Chapter IV: Theory of the Steady Motion of Fluids

§ 28. The general equation of the steady motion of a fluid given under
Hydrodynamics furnishes immediately three results as to the
distribution of pressure in a stream which may here be assumed.

(a) If the motion is rectilinear and uniform, the variation of
pressure is the same as in a fluid at rest. In a stream flowing in an
open channel, for instance, when the effect of eddies produced by the
roughness of the sides is neglected, the pressure at each point is
simply the hydrostatic pressure due to the depth below the free
surface.

(b) If the velocity of the fluid is very small, the distribution of
pressure is approximately the same as in a fluid at rest.

(c) If the fluid molecules take precisely the accelerations which they
would have if independent and submitted only to the external forces,
the pressure is uniform. Thus in a jet falling freely in the air the
pressure throughout any cross section is uniform and equal to the
atmospheric pressure.

(d) In any bounded plane section traversed normally by streams which
are rectilinear for a certain distance on either side of the section,
the distribution of pressure is the same as in a fluid at rest.

DISTRIBUTION OF ENERGY IN INCOMPRESSIBLE FLUIDS.

§ 29. _Application of the Principle of the Conservation of Energy to
Cases of Stream Line Motion._--The external and internal work done on
a mass is equal to the change of kinetic energy produced. In many
hydraulic questions this principle is difficult to apply, because from
the complicated nature of the motion produced it is difficult to
estimate the total kinetic energy generated, and because in some cases
the internal work done in overcoming frictional or viscous resistances
cannot be ascertained; but in the case of stream line motion it
furnishes a simple and important result known as Bernoulli's theorem.

Let AB (fig. 25) be any one elementary stream, in a steadily moving
fluid mass. Then, from the steadiness of the motion, AB is a fixed
path in space through which a stream of fluid is constantly flowing.
Let OO be the free surface and XX any horizontal datum line. Let
[omega] be the area of a normal cross section, v the velocity, p the
intensity of pressure, and z the elevation above XX, of the elementary
stream AB at A, and [omega]1, p1, v1, z1 the same quantities at B.
Suppose that in a short time t the mass of fluid initially occupying
AB comes to A´B´. Then AA´, BB´ are equal to vt, v1t, and the volumes
of fluid AA´, BB´ are the equal inflow and outflow = Qt = [omega]vt =
[omega]1v1t, in the given time. If we suppose the filament AB
surrounded by other filaments moving with not very different
velocities, the frictional or viscous resistance on its surface will
be small enough to be neglected, and if the fluid is incompressible no
internal work is done in change of volume. Then the work done by
external forces will be equal to the kinetic energy produced in the
time considered.

The normal pressures on the surface of the mass (excluding the ends A,
B) are at each point normal to the direction of motion, and do no
work. Hence the only external forces to be reckoned are gravity and
the pressures on the ends of the stream.

The work of gravity when AB falls to A´B´ is the same as that of
transferring AA´ to BB´; that is, GQt(z - z1). The work of the
pressures on the ends, reckoning that at B negative, because it is
opposite to the direction of motion, is (p[omega] × vt) - (p1[omega]1
× v1t) = Qt(p - p1). The change of kinetic energy in the time t is the
difference of the kinetic energy originally possessed by AA´ and that
finally acquired by BB´, for in the intermediate part A´B there is no
change of kinetic energy, in consequence of the steadiness of the
motion. But the mass of AA´ and BB´ is GQt/g, and the change of
kinetic energy is therefore (GQt/g) (v1²/2 - v²/2). Equating this to
the work done on the mass AB,

GQt(z - z1) + Qt(p - p1) = (GQt/g)(v1²/2 - v²/2).

