Chapter III: Phenomena of the Discharge of Liquids From Orifices as

ASCERTAINABLE BY EXPERIMENTS

§ 16. When a liquid issues vertically from a small orifice, it forms a
jet which rises nearly to the level of the free surface of the liquid
in the vessel from which it flows. The difference of level h_r (fig.
14) is so small that it may be at once suspected to be due either to
air resistance on the surface of the jet or to the viscosity of the
liquid or to friction against the sides of the orifice. Neglecting for
the moment this small quantity, we may infer, from the elevation of
the jet, that each molecule on leaving the orifice possessed the
velocity required to lift it against gravity to the height h. From
ordinary dynamics, the relation between the velocity and height of
projection is given by the equation

v = [root](2gh). (1)

As this velocity is nearly reached in the flow from well-formed
orifices, it is sometimes called the theoretical velocity of
discharge. This relation was first obtained by Torricelli.

If the orifice is of a suitable conoidal form, the water issues in
filaments normal to the plane of the orifice. Let [omega] be the area
of the orifice, then the discharge per second must be, from eq. (1),

Q = [omega]v = [omega][root](2gh) nearly. (2)

This is sometimes quite improperly called the theoretical discharge
for any kind of orifice. Except for a well-formed conoidal orifice the
result is not approximate even, so that if it is supposed to be based
on a theory the theory is a false one.

_Use of the term Head in Hydraulics._--The term _head_ is an old
millwright's term, and meant primarily the height through which a mass
of water descended in actuating a hydraulic machine. Since the water
in fig. 14 descends through a height h to the orifice, we may say
there are h ft. of head above the orifice. Still more generally any
mass of liquid h ft. above a horizontal plane may be said to have h
ft. of elevation head relatively to that datum plane. Further, since
the pressure p at the orifice which produces outflow is connected with
h by the relation p/G = h, the quantity p/G may be termed the pressure
head at the orifice. Lastly, the velocity v is connected with h by the
relation v²/2g = h, so that v²/2g may be termed the head due to the
velocity v.

§ 17. _Coefficients of Velocity and Resistance._--As the actual
velocity of discharge differs from [root]2gh by a small quantity, let
the actual velocity

= v_a = c_v [root](2gh), (3)

where c_v is a coefficient to be determined by experiment, called the
_coefficient of velocity_. This coefficient is found to be tolerably
constant for different heads with well-formed simple orifices, and it
very often has the value 0.97.

The difference between the velocity of discharge and the velocity due
to the head may be reckoned in another way. The total height h causing
outflow consists of two parts--one part h_e expended effectively in
producing the velocity of outflow, another h_r in overcoming the
resistances due to viscosity and friction. Let

h_r = c_r h_e,

where c{r} is a coefficient determined by experiment, and called the
_coefficient of resistance_ of the orifice. It is tolerably constant
for different heads with well-formed orifices. Then

v_a = [root](2gh_e) = [root]{2gh/(1 + c_r)}. (4)

The relation between c_v and c_r for any orifice is easily found:--

v_a = c_v[root](2gh) = [root]{2gh/(1 + c_r)}

c_v = [root]{1/(1 + c_r)} (5)

c_r = 1/c_v² - 1 (5a)

Thus if c_v = 0.97, then c_r = 0.0628. That is, for such an orifice
about 6¼% of the head is expended in overcoming frictional resistances
to flow.

_Coefficient of Contraction--Sharp-edged Orifices in Plane
Surfaces._--When a jet issues from an aperture in a vessel, it may
either spring clear from the inner edge of the orifice as at a or b
(fig. 15), or it may adhere to the sides of the orifice as at c. The
former condition will be found if the orifice is bevelled outwards as
at a, so as to be sharp edged, and it will also occur generally for a
prismatic aperture like b, provided the thickness of the plate in
which the aperture is formed is less than the diameter of the jet. But
if the thickness is greater the condition shown at c will occur.

When the discharge occurs as at a or b, the filaments converging
towards the orifice continue to converge beyond it, so that the
section of the jet where the filaments have become parallel is smaller
than the section of the orifice. The inertia of the filaments opposes
sudden change of direction of motion at the edge of the orifice, and
the convergence continues for a distance of about half the diameter of
the orifice beyond it. Let [omega] be the area of the orifice, and
c_c[omega] the area of the jet at the point where convergence ceases;
then c_c is a coefficient to be determined experimentally for each
kind of orifice, called the _coefficient of contraction_. When the
orifice is a sharp-edged orifice in a plane surface, the value of c_c
is on the average 0.64, or the section of the jet is very nearly
five-eighths of the area of the orifice.

