IDEAL-FLUID FLOW
4.21 FLOW THROUGH AN APERTURE
semi-in¢nite channel. The foregoing £ow con¢gurations are clearly related to the largest £ow ¢eld, shown in Fig. 4.20b, by symmetry. The total quantity of £uid leaving the source is 4lU, so that the source strengthmin Eq. (4.26) should be 4lU in order that the channel velocity in the four con¢gurations shown in Fig. 4.20b and c will beU.
Figure 4.20d depicts an in¢nite array of line sources spaced a distance 2lapart. The horizontal lines that pass midway between each pair of sources will obviously be streamlines for such an array of sources. It follows that the case of a source located in a horizontal channel may be thought of as only one component of an in¢nite number of such channels stacked on top of each other in the vertical direction. Mathematically, the fact that Eq. (4.26) represents an in¢nite number of sources spread in they direction follows from the fact that the hyperbolic sine function repeats itself for imaginary values of its argument.
FIGURE4.21 (a) Mapping planes for flow through a slit, and (b) geometry of one of the free bounding streamlines.
The ¢rst transformation will be to thehodograph plane, which will be designated thezplane here. The transformation will be taken to be
z¼U dz dF ¼U
W ¼ U
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2þv2
p eiy ð4:27aÞ
That is, thezplane is de¢ned by the nondimensional reciprocal of the com- plex velocity and the last equality follows from the fact thatffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W ¼uiv¼
u2þv2
p eiy. The signi¢cance of this transformation is that the free stream- lines, whose positions are unknown, are mapped onto the unit circle in thez plane, as will now be shown. In so doing, it should be noted that, by the fore- going de¢nition,yis the angle subtended by the velocity vector in thezplane.
Along the free streamlinesBCandB0C0the pressure will be constant, typically atmospheric pressure, so that, from Bernoulli’s equation, the quan- tityu2þv2will be constant.The value of this constant may be determined by noting that away from the edge e¡ects, the velocity in the jet isU. Hence the value ofu2þv2along the free streamlines isU2. Then, along the free stream- lines,z¼eiy,which is the equation of the unit circle in thezplane. To ¢nd the portion of this unit circle that represents the free streamlines, it is observed that along the streamlineA0B0the angleyof the velocity vector is 0 or 2p,while the value ofyalongABisp. Also, along the streamlineII0the angleyof the velocity vector is 3p=2. From these observations it is evident that the lower half of the unit circle in thezplane represents the streamlinesBCandB0C0, as shown in Fig. 4.21a. The other principal streamlines may be identi¢ed as fol- lows: AlongA0B0the value ofyis 0 or 2p,whileu2þv2varies from 0 atA0toU2 atB0; hencejzjvaries from in¢nity atA0to unity atB0. Likewise, alongABthe value ofjzjvaries from in¢nity atAto unity atB, with the value ofybeingp.
Finally, along the streamlineII0the value ofyis 3p=2,whileu2þv2varies from zero atItoU2atI0, makingjzjin¢nity atIand unity atI0.This establishes the
£ow con¢guration shown in thezplane of Fig. 4.21a. Since the £ow is toward the pointz¼ i,which is identi¢ed byC;C0, andI0, there is a £uid sink there.
Since the principal streamlines in thezplane are either radial lines or the unit circle, the £ow pattern may be mapped into a plane con¢guration by means of the logarithmic transformation. Then a second mapping is pro- posed to thez0plane, wherez0is de¢ned by
z0¼logz ð4:27bÞ
If a point in thezplane is represented by its polar coordinatesRandy, where R¼U=ðu2þv2Þ1=2, thenz¼Reiy, so that
z0¼logz¼logRþiy
Thus the radial lines in thezplane become the horizontal lines de¢ned by z0¼logRþiconstant in thez0plane, while the unit circleR¼1 becomes the vertical linez0¼iy. Noting that the angleyis the angle subtended by the velocity vector in thezplane, it follows that the value ofyalongA0B0is 0 or 2p.
This gives the £ow con¢guration shown in Fig. 4.21a in thez0plane, which corresponds to the £ow in a semi-in¢nite channel due to a sink located at the center of the end of the channel. But it was seen in the previous section that such a con¢guration could be mapped into that of a simple source. Using the results of the previous section, a simple source £ow will result in thez00plane through the mapping
z00¼coshðz0ipÞ i.e.,
z00¼ coshz0 ð4:27cÞ Here the rectangleABCC0B0A0has been taken as the equivalent of the half channel of widthl that was considered in the previous section. Then the quantitylthat appeared in the transformation ispin this case, and in order to bring the cornerBto the origin in thez0plane, the quantityz0iprather than z0is the appropriate variable.