Dividing by GQt and rearranging the terms,

v²/2g + p/G + z = v1²/2g + p1/G + z1; (1)

or, as A and B are any two points,

v²/2g + p/G + z = constant = H. (2)

Now v²/2g is the head due to the velocity v, p/G is the head
equivalent to the pressure, and z is the elevation above the datum
(see § 16). Hence the terms on the left are the total head due to
velocity, pressure, and elevation at a given cross section of the
filament, z is easily seen to be the work in foot-pounds which would
be done by 1 lb. of fluid falling to the datum line, and similarly p/G
and v²/2g are the quantities of work which would be done by 1 lb. of
fluid due to the pressure p and velocity v. The expression on the left
of the equation is, therefore, the total energy of the stream at the
section considered, per lb. of fluid, estimated with reference to the
datum line XX. Hence we see that in stream line motion, under
the restrictions named above, the total energy per lb. of fluid is
uniformly distributed along the stream line. If the free surface of
the fluid OO is taken as the datum, and -h, -h1 are the depths of A
and B measured down from the free surface, the equation takes the form

v²/2g + p/G - h = v1²/2g + p1/G - h1; (3)

or generally

v²/2g + p/G - h = constant. (3a)

§ 30. _Second Form of the Theorem of Bernoulli._--Suppose at the two
sections A, B (fig. 26) of an elementary stream small vertical pipes
are introduced, which may be termed pressure columns (§ 8), having
their lower ends accurately parallel to the direction of flow. In such
tubes the water will rise to heights corresponding to the pressures at
A and B. Hence b = p/G, and b´ = p1/G. Consequently the tops of the
pressure columns A´ and B´ will be at total heights b + c = p/G + z
and b´ + c´ = p1/G + z1 above the datum line XX. The difference of
level of the pressure column tops, or the fall of free surface level
between A and B, is therefore

[xi] = (p - p1)/G + (z - z1);

and this by equation (1), § 29 is (v1² - v²)/2g. That is, the fall of
free, surface level between two sections is equal to the difference of
the heights due to the velocities at the sections. The line A´B´ is
sometimes called the line of hydraulic gradient, though this term is
also used in cases where friction needs to be taken into account. It
is the line the height of which above datum is the sum of the
elevation and pressure head at that point, and it falls below a
horizontal line A´´B´´ drawn at H ft. above XX by the quantities a =
v²/2g and a´ = v1²/2g, when friction is absent.

§ 31. _Illustrations of the Theorem of Bernoulli._ In a lecture to the
mechanical section of the British Association in 1875, W. Froude gave
some experimental illustrations of the principle of Bernoulli. He
remarked that it was a common but erroneous impression that a fluid
exercises in a contracting pipe A (fig. 27) an excess of pressure
against the entire converging surface which it meets, and that,
conversely, as it enters an enlargement B, a relief of pressure is
experienced by the entire diverging surface of the pipe. Further it is
commonly assumed that when passing through a contraction C, there is
in the narrow neck an excess of pressure due to the squeezing together
of the liquid at that point. These impressions are in no respect
correct; the pressure is smaller as the section of the pipe is smaller
and conversely.

Fig. 28 shows a pipe so formed that a contraction is followed by an
enlargement, and fig. 29 one in which an enlargement is followed by a
contraction. The vertical pressure columns show the decrease of
pressure at the contraction and increase of pressure at the
enlargement. The line abc in both figures shows the variation of free
surface level, supposing the pipe frictionless. In actual pipes,
however, work is expended in friction against the pipe; the total head
diminishes in proceeding along the pipe, and the free surface level is
a line such as ab1c1, falling below abc.

Froude further pointed out that, if a pipe contracts and enlarges
again to the same size, the resultant pressure on the converging part
exactly balances the resultant pressure on the diverging part so that
there is no tendency to move the pipe bodily when water flows through
it. Thus the conical part AB (fig. 30) presents the same projected
surface as HI, and the pressures parallel to the axis of the pipe,
normal to these projected surfaces, balance each other. Similarly the
pressures on BC, CD balance those on GH, EG. In the same way, in any
combination of enlargements and contractions, a balance of pressures,
due to the flow of liquid parallel to the axis of the pipe, will be
found, provided the sectional area and direction of the ends are the
same.

