)AMORANDUM REPORT
TBE ANALOGY BETWEEN TH FLOW. OF A LIQUID WTTH A FREE SURFACE AND TEE TWO.DIMENS ION&L FLOW OF A GAS
O.N.R
- Fluid Mechanics Branch
Project NR-062-059
Prepared by: Approved by:
Forrest R. Gilmore
M.S. Plesset
Hydrodynamics Laboratory
California institute of Technology
Pasadena, California
Report No. M-54.l
March 1949
Tech&sche Hogesc.hool
AND THE TYO-DThIENSIONAL FLOW OF A GAS
by Forrest R. Gilmoro*
ABSTRACT
Generalizing the methods of Riabouchinaky and Preiswerk, the
equa-tions for the three-dimensional, rotatibnal, isentropic flow of a liquid
with a free surface are derived, and their similarity to the
correspond-ing equations for two-dimensional compressible flow of a perfect gas is
shown. Theoretical and practical applications of this analogy, and its
limitations, are discussed.
Similarly, the equations for hythaulio jms and compression shocks
are derived and compared. The analogy is shown to be less accurate in
this cage.
Finally, the application of the analogy to shock-wave or
hydraulic-jump intersections is discussed, with particular reference to the
disa-greement of theory with experiment for the "Mach-type" intersections.
*Graduate Assistant, Hydrodynamics Laboratory, California Institute of Technology, Pasadena, California.
t
AND TEE TWO-Dfl!ENS ION&L. FLOW OF A GAS
I INTRODUCT ION
It has been Iown for over a hundred years that liquid flow
in open
channels is of two different typess relatively smooth "streaming" flow atlow speeds, and high-speed "shooting" flow characterized by standing waves
and frequently by sudden changes in depth aiown as "hydraulic jumps".
the early twentieth oentury, the increased study of compressible flow
phenomena led investigators to note the resemblenoes between Streaming
ohAluiel flow and subsonic compressible flow, and between shooting channel flow and supersonic compressible flow.
This similarity was first plac4 on a mathematical basis by Jouguet
for t o-diin,vuiona1 motion, and by Riabouchinsky2 for three.-dimensional
lotion. Further investigations were made by Ippen3, Bimnie and Hooker4,
and von Krm.n5. A comprehensive treatment of the analogy was given by
Preiawerk6, who carried out a lengthy mathematical analysis of the flow
process and also included some experimental verifications of the theory.
Most
later work has
beenconcerned
either with numerical flowoalou].a-tione7'8 or with the practical application of
the analogy to model
test-ing9'°. Recently, an important paper on liquid free-surface flow by
StokerU, with
an
appendix by Friedrichs2, has appeared.In the meantime,
new discoveries have been made concerning the experimental reflection of 13,14
shook waves in compressible flow , and some studies have been made at
the Hydrodynamics Laboratory of the California Institute of Technology On
the similar problem of hydraulic-jump intersections15'6.
-2-The object of the present paper is to derive the
inathertaticalanal-ogy between open-channel liquid flow
andcompressible gas flow in a manner
sontewhatsxapler and more general than thct
ed by Preiswerk, to discuss
when and. how the theory may diverge front the real phy3ical situation, and
to treat in detail the application of the analogy to ahóck-intersection
problems.
II.
T ISENTROPIC FLOW .A1(&LOGYVarious mathematical derivations of the hydraulic analogy have
ap-peared in the
literature'2'6'11'2, but they are either incomplete or not
general enough for our purposes.
Therefore, anew derivation,
emphasizingthe basic assiptiona made, will be given below.
The Bernoulli Equation
Let AL
(Pig. 1)be an arbitrary stream tube in a
flowingfluid (liquid
or gas).
Take the cross-section of the stream tube infinitesimally small,
so that the fluid velocity, pressure and other parameters are
sensibly
cànstant over any- one cross-section. Now, to simplify the problem, assume
that
The flow is steady, i.e .., the velocity, density, eta., at. any
point is indepndeut of time
In steady flow, the stream-tube boundaries are motioi3ess, so
that the fluid
Within the tube oanot do mechanical vork on the outside fluid. If one
further assumes that
There is no viscous or thermal transfer of' energy across the
tube boundaries,
. .-then the rate of ezergy flow through the stream tibe must be the same for
every point along the tube
When a unit mass of fluid flows past a given
chanical work p/p, kinetic energy 1/2 U2, and graitationa1 potential energy
gz. Hence, one obtains the familiar Bernoulli eq*tion
+ .. + .. + gz constant (3.)
In
general the constant in Bq. (1) may vary from stróam tube to stream tube.Eq.(l) can be used
in
flows where there, is energy transfer upstream ordovv-stream (but not cross-dovv-stream) by viscous orthermal action, such as occurs
in shock waves, (except right in the shock, where extra energy-transfer
terms would have to be added to the equation).
Vorticity
The vorticity at any point in a flowing fluid is a vector defined by
in
a flow field 'where(iii) There are no viscous forces or other non-conservative forces,
and
(iv) The fluid density is either constant, or a function of
pressure only,
then it is well ]own that
J'curi
= constaut (3)
S
where the surface S is any surface fixed physically in the fluid (moving
with the fluid) and the constant depends on the particular surface This
is Kelvin's circulation theorem; a proof' may be found in Ref. 17, pp 112-U6.
fu
eu
1fu
u 1f?ü
au_'
(2)
Open-ohane1 Flow of a Liquid without Losses (Isentropic Flow)
The Bernoulli equation will now be applied to the flow of liquid
having a free surface and bounded below by a horizontal bed Make the
fol-lowing simplifying aastiona:
The density of the liquid, p, is constant.
The pressure on the free suzface of the liquid, p5,is constant.
Surface tension forces are negligible.
