SPECIAL CASES
§ 155. (I) A Jet impinges on a plane surface at rest, in a direction normal to the plane (fig. 154).Let a jet whose section is w impinge with a velocity v on a plane surface at rest,
in a direction normal to the plane. The
particles approach the plane, are gradually
deviated, and finally flow away parallel to the plane, having then no velocity in the original direction of the jet. The quantity of water impinging per second is wv. The pressure on the plane, which is equal to the change of momentum per second, is P = (G/g)wva.
(2) If the plane is moving in the direction of the jet with the velocity tu, the quantity impinging per second is w(vsu). The momentum of this quantity before impact
is (G/g)w(v= u)v. After impact, the water
still possesses the velocity to in the
direction of the jet; and the momentum,
in that direction, of so much water as
impinges in one second, after impact, is FIG. 154.
1 t((G/g)w(v=u)u. The pressure on the
plane, which is the change of momentum
per second, is the difference of these quantities or P = (G/g)w(v= u)2. This differs from the expression obtained in the previous case, in that the relative velocity of the water and plane v=u is substituted for v. The expression maybe written P =2 XGXw(v= u)2/2g, where the last two terms are the volume of a prism of water whose section is the area of the jet and whose length is the head due to the relative velocity. The pressure on the plane is twice the weight of that prism of water. The work done when the plane
V
z ;»> >
OF WATER
is moving in the same direction as the jet is Pu= (G/g)w(vu)2u footpounds per second. There issue from the jet wv cub. ft. per second, and the energy of this quantity before impact is (G/2g)wv3. The efficiency of the jet is therefore n=2(vu)2it/v3. The value of u which makesthisa maximum isfound bydifferentiating and equating the differential coefficient to zero:
do/du = 2 (v24Vu+3112)/v3 =o;
u=v or 3v.
The former gives a minimum, the latter a maximum efficiency. Putting u= 3v in the expression above,
n max. = 287•
(3) If, instead of one plane moving before the jet, a series of planes are introduced at short intervals at the same point, the quantity of water impinging on the series will be wv instead of w(vu), and the whole pressure =(G/g)wv(vu). The work done is (G/g)wvu(vu). The efficiency n= (G/g)wvu(vu) = (G/2g)wv1=2u(vu)/v2. This becomes a maximum for do/du=2(v2u)=o, or u=av, and the n=i. This result is often used as an approximate expression for the velocity of greatest efficiency when a jet of water strikes the floats of a water wheel. The work wasted in this case is half the whole energy of the jet when the floats run at the best speed.
§ 156. (4) Case of a Jet impinging on a Concave Cup Vane. velocity of water v, velocity of vane in the same direction u (fig. 155), weight impinging per second = Gw(v  u).
If the cup is hemispherical, the water leaves the cup in a
direction parallel to the jet. Its relative velocity is vu when approaching the cup, and  (v u) when leaving it. Hence its absolute velocity when leaving the cup is
wee t.
u  (v  u) = 2U  v. The
u, change of momentum ~e r
l~ second = (G/g)w(v  u) iv (2u—v)) = 2(G/g)w(vu)2. Comparing this with case 2, it is seen that the pressure
,ee on a hemispherical cup is
2 uv double that on a flat plane.
cup=2(G/g)w (vu)2u foot
pounds per second. The efficiency of the jet is greatest when v=3u; in that case the efficiency =1 .
If a series of cup vanes are introduced in front of the jet, so that the quantity of water acted upon is wv instead of w(vu), then the whole pressure on the chain of cups is (G/g)wv{v(2uv)l =2(G/g)wv(vu). In this case the efficiency is greatest when v=2u, and the maximum efficiency is unity, or all the energy of the water is expended on the cups.