_Coefficient of Discharge._--In applying the general formula Q =
[omega]v to a stream, it is assumed that the filaments have a common
velocity v normal to the section [omega]. But if the jet contracts, it
is at the contracted section of the jet that the direction of motion
is normal to a transverse section of the jet. Hence the actual
discharge when contraction occurs is

Q_a = c_vv × c_c[omega] = c_c c_v[omega][root](2gh),

or simply, if c = c_vc_c,

Q_a = c[omega][root](2gh),

where c is called the _coefficient of discharge_. Thus for a
sharp-edged plane orifice c = 0.97 × 0.64 = 0.62.

§ 18. _Experimental Determination of c_v, c_c, and c._--The
coefficient of contraction c_c is directly determined by measuring the
dimensions of the jet. For this purpose fixed screws of fine pitch
(fig. 16) are convenient. These are set to touch the jet, and then the
distance between them can be measured at leisure.

The coefficient of velocity is determined directly by measuring the
parabolic path of a horizontal jet.

Let OX, OY (fig. 17) be horizontal and vertical axes, the origin being
at the orifice. Let h be the head, and x, y the coordinates of a point
A on the parabolic path of the jet. If v_a is the velocity at the
orifice, and t the time in which a particle moves from O to A, then

x = v_a t; y = ½gt².

Eliminating t,

v_a = [root](gx²/2y).

Then

c_v = v_a [root](2gh) = [root](x²/4yh).

In the case of large orifices such as weirs, the velocity can be
directly determined by using a Pitot tube (§ 144).

The coefficient of discharge, which for practical purposes is the most
important of the three coefficients, is best determined by tank
measurement of the flow from the given orifice in a suitable time. If
Q is the discharge measured in the tank per second, then

c = Q/[omega][root](2gh).

Measurements of this kind though simple in principle are not free from
some practical difficulties, and require much care. In fig. 18 is
shown an arrangement of measuring tank. The orifice is fixed in the
wall of the cistern A and discharges either into the waste channel BB,
or into the measuring tank. There is a short trough on rollers C which
when run under the jet directs the discharge into the tank, and when
run back again allows the discharge to drop into the waste channel. D
is a stilling screen to prevent agitation of the surface at the
measuring point, E, and F is a discharge valve for emptying the
measuring tank. The rise of level in the tank, the time of the flow
and the head over the orifice at that time must be exactly observed.

For well made sharp-edged orifices, small relatively to the water
surface in the supply reservoir, the coefficients under different
conditions of head are pretty exactly known. Suppose the same quantity
of water is made to flow in succession through such an orifice and
through another orifice of which the coefficient is required, and when
the rate of flow is constant the heads over each orifice are noted.
Let h1, h2 be the heads, [omega]1, [omega]2 the areas of the orifices,
c1, c2 the coefficients. Then since the flow through each orifice is
the same

Q = c1[omega]1 [root](2gh1) = c2[omega]2 [root](2gh2).

c2 = c1([omega]1/[omega]2) [root](h1/h2).

§ 19. _Coefficients for Bellmouths and Bellmouthed Orifices._--If an
orifice is furnished with a mouthpiece exactly of the form of the
contracted vein, then the whole of the contraction occurs within the
mouthpiece, and if the area of the orifice is measured at the smaller
end, c_c must be put = 1. It is often desirable to bellmouth the ends
of pipes, to avoid the loss of head which occurs if this is not
done; and such a bellmouth may also have the form of the contracted
jet. Fig. 19 shows the proportions of such a bellmouth or bell-mouthed
orifice, which approximates to the form of the contracted jet
sufficiently for any practical purpose.

For such an orifice L. J. Weisbach found the following values of the
coefficients with different heads.

+--------------------------------+------+------+------+------+-------+
| Head over orifice, in ft. = h | .66 | 1.64 |11.48 |55.77 |337.93 |
+--------------------------------+------+------+------+------+-------+
| Coefficient of velocity = c_v | .959 | .967 | .975 | .994 | .994 |
| Coefficient of resistance = c_r| .087 | .069 | .052 | .012 | .012 |
+--------------------------------+------+------+------+------+-------+

As there is no contraction after the jet issues from the orifice, c_c
= 1, c = c_v; and therefore

Q = c(v)[omega][root](2gh) = [omega][root]{2gh/(1 + c_r}.