The vertical acceleration of the liquid can be neglected
corn-prod to
ths
acoelsre.tion of gravity.Then the pressure at a point of elevation z (Fig. 2) can be -written in terms of the liquid depth h:
p K- z) gp (4)
Thus Eq. (1) simplifies to
+ gh + .
(u2
+ u +u2)
constant (5)where the
ext. term 1*s been incorporated in the constant.
Moreover, if, according to (iii),. viscous forces are negligible,
then the internal energy E dOes not change, so that it also can be included
in the constant, leaving,
gh + ..
(u2
+u2
+u2)
constant (6)In general, this constazxt nay vary from streamline to streamline.
We shall noW look for solutions of Eqs. (3) and (6) which are nearly
independent of z, that is, which have
au
hx
hy
r-
r-"
where £(( 1 The vertical velocity u approaches zero atthe horizontal
U
the sine of the angle between the surface velocity and a horizontal plane.
For u/u
, this angle must be small, i.e.,The slope of the free surface, considered in the direction of
liquid motion, must be of order C.
Anàthor condition necessary in order that relations (7) may hold. is that
The boundary conditions must be independent of z; in other words,
the constants in Eqs. (3) and (6) must be independent of z, and all
boundarywalls must be vertioal.
Neglecting quantities of order £, u and. un,, are independent of z,
can be dropped, and the vorticity given by Eq.. (2) has only a vertical
component. Eq. (3) can thus be written
.
r
au1
--constant (8)
Now Choose the surface Sto coincide with the free surface of the liquid, 1
and let it be. very: small, so that
_.j is
constant over, the surface.Then
Moreover, the vertical 'cylinder of f]tiid bounded by S and
the
horizontalbe (Fig. 3) remains cylindrical (although its shape changes) as the fluid
moves, since and u. are independent of z, and thO volume of this CylindOr
remains constart since 'the liquid is incompressible, so: that
Division of Eq. (8) by Eq. (9) gives
ru
au
)
constant. constant (10) (11).hay
8x rauIx
ou
y
ax
.!
constant (9)L8r
Under the same approximations, Eq.(6) becomes
+
(u2
+u2)
constant (12)Moreover, the continuity condition for conservation of mass can. be written
$ imply:
a
(hu)
ar (13)
Th..three .quationa (ii), (12) and (13 for the thee unknowns u1, u, and
k, t.g.th.r witk the appropriate boundary onditions, ao1et.17 dst.r4,,*
t
Ud flw fiel4.
Isentropio Two-dimensional Flow of a Perfeet Gas. In a perfect gas,
Moreover, at all ordinary temperatures, and distances up to a few hundred
feet, gz is entiróly negligible compared to cDT. Hence Eq.(l) simplifies
to
oT + a. u2 = constant (15)
Consider a completely two-dimensional gas flow, ere all flow
parameters and boundary conditions are independent of z, and u Ø_ Eq. (15)
then becomes
o T + 2 + u 2) . constant
p.2
x yThe vo rt icity
(sq. (2))
has only a vertical component, and by reasoning similar to that used in deriving Eq.(]1) for a liquid, it is easily shownthat for a gas
r an ii x p
Lay
(16) constant (17) p (14)açPu
epu1)
(18)
In flowwithout viscous or thermal losses, the isentropio* relation
for a perfect gas holds
pp'
constant (19)where Y is the ratióóf specific heats for the gas.
This can be written alteruatively
1
p = T x. constant
Substitution of
Eq.(20)
in Eqa.(17) and (18) yields, respeotively,T
-
constant anda&'u)
O(TSu)
x +y=o
aThe three equations (16), (21) and (22), for the three unknowns,
U,
and T, together with the appropriate boundary conditions, completelydetermine the gas flaw. The related quantities p and p can be evaluated
by means of Eqs. (19) and. (20).
The Analogy
between Isàntropio Free-Surface Liquid Flow and Isentropic Perfect-Gas FlowWe wish to compare Eqs. (ii),
(12)
and (13) for liquid flow withEqs. (16),
(21) and(22)
for gas flow. Eq.(12) is
equivalent to Eq.(16)designated by "adiabatic-reversible", sometimes inaccurately shortened
if gl is replaced by oT. Eq. (ii.) is equivalent to Eq. (21) and Eq. (is)
to Eq. (22) if two cond.itions are met:
gh
oT and
S (23)The analogy between the motion of a free-surface liquid and a gas having
'V = 2 is evidently complete,
withinhe
limits of assumptions (i) to (x),provided the boundary conditions are ana1cgus. The analogy has thus been
proved to hold even for rotational flows, e.s long as the circulation is
constant (no viscous forces in the region of interest), a more general case
than that treated by previous investi.gatos.
-According to well-known dimnsiona1 arguments, similarity between
tVo different.physioal sItuations must occur if all the corresponding
di-mensionless ratios of the .re]evsnt parameters are equal. Thusq. (23) can
be written
h
T(24)
where It and T are the height and temperature
occurring
at some referencepoint, usually taken as a point of zero velocity. According to Eqs. (19)
and (20) one can also write
(y=2)
.. (25)
Because of relations (25) and (26)., the hydraulic analogy is often derived in terms of h and p, or h2 and p. However, the derivation given above in
In the special case of flaws starting from rest or from a state of
uniform motion, the vorticity is everywhere zero and th. constant in Eqs.
(Ii) and (21) vanishes. It i Co*vemi.nt to satiety thesS .quation.s by
'introducing a v.1cc it1 potential, .0', th.re
u.'lt,
x 8x.y
u,.!