§ 157. (5) Case of a Flat Vane oblique to the Jet (fig.156).—This case presents some difficulty. The water spreading on the plane in all
directions from the point of impact, different particles leave the plane with different absolute velocities. Let AB =v =velocity of water, AC =u =velocity of plane. Then, completing the parallelogram, AD represents in magnitude and direction the relative velocity of water and plane. Draw AE normal to the plane and DE parallel to the plane. Then the relative velocity AD may be regarded as consisting of two components, one AE normal, the other DE parallel to the plane. On the assumption that friction is insensible, DE is unaffected by impact, but AE is destroyed. Hence AE represents the entire change of velocity due to impact and the direction of that change. The pressure on the plane is in the direction AE, and its amount is = mass of water impinging per second X AE.
Let D AE =8, and lei AD =vr. Then AE =vr cos B; DE =vr sin B. If Q is the volume of water impinging on the plane per second, the change of momentum is (G/g)Qvr cos O. Let AC=u=velocity of the plane, and let AC make the angle CAE =6 with the normal to the plane. The velocity of the plane in the direction AE= u cos S. The work of the jet on the plane= (G/g)Qvr cos 0 u cos S. The same problem may be thus treated algebraically (fig. 157). Let BAF =a, and CAF =S. The velocity v of the water may be decomposed into AF =v cos a normal to the plane, and FB =v sin a parallel to the plane. Similarly the velocity of the plane =u =AC = BD can be decomposed into BG=FE=u cos S normal to the plane, and DG =u sin S parallel to the plane. As friction is neglected, the velocity of the water parallel to the plane is unaffected by the impact, but its component v cos a normal to the plane becomes after87
impact the same as that of the plane, that is, u cos S. Hence the change of velocity during impact =AE =v cos au cos S. The change of momentum per second, and consequently the normal
q FIG. 157.
pressure on the plane is N = (Gig) Q (v cos aucos S). The pressure in the direction in which the plane is moving is P = N cos S = (G/g)Q (v cos au cos S) cos S, and the work done on the plane is Pu= (G/g)Q(v cos au cos S) u cos S, which is the same expression as before, since AE =vr cos B =v cos au cos S.
In one second the plane moves so that the point A (fig. 158) comes to C, or from the position
shown in full lines to the position shown in dotted lines. If the plane remained stationary, a length AB =v of the jet would impinge on the plane, but, since the plane moves in the same direction as the jet, only the length HB=ABAH impinges on the plane.
But AH =AC cos S/ cos a = u cos S/ cos a, and therefore HB =vu cos S/ cos a. Let w=sectional area of jet; volume impinging on plane per second=Q=w(vu cos
3/cos a) =w(v cos au cos S)/ cos a. Inserting this in the formulae above, we get
N=G (v cos au cos 3)2; g cos a
G wcos3
P= (v cos au cos 6)2; cos a
Pu = w u cos S(v cos au cos 3)2.
g cos a
Three cases may be distinguished:
(a) The plane is at rest. Then u =o, N = (G/g)wv2 cos a; and the work done on the plane and the efficiency of the jet are zero.
(b) The plane moves parallel to the jet. Then 5= a, and Pu = (G/g)wucos2a(vu)2,which is a maximum when u=;v.
When u =iv then Pu max. = (G/g)wv3 cos 2a, and the efficiency =n= cos 2a.
(c) The plane moves perpendicularly to the jet. Then 3 =90°a;
cos S =sin a; and Pu=Gwusin 5(v cos au sin a)2. This is a maxi
g cos a
mum when u = iv cos a.
When u= 3v cos a, the maximum work and the efficiency are the same as in the last case.
§ 158. Best Form of Vane to receive Water.—When water impinges normally or obliquely on a plane, it is scattered in all directions after impact, and the work carried away by the water is then generally lost, from the impossibility of dealing afterwards with streams of water deviated in so many directions. By suitably forming the vane,
however, the water may be entirely deviated in one direction, anti the loss of energy from agitation of the water is entirely avoided.
Let AB (fig. 159) be a vane, on which a jet of water impinges at the point A and in the direction AC. Take AC=v=velocity of
(I)
(2) (3)
water, and let AD represent in magnitude and direction the velocity of the vane. Completing the parallelogram, DC or AE represents the direction in which the water is moving relatively to the vane. If the lip of the vane at A is tangential to AE, the water will not have its direction suddenly changed when it impinges on the vane, and will therefore have no tendency to spread laterally. On the contrary it will be so gradually deviated that it will glide up the vane in the direction AB. This is sometimes expressed by saying that the vane
receives the water without shock.