The following experiment is interesting. Two cisterns provided with
converging pipes were placed so that the jet from one was exactly
opposite the entrance to the other. The cisterns being filled very
nearly to the same level, the jet from the left-hand cistern A entered
the right-hand cistern B (fig. 31), shooting across the free space
between them without any waste, except that due to indirectness of aim
and want of exact correspondence in the form of the orifices. In the
actual experiment there was 18 in. of head in the right and 20½ in. of
head in the left-hand cistern, so that about 2½ in. were wasted in
friction. It will be seen that in the open space between the orifices
there was no pressure, except the atmospheric pressure acting
uniformly throughout the system.

§ 32. _Venturi Meter._--An ingenious application of the variation of
pressure and velocity in a converging and diverging pipe has been made
by Clemens Herschel in the construction of what he terms a Venturi
Meter for measuring the flow in water mains. Suppose that, as in fig.
32, a contraction is made in a water main, the change of section being
gradual to avoid the production of eddies. The ratio [rho] of the
cross sections at A and B, that is at inlet and throat, is in actual
meters 5 to 1 to 20 to 1, and is very carefully determined by the
maker of the meter. Then, if v and u are the velocities at A and B, u
= [rho]v. Let pressure pipes be introduced at A, B and C, and let H1,
H, H2 be the pressure heads at those points. Since the velocity at B
is greater than at A the pressure will be less. Neglecting friction

H1 + v²/2g = H + u²/2g,

H1 - H = (u² - v²)/2g = ([rho]² - 1)v²/2g.

Let h = H1 - H be termed the Venturi head, then

u = [root]{[rho]²·2gh/([rho]² - 1)},

from which the velocity through the throat and the discharge of the
main can be calculated if the areas at A and B are known and h
observed. Thus if the diameters at A and B are 4 and 12 in., the areas
are 12.57 and 113.1 sq. in., and [rho] = 9,

u = [root]81/80 [root](2gh) = 1.007 [root](2gh).

If the observed Venturi head is 12 ft.,

u = 28 ft. per sec.,

and the discharge of the main is

28 × 12.57 = 351 cub. ft. per sec.

Hence by a simple observation of pressure difference, the flow in the
main at any moment can be determined. Notice that the pressure height
at C will be the same as at A except for a small loss h_f due to
friction and eddying between A and B. To get the pressure at the
throat very exactly Herschel surrounds it by an annular passage
communicating with the throat by several small holes, sometimes formed
in vulcanite to prevent corrosion. Though constructed to prevent
eddying as much as possible there is some eddy loss. The main effect
of this is to cause a loss of head between A and C which may vary from
a fraction of a foot to perhaps 5 ft. at the highest velocities at
which a meter can be used. The eddying also affects a little the
Venturi head h. Consequently an experimental coefficient must be
determined for each meter by tank measurement. The range of this
coefficient is, however, surprisingly small. If to allow for friction,
u = k[root]{[rho]²/([rho]² - 1)}[root](2gh), then Herschel found
values of k from 0.97 to 1.0 for throat velocities varying from 8 to
28 ft. per sec. The meter is extremely convenient. At Staines
reservoirs there are two meters of this type on mains 94 in. in
diameter. Herschel contrived a recording arrangement which records the
variation of flow from hour to hour and also the total flow in any
given time. In Great Britain the meter is constructed by G. Kent, who
has made improvements in the recording arrangement.

In the Deacon Waste Water Meter (fig. 33) a different principle is
used. A disk D, partly counter-balanced by a weight, is suspended in
the water flowing through the main in a conical chamber. The
unbalanced weight of the disk is supported by the impact of the water.
If the discharge of the main increases the disk rises, but as it rises
its position in the chamber is such that in consequence of the larger
area the velocity is less. It finds, therefore, a new position of
equilibrium. A pencil P records on a drum moved by clockwork the
position of the disk, and from this the variation of flow is inferred.