§ 20. _Coefficients for Sharp-edged or virtually Sharp-edged
Orifices._--There are a very large number of measurements of discharge
from sharp-edged orifices under different conditions of head. An
account of these and a very careful tabulation of the average values
of the coefficients will be found in the _Hydraulics_ of the late
Hamilton Smith (Wiley & Sons, New York, 1886). The following short
table abstracted from a larger one will give a fair notion of how the
coefficient varies according to the most trustworthy of the
experiments.

_Coefficient of Discharge for Vertical Circular Orifices, Sharp-edged,
with free Discharge into the Air._ Q = c[omega][root](2gh).

+-----------+------------------------------------------------+
| Head | Diameters of Orifice. |
|measured to+------+------+------+------+------+------+------+
| Centre of | .02 | .04 | .10 | .20 | .40 | .60 | 1.0 |
| Orifice. +------+------+------+------+------+------+------+
| | Values of C. |
+-----------+------+------+------+------+------+------+------+
| 0.3 | .. | .. | .621 | .. | .. | .. | .. |
| 0.4 | .. | .637 | .618 | .. | .. | .. | .. |
| 0.6 | .655 | .630 | .613 | .601 | .596 | .588 | .. |
| 0.8 | .648 | .626 | .610 | .601 | .597 | .594 | .583 |
| 1.0 | .644 | .623 | .608 | .600 | .598 | .595 | .591 |
| 2.0 | .632 | .614 | .604 | .599 | .599 | .597 | .595 |
| 4.0 | .623 | .609 | .602 | .599 | .598 | .597 | .596 |
| 8.0 | .614 | .605 | .600 | .598 | .597 | .596 | .596 |
| 20.0 | .601 | .599 | .596 | .596 | .596 | .596 | .594 |
+-----------+------+------+------+------+------+------+------+

At the same time it must be observed that differences of sharpness in
the edge of the orifice and some other circumstances affect the
results, so that the values found by different careful experimenters
are not a little discrepant. When exact measurement of flow has to be
made by a sharp-edged orifice it is desirable that the coefficient for
the particular orifice should be directly determined.

The following results were obtained by Dr H. T. Bovey in the
laboratory of McGill University.

_Coefficient of Discharge for Sharp-edged Orifices._

+----+------------------------------------------------------------------+
| | Form of Orifice. |
| +------+----------------+-----------------+-----------------+------+
| | | Square. |Rectangular Ratio|Rectangular Ratio| |
|Head| | | of Sides 4:1 | of Sides 16:1 | |
| in | Cir- +------+---------+---------+-------+---------+-------+ Tri- |
| ft.|cular.|Sides | | Long | Long | Long | Long |angu- |
| | |Verti-|Diagonal | Sides | Sides | Sides | Sides | lar. |
| | | cal. |Vertical.|Vertical.| hori- |Vertical.| Hori- | |
| | | | | |zontal.| |zontal.| |
+----+------+------+---------+---------+-------+---------+-------+------+
| 1 | .620 | .627 | .628 | .642 | .643 | .663 | .664 | .636 |
| 2 | .613 | .620 | .628 | .634 | .636 | .650 | .651 | .628 |
| 4 | .608 | .616 | .618 | .628 | .629 | .641 | .642 | .623 |
| 6 | .607 | .614 | .616 | .626 | .627 | .637 | .637 | .620 |
| 8 | .606 | .613 | .614 | .623 | .625 | .634 | .635 | .619 |
| 10 | .605 | .612 | .613 | .622 | .624 | .632 | .633 | .618 |
| 12 | .604 | .611 | .612 | .622 | .623 | .631 | .631 | .618 |
| 14 | .604 | .610 | .612 | .621 | .622 | .630 | .630 | .618 |
| 16 | .603 | .610 | .611 | .620 | .622 | .630 | .630 | .617 |
| 18 | .603 | .610 | .611 | .620 | .621 | .630 | .629 | .616 |
| 20 | .603 | .609 | .611 | .620 | .621 | .629 | .628 | .616 |
+----+------+------+---------+---------+-------+---------+-------+------+

The orifice was 0.196 sq. in. area and the reductions were made with g
= 32.176 the value for Montreal. The value of the coefficient appears
to increase as (perimeter) / (area) increases. It decreases as the
head increases. It decreases a little as the size of the orifice is
greater.