(27)y
Th, gas continuity Eq. (22) can then be written in the form
20
u1 .2%(y-l)T x
x
fr-l)T a1
6yEq. (16) can be differentiated either with respect to x or With respect to
y, giving expressions for and , respectively. These can then be
substitutod in Eq. (28) to yield
u 2 .2% 2u u .2% u 2 2
0 (29)
where a
{(_i)oT
/RT
local velocity of sound. (30)The character, of the solutions of Eq. (29) depends upon whether the
equa-tion is elliptic or hyoerbolic, that is, whether B2 is smaller or larger
than 4 AC, where A, B and C are the three coefficients in brackets This
condition is therefore 2 f2u u 1 11_'XYI
[2
J , u 2 2 a2 xThus the veloóity of sound plays a critical role, separat.ng "subsonic"
from "Supersonic" regimes.
L
U2
'-''C I
rj
L'
The oorezponding critical velocity for 1iqid flow y be fbund
in a similar nner. By using relitions (23) we can jm at onoe. to the
answer:
a
V'(y-l)cT ..i.(2-l)gh
Vi
(32)The transition between "strenniing" and "shooting" water thun occurs When the Velocity reaches
V.
The velocity of infinitesimal surface waves, unlike the velocity of
18
acoustic waves, varies with the wavelength (Lamb , p. 367):
a()
=y/tanh!
(33)This velocity increases as the ratio/h increases, end approaohes.an
asymptotic value
a()
a (34)The critical velocity thus equals the maxiflium velocity of surf.ce waves.
The actual method of solubion of Eq. (2) or the corresponding
liquid -flow equation for specified boundary conditions will not be dis-Cussed in this paper. Th method of characteristics has been treated in
detail by Preiswmrk6; other usefi1 methods may be found in textbooks on
compressible flow (see Liepmenn and Puckett17 or Courant and Priedriohs).
Limitations of the Isentropic Analogy
The assumptions which were made in the course of deriving the
analo-gy were numbered (i) to (x). Assumption (i) has restricted the analy8isto
steady flows; this includes, of course, flows Which can be made steady by
choosing coordinates fiCed on a moving body, such as a projectile moving
through a fluid at constant velocity. Stoker11 and Priedrichs12 have
two-dimensional flow. Probc.bly the analogy holds for rotational
thróe-dimensional non-steady flow, but this has not yet been proved.
The isentropic conditions (ii) and (iii) are usually quite well
satisfied except near the channel bed or walls, and in shock waves or
hydraulic jumps. The latter are discussed later in this report. It is
found that shocks or jumps of snail anlitude are approximately isentropic.
The effects of the channel walls are generally confined to a"'boundary layer" close to the wall. For gasesand. the less-viscous liquids, this layer is
uSually only a snail fraction of an inch thick; thus for water flowing
through a typical nozzle, Préiswerk6 calculated a boundary layer which
built up at the rate of 0.01 inch in thiclthess per foot distance downstream.
Viscous losses in the main flow field are usually neg1iible, especially
since it has been shown that velocity gradients in the z direction are
smu. However, sharp bends in the boundary walls could cause large
hori-zontal velocity gradients and thus appreciable- viscous effects.
Assumption (v) of;constant liquid density is well satisfied by most
liquids at ordinary pressures. The assumption of constant external
pres-sure, (vi), is usually accurate in problems. of physical interest.
The validity of assumption (vii), that surface tension forces are
negligible1 depends upon the surface tension , , of the liquid under
consideration, and on the radius of curvature, rc, of the liquid surface.
Specifically, the condition is
-12-In order to satisfy ass )tios (vili) and (ix), to be discussed below,
we must have r0>> h, so that Eq. (35) is satisfied if
h2> L
(36)pg
For water at ordinary temperatures, Eq. (36) is satisfied when h 0.3 cm.
In
borderline cases, the accuracy of assumption (vii) might be determined experimentally by adding a surface-tension reducing solute to the liquidand repeating the experiment.
Assumption (viii),
that the vertical
acceleration is negligiblecom-pared to
the
acceleration of gravity, is closely related to asstion (ix)that
the
free surface does not slope much inthe
direction of liquid motion. To analyze theseassumptions,
choose coordinates so thatthe surface
v-looitr at the point f interest lies inthe
xzp].ane, i.e., u 0.Than at the surface
ye rt ye 1. u u -z ax
13u
3u h! z Z vert accej.u - -
---.
zcJ
1 zusing --
But from Eq. (6).,so. that Eq. (38) becomes
ye rt aoce 1 2 au = g
E
u[u
+ (gf\2
18x/
2h&a8h
u/5h21+ - - +
--ahc
For velocities of the order of -/i, Eq. (40) can be written
2 2 2
B h vert. acoel.
I
\ fBI,.h'
+ 1.'..--j 1 8x'ghI
(37) (39) (49) (41) (38)According to assumptions. (viii) and (ix), the right hand side of Eq. (41)
is small compared to 1, hence It 1! is small Compared to 1, and thua:
ax
(xi) The radius of curvature of the free surface, r
[1 21 O/GJ
Ia
V5j1 I/ -'
If
1 c y a large compared to h.Condition (xi) has been deduced from assumptions (viii) and (ix).
Recently Friedriohe has shown that starting with condition (xi) alone,
the analogy may be derived for liquid flows that are steady or non-steady,
irrotational arid two-dimensional.
Another limitatii of the analogy between liquid and gas flo* is.
that it is striet]y true only for a gas
having 'V
.2, i.e..,
Jouguet's "hydrodynami a gas". Kineto theory shows that real gases are limited by5 3
Theoretical gas flow relations are usually derived for arbitrary f, and
thus can readily be applied to liquid flow prOblemS. The diffioulty comes
When. direct comparisons are to be made between liquid-flow and gas-flow ex-perimonts. However, it. has been found theoretically or experimentally that
the solutions to soie flow problems do not vary greatly with 'V , so that
a useable experimental analogy holds approximately. In other oases, the
ariations with I may be so large as to make. the experimental analogy
'worthless.