§ 33. _Pressure, Velocity and Energy in Different Stream Lines._--The
equation of Bernoulli gives the variation of pressure and velocity
from point to point along a stream line, and shows that the total
energy of the flow across any two sections is the same. Two other
directions may be defined, one normal to the stream line and in the
plane containing its radius of curvature at any point, the other
normal to the stream line and the radius of curvature. For the
problems most practically useful it will be sufficient to consider the
stream lines as parallel to a vertical or horizontal plane. If the
motion is in a vertical plane, the action of gravity must be taken
into the reckoning; if the motion is in a horizontal plane, the terms
expressing variation of elevation of the filament will disappear.[3]

Let AB, CD (fig. 34) be two consecutive stream lines, at present
assumed to be in a vertical plane, and PQ a normal to these lines
making an angle [phi] with the vertical. Let P, Q be two particles
moving along these lines at a distance PQ = ds, and let z be the
height of Q above the horizontal plane with reference to which the
energy is measured, v its velocity, and p its pressure. Then, if H is
the total energy at Q per unit of weight of fluid,

H = z + p/G + v²/2g.

Differentiating, we get

dH = dz + dp/G + vdv/g, (1)

for the increment of energy between Q and P. But

dz = PQ cos [phi] = ds cos [phi];

.: dH = dp/G + v dv/g + ds cos [phi], (1a)

where the last term disappears if the motion is in a horizontal plane.

Now imagine a small cylinder of section [omega] described round PQ as
an axis. This will be in equilibrium under the action of its
centrifugal force, its weight and the pressure on its ends. But its
volume is [omega] ds and its weight G[omega]ds. Hence, taking the
components of the forces parallel to PQ--

[omega]dp = Gv²[omega] ds/g[rho] - G[omega] cos [phi] ds,

where [rho] is the radius of curvature of the stream line at Q.
Consequently, introducing these values in (1),

dH = v² ds/g[rho] + v dv/g = (v/g)(v/[rho] + dv/ds) ds. (2)

CURRENTS

§ 34. _Rectilinear Current._--Suppose the motion is in parallel
straight stream lines (fig. 35) in a vertical plane. Then [rho] is
infinite, and from eq. (2), § 33,

dH = v dv/g.

Comparing this with (1) we see that

dz + dp/G = 0;

.: z + p/G = constant; (3)

or the pressure varies hydrostatically as in a fluid at rest. For two
stream lines in a horizontal plane, z is constant, and therefore p is
constant.

_Radiating Current._--Suppose water flowing radially between
horizontal parallel planes, at a distance apart = [delta]. Conceive
two cylindrical sections of the current at radii r1 and r2, where the
velocities are v1 and v2, and the pressures p1 and p2. Since the flow
across each cylindrical section of the current is the same,

Q = 2[pi]r1[delta]v1 = 2[pi]r2[delta]v2

r1v1 = r2v2

r1/r2 = v2/v1. (4)

The velocity would be infinite at radius 0, if the current could be
conceived to extend to the axis. Now, if the motion is steady,

H = p1/G + v1²/2g = p2/G + v2²/2g;
= p2/G + r1² + v1²/r2²2g;

(p2- p1)/G = v1²(1 - r1²/r2²)/2g; (5)

p2/G = H - r1²v1²/r2²2g. (6)

Hence the pressure increases from the interior outwards, in a way
indicated by the pressure columns in fig. 36, the curve through the
free surfaces of the pressure columns being, in a radial section, the
quasi-hyperbola of the form xy² = c³. This curve is asymptotic to a
horizontal line, H ft. above the line from which the pressures are
measured, and to the axis of the current.

_Free Circular Vortex._--A free circular vortex is a revolving mass of
water, in which the stream lines are concentric circles, and in which
the total head for each stream line is the same. Hence, if by any slow
radial motion portions of the water strayed from one stream line to
another, they would take freely the velocities proper to their new
positions under the action of the existing fluid pressures only.

For such a current, the motion being horizontal, we have for all the
circular elementary streams

H = p/G + v²/2g = constant;

.: dH = dp/G + v dv/g = 0. (7)

Consider two stream lines at radii r and r + dr (fig. 36). Then in
(2), § 33, [rho] = r and ds = dr,

v² dr/gr + v dv/g = 0,

dv/v = -dr/r,

v [oo] 1/r, (8)

precisely as in a radiating current; and hence the distribution of
pressure is the same, and formulae 5 and 6 are applicable to this
case.