Very careful experiments by J. G. Mair (_Proc. Inst. Civ. Eng._
lxxxiv.) on the discharge from circular orifices gave the results
shown on top of next column.

The edges of the orifices were got up with scrapers to a sharp square
edge. The coefficients generally fall as the head increases and as the
diameter increases. Professor W. C. Unwin found that the results agree
with the formula

c = 0.6075 + 0.0098/[root]h - 0.0037d,

where h is in feet and d in inches.

_Coefficients of Discharge from Circular Orifices. Temperature 51° to
55°._

+-------+--------------------------------------------------------------+
|Head in| Diameters of Orifices in Inches (d). |
| feet +------+------+------+------+------+------+------+------+------+
| h. | 1 | 1¼ | 1½ | 1¾ | 2 | 2¼ | 2½ | 2¾ | 3 |
+-------+------+------+------+------+------+------+------+------+------+
| | Coefficients (c). |
| +------+------+------+------+------+------+------+------+------+
| .75 | .616 | .614 | .616 | .610 | .616 | .612 | .607 | .607 | .609 |
| 1.0 | .613 | .612 | .612 | .611 | .612 | .611 | .604 | .608 | .609 |
| 1.25 | .613 | .614 | .610 | .608 | .612 | .608 | .605 | .605 | .606 |
| 1.50 | .610 | .612 | .611 | .606 | .610 | .607 | .603 | .607 | .605 |
| 1.75 | .612 | .611 | .611 | .605 | .611 | .605 | .604 | .607 | .605 |
| 2.00 | .609 | .613 | .609 | .606 | .609 | .606 | .604 | .604 | .605 |
+-------+------+------+------+------+------+------+------+------+------+

The following table, compiled by J. T. Fanning (_Treatise on Water
Supply Engineering_), gives values for rectangular orifices in
vertical plane surfaces, the head being measured, not immediately over
the orifice, where the surface is depressed, but to the still-water
surface at some distance from the orifice. The values were obtained by
graphic interpolation, all the most reliable experiments being plotted
and curves drawn so as to average the discrepancies.

_Coefficients of Discharge for Rectangular Orifices, Sharp-edged, in
Vertical Plane Surfaces._