III TEE APPROXBTE ANALOGY BETWET HYDRAULIC
AND SHOCK IVAVES
The. Formation of Kydraulic Js:
If an elevation wave of finite amplitude is produced on. the surface
of a .liquid Stoker11
has shown
from the equations. of unsteady motion thatthe wave front will grow continually steeper as it progresses, until
final-ly the wave leans forward. At this point experiments show that the wave
form deviates from the. theOretical form, with the formation of a breaker or uroflertt and the establishment of a turbulent wave of constant shape
(except for minor fluctuations of short period). Such a steady finite
wave is called a hydraulic jump. A moving hydraulic j may be produced
by a sudden disturbance of the surface of a liquid. A stationary hydraulic
jump may be produced by an obstacle placed in a rapidly flowing liquid.
The difference in these two types of jumps is purely one of relative motion;
the two situations are hydrodynamically identical.
Normal Hydraulic Jumps
We shall first treat a normal hydraulic jump occurring in a region
of uniform parallel flow. It is convenient to choose coordinates
sta-tionary with respect to the jump, thus reducing the problem to one of
steady flow. Let the fluid be flowing with height h, and uniform velocity
u, up to the plane ABCD (Figure 4) where the jump starts, and assume that
at the plane EFGE, a distance w behind the start of the jump, the flow again
has a uniform velocity, u2, and height, h2.
The energy balance for steady flow, Eq. (5), gives
+ gh
+
4
2
+ gh2
+
4
U22 (4The two interñál energy tOrmâ cannot be cancelled, becaise viScous, losses
which increase the internal energy
2 over are necessary to maintain
the jump, áswill be shown latere
The Oontinuity relation is simply
If E1., h1 and U1 are specified, then E2, h2 and u2 become the un-knowns. Evidently a third equation is needed to determine these three
unknowns. This equation, can be obtained from the principle that
the-in-crease in momentum of the liquid per unit time equals the net force acting
on the liquid. The mean force in the direction of flow on the surface
.ABCD is .gp h1 per uni.t' area, or a. total force of gp
h2
if the lengthABCD=EFGB is taken as unity. The force on EFGH is .gj h22. Since the
sides AE and BFGH are taken. parallel to the flow velocity, they contribute
no net force in the flow direction. The effect of any external pressure also
cancels out In a unit tine, an amount of liquid p h.u1 = p h2u2 eeriences
an increase in momentum equal to p(h2u22 -
hu12).
Hence the momentumequation can be written
2
. li1u 2
2 + h2u22 (45)
Eqs. (43), (44) and (45) determine any three of the variables B1, B2,
h1, h2 if the other three are specified.' Moreover, B1 and B2 do
not appear in the latter two equations, so they can be solved without the
use of Eq.,(43). Thus, solving Eq. (44) for u2 and substituting in Eq.(45)
yields
UL = g (h1.i. h2) h,43
(46)
dropping the trivial solution, h1 =-h2. By considering coordinates fixed
with respect to the fluid ahead of the jump, it is immediately seen that
u1 is the velocity with which a hydraulic jump will move into a still body
of liquid.
ratio h
fzu1
+1. (47)
h1 1Jgh1
The equality of the first and last members of Eq. (47) follows from Eq:.(44).
Agraph Of Eq. (47) is shown in Fig. 7.
The gain in internal energy of the fluid may be found by first
eliminating u2 from qs. (37) and (38), and then substituting for u1 from
Eq. (46), which gives on simplification
3
g.(hh)
-, E1 =
- 4h1h2 (48)
It should be mentioned that in hydraulics it is frequently
custom-ary to denote by "flow energy" only the potential plus kinetic energy,
gh + u2 Thus the gain in internal (thermal) energy given by Eq. (48)
becomes a loss in the "flow energy", in aocordance with the conservation
of energy principle, Eq. (43). The symbol E is used throughout this paper
for the thermodynamic E, not the hydraulic E.
If h2 were less tha.ti h1, Eq. (48) would give a decrease in
internal
energy for the liquid passing through the jump. For an incompressible
fluid, this means a decrease in entropy, which violates the
second law of
thermodynamics. Therefore, it is physically impossible to have a steadrstate
"hydraulic drop" in parallel flow.Since h1 I12 Eq. (35) shows that U1 in other words,
stationary hydraulic jumps can occur only in "shooting" flow.
hen li2 is greater than h1, viscous or turbulent processes must be
Eq. (48). Since the pressure forces on surfaces.BCDand EFGH (Fig. 4)
are not only unequal, producing the change in velocity shown by Eq. (45),
but also do not have the same effective point of application, they result
in a torque which will cause rotation of the fluid. It is found
experi-mentally'9 that this rotation oonslsts primarily of a "roller" immediately
below the slanting section of the water surface, while the motion is
sensi-bly uniform and irrotational at a moderate distance behind the jump.
Bakhmeteff and Matzke'9 have found that strong hydraulic jumps,
h2/h1> 2, have the simple profile shown in Figure 5a, where the Width w
is 4 to 5 times the jump height, h2h1. Weak jumps, l<b2/h.12,
general-ly have the undulatory form shown in Figure 5b. The surface first rises
above the final level and then oscillates with diminishing amplitude
about the final level. The width of the undulatory jump is not
well-defined, but the undulations generally die out at distances of 3 to 10
times the initial water height, K1, the distance being greater for the
smaller jumps.