§ 159. Floats of Poncelet Water Wheels.—Let AC (fig. 16o) represent the direction of a thin horizontal stream of water having the
velocity v. Let AB be a curved float moving horizontally with velocity u. The relative motion of water and float is then initially horizontal, and equal to v—u.
In order that the float may receive the water without shock, it is necessary and sufficient that the lip of the float at A should be tangential to the direction AC of relative motion. At the end of (v—u)/g seconds the float moving with the velocity u comes to the position AIBI, and during this time a particle of water received at A and gliding up the float with the relative velocity v—u, attains a height DE = (v—u)2/2g. At E the water comes to relative rest. It then descends along the float, and when after 2(v—u)/g seconds the float has come to A2B2 the water will again have reached the lip at A2 and will quit it tangentially, that is, in the direction CA2, with a relative velocity — (v— u) = —Al (2gDE) acquired under the influence of gravity. The absolute velocity of the water leaving the float is therefore u—(v—u) =2n—V. If u= 2 v, the water will drop off the bucket deprived of all energy of motion. The whole of the work of the jet must therefore have been expended in driving the float. The water will have been received without shock and discharged without velocity. This is the principle of the Poncelet wheel, but in that case the floats move over an arc of a large circle; the stream of water has considerable thickness (about 8 in.); in order to get the water into and out of the wheel, it is then necessary that the lip of the float should make a small angle (about 15°) with the direction of its motion. The water quits the wheel with a little of its energy of motion remaining.
§ 16o. Pressure on a Curved Surface when the Water is deviated wholly in one Direction.—When a jet of water impinges on a curved
surface in such a direction that it is received without shock, the pressure on the surface is due to its gradual deviation from its first direction. On any portion of the area the pressure is equal and opposite to the force required to cause the deviation of so much water as rests on that surface. In common language, it is equal to the centrifugal force of that quantity of water.
Case I. Surface Cylindrical and Stationary.—Let AB (fig. 161)
be the surface, having its axis at 0 and its radius =r. Let the water impinge at A tangentially, and quit the surface tangentially at B. Since the surface is at rest, v is both the absolute velocity of the water and the velocity relatively to the surface, and this remains unchanged during contact with the surface, because the deviating force is at each point perpendicular to the direction of motion. The water is deviated through an angle BCD =AOB =4. Each particle of water of weight p exerts radially a centrifugal force pv2/rg. Let the thickness of the stream =et ft. Then the weight of water resting on unit of surface=Gt lb; and the normal pressure per unit of surface =n=Gtv2/gr. The resultant of the radial pressures uniformly distributed from A to B will be a force acting in the direction OC bisecting AOB, and its magnitude will equal that of a force of intensity = n, acting on the projection of AB on a plane perpendicular to the direction OC. The length of the chord AB = 2r sin 2¢; let b =breadth of the surface perpendicular to the plane of the figure. The resultant pressure on surface
=R =2rb sin X gt• r =2 G btv2sin,
which is independent of the radius of curvature. It may be inferred that the resultant pressure is the same for any curved surface of the same projected area, which deviates the water through the same angle.
Case 2. Cylindrical Surface moving in the Direction AC with Velocity u.—The relative velocity =v—u. The final velocity BF (fig. 162) is found by combining the relative velocity BD=v—u tangential to the surface with the velocity BE =u of the surface. The intensity of normal pressure, as in the last case, is (G/g)t(v—u)2/r. The resultant
0 P UN F e.g
normal pressureR=2(G/g)bt(v—u)2sin 2 4. This resultant pressure may be resolved into two components P and L, one parallel and the other perpendicular to the direction of the vane's motion. The former is an effort doing work on the vane. The latter is a lateral force which does no work.
P = R sin 2¢=(G/g)bt(v—u)2(1 cos 4);
L = R cos ~=(G/g)bt(v—u)2sin0.