+--------+----------------------------------------------------------------+
| Head | Ratio of Height to Width. |
| to | |
| Centre +------+------+------+------+--------+--------+--------+---------+
| of | | | | | | | | |
|Orifice.| 4 | 2 | 1½ | 1 | ¾ | ½ | ¼ | 1/8 |
+--------+------+------+------+------+--------+--------+--------+---------+
| | 4 ft.| 2 ft.|1½ ft.| 1 ft.|0.75 ft.|0.50 ft.|0.25 ft.|0.125 ft.|
| | high.| high.| high.| high.| high. | high. | high. | high. |
| Feet. | | | | | | | | |
| | 1 ft.| 1 ft.| 1 ft.| 1 ft.| 1 ft. | 1 ft. | 1 ft. | 1 ft. |
| | wide.| wide.| wide.| wide.| wide. | wide. | wide. | wide. |
+--------+------+------+------+------+--------+--------+--------+---------+
| 0.2 | .. | .. | .. | .. | .. | .. | .. | .6333 |
| .3 | .. | .. | .. | .. | .. | .. | .6293 | .6334 |
| .4 | .. | .. | .. | .. | .. | .6140 | .6306 | .6334 |
| .5 | .. | .. | .. | .. | .6050 | .6150 | .6313 | .6333 |
| .6 | .. | .. | .. |.5984 | .6063 | .6156 | .6317 | .6332 |
| .7 | .. | .. | .. |.5994 | .6074 | .6162 | .6319 | .6328 |
| .8 | .. | .. |.6130 |.6000 | .6082 | .6165 | .6322 | .6326 |
| .9 | .. | .. |.6134 |.6006 | .6086 | .6168 | .6323 | .6324 |
| 1.0 | .. | .. |.6135 |.6010 | .6090 | .6172 | .6320 | .6320 |
| 1.25 | .. |.6188 |.6140 |.6018 | .6095 | .6173 | .6317 | .6312 |
| 1.50 | .. |.6187 |.6144 |.6026 | .6100 | .6172 | .6313 | .6303 |
| 1.75 | .. |.6186 |.6145 |.6033 | .6103 | .6168 | .6307 | .6296 |
| 2 | .. |.6183 |.6144 |.6036 | .6104 | .6166 | .6302 | .6291 |
| 2.25 | .. |.6180 |.6143 |.6029 | .6103 | .6163 | .6293 | .6286 |
| 2.50 |.6290 |.6176 |.6139 |.6043 | .6102 | .6157 | .6282 | .6278 |
| 2.75 |.6280 |.6173 |.6136 |.6046 | .6101 | .6155 | .6274 | .6273 |
| 3 |.6273 |.6170 |.6132 |.6048 | .6100 | .6153 | .6267 | .6267 |
| 3.5 |.6250 |.6160 |.6123 |.6050 | .6094 | .6146 | .6254 | .6254 |
| 4 |.6245 |.6150 |.6110 |.6047 | .6085 | .6136 | .6236 | .6236 |
| 4.5 |.6226 |.6138 |.6100 |.6044 | .6074 | .6125 | .6222 | .6222 |
| 5 |.6208 |.6124 |.6088 |.6038 | .6063 | .6114 | .6202 | .6202 |
| 6 |.6158 |.6094 |.6063 |.6020 | .6044 | .6087 | .6154 | .6154 |
| 7 |.6124 |.6064 |.6038 |.6011 | .6032 | .6058 | .6110 | .6114 |
| 8 |.6090 |.6036 |.6022 |.6010 | .6022 | .6033 | .6073 | .6087 |
| 9 |.6060 |.6020 |.6014 |.6010 | .6015 | .6020 | .6045 | .6070 |
| 10 |.6035 |.6015 |.6010 |.6010 | .6010 | .6010 | .6030 | .6060 |
| 15 |.6040 |.6018 |.6010 |.6011 | .6012 | .6013 | .6033 | .6066 |
| 20 |.6045 |.6024 |.6012 |.6012 | .6014 | .6018 | .6036 | .6074 |
| 25 |.6048 |.6028 |.6014 |.6012 | .6016 | .6022 | .6040 | .6083 |
| 30 |.6054 |.6034 |.6017 |.6013 | .6018 | .6027 | .6044 | .6092 |
| 35 |.6060 |.6039 |.6021 |.6014 | .6022 | .6032 | .6049 | .6103 |
| 40 |.6066 |.6045 |.6025 |.6015 | .6026 | .6037 | .6055 | .6114 |
| 45 |.6054 |.6052 |.6029 |.6016 | .6030 | .6043 | .6062 | .6125 |
| 50 |.6086 |.6060 |.6034 |.6018 | .6035 | .6050 | .6070 | .6140 |
+--------+------+------+------+------+--------+--------+--------+---------+

§ 21. _Orifices with Edges of Sensible Thickness._--When the edges of
the orifice are not bevelled outwards, but have a sensible thickness,
the coefficient of discharge is somewhat altered. The following table
gives values of the coefficient of discharge for the arrangements of
the orifice shown in vertical section at P, Q, R (fig. 20). The plan
of all the orifices is shown at S. The planks forming the orifice and
sluice were each 2 in. thick, and the orifices were all 24 in. wide.
The heads were measured immediately over the orifice. In this case,

Q = cb(H - h) [root]{2g(H + h)/2}.

§ 22. _Partially Suppressed Contraction._--Since the contraction of
the jet is due to the convergence towards the orifice of the issuing
streams, it will be diminished if for any portion of the edge of the
orifice the convergence is prevented. Thus, if an internal rim or
border is applied to part of the edge of the orifice (fig. 21), the
convergence for so much of the edge is suppressed. For such cases G.
Bidone found the following empirical formulae applicable:--

_Table of Coefficients of Discharge for Rectangular Vertical Orifices
in Fig. 20._