Oblique Hydraulic Jumps,
The results obtained thus far for straight hydraulic jumps normal
to the liquid flow can be easily generalized to apply to jumps oblique to
the flow. Let the double line in Figure 6 represent the top view of a
hydraulic jump making an angle 9 with the initial liquid velocity ç. it
is convenient to resolve into components u1 u1 sin 9 and u.= U1 cos
-which are respectively normal and tangential to the hydraulic jump. If
every cross-section of the jump is identical, there can be no pressure
gradient in a direction parallel to the jump, and thus no tangential force
There fore
U1T
U1 005
9 = (49)The change in height across the jump produces a pressure gradient
normal to
the jimip, which changes the normal, velocity of the fluid passingthrough the jump. The relation for conservation of momentum in this
di-rection is identical with, Eq. (45) if u1 and u2 are replaced by U1f
sin 9 and UZN respectively. Moreover, the continuity relation is just
Eq. (44) with the same substitution. Hence the, solutions to these
equa-tions can be obtained by modification of Eqs. (46) and (47):
u = U1 sin 0 g (h1+ h2) h/h1 (50) h2 1 2 2 ' 1 u_1sin9 = 'I 211 sin 9 + - .=. = fl1 Y 1. 6 U2N
where M1, the initial Mach
number, is
defined as u1/*'ç. Eqs. (49) and(51) for u
and U2N can be changed to give (u2) and (u2) thecorn-ponents parallel arid normal to the initial velocity , along which the
x-axis is taken. Thus
(u2)
U1 cos
9 +(u2) u1 sin 9 cos 9
[ U1 sin2e
1/
2M12sjn29 + 1-.
(51)
(52) 1 I2.2
i 11(53.)y2M1 sinO
+._.j
For a given u.1 and M1, Eqs. (52) and (53) give in parametric form a
rela-tion between (u2) and (u2)1. If a curve of (u2) vs. (u2) is plotted,
the ourv with the x-axie represents the velocity ahèaô. of the jump,
while a vector from, the origin to any point on the curve represents a
possible velooity çbehind the shook. Such a figure is called a shock
polar. InFigure 8, all velocities have been divided by the initial
acoustic velocity, a1, to give a more general dimensionless
representa-tion *
If Eq. (45) te applied to oblique jumps, substituting (UN2
2)
for u2 on each side, the tvio terms cancel by virtue of Eq. (49). On
applying Eq. (50) and the continuity condition, Eq. (48) is obtained as
before...
Thus
Eq. (48) for the internal energy gain (or "flow energy"loss) holds for both normal and oblique hydraulic jumps.
All of the above results for oblique jumps can also be obtained
by a strategem vhich Will be of use later in the analysis. To treat the
hydraulic jump of Fig. 5, choose coordinates moving with the constant
velocity
"iT parallel to the jump.
In such a coordinate system, the fluid
ahead of the jump will appear to be moving into it at right angles, as in
a normal jump. Since the equations of
mechanics
a-re invarient to such a0Galilean transformation" of coordinates, the equations of a normal
hy-draulic jump in these coordinates are applicable. The results maybe
referred back to stationary coordinates simply by adding the uniform
velocity It is easily seen that this
method. Will
again yield Eqs. (48) to (50).Curved Hydraulic Jumps.
If a hydraulic j tanp is curved (as vievjed from above), but the radius
of curvature is large compared to the Width of the jump, the jump can be
-divided into small segments each of which is sensibly straight The
In compressible flow problems it is customary to use a*, the acoustie
velocity at Mach number one, instead of a1 in the denominator.. The value
of ft is constant across a compression shook, but not across hydraulic jtmp; hence its use is inconvenient when both cases are to be treated.
equations developed for oblique jumps can then be applied to each segment,
with only the additional complication that the angle 9 varies from
seg-ment to segseg-ment.
On the other hand, if the radius of curvature is not large compared
to the jump width, a short segment is not comparable to a straight jump,
for the sides AEHD and. BFGC of a stream tube (Fig. 4) can never be made
quite parallel, and the pressure forces on these sides will always contribute
a'reeultant in the flow direction iich is appreciable compared to the forces
on ABCD and EFGE. (By taking AB shorter, the sides could be nade more
near-ly parallel, but the decrease in the force contribution from the sides is
offset by the decreased forces on the diminished faces ABCD and EFGH.)
An accurate solution of the curved hydraullo jump problem would
re-quire knowledge of the pressure distribution within the jump. However, a
first approximation might be obtained by assuming a constant pressure
gradient between points C and G. Such a procedure will not be carried out
here.
The FormatiOn of Shock Waves in Comprossible Flow
If a compression wave of finite amplitude is produced in a gas, the
wave front
Will
grow continually steeper (higher pressure gradient) as thewave progresses. This was shown byRienann in 1860. As the pressure
gradi-ent, temperature gradient and velocity gradient approach infinity, viscosity
and heat conduction, which at other points have negligible influence, become
important at the wave front. The final result is the formation of a steady
wave of large but finite steepness, known as a shock wave. The process of
formation. of a compression shock is thus similar to that for a hydraulic jump. However, the shock wave involves simple one-dimensional viscous and
turbulence. phenomena. Analysis of the interior details of a compression
shook has been carried out successfully, while that for a hydraulic jump
has not. Taylor (see Ref. 17, p. 48) has found that, for a normal shock
wave in air, the thickness over which 80% of the velocity change occurs is
approximately:
thickness (in inches)
Normal Compression Shocks
-AS in the case of a hydraulic jump, flow relations across a straight
shock wave can be derived from simple momentum and. energy considerations,
without inquiring into óonditions inside the shock. Choosing coordinates
to make the shock stationary and normal to the flOw, the conservation of
energy Eq. (15) becomes
oT +.u2
p1
.1.p =RpT
o pT Y PcT+1u2
p2
. 2 io_2 - u2) in ft/sec(M)
(55) (58)The continuity relation Is.
p1u1 p2u2
(56)
The momentum equation is
2 2
p1 +p1u
= p2 +p2u2 (57)Eqs. (55), (56') and (57) are the well known Ra1dne-Hugoniot equations for
a shock wave. Combined with the perfect gas relation, they can be solved
for any four of the variables, such as p2, p2, T2 and U2.