The work done by the jet on the vane is Pu=(G/g)btu(v—u)2(icos 4), which is a maximum when u =Iv. This result can also be obtained by considering that the work done on the plane must be equal to the energy lost by the water, when friction is neglected.
If 4=18o°, cos cis= I—cos 0=2; then P=2(G/g)bt(vu)2, the same result as for a concave cup.
§ 161. Position which a Movable Plane takes in Flowing Water.
When a rectangular plane, movable about an axis parallel to one of its sides, is placed in an in
definite current of fluid, it takes a position such that the resultant of the normal pres
sures on the two sides of the t axis passes through the axis. If, therefore, planes pivoted
'
so that the ratio a/b (fig. 163)
is varied are placed in water,
and the angle they make with
the direction of the stream is
observed, the position of the
resultant of the pressures on the plane is determined for
different angular positions. Experiments of this kind have been made by Hagen. Some of his results are given in the following table:
Larger plane. Smaller Plane.
a/b =1.o ~_.... 4=90°
0. 3 ° 722°
6o° 57°
oo•6 25 ° 29°
29
0.5 13° 13°
0.4 8° 61°
0.3 62°
0.2 40
§ 162. Direct Action distinguished from Reaction (Rankine, Steam Engine, § 147).
The pressure which a jet exerts on a vane can be distinguished into two parts, viz. :
(f) The pressure arising from changing the direct component of the velocity of the water into the velocity of the vane. In fig. 153, § 154, ab cos bae is the direct component of the water's velocity, or component in the direction of motion of vane. This is changed into the velocity ae of the vane. The pressure due to direct impulse is then
PI=GQ(ab cos bae—ae)/g.
For a flat vane moving normally, this direct action is the only action producing pressure on the vane.
(2) The term reaction is applied to the additional action due to the direction and velocity with which the water glances off the vane. It is this which is diminished by the friction between the water and the vane. In Case 2, § 16o, the direct pressure is
PI =Gbt(v—u)2/g.
That due to reaction is
P2 = —Gbt(v—u)2cos 4/g.
If 0<9o'', the direct component of the water's motion is not wholly converted into the velocity of the vane, and the wholes
Al FIG. 160.
0
Pressure due to direct impulse is not obtained. If h> 90°, cos y5 is negative and an additional pressure due to reaction is obtained.
§ 163. Jet Propeller.—In the case of vessels propelled by a jet of water (fig. :64), driven sternwards from orifices at the side of the vessel, the water, originally at rest outside the vessel, is drawn into the ship and caused to move with the forward velocity V of the ship. Afterwards it is projected sternwards from the jets with a velocity v relatively to the ship, or v—V relatively to the earth. If S2 is the total sectional area of the jets, 52v is
L J 1 J the quantity of water discharged per second. The momentum generated per second in a sternward direction is (G/g)Slv(v—V), and this is equal to the forward acting reaction P which propels the ship.
The energy carried away by the water
(G/g)1s(v—V)2• (I) The useful work done on the ship
PV = (G; g)S2v(v — V) V. (2) Adding (1) and (2), we get the whole work expended on the water, neglecting friction: —
W = z (G/g)S2v(v2 — V2).
Hence the efficiency of the jet propeller is
PV/W=2V/(v+V). (3)
This increases towards unity as v approaches V. In other words, the less the velocity of the jets exceeds that of the ship, and therefore the greater the area of the orifice of discharge, the greater is the efficiency of the propeller.
In the " \V'aterwitch " v was about twice V. Hence in this case the theoretical efficiency of the propeller, friction neglected, was about f.
