§ 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.
[Illustration: FIG. 38.]
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)