_Free Spiral Vortex._--As in a radiating and circular current the
equations of motion are the same, they will also apply to a vortex in
which the motion is compounded of these motions in any proportions,
provided the radial component of the motion varies inversely as the
radius as in a radial current, and the tangential component varies
inversely as the radius as in a free vortex. Then the whole velocity
at any point will be inversely proportional to the radius of the
point, and the fluid will describe stream lines having a constant
inclination to the radius drawn to the axis of the current. That is,
the stream lines will be logarithmic spirals. When water is delivered
from the circumference of a centrifugal pump or turbine into a
chamber, it forms a free vortex of this kind. The water flows spirally
outwards, its velocity diminishing and its pressure increasing
according to the law stated above, and the head along each spiral
stream line is constant.

§ 35. _Forced Vortex._--If the law of motion in a rotating current is
different from that in a free vortex, some force must be applied to
cause the variation of velocity. The simplest case is that of a
rotating current in which all the particles have equal angular
velocity, as for instance when they are driven round by radiating
paddles revolving uniformly. Then in equation (2), § 33, considering
two circular stream lines of radii r and r + dr (fig. 37), we have
[rho] = r, ds = dr. If the angular velocity is [alpha], then v =
[alpha]r and dv = [alpha]dr. Hence

dH = [alpha]²r dr/g + [alpha]²r dr/g = 2[alpha]²r dr/g.

Comparing this with (1), § 33, and putting dz = 0, because the motion
is horizontal,

dp/G + [alpha]²r dr/g = 2[alpha]²r dr/g,

dp/G = [alpha]²rdr/g,

p/G = [alpha]²/2g + constant. (9)

Let p1, r1, v1 be the pressure, radius and velocity of one cylindrical
section, p2, r2, v2 those of another; then

p1/G - [alpha]²r1²/2g = p2/G - [alpha]²r2²/2g;

(p2 - p1)/G = [alpha]²(r2² - r1²)/2g = (v2² - v1²)/2g. (10)

That is, the pressure increases from within outwards in a curve which
in radial sections is a parabola, and surfaces of equal pressure are
paraboloids of revolution (fig. 37).

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 re-established.
Let [omega], p, v be the area of section, pressure and velocity at AB,
and [omega]1, p1, v1 corresponding quantities at CD. Then if no work
were expended internally, and assuming the stream horizontal, we
should have

p/G + v²/2g = p1/G + v1²/2g. (1)

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

p[omega] + p0([omega]1 - [omega]) - p1[omega]1,

where p0 is put for the unknown pressure on the annular space between
AB and EF. The impulse of that force is

{p[omega] + p0([omega]1 - [omega]) - p1[omega]1} t.

The horizontal change of momentum in the same time is the difference
of the momenta of CDC´D´ and ABA´B´, because the amount of momentum
between 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
t, is Qt = [omega]vt = [omega]1v1t, and the momentum of these masses
is (G/g)Qvt and (G/g)Qv1t. The change of momentum is therefore
(G/g)Qt(v1 - v). Equating this to the impulse,

{p[omega] + p0([omega]1 - [omega]) - p1[omega]1}t = (G/g)Qt(v1 - v).

Assume that p0 = 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 = [omega]1v1)

(p - p1) = (G/g) v1 (v1 - v);

p/G - p1/G = v1 (v1 - v)/g; (2)

p/G + v²/2g = p1/G + v1²/2g + (v - v1)²/2g. (3)

This differs from the expression (1), § 29, obtained for cases where
no sensible internal work is done, by the last term on the right. That
is, (v - v1)²/2g has to be added to the total head at CD, which is
p1/G + v1²/2g, to make it equal to the total head at AB, or (v -
v1)²/2g is the head lost in shock at the abrupt change of section. But
(v - v1) 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 - v1)²/2g foot-pounds of
energy is wasted for each pound of fluid. Experiment verifies this
result, so that the assumption that p0 = p appears to be admissible.

If there is no shock,

p1/G = p/G + (v² - v1²)/2g.

If there is shock,

p1/G = p/G - v1(v1 - v)/g.

Hence the pressure head at CD in the second case is less than in the
former by the quantity (v - v1)²/2g, or, putting [omega]1v1 =
[omega]v, by the quantity

(v²/2g)(1 - [omega]/[omega]1)². (4)