+--------+-----------------------------------------------------------------------------------------------+
|Head h | |
|above | Height of Orifice, H - h, in feet |
|upper +-----------------------+-----------------------+-----------------------+-----------------------+
|edge of | 1.31 | 0.66 | 0.16 | 0.10 |
|Orifice +-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
|in feet.| P | Q | R | P | Q | R | P | Q | R | P | Q | R |
+--------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
| 0.328 | 0.598 | 0.644 | 0.648 | 0.634 | 0.665 | 0.668 | 0.691 | 0.664 | 0.666 | 0.710 | 0.694 | 0.696 |
| .656 | 0.609 | 0.653 | 0.657 | 0.640 | 0.672 | 0.675 | 0.685 | 0.687 | 0.688 | 0.696 | 0.704 | 0.706 |
| .787 | 0.612 | 0.655 | 0.659 | 0.641 | 0.674 | 0.677 | 0.684 | 0.690 | 0.692 | 0.694 | 0.706 | 0.708 |
| .984 | 0.616 | 0.656 | 0.660 | 0.641 | 0.675 | 0.678 | 0.683 | 0.693 | 0.695 | 0.692 | 0.709 | 0.711 |
| 1.968 | 0.618 | 0.649 | 0.653 | 0.640 | 0.676 | 0.679 | 0.678 | 0.695 | 0.697 | 0.688 | 0.710 | 0.712 |
| 3.28 | 0.608 | 0.632 | 0.634 | 0.638 | 0.674 | 0.676 | 0.673 | 0.694 | 0.695 | 0.680 | 0.704 | 0.705 |
| 4.27 | 0.602 | 0.624 | 0.626 | 0.637 | 0.673 | 0.675 | 0.672 | 0.693 | 0.694 | 0.678 | 0.701 | 0.702 |
| 4.92 | 0.598 | 0.620 | 0.622 | 0.637 | 0.673 | 0.674 | 0.672 | 0.692 | 0.693 | 0.676 | 0.699 | 0.699 |
| 5.58 | 0.596 | 0.618 | 0.620 | 0.637 | 0.672 | 0.673 | 0.672 | 0.692 | 0.693 | 0.676 | 0.698 | 0.698 |
| 6.56 | 0.595 | 0.615 | 0.617 | 0.636 | 0.671 | 0.672 | 0.671 | 0.691 | 0.692 | 0.675 | 0.696 | 0.696 |
| 9.84 | 0.592 | 0.611 | 0.612 | 0.634 | 0.669 | 0.670 | 0.668 | 0.689 | 0.690 | 0.672 | 0.693 | 0.693 |
+--------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+

For rectangular orifices,

C_c = 0.62(1 + 0.152n/p);

and for circular orifices,

C_c = 0.62(1 + 0.128n/p);

when n is the length of the edge of the orifice over which the border
extends, and p is the whole length of edge or perimeter of the
orifice. The following are the values of c_c, when the border extends
over ¼, ½, or ¾ of the whole perimeter:--

+--------+-----------------------+--------------------+
| | C_c | C_c |
| n/p | Rectangular Orifices. | Circular Orifices. |
+--------+-----------------------+--------------------+
| 0.25 | 0.643 | .640 |
| 0.50 | 0.667 | .660 |
| 0.75 | 0.691 | .680 |
+--------+-----------------------+--------------------+

For larger values of n/p the formulae are not applicable. C. R.
Bornemann has shown, however, that these formulae for suppressed
contraction are not reliable.

§ 23. _Imperfect Contraction._--If the sides of the vessel approach
near to the edge of the orifice, they interfere with the convergence
of the streams to which the contraction is due, and the contraction is
then modified. It is generally stated that the influence of the sides
begins to be felt if their distance from the edge of the orifice is
less than 2.7 times the corresponding width of the orifice. The
coefficients of contraction for this case are imperfectly known.

§ 24. _Orifices Furnished with Channels of Discharge._--These external
borders to an orifice also modify the contraction.

The following coefficients of discharge were obtained with openings 8
in. wide, and small in proportion to the channel of approach (fig. 22,
A, B, C).

+-----------+-------------------------------------------------------+
| h2--h1 | h1 in feet. |
| in feet |------+-----+-----+-----+------+-----+-----+-----+-----+
| |.0656 |.164 |.328 |.656 |1.640 |3.28 |4.92 |6.56 |9.84 |
+-----------+------+-----+-----+-----+------+-----+-----+-----+-----+
| A\ | .480 |.511 |.542 |.574 | .599 |.601 |.601 |.601 |.601 |
| B > 0.656 | .480 |.510 |.538 |.506 | .592 |.600 |.602 |.602 |.601 |
| C/ | .527 |.553 |.574 |.592 | .607 |.610 |.610 |.609 |.608 |
| | | | | | | | | | |
| A\ | .488 |.577 |.624 |.631 | .625 |.624 |.619 |.613 |.606 |
| B > 0.164 | .487 |.571 |.606 |.617 | .626 |.628 |.627 |.623 |.618 |
| C/ | .585 |.614 |.633 |.645 | .652 |.651 |.650 |.650 |.649 |
+-----------+------+-----+-----+-----+------+-----+-----+-----+-----+

