DISSIPATION OF HEAD IN SHOCK
§ 36. Relation of Pressure and Velocity in a Stream in Steady Motion when the Changes of Section of the Stream are Abrupt.—When a stream changes section abruptly, rotating eddies are formed which dissipate energy. The energy absorbed in producing rotation is at once abstracted from that effective in causing the flow, and sooner or later it is wasted by frictional resistances due to the rapid relative motion of the eddying parts of the fluid. In such cases the work thus expended internally in the fluid is too important to be neglected, and the energy thus lost is commonly termed energy lost in shock. Suppose fig. 38 to represent a stream having such an abrupt change of section. Let AB, CD be normal sections at points where ordinary stream line motion has not been disturbed and where it has been reestablished. Let w, p, v be the area of section. pressure and velocity at AB, and PI, vi corresponding quantities at CD. Then if no work were expended internally, and assuming the stream horizontal, we should have
p/G+v2/2g = p,/G+v,2/2g• (I)
But if work is expended in producing irregular eddying motion, the head at the section CD will be diminished.
Suppose the mass ABCD comes in a short time t to A'B'C'D'. The resultant force parallel to the axis of the stream is
pw+Ps(cei—w)—piwi,
where po is put for the unknown pressure on the annular space between AB and EF. The impulse of that force is
Ipw+po(wi—w)—piwi}t.
The horizontal change of momentum in the same time is the difference of the momenta of
rI c ,CDC'D' and ABA'B',
 . because the amount of momentum be
t / tween A'B' and CD
remains unchanged
if the motion is
steady. The volume
~ — of ABA'B' or CDC'D',
being the inflow and outflow in the time
, ) Qt=wvt=wivit,
and the momentum of
t
these masses is
The change of momentum is therefore (G/g)Qt(vi—v). Equating this to the impulse,
t pw +po (i—w)—piwi } t = (G/g)Qt(viv).
Assume that po = p, the pressure at AB extending unchanged through the portions of fluid in contact with AE, BF which lie out of the path of the stream. Then (since Q=wivi)
(ppi) = (G/g)vi (viv) ;
p/G—pi/G =v1(vi—v)/g; (2)
p/G+v2/2g =pi/G+v12/2g+(vvi)2/2g• (3) This differs from the expression (I), § 29, obtained for cases where no sensible internal work is done, by the last term on the right. That is, (v—v1)"/2g has to be added to the total head at CD, which is piiG+v12/2g, to make it equal to the total head at AB, or (v—vi)2/2g is the head lost in shock at the abrupt change of section. But v—vi is the relative velocity of the two parts of the stream. Hence, when an abrupt change of section occurs, the head due to the relative velocity is lost in shock, or (v—vi)2/2g footpounds of energy is wasted for each pound of fluid. Experiment verifies this result, so that the assumption that po = p appears to be admissible.
If there is no shock,
p1 /G = p/G + (v2—vi2) /2g•
It there is shock,
pi/G = p/Gvi (viv)/g•
Hence the pressure head at CD in the second case is less than in the former by the quantity (t!—vi)2/2g, or, putting wive=wv, by the quantity
(v2/2g) (1—w/wi)2. (4)
V. THEORY OF THE DISCHARGE FROM ORIFICES AND MOUTHPIECES
37. Minimum Coefficient of Contraction. Reentrant Mouth
piece of Borda.—In one special case the coefficient of contraction can be determined theoretically, and, as it is the case where the convergence of the streams approaching the orifice takes place through the greatest possible angle, the coefficient thus determined is the minimum coefficient.
Let fig. 39 represent
a vessel with vertical
sides, 00 being the
free water surface, at
which the pressure is
pa. Suppose the liquid
issues by a horizontal
mouthpiece, which is
reentrant and of the
greatest length which
permits the jet to
spring clear from the
inner end of the
orifice, without adher
ing to its sides. With
such an orifice the
velocity near the
points CD is negligible,
and the pressure at those points may be taken equal to the hydro
static pressure due to the depth from the free surface. Let 12 be
the area of the mouthpiece AB, w that of the contracted jet aa
Suppose that in a short time t, the mass OOaa comes to the position 0'O' a'a'; the impulse of the horizontal external forces acting on the mass during that time is equal to the horizontal change of momentum.
The pressure on the side OC of the mass will be balanced by the pressure on the opposite side OE, and so for all other portions of the vertical surfaces of the mass, excepting the portion EF opposite the mouthpiece and the surface AaaB of the jet. On EF the pressure is simply the hydrostatic pressure due to the depth, that is, (pa+Gh)St. On the surface and section AaaB of the jet, the horizontal resultant of the pressure is equal to the atmospheric pressure pa acting on the vertical projection AB of the jet; that is, the resultant pressure is —pat2. Hence the resultant horizontal force for the whole mass OOaa is (pa+Gh)S2pa[2=Ghat. Its impulse in the time t is Ghst t. Since the motion is steady there is no change of momentum between O'O' and aa. The change of horizontal momentum is, therefore, the difference of the horizontal momentum lost in the space 000'0' and gained in the space aaa'a'. In the former space there is no horizontal momentum.
The volume of the space aaa'a' is wvt; the mass of liquid in that space is (G/g)wvt; its momentum is (G/g)wv2t. Equating impulse to momentum gained,
Ght2t= (G/g)wv2t;
.'.w/it = gh/v2.
v2=2gh, and w/S2=es;
w/It ==eal
a result confirmed by experiment with mouthpieces of this kind. A similar theoretical investigation is not possible for orifices in plane surfaces, because the velocity along the sides of the vessel in the neighbourhood of the orifice is not so small that it can be neglected. The resultant horizontal pressure is therefore greater than Ght , and the contraction is less. The experimental values of the coefficient of discharge for a reentrant mouthpiece are 0.5149 (Borda), 0.5547 (Bidone), 0.5324 (Weisbach), values which differ little from the theoretical value, 0.5, given above.
§ 38. Velocity of Filaments issuing in a Jet.—A jet is composed of fluid filaments or elementary streams, which start into motion at some point in the
interior of the vessel from which the fluid is discharged, and gradually acquire the velocity of the jet. Let Mm, fig. 40 be such a filament, the point M being taken where the velocity is insensibly small, and m at the most contracted section of the jet, where the filaments have be
come parallel and
exercise uniform mutual pressure. Take the free surface AB for datum line, and let pi, vi, hi, be the pressure, velocity and depth below datum at M ; p, v, h, the corresponding quantities at m. Then § 29, eq. (3a),
vi2/2g + pi /G—hi = v2/2g+p/Gh. (I) But at M, since the velocity is insensible, the pressure is the hydrostatic pressure due to the depth; that is, vi = o, pi = P +Ghi. At m, p=pa, the atmospheric pressure round the jet. Hence, inserting these values,
or
That is, neglecting the viscosity of the fluid, the velocity of filaments at the contracted section of the jet is simply the velocity due to the difference of level
of the free surface in the reservoir and the orifice.
End of Article: DISSIPATION OF HEAD IN 

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