Substitution of Eq. (58) in Eq. (57), and elimination of p2 by means of
Eq. (56) yield
O[T2_1]
Substituting for T2 from Eq. (58), one obtains after simplification
U2
(Yi)M2
+ 2(Y+)12
where the initial Mach number, is defined by
Ui
Ui
(y+].)2M12
Use of Eq. (58) then yields
p2
2YM12
- (v -1)p1
2 T2 1P2
Values of-,
y-
-
from the above equations are plotted in Figure11
P17 both for y 2 and for Y 1.4. This graph will be discussed later.
A shook is physically possible only when the entropy of the gas
remains constant or increases across the shook. By expressing the entropy
differenCe in terms of p2/p1 and p2/p1, and then using Eqs. (60) and (63),
one finds (see Liepran and Puokett17, p. 41), that this condition is
atisfie4 only when 1, whence p2/p1 1, p2/p1 u1/u2 1, 1.
I(Y-l) oT1
Eqs. (55) and (60) can be combined to give
{y
2(y1)I}
[(Y_l)M,L2 + 2)Oblique.Compression Shocks
An oblique shOok can be resolved into a normal shook plus a velocity (59)
shock) in a manner precisely similar to that used for the oblique hydraulic
jumpo The results corresponding to Eqs.(52) and, (53) are
[4..l)&2
sjn29 + 2] u cos2o. + 2 (64). (y+i) 2 U1005 e
(M2 sin2g -(65) (y+ ) M] sin 9The dotted lines of Fig. 8 show (u2)/a1 plotted against (u2)/a1 for
3.0.,: with .,Y= 1.4 and Y= 2, 9 being varied as the parameter.
Curved Compression Shocks
If the radius of curvature of a compression shook is large compared
to the thickness of the shock, Eqs..(64) and (65) for a straight shook can
be applied to each infinitesimal segment of the shock, the only complexity
being that the é.ngle 9 varies from segment to segment. According to Eq.
a typical shook wave has an effective thickness of lO to 1O
inches; thus all known practical cases of curved shocks are included in
the above statement
The Analogy between Hydraulic Jumps and Compresion Shocks.
One method of finding an analogy between jumps and shocks is to
com-pare the basic equations for the two cases. The two continuity conditions,
Eqs. (44) and (56), are identical if h is taken to be equivalent to p.
The two momentum relations, Eqs. (45) and (57), are equivalent if h is
2.
equivalent to p and h is equivalent to 2p/g. In order for this to be
must have
2
fP1\ (P2'\
çi k;)=
1 (66).Physically, not Eq. (66) but the energy conservation Eq. (55) must
be satisfied. To determine whether the two conditions are coiratib1e,
one can use shock-Wave Eqs. (60) and (63) to obtain
2 (P2\
r)2 + 2}
1z2
'2içì
(Y )3 M14r
2r
2Y (67) ,i'srJ
L'J
(i+r)
where the convenient parameter,t 2_i, has been introduced. The right
hand side of Eq. (67) cannOt equal 1 as f approaches infinity, except when
Y 0, 0 0, which is physically impossible. Since Eq. (67) thus cannot be satisfied for large f, the next best possibility is an approximate
validity for small f. Expanding Eq. (67) in a power series,
V = 1 + 2(y -2)
-
:4(y -2) V P2 V'r +.
(y +1)2 V V V V4(Y -4.--5)
3 V-
f +...
(68) V(vi)'
VThe beet choice of Y is 1 2, which eliminates the terms in f and f2,
giv-ing
2
1P1\1r'2
;)
4 3 + f + ... ( Y= 2) (69)shock3 it becomes increasingly inaccurate.
Condition (66) is identical with the isentropic condition, Eq. (20),
when 1Y = 2. It is seen that real hydraulic jtnps (with losses) are
anQo-gous to fictitious isentropic comDression shook (without losses) in a gas
having Y = 2. Since actual weak shocks are nearly isentropic, while strong
shocks have large losses, the analogy should be more accurate for weak than
for strong shocks.
The same results are shown qua.ntitatively by Fig. 7, which compares
the values of h/h, = u/u2 for normal hydraulic jumps with the values of
analogous parameters for normal shocks When Y 2, the shock parameters
are asymptotic to the jump parameter at = 1, but diverge for stronger
shocks. Values for Y= 1.4, typical of air and diatonic gases, diverge even
more, except for values of p/p1 in the region near N1 = 3. (The latter
agreement would be of value only in flow situations consisting erirely of
shocks having M) 3, or p2/p1 4).
- A similar situation appears in Fig. 8, which depicts initial and
final velocities for fluid passing through an oblique shock or jump. Only
one curve from each of the curve families fbr 'y= 2 and y= 1.4 is plotted.
The typical y . 2 curve is asymptotic to the hydraulic jinp curve for weak shocks i) but diverges considerably for strong shocks. The curve for
Y = 1.4 is,-gnal1y,,.' even less close to the hydraulic curve but may be
closer for a limited range of strong shocks.
IV. INTERSECTIONS AND REFLECTIONS OF HYDRAULIC
JT.ThflPS RD SHOCK WAVES
The discussion beloyr will apply to both hydraulic jumps and com-pression shock waves; hence. the term "shock" will be used to signify both
Regular Intersections or Reflections
Figure 9 depicts the intersection of two straight shocks of' equal
amplitude. The' fluid
in Region I,ahead of the shocks, is assumed to be
either at rest or in a state of uniform motion, in physical coordinates.
It is now convenient to change to coordinates moving with
theshook
inter-section, so that
the problem is reduced to
oneof steady flow.
In
thiscoordinate system, the fluid
inRegion I is moving with a velocity U.1
toward
the twostationary shocks, S and S.