§ 25. _Inversion of the Jet._--When a jet issues from a horizontal
orifice, or is of small size compared with the head, it presents no
marked peculiarity of form. But if the orifice is in a vertical
surface, and if its dimensions are not small compared with the head,
it undergoes a series of singular changes of form after leaving
the orifice. These were first investigated by G. Bidone (1781-1839);
subsequently H. G. Magnus (1802-1870) measured jets from different
orifices; and later Lord Rayleigh (_Proc. Roy. Soc._ xxix. 71)
investigated them anew.

Fig. 23 shows some forms, the upper figure giving the shape of the
orifices, and the others sections of the jet. The jet first contracts
as described above, in consequence of the convergence of the fluid
streams within the vessel, retaining, however, a form similar to that
of the orifice. Afterwards it expands into sheets in planes
perpendicular to the sides of the orifice. Thus the jet from a
triangular orifice expands into three sheets, in planes bisecting at
right angles the three sides of the triangle. Generally a jet from an
orifice, in the form of a regular polygon of n sides, forms n sheets
in planes perpendicular to the sides of the polygon.

Bidone explains this by reference to the simpler case of meeting
streams. If two equal streams having the same axis, but moving in
opposite directions, meet, they spread out into a thin disk normal to
the common axis of the streams. If the directions of two streams
intersect obliquely they spread into a symmetrical sheet perpendicular
to the plane of the streams.

Let a1, a2 (fig. 24) be two points in an orifice at depths h1, h2 from
the free surface. The filaments issuing at a1, a2 will have the
different velocities [root](2gh1) and [root](2gh2). Consequently they
will tend to describe parabolic paths a1cb1 and a2cb2 of different
horizontal range, and intersecting in the point c. But since two
filaments cannot simultaneously flow through the same point, they must
exercise mutual pressure, and will be deflected out of the paths they
tend to describe. It is this mutual pressure which causes the
expansion of the jet into sheets.

Lord Rayleigh pointed out that, when the orifices are small and the
head is not great, the expansion of the sheets in directions
perpendicular to the direction of flow reaches a limit. Sections taken
at greater distance from the orifice show a contraction of the sheets
until a compact form is reached similar to that at the first
contraction. Beyond this point, if the jet retains its coherence,
sheets are thrown out again, but in directions bisecting the angles
between the previous sheets. Lord Rayleigh accepts an explanation of
this contraction first suggested by H. Buff (1805-1878), namely, that
it is due to surface tension.

§ 26. _Influence of Temperature on Discharge of Orifices._--Professor
VV. C. Unwin found (_Phil. Mag._, October 1878, p. 281) that for
sharp-edged orifices temperature has a very small influence on the
discharge. For an orifice 1 cm. in diameter with heads of about 1 to
1½ ft. the coefficients were:--

Temperature F. C.
205° .594
62° .598

For a conoidal or bell-mouthed orifice 1 cm. diameter the effect of
temperature was greater:--

Temperature F. C.
190° 0.987
130° 0.974
60° 0.942

an increase in velocity of discharge of 4% when the temperature
increased 130°.

J. G. Mair repeated these experiments on a much larger scale (_Proc.
Inst. Civ. Eng._ lxxxiv.). For a sharp-edged orifice 2½ in. diameter,
with a head of 1.75 ft., the coefficient was 0.604 at 57° and 0.607 at
179° F., a very small difference. With a conoidal orifice the
coefficient was 0.961 at 55° and 0.98l at 170° F. The corresponding
coefficients of resistance are 0.0828 and 0.0391, showing that the
resistance decreases to about half at the higher temperature.

§ 27. _Fire Hose Nozzles._--Experiments have been made by J. R.
Freeman on the coefficient of discharge from smooth cone nozzles used
for fire purposes. The coefficient was found to be 0.983 for ¾-in.
nozzle; 0.982 for 7/8 in.; 0.972 for 1 in.; 0.976 for 1(1/8) in.;
and 0.971 for 1¼ in. The nozzles were fixed on a taper play-pipe, and
the coefficient includes the resistance of this pipe (_Amer. Soc. Civ.
Eng._ xxi., 1889). Other forms of nozzle were tried such as ring
nozzles for which the coefficient was smaller.