§ 164. Pressure of a Steady Stream in a Uniform Pipe on a Plane normal to the Direction of Motion.—Let CD (fig. 165) be a plane
A0 Al i Aa
_
z , ~
\ten se%/% 
A0 AI A
placed normally to the stream which, for simplicity, may be supposed to flow horizontally. The fluid filaments are deviated in front of the plane, form a contraction at AIAI, and converge again, leaving a mass of eddying water behind the plane. Suppose the section AoAo taken at a point where the parallel motion has not begun to be disturbed, and A2A2 where the parallel motion is reestablished. Then since the same quantity of water with the same velocity passes AoAo, A2A2 in any given time, the external forces produce no change of momentum on the mass A0A0A2A2, and must therefore be in equilibrium. If 12 is the section of the stream at AOA0 or A2A2, and w the area of the plate CD, the area of the contracted section of the stream at AIAI will be cs(S2— w), where cs is the coefficient of contraction. Hence, if v is the velocity at AoAo or A2A2, and vi the velocity at A1A1,
v12 =cw (12—w);
..ill=vS2/cc(S2—w)• (I)
Let pis p,, p2 be the pressures at the three sections. Applying Bernoulli's theorem to the sections AoAo and AIAI,
Po v2 p, rI2
G+2g G 2g.
Also, for the sections AIAI and A2A2, allowing that the head due to the relative velocity vi —v is lost in shock:
pl vl2 p2 v2 (v, —v)2. G+2g=G+2g+ 2g ...po—p2=G(vi—v)2/2g; or, introducing the value in (1),
G S2
po—p2=2g(c (S2—w)—1)20 (3)
Now the external forces in the direction of motion acting on the mass A0A0A2A2 are the pressures port, —NI at the ends, and the reaction —R of the plane on the water, which is equal and opposite to the pressure of the water on the plane. As these are in equilibrium,
(Po—p2)1—R=0;
R=GS2 (c0(12—w)—) 2g;If S2/w = p, the equation (4) becomes
u
R =Gw 2g p P cc(p—I)—I
R = Gw (v2/2g) K,
where K depends only on the ratio of the sections of the stream and plane.
For example, let ca=o.85, a value which is probable, if we allow that the sides of the pipe act as internal borders to an orifice. Then
K =p (I'176 P I I2• P —
P= K=
I o0
2 3.66
3 1'75
4 1.29
5 1•Io
to •94
50 2.00
loo 3.50
The assumption that the coefficient of contraction cs is constant for different values of p is probably only true when p is not very large. Further, the increase of K for large values of p is contrary to experience, and hence it may be inferred that the assumption that all the filaments have a common velocity vI at the section ALAI and a common velocity v at the section A2A2 is not true when the stream is very much larger than the plane. Hence, in the expression
R = KGwv2/2g,
K must be determined by experiment in each special case. For a cylindrical body putting w for the section, ca for the coefficient of contraction, c0(12—w) for the area of the stream at AIAI,
vl = v12/ca (12 — w) ; v2 = vS2/ (S2 —w) ;
or, putting p=12/w,
vl=vp/c,(P—1), v2=vp/(P—I). Then
R = KIGwv2/2g,
where l l )
KI=P (PPI)2(cc—1)2+ (p'2 PI1)2
Taking ca=o•85 and p=4, KI=o•467, a value less than before. Hence there is less pressure on the cylinder than on the thin plane.
§ 165. Distribution of Pressure on a Surface on which a Jet impinges normally.—The principle of momentum gives readily enough the total or resultant pressure of a jet impinging on a plane surface, but in some cases it is useful to know the distribution of the pressure.
(2)
(4)
an expression like that for the pressure of an isolated jet on an indefinitely extended plane, with the addition of the term in brackets, which depends only on the areas of the stream and the plane. For a given plane, the expression in brackets diminishes as 12 increases.
which is of the form
/ 2
(4a)
plane must be at the axis of the jet, and the pressure must decrease from the axis outwards, in some such way as is shown by the curve of pressure in fig. 167, the branches of the curve being probably asymptotic to the plane.
For simplicity suppose the jet is a vertical one. Let hl (fig. 167) be the depth of the orifice from the free surface, and vI the velocity of discharge. Then, if w is the area of the orifice, the quantity of water impinging on the plane is obviously
Q= wv, = w a/ (2ghl) ;
that is, supposing the orifice rounded, and neglecting the coefficient of discharge.
The velocity with which the fluid reaches the plane is, however, greater than this, and may reach the value
v — I (2gh) ;
where h is the depth of the plane below the free surface. The
external layers of fluid subjected throughout, after leaving the
orifice, to the atmospheric pressure will attain the velocity v, and
will flow away with this velocity unchanged except by friction.
