction.
This last result, of course, contains the two previous results as special cases corresponding to a¼ 0 and a ¼ p=2.
4.4
SOURCE, SINK, AND VORTEX FLOWS
Complex potentials that correspond to the £ow ¢elds generated by sources, sinks, and vortices are obtained by considering FðzÞ to be proportional to log z.When considering log z, we consider the principal part of this multivalued function corresponding to 0< y < 2p.
Consider, ¢rst, the constant of proportionality to be real. Then FðzÞ ¼ c log z
¼ c log Reiy
¼ c log R þ icy
Hence, from Eq. (4.3),
f¼ c log R c¼ cy
That is, the equipotential lines are the circles R¼ constant and the stream- lines are the radial lines y¼ constant. This gives a £ow ¢eld as shown in Fig. 4.4a in which the streamlines are shown solid and the direction of the £ow is shown for c> 0. The direction of the £ow is readily con¢rmed by evaluating the velocity components. In view of the geometry of the £ow, cylindrical coordinates are preferred, so that
WðzÞ ¼c z¼
c Re
iy
Comparison with Eq. (4.6) shows that the velocity components are
uR¼
c R uy¼ 0
which con¢rms the directions indicated in Fig. 4.4a for c> 0.
The £ow ¢eld indicated in Fig. 4.4a is called a source. The velocity is purely radial and its magnitude decreases as the £ow leaves the origin. In fact, the origin is a singular point corresponding to in¢nite velocity, and as the £uid £ows radially outwards, its velocity is decreased in such a way
that the volume of £uid crossing each circle is constant, as required by the continuity equation.
Sources are characterized by their strength, denoted by m, which is de¢ned as the volume of £uid leaving the source per unit time per unit depth of the £ow ¢eld. From this de¢nition it follows that
m¼ Z 2p 0 uRR dy ¼ Z 2p 0 c dy¼ 2pc
Here, the result uR¼ c=R has been used. Then c may be replaced by m=2p,
giving the following complex potential for a source of strength m: FðzÞ ¼ m
2plog z
The source corresponding to this complex potential is located at the origin, the location of the singularity.Then the complex potential for a source of strength m located at the point z¼ z0will be
FðzÞ ¼ m
2plogðz z0Þ ð4:8Þ
FIGURE4.4 Streamlines (shown solid) and equipotential lines (shown dashed) for (a) source flow and (b) vortex flow in the positive sense.
Clearly, the complex potential for a sink, which is a negative source, is obtained by replacing m bym in Eq. (4.8).
Now consider the constant of proportionality in the logarithmic com- plex potential to be imaginary. That is, consider
FðzÞ ¼ ic log z
where c is real and the minus is included to give a positive vortex.Then, using cylindrical coordinates,
FðzÞ ¼ ic log Reiy ¼ cy ic log R
Then, from Eq. (4.3), the velocity potential and the stream function are f¼ cy
c¼ c log R
That is, the equipotential lines are the radial lines y¼ constant and the streamlines are the circles R¼ constant as shown in Fig. 4.4b. The velocity components may be evaluated by use of the complex velocity.
WðzÞ ¼ ic z¼ i
c Re
iy
Comparison with Eq. (4.6) shows that the velocity components are uR¼ 0
uy¼
c R
Hence the direction of the £ow is positive (counterclockwise) for c> 0, and the resulting £ow ¢eld is called a vortex.
Avortex is characterized by its strength,which may be measured by the circulationG associated with it. From Eq. (2.3), the circulation G associated with the singularity at the origin is
G ¼ I ^ u dl ¼ Z 2p 0 uyR dy ¼ Z 2p 0 c dy¼ 2pc
Here, the result uy ¼ c=R has been used. Then c may be replaced by G=2p,
giving the following complex potential for a positive (counterclockwise) vortex of strengthG.
FðzÞ ¼ i G 2plog z
The singularity in this expression is located at z¼ 0.That is,the line vortex is located at z¼ 0. Then the complex potential for a positive vortex located at z¼ z0will be
FðzÞ ¼ i G
2plogðz z0Þ ð4:9Þ
The complex potential for a negative vortex would be obtained by replacing G by G in Eq. (4.9).Note, however,that the negative coe⁄cient is associated with the positive vortex.
The £ow ¢eld represented by Eq. (4.9), which is shown in Fig. 4.4b for z0¼ 0, corresponds to a so-called free vortex. That is, for any closed contour
that does not include the singularity, the circulation will be zero and the £ow will be irrotational. All the circulation and vorticity associated with this type of vortex is concentrated at the singularity. This is in contrast with the solid- body rotation vortex mentioned in Chap. 2.
The principal application of the source, the sink, and the vortex is in the superposition with other £ows to yield more practical £ow ¢elds.
4.5
FLOW IN A SECTOR
The £ows in sharp bends or sectors are represented by complex potentials that are proportional to zn,where n 1. A special case of such complex potentials would be n¼ 1, which represents a uniform rectilinear £ow. Then, in order that this special case will reduce to Eq. (4.7a),consider the complex potentials
FðzÞ ¼ Uzn
Substituting z¼ Reiy and separating the real and imaginary parts of this
function gives
FðzÞ ¼ URncos nyþ iURnsin ny Then the velocity potential and the stream function are
f¼ URncos ny c¼ URnsin ny
From this it is evident that when y¼ 0 and when y ¼ p=n, the stream func- tion c is zero. That is, the streamline c¼ 0 corresponds to the radial lines y¼ 0 and y ¼ p=n. Between these two lines, the streamlines are de¢ned by Rnsin ny¼ constant.This gives the £ow ¢eld shown in Fig. 4.5.The direction
of the £ow along the streamlines may be determined from the complex velo- city as follows:
WðzÞ ¼ nUzn1¼ nURn1eiðn1Þy
¼ ðnURn1cos nyþ inURn1sin nyÞeiy
Thus, by comparison with Eq. (4.6), the velocity components are uR¼ nURn1cos ny
uy ¼ nURn1sin ny
Then, for 0< y < ðp=2nÞ; uR is positive while uy is negative and for
ðp=2nÞ < y < ðp=nÞ; uRis negative and uyremains negative. This establishes
the £ow directions as indicated in Fig. 4.5.