In order for these two shocks
to have equal amplitudes, the oononents of
noimal
toeach of the shocks
must be equal.
This is possible only if
is parallel to the bisector,
BB, of the angle between
theshocks. By symmetry, BB is not only a
stream-line in Region I, but also in Region III.
Since any streamline can be
re-placed by a rigid wall
withoutaltering the flow, provided, boundary-layer
effects can be neglected, the problem is equivalent to the reflection of a
hock wave from a flat wall, depic1ed in Fig. 10.
Obviously, this
equiva-lence between shock intersections and shock reflections holds only for
iderseotions of shocks of equal amplitude.
Intersections of unequal shocks
will not be treated in this paper.
Referring now to Figure 10, the incident shock S
inc.
will both slow
doni and deflect the flow to a new velocity
Uin Region II
This is
'represented dimensionlessly in Fig. 8 by the two vectors c/a1 and
The second shock, S
ref.
parallel to the boundary wall in Region Illo The angles of deflection of
'the two shocks must evidently be equal and opposite.
If Fig
S is rotated
so that c/a2 falls on the x-axis as shown (note that u/a2 < u/a1 sInce
a2> a1), then
/a2 must fall on the' same shock polar curve at an angle equal
, must deflect the flow from
to a velocity ii
and opposite to the previous deflection angle. Thus there are two
Solu-tions for c/a2, namely OP and 0?, corresponding respectively to large and
small velocities,
,
or weak and strong reflected shocks, 5ref'
Experi-mentally, only the weak-shock situation is found; the theoretical reason
for this limitation is not known.
Fig. 8 shows that for a s irrnt larger angle of deflection between
and such as shown by the dotted arrow, there would be no possible
shook Sref. which would give the necessary equal and opposite deflection
to fit the boundary condition in Region III. Hence, for incident shocks too
strong or making too great, an angle a with the mall, there can be no
"regular" reflection according to the scheme of Fig. 10.
Mach Intersections or Reflections
When 'regular reflection is not possible, the actual picture is found
experimentally to look like Fig. 11. A third shook, the "Mach",
8M'
appears, and its length, i.e., the distance PP', increases with time at a
fairly constant rate. The Mach is frequently curved, and also the
re-flected shock 5ref is often curved near the triple shook intersection.
The simplest theory of the Mach reflection assumes that all three
shocks are straight. Coordinates are chosen fixed with respect
to the
shock intersection P, thus reduoing the problem to one of steady flow. In theSe coordinates, the wall (or line of syimnetry) is moving downwardat a constant velocity equal to the component of or tangential to
SM. A line PD can be drawn separating the fluid which has passed through
S. and 5:,: from that which has passed through SM. The gas pressure
inc. ref.
or liquid height must be continuous across PD, since it is not . shock.
flow energy than that which has passed through two smaller shocks v4th
the same total change in. pressure or height (see Eq.(48)); hence the
velocity of a liquid, or the velocity and density of a gas, will be less
below PD than above it. Thus PD is a slip-stream. (Sohlieren photographs
of gas flows, actually showthe density discontinuity at PD).
Quantitative relations for such three-shock intersections can be
obtained by applying the oblique shock equations to each of the shocks.
This procedure yields a complicated set of simultaneous equations, which
oan be solved by numerical methods. The results for some ranges of interest
have been tabulated by Polachek and Seeger both for compression shocks7 and
for hydraulic jumps8. Einstein and Bair416 have alsO solved the hydraulic
jump problem, u8ing different coordinates.
Comparison of Theory with Eeriment:.
Harrison and Bleakney14 and. Einstein and Baird15'16 have found that
theory and experiment agree closely for regular reflections up to the limit
of regular reflection, for both compression shocks and h,draulic jumps.
Beyond this limit, the expected transition to Mach reflections occurs, but
there are discrepancies between theory and experiment which are largest
for the weakest incident shocks. 4s presented in these reports, the
dis-crepancies are not directly comparable, so they have been recalculated to
a conunon parameter, and plotted in Figs. 12, 13 and 14. The parameter
chosen was a - a' (see Figs. 10 and ii), which represents the difference
between the angle of shock incidence and the angle of reflection. These
values are plotted against a, for a given shock strength, using circles
for experimental points and curves for theoretical values. The curves
the limit of regular solutions, are shown dotted since they are found to
be not realized physically. The theoretical curves for liquids have a
limit beyond which neither regular nor simple IiIaoh reflection Is possible;
no such limit is found, for gases7'8.
Figures 12, 13 and 14 present data for weak, medium and strong
shocks, respectively.
The agreement of theory with experiment is good for
regular reflections of both shocks and hydraulic jumps.
In the. region of
Mach reflection, there are considerable discrepancies between the simple
theory and experiment, the discrepancies being greater for the weaker
shocks and jumps
However, the discrepancies are frequently in opposite
directions in the
cases.
At least part of these discrepancies can be explained by the
curva-ture of the shocks. As long as the radii of curvacurva-ture of S
ref.
and S
are
M
large compared to the Shook thicbiess, the short seents of these shocks
near the triple intersection should behave like straight shockS, and the
flow in this neighborhood should approximate that given by the simple
theory. The theory and experiment should thus be comparable provided
that the shock angles are measured at the triple intersection, and the
wall is replaced by an "effective wall" perpendicular to the direction of
SM at the triple, intersection (Fig. ii).
In other words, if the Mach shook,
is. convex and changes its direction by an. angle 4, between its two ends,
the experimental value of a plotted in Figs. 12 to 14 should be decreased
by
4,, and the value of (a - a') by 24r
, for comparison
with the
straight-shock theory.