The layers towards the interior of the jet, being subjected to a pressure
greater than atmospheric pressure, will attain a less velocity, and so
much less as they are nearer the centre of the jet. But the pressure
itr FIG. 166.
The problem in the case in which the plane is struck normally, and the jet spreads in all directions, is one of great complexity, but even in that case the maximum intensity of the pressure is easily assigned. Each layer of water flowing from an orifice is gradually deviated (fig. 166) by contact with the surface, and during deviation exercises a centrifugal pressure towards the
axis of the jet. The force exerted by each small mass of water is
normal to its path and inversely as the radius of curvature of the path. Hence the greatest pressure on the
can in no case exceed the pressure v2/2g or h measured in feet of water, or the direction of motion of the water would be reversed, and there would be reflux. Hence the maximum intensity of the pressure
of the jet on the plane is h ft. of water. If the pressure curve is drawn with pressures represented by feet of water, it will touch the free water surface at the centre of the jet.
Suppose the pressure curve rotated so as to form a solid of revolution. The weight of water contained in that solid is the total pressure of the jet on the surface, which has already been determined. Let V = volume of this solid, then GV is its weight in pounds. Consequently
GV = (G/g)wvly;
V =20)11 (hhr).
We have already, therefore, two conditions to be satisfied by the pressure curve.
Some very interesting experiments on the distribution of pressure on a surface struck by a jet have been made by J. S. Beresford (Prof. Papers on Indian Engineering, No. cccxxii.), with a view to afford information as to the forces acting on the aprons of weirs. Cylindrical jets a in. to 2 in. diameter, issuing from a vessel in which the water level was constant, were allowed to fall vertically on a brass plate 9 in. in diameter. A small hole in the brass plate communicated by a flexible tube with a vertical pressure column. Arrangements were made by which this aperture could be moved t~ in. at a time across the area struck by the jet. The height of the pressure column, for each position of the aperture, gave the pressure at that point of the area struck by the jet. When the aperture was
,
caner e,&o..Li s.rterr/ r Oian1:99' ,
=_  IYpter Level in Reserooir // •
1.0 o,s o o•s ,•o
Distance from axis of jet in inches.
exactly in the axis of the jet, the pressure column was very nearly level with the free surface in the reservoir supplying the jet; that is, the pressure was very nearly v2/2g. As the aperture moved away from the axis of the jet, the pressure diminished, and it became insensibly small at a distance from the axis of the jet about equal to the diameter of the jet. Hence, roughly, the pressure due to the jet extends over an area about four times the area of section of the jet.
Fig. 168 shows the pressure curves obtained in three experiments with three jets of the sizes shown, and with the free surface level in the reservoir at the heights marked.
Experiment r. Experiment 2. Experiment 3.
Jet'988in. diameter. .
Jet 19'5 in. diameter.
Jet'475ax5in. diameter. l u Q.8 °
o 0.g aa, n 1
m'0°..3 y~ yo
w& 9 o w
q
W u ' .
o n.E 1 a
non .1 •0
x °w N o p
q a
p_to~ s a .