FIGURE4.5 Streamlines (shown solid) and equipotential lines (shown dashed) for flow in a sector.
From the foregoing, the complex potential for the £ow in a corner or sector of angle p=n is
FðzÞ ¼ Uzn ð4:10Þ
For n¼ 1, Eq. (4.10) gives the complex potential for a uniform rectilinear £ow, and for n¼ 2, it gives the complex potential for the £ow in a right- angled corner.
4.6
FLOW AROUND A SHARP EDGE
The complex potential for the £ow around a sharp edge, such as the edge of a £at plate, is obtained from the function z1=2. Then consider the complex potential
FðzÞ ¼ cz1=2
where c is real and 0< y < 2p.Then, in cylindrical coordinates, FðzÞ ¼ cR1=2eiy=2
so that the velocity potential and stream function are f¼ cR1=2cosy
2 c¼ cR1=2siny
2
Thus the lines y¼ 0 and y ¼ 2p correspond to the streamline c ¼ 0. The other streamlines are de¢ned by the equation R1=2sin y=2 ¼ constant,which yields the £ow pattern shown in Fig. 4.6.The direction of the £ow is obtained from the complex velocity as follows:
WðzÞ ¼ c 2z1=2¼ c 2R1=2e iy=2 ¼ c 2R1=2 cos y 2þ i sin y 2 eiy
Hence the velocity components are uR¼ c 2R1=2cos y 2 uy¼ c 2R1=2sin y 2
Then, for 0< y < p; uR> 0 and uy< 0. Also, for p < y < 2p; uR< 0 and
uy < 0.This gives the direction of £ow as indicated in Fig. 4.6.
The £ow ¢eld shown in Fig. 4.6 corresponds to the £ow around a sharp edge, and so the complex potential for such a £ow is
FðzÞ ¼ cz1=2 ð4:11Þ
An important feature of this result is that the corner itself is a singular point at which the velocity components become in¢nite. Since both uRand uyvary
as the inverse of R1=2, it follows that the velocity is singular as the square root
of the distance from the edge. This result will be discussed in Sec. 4.15.
4.7
FLOW DUE TO A DOUBLET
The function 1=z has a singularity at z ¼ 0, and in the context of complex potentials, this singularity is called a doublet. The quickest way of establish- ing the £ow ¢eld which corresponds to the complex potentials that are proportional to 1=z would be to follow the methods used in the previous
FIGURE4.6 Streamlines (shown solid) and equipotential lines (shown dashed) for flow around a sharp edge.
sections. However, it turns out that the doublet may be considered to be the coalescing of a source and a sink, and the required complex potential may be obtained through a limiting procedure that uses this fact.This interpretation leads to a better physical understanding of the doublet, and for this reason it will be followed here before studying the £ow ¢eld.
Referring to the geometry indicated in Fig. 4.7a, consider a source of strength m and a sink of strength m , each of which is located on the real axis a small distance e from the origin. The complex potential for such a con¢g- uration is, from Eq. (4.8),
FðzÞ ¼ m 2plogðz þ eÞ m 2plogðz eÞ ¼ m 2plog zþ e z e ¼ m 2plog 1þ e=z 1 e=z
If the nondimensional distance e=jzj is considered to be small, the argument of the logarithm may be expanded as follows:
FðzÞ ¼ m 2plog 1þ e z 1þe zþ O e2 z2 ¼ m 2plog 1þ 2 e zþ O e2 z2
where the designation Oðe2=z2Þ means terms of order e2=z2or smaller. The
logarithm is now in the form logð1 þ gÞ, where g 1, so that the equivalent expansion gþ Oðg2Þ may be used.Then
FðzÞ ¼ m 2p 2 e zþ O e2 z2
It is now proposed to let e! 0 and m ! 1 in such a way that lime!0ðmeÞ ¼
pm, where m is a constant. Then the complex potential becomes FðzÞ ¼m
z
Thus the complex potential m=z may be thought of as being the equivalent of the superposition of a very strong source and a very strong sink that are very close together.
In order to establish the £ow ¢eld that the above complex potential represents, the stream function will be established as follows:
FðzÞ ¼ m xþ iy ¼ m x iy x2þ y2 ::: c ¼ m y x2þ y2
Thus the equation of the streamlines c¼ constant is x2þ y2þm cy¼ 0 or x2þ y þ m 2c 2 ¼ m 2c 2
FIGURE4.7 (a) Superposition of a source and a sink leading to (b) streamline pat-
But this is the equation of a circle of radius m=ð2cÞ whose center is located at y¼ m=ð2cÞ. This gives the streamline pattern shown in Fig. 4.7b. Although the direction of the £ow along the streamlines may be deduced from the source and sink interpretation, it will be checked by evaluating the velocity components. The complex velocity for this complex potential is
WðzÞ ¼ m z2¼ m R2e i2y ¼ m
R2ðcos y i sin yÞe iy
Hence the velocity components are uR¼ m R2cos y uy¼ m R2sin y
These expressions for uR and uy con¢rm the £ow directions indicated in
Fig. 4.7b.
The £ow ¢eld illustrated in Fig. 4.7b is called a doublet £ow, and the