If the Mach is concave, similar corrections should be made
in the opposite direction.
the correction is then to move the experimental points that represent
Mach intersections downward and to the left, thus improving the agreement
of experiment with theory. Binstein and Baird included several photographs
in their reports15'16 from thich it was possible to approximate the
cor-rection angle * for a few cases. These corrected points are plotted in
Figs. 12A, l3A and 14A, with dotted arrows showing the amount of àorrection.
The corrections are all in the riit direction, but the final results still
show considerable deviations from the theoretical curves These may be
caused simply by errors in the measurement of r , or they may be due to
the finite width of the jumps, or to other unknown effects.
The compression-shock Maóh usually appears either straight or
slightly concave. In the latter case, the 41-correction will be in the
proper direction to improve the agreement of the experimental points with
the theoretical curves shown in Figures 12B, 13B and 143. Quantitative
data, ho-waver, are not available, so this correction has not been made.
Harrison and Bleakney4 have allowed for curvature of the shocks
In a different mariner. If the shock.angles are measured with respect to
the direction of motion of the triple intersection, the results should be
independent of the shook curvature Harrison. and Bleakney14 have plotted
such data, and find definite disagreement of theory with experiment for
weak compression shocks. The reason for this discrepancy s still unknown
(see Courant and Friedrichs20, pp. 342-346). Data to determine if a
cor-responding discrepancy existS
in
the hydraulic case are not yet available. Even if the local behavior of the triple shock intersection cax beexplained, -there remains the theoretical problem of shock curvature. If
PD of Fig. 11, would also be straight, and would intersect the wall. This
is not permissible because the boundary conditions at the wall cannot be
satisfied on both sides of the slip stream. Consequently, at least one
of the shocks must be curved. However, no theory has yet been developed
to explain the curvature quantitatively.
According to the theory of hydraulic. jumps, straight-Mach
refleà-tions cannot occur when the incident angle exceeds a certain limit
(dé-pending on the incident jtip strength). Experiments show that in such
cases the Mach becomes convex, thus decreasing the "effective incident
angle." The compression-shock theory exhibits no such limiting angles;
this may be connected with, the fot that compression-shock Mache are
usually concave rather than convex, but the details are. obscure.
V AC OWLEDGMEI1TS
This study was supported by the Fluid Mechanics Branch of the Orfice
of Naval Research. The author wishes to thank Professors R. Ladenburg and
R. T. Knapp for their discussions with him regarding experimental data.
This problem was proposed to the author by Professor M. S.. Plesse't, to
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math. pures et appliq. (Series 8), 3, 1 (1920).
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fluide compressibl&', Conipt. Rend. do 1!acad. des sciences, 19S, 998
(1932); 199, 632 (1931i). 202, 172S (1936).
3., Ippen, A., "An Analytical and Experimental Study of High Velocity Flow
in Curved Sections of Open Channels",
Thesis
for PhD, Call Inst. ofTechnology, 1936; Ippen, A., and. Knapp, R. T., "A Study of High Velocity Flow in Curved Channels," Trans. Am. Geophy. Soc., Jan., 1936.
Binnie, A. H., and Hooker, S. G., "The flow under G'avity of an
Ircom-pressible and Inviscid Fluid through a Constriction in a Horizontal
Channel", Proc. Roy. Soc., 19, 592 (1937).
von Karman, Th., "Eine praktizche Anwendung der analogie zwischen
Uberschafl-Str6mung
in
Gasen und iberkritischer Str°ómung in offenen Gerinnen", Z. fur angew, iiath. u. LIech., 18, L9 (1938).6.: Preiswerk, E.,
"Application of
the Hethods of Gas Dynamics to WaterFlaws with Free Surface; I. Flaws with no Energy Dissipatioñ;
II. Flows with Momentum Discontinuitiès. (Hydraulic Jumps)", NACA T.M. 93L and 93S (191O); translated from "Mitteilungen aus deni
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and Seeger, R. J., "Regular Reflection of Shocks in Ideal Gases", Explosives Research Report No. 13, Navy Dept. BuOrd, WashingtonPolachék, H., and Seeger, H. J., "Interaction of Shock Waves in
Water-like Substances", Explosives Research Report No. 14, Navy
Dept., BuOrd, Washington, D. C., Aug. 1944.
Bran, J. H., "AppliOation of the Water Channel - Compressible Gas
Analogy", Report NA-47-87, North American Aviation, Inglewood, Calif.,
March 3, 1947.
Johnson, R. H., Nial., W., and Witbeok, N., "Water Analogy to Two-Dimensional Air Flow" Report No. 55218, General Electric Co.,
Schenectady, N. Y., August28, 1947.
Stoker, J. J., "The Formation of Breakers and Bores", C nun. on
Applied Math., 1, 1 (i9'8).
Friedrichs, K. 0., "On the Derivation of the Shallow Water Theor?'
(appendix to Ref. ii), Coininun. On Applied Math., 1, 81 (1948).
von Neumann, J., "Oblique Refleotion of Shocks", Explosives
Research
Report No. 12, Navy Dept., BuOrd, Washington, D. C., Oct. 1943.Harrison, F. B., and
Bleakney, W., "Remeasurement of ReflectionAngles in Regular and Mach Reflection of Shock Waves", Physics Dept.,
Princeton Univ., March, 1947.
Einstein., H. A., and Baird, B. G., "Progress Report of the Analogy
between Surface Shock Waves on Liquids and Shocks in Compressible
Gases", Hydrodynamics, Laboratory, Calif. Institute Of Technology,
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between Surface. Shook Waves on Liquids and Shocks in Conipressible
Gases", Hydrodynamics Laboratory, Calif. Institute of Technology,
July 30, 1947.
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and Puokett, A.B., Introduction to
Aerodynamicsof a Compressible Fluid, J. Wiley
andSons,. New York, 1947.
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inTerms of
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andPriedricha, K. 0., Supersonic Flow
and ShookOLLE
z
REFERENOE-LEVL
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