The £ow ¢eld in the z00 plane is shown in Fig. 4.21a. The complex potential in this plane will be that for a simple sink located atz00¼0, so that
Fðz00Þ ¼ m
2plogz00þK ð4:27dÞ where the constantKhas been added to permit the streamlinec¼0 and the equipotential linef¼0 to correspond to a chosen streamline and equipo- tential line, respectively. Referring to Fig. 4.21b, it will now be speci¢ed that the streamlinec¼0 be the streamlineII0 and that the equipotential line f¼0 passes through the points B0andB. Then, using the property of the stream function that the di¡erence of the values ofcbetween two stream- lines equals the volume of £uid £owing between these two streamlines, the value ofcalongA0B0C0, which will be denoted bycA0B0C0, may be identi¢ed.
Considering the £ow between the streamlinesII0andA0B0C0, it follows that 0cA0B0C0 ¼CclU
whereCcis the contraction coe⁄cient of the jet. Similarly, ifcABCis the value ofcalong the streamlineABC, it follows by considering the £ow between the streamlinesABCandII0that
cABC0¼CclU That is,
cABC ¼CclU and
cA0B0C0¼ CclU
Then, at the point,B0;f¼0 andc¼ CclU. Hence the value of the complex potential there is 0iCclU. Applying this result to Eq. (4.27d) and noting thatz00¼ 1 at the pointB0gives
0iCclU ¼ m
2plogð1Þ þK iCclU ¼ im
2þK
Likewise at the pointBthe complex potential is 0þiCclUand the value ofz00 is unity; hence
0þiCclU ¼ m
2plog 1þK or
iCclU ¼K
These two equations show thatK ¼iCclUandm¼4CclU, so that the com- plex potential (4.27d) becomes
Fðz00Þ ¼ 2CclU
p logz00þiCclU
The corresponding complex potential in thezplane may be obtained by use of the transformations (4.27a), (4.27b), and (4.27c). This gives
FðzÞ ¼ 2CclU
p log cosh log U dz dF
ip
þiCclU ð4:27eÞ
This result is an implicit expression forF(z)rather than an explicit expres- sion, sincedF=dzappears inside the expression forFðzÞ. However, the £ow problem has, in principle, been solved, and it is possible to obtain useful information from the result. The quantity that is of prime interest in this problem is the value of the contraction coe⁄cientCc, so this value will be determined below.
In order to evaluate the contraction coe⁄cientCc, the equation of the free streamlineB0C0will be established. From this result the value ofxat the pointC0should be numerically equal to the half-jet dimensionCcl. This will enable the quantityCc to be evaluated. The equation of the free streamline B0C0is most readily established in terms of a coordinateswhose value is zero at the pointB0and whose magnitude increases alongB0C0. Then, considering a small element of a curve, such as the streamlineB0C0, whose slope is posi- tive, it follows that
dx
ds ¼cosy ::: x¼x0þ
Z s
0
cosyds
where the constantx0has been added to permit the conditionx¼ lwhen s¼0 to be applied. The variation ofdswithyis now required and, owing to the implicit nature of the mapping function (4.27a), this variation must be obtained by indirect methods as follows: The preceding expression for the lateral displacementxof the jet surface may be written
x¼x0þ Z y
2p
cosy ds dz00
dz00 dy dy
where the quantitiesds=dz00 and dz00=dy must be expressed in terms ofy before the integration may be performed. The value ofdz00=dyonB0C0will be obtained from the equations of the mappings, whileds=dz00will be obtained from the complex potentialFðz00Þ. Considering ¢rst the value ofds=dz00, it may be stated that, on the streamlineB0C0
1¼j j ¼z U W
¼ U dz dF
¼ U dz dz00
dz00 dF
But from Eq. (4.27d) withm¼4CclU, dF
dz00¼ 2CclU p
1 z00 ::: 1¼ U dz
dz00 p 2CclUz00
OnB0C0,z00<0, so that
dz dz00
¼2Ccl p
1 z00
Now on the streamlineB0C0the value ofdzmay be represented byds eiy, wheredsis an element of the coordinates, which was previously introduced.
Also, alongB0C0,z00is increasing so thatdz00>0. Hence ds
dz00¼ 2Ccl p
1 z00
The equations of the various mappings may now be used to evaluatez00in terms ofy. On the streamlineB0C0the value ofzis
z¼eiy ::: z0¼logz¼iy and
z00¼ coshz0¼ cosy ::: dz00
dy ¼siny and
ds
dz00¼2Ccl p
1 cos hy
Using these last two equations, the expression for the lateral displacementx of the free streamline becomes
x¼x0þ Z y
2p
cosy 2Ccl
pcosysinydy
¼x0þ2Ccl p
Z y 2p
sinydy
¼x0þ2Ccl
p ðlcosyÞ
But wheny¼2p, that is, at the pointB0,x¼ l. Hencex0¼ l, so that x¼ lþ2Ccl
p ð1cosyÞ
Also, the value ofxat the pointC0isCcl, while the value ofyis 3p=2. Thus Ccl¼ lþ2Ccl
p ::: Cc¼ p
pþ2 ð4:27fÞ
Equation (4.27f) predicts that the free jet that emerges from the aperture will assume a width that is 0.611 of the width of the slit. This result is well estab- lished experimentally, and the ¢gure of 0.611 has been con¢rmed for open- ings under deep liquids.