0 O.0 C
'a as °o
,sti w
o
q
43 0 40.5 42.15 0 42 27.15 0 26.9
'05 3940 .9 •05 41.9 „ •o8 26.9
+, '1 37.539'5 ,, •1 41.541.8 ,, •13 26.8
'15 35 „ .15 41 „ •18 26.526.6
'2 33.537 „ 2 40.3 „ •23 26.426.5
++ '25 31 .25 39.2 •28 26.326.6
„ •3 2127 „ '3 37.5 27 33 26.2
„ '35 21 „ 35 34'8 „ •38 25'9
.4 14 '45 27 '43 25'5
'45 8 42.25 .5 23 „ '48 25
.5 3.5 „ 55 18.5 ,, 53 24'5
.55 1 •6 13 „ .58 24
„ •6 0.5 „ '65 8.3 „ '63 23.3
,, •65 0 „ .7 5 •68 22.5
'75 3 „ •73 21.8
•8 2.2 „ •78 21
42.15 •85 1.6 „ .83 20.3
.95 1 •88 19.3
'93 18
'98 17
' 1'13 13.5
265
„ 1.18 12.5
1'23 io•8
„ 1.28 9.5
33 8
1.38 7
1.43 6.3
9, 1.48 5
„ 1.53 4'3
158 35
„ 1.9 2
As the general form of the pressure curve has been already indicated, it may be assumed that its equation is of the form
y = abx. (1)
But it has already been shown that for x = o, yoh, hence a = h. To determine the remaining constant, the other condition may be used, that the solid formed by rotating the pressure curve represents the total pressure on the plane The volume of the solid is
00
V =f 0 25rxydx
where f is a coefficient having about the value 1.3 for a plate moving in still fluid, and 18 for a current impinging on a fixed plane, whether the fluid is air or water. The difference in the value of the coefficient in the two cases is perhaps due to errors of experiment. There is a similar resistance to motion in the case of all bodies of " unfair +' form, that is, in which the surfaces over which the water slides are not of gradual and continuous curvature.
The stress between the fluid and plate arises chiefly in this way.
_ 2
ao' o = 27rh 0 b xdx
. 2 00
o = (7rh/logeb) [ _ b_: ]o
5 = arh/logeb. so Using the condition already stated,
2w 1(hhi) =rrh/log0b, loge b = (,r/2w) ' (h/hl).
Putting the value of bin (2) in eq. (I), and also r for the radius of the jet at the orifice, so that w=irr2, the equation to the pressure curve is
y=he l"\Ihr
§ 166. Resistance of a Plane moving through a Fluid, or Pressure 0 of a Current on a Plane.When a thin plate moves through the air, or through an indefinitely large mass of still water, in a direction normal to its surface, there is an excess of pressure on the anterior face and a dirninution of pressure on the posterior face. Let v be the relative velocity of the plate and fluid, 12 the area of the plate, G the density of the fluid, h the height due to the velocity, then the total resistance is expressed by the equation
R=fGc v'/2g pounds =fGf2h;
The streams of fluid deviated in front of the plate, supposed for definiteness to be moving through the fluid, receive from it forward momentum. Portions of this forward moving water are thrown off laterally at the edges of the plate, and diffused through the surrounding fluid, instead of falling to their original position behind the plate. Other portions of comparatively still water are dragged into motion to fill the space left behind the plate; and there is thus a pressure less than hydrostatic pressure at the back of the plate. The whole resistance to the motion of the plate is the sum of the excess of pressure in front and deficiency of pressure behind. This resistance is independent of any friction or viscosity in the fluid, and is due simply to its inertia resisting a sudden change of direction at the edge of the plate.
Experiments made by a whirling machine, in which the plate is fixed on a long arm and moved circularly, gave the following values of the coefficient f. The method is not free from objection, as the centrifugal force causes a flow outwards across the plate.
Approximate Values of f.
Area of Plate Borda. Hutton. Thibault.
in sq. ft.
0'13 I'39 P24
0.25 1'49 1'43 1.525
0.63 I.64
I•II .. .. 1.784
There is a steady increase of resistance with the size of the plate, in part or wholly due to centrifugal action.
P. L. G. Dubuat (1734–1809) made experiments on a plane i ft. square, moved in a straight line in water at 3 to 6% ft. per second. Calling m the coefficient of excess of pressure in front, and n the coefficient of deficiency of pressure behind, so that f =m+n, he found the following values:
m=1; n=0.433;f=1.433.
The pressures were measured by pressure columns. Experiments by A. J. lorin (1795–1880), G. Piobert (1793–1871) and I. Didion (1798–1878) on plates of 0.3 to 2.7 sq. ft. area, drawn vertically through water, gave f=2.18; but the experiments were made in a reservoir of comparatively small depth. For similar plates moved through air they found f =1.36, a result more in accordance with those which precede.
For a fixed plane in a moving current of water E. Mariotte found
f =1.25. Dubuat, in experiments in a current of water like those mentioned above, obtained the values m=1.186; n =o•67o; f = 1.856. Thibault exposed to wind pressure planes of 1.17 and 2.5 sq. ft. area, and found f to vary from 1.568 to 2.125, the mean value being f =1.834, a result agreeing well with Dubuat.
§ 167. Stanton's Experiments on the Pressure of Air on Surfaces.
At the National Physical Laboratory, London, T. E. Stanton carried
out a series of experiments on the distribution of pressure on surfaces
in a current of air passing through an air trunk. These were on a
small scale but with exceptionally accurate means of measurement.
These experiments differ from those already given in that the plane
is small relatively to the cross section of the current (Prot. Inst.
Civ. Eng. clvi., 1904). Fig. 169 shows the distribution of pressure
on a square plate. ab is the plate in
vertical section. acb the distribution
4f, of pressure on the windward and alb
that on the leeward side of the central
section. Similarly aeb is the distribu
tion of pressure on the windward and
afb on the leeward side of a diagonal
' section. The intensity of pressure at
the centre of the plate on the windward side was in all cases p=Gv2/2g lb per sq. ft., where G is the weight of a cubic foot of air and v the velocity of the current in ft. per sec. On the leeward side the negative pressure is uniform except near the edges, and its value depends on the form of the plate. For a circular plate the pressure on the leeward side was o•48 Gv2/2g and for a rectangular plate o•66 Gv2/2g. For circular or square plates the resultant pressure on the plate was P=o•oo,26 v2 lb per sq. ft. where v is the velocity of the current in ft. per sec. On a lon narrow rectangular plate the resultant pressure was nearly 6o% greater than on a circular plate. In later tests on larger planes in free air, Stanton found resistances 18 % greater than those observed with small planes in the air trunk.
§ 168. Case when the Direction of Motion is oblique to the Plane. —
The determination of the pressure between a fluid and surface in this case is of importance in many practical questions, for instance, in assigning the load due to wind pressure on sloping and curved roofs, and experiments have been made by Hutton, Vince, and Thibault on planes moved circularly through air and water on a whirling machine.
Let AB (fig. 170) be a plane moving in the direction R making an angle ¢ with the plane. The resultant pressure between the fluid and the plane will be a normal
pressure N. The component R of this normal pressure is the resistance to the motion of the plane and the other component L is a lateral force resisted by the guides which support the plane. Obviously
R=N sink;
L = N cos 0.
In the case of wind pressure on
a sloping roof surface, R is the L
horizontal and L the vertical
component of the normal press FIG. 170. sure.
In experiments with the whirling machine it is the resistance to motion, R, which is directly measured. Let P be the pressure on a plane moved normally through a fluid. Then, for the same plane inclined at an angle cis to its direction of motion, the resistance was found by Hutton to be
R=P(sin 0)1.842 COS 4.
A simpler and more convenient expression given by Colonel Duchemin is
R=2P sin2 ¢/(I+sin' 4)).
Consequently, the total pressure between the fluid and plane is N=2P sin ¢/(I+sin' ¢) =2P/(cosec c + sin ¢), and the lateral force is
L=2P sin (A cos 4'/(I+sine p).
In 1872 some experiments were made for the Aeronautical Society ' on the pressure of air on oblique planes. These plates, of i to 2 ft. square, were balanced by ingenious mechanism designed by F. H. Wenham and Spencer Browning, in such a manner that both the pressure in the direction of the air current and the lateral force were separately measured. These planes were placed opposite a blast from a fan issuing from a wooden pipe 18 in. square. The pressure of the blast varied from l% to I in. of water pressure. The following are thet~results given in pounds per square foot of the plane, and a comparison of the experimental results with the pressures given by Duchemin's rule. These last values are obtained by taking P=3.31, the observed pressure on a normal surface:
Angle between Plane and Direction 150 200 600 900
of Blast . . . .
Horizontal pressure R . . . 0.4 o•61 2.73 3.31
Lateral pressure L 1.6 1.96 1.26 ..
Normal pressures/ L2+R2 1.65 2.05 3.01 3.31
Normal pressure by Duchemin 's rule 1.605 2.027 3.276 3.31
End of Article: SPECIAL 

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