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(1)

)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

(2)

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.

(3)

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 at

low 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

been

concerned

either with numerical flow

oalou].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.

(4)

-2-The object of the present paper is to derive the

inathertaticalanal-ogy between open-channel liquid flow

and

compressible 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(&LOGY

Various 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,

emphasizing

the basic assiptiona made, will be given below.

The Bernoulli Equation

Let AL

(Pig. 1)

be an arbitrary stream tube in a

flowing

fluid (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

(5)

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 or

dovv-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

1

fu

u 1

f?ü

au_'

(2)

(6)

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

h

y

r-

r-"

where £(( 1 The vertical velocity u approaches zero atthe horizontal

U

(7)

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

horizontal

be (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 rau

Ix

ou

y

ax

.!

constant (9)

L8r

(8)

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 y

The 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 shown

that for a gas

r an ii x p

Lay

(16) constant (17) p (14)

(9)

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 and

a&'u)

O(TSu)

x +

y=o

a

The three equations (16), (21) and (22), for the three unknowns,

U,

and T, together with the appropriate boundary conditions, completely

determine 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 Flow

We wish to compare Eqs. (ii),

(12)

and (13) for liquid flow with

Eqs. (16),

(21) and

(22)

for gas flow. Eq.

(12) is

equivalent to Eq.(16)

designated by "adiabatic-reversible", sometimes inaccurately shortened

(10)

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 reference

point, 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

(11)

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

6y

Eq. (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 x

Thus the veloóity of sound plays a critical role, separat.ng "subsonic"

from "Supersonic" regimes.

L

U2

'-''C I

rj

L'

(12)

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

(13)

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

(14)

-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 liquid

and repeating the experiment.

Assumption (viii),

that the vertical

acceleration is negligible

com-pared to

the

acceleration of gravity, is closely related to asstion (ix)

that

the

free surface does not slope much in

the

direction of liquid motion. To analyze these

assumptions,

choose coordinates so that

the surface

v-looitr at the point f interest lies in

the

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 - -

---.

z

cJ

1 z

using --

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

+ - - +

--a

hc

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)

(15)

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

/ -'

I

f

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 by

5 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 that

(16)

the 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 (4

The 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

(17)

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 length

ABCD=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 momentum

equation 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.

(18)

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 steadr

state

"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

(19)

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

(20)

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 passing

through 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) the

corn-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 I

2.2

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,

(21)

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 a

0Galilean 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.

(22)

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 the

wave 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

(23)

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 P

cT+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.

(24)

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 Figure

11

P1

7 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)

(25)

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 U1

005 e

(M2 sin2g

-(65) (y+ ) M] sin 9

The 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

(26)

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

'2

içì

(Y )3 M14

r

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 V

4(Y -4.--5)

3 V

-

f +...

(68) V

(vi)'

V

The 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)

(27)

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

(28)

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

the

shook

inter-section, so that

the problem is reduced to

one

of steady flow.

In

this

coordinate system, the fluid

in

Region I is moving with a velocity U.1

toward

the two

stationary shocks, S and S.

In order for these two shocks

to have equal amplitudes, the oononents of

noimal

to

each of the shocks

must be equal.

This is possible only if

is parallel to the bisector,

BB, of the angle between

the

shocks. 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

without

altering 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

U

in 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

(29)

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 downward

at 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.

(30)

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

(31)

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.

(32)

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 be

explained, -there remains the theoretical problem of shock curvature. If

(33)

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

(34)

.BIJI0GRA.PHY

Jouguet, E, "(ue1ques prob1mes d'Hydrodynaique g6nrale", J. de

math. pures et appliq. (Series 8), 3, 1 (1920).

Riabouchinsky, D.., "Sur l'analogie hydraulique des niouvementa d' un

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. of

Technology, 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 Water

Flaws 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

Institut fir Aerodynathik", No. 7, Eid. Tech. Hochsch., zurich (1938).

POlachek, H..,

and Seeger, R. J., "Regular Reflection of Shocks in Ideal Gases", Explosives Research Report No. 13, Navy Dept. BuOrd, Washington

(35)

Polachék, 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 Reflection

Angles 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,

(36)

Einstein, H. A., and Baird, B. G., "Progress Report of the Analogy

between Surface. Shook Waves on Liquids and Shocks in Conipressible

Gases", Hydrodynamics Laboratory, Calif. Institute of Technology,

July 30, 1947.

Liopmenn, H. L,

and Puokett, A.

B., Introduction to

Aerodynamics

of a Compressible Fluid, J. Wiley

and

Sons,. New York, 1947.

Lamb, Horace, Hyd.rodyrla!nics, Sixth Edition, Dover Publications,

New Yàrk, 1945.

Bakhmeteff, B.,. and Matzke, A., "The Hydraulic Jump

in

Terms of

Dynamic Similarity",

A.S .C.E., 101, 630 (1936).

Courant, R.,

and

Priedricha, K. 0., Supersonic Flow

and Shook

(37)

OLLE

z

REFERENOE-LEVL

o;

'QU/D

/7/I/I

/1)/lI//I/I/ft

C,

F. 3

TPir'e.- Drne.nio

FIq. 4

A HydrcuC J.rrp

Lj.i%c

FIo#

-F.5A TIpc41

Profile

o

5B T-jpcaA Pr4e.

'

5tov

4yrAfl. Jurrtp

:"-'

"aJ't

Z

2.

Ltcjud Flow

Fee.

Genercd F'ow Fe.d

4orz.ont

2ed

-I

(38)

Fig.. 6 Qbe

(.0 .5 4 3 2

Fi9. 7 Ret,r

Ac..ro

NOp-rnai Shoc,c

OP.

'Jurnp

a

4

M1.NIACW NUI-iBR AWE.AD OF SHOCK /

/

h1

--::

1.4

--7,'

... ... i ... Ta 1.;.

ALIQUID

--GAS1'-2.O

C,&S:T.

LIQUID 0 .6 0 (5 2.0 25 30 36 (U2) /e

Fi9

hc.V(b P'r

j

.5

(39)

9.

Q Pt9MP

e+.%ec.tj*r

+ a SPtc +rom

R%gld W

U1

-

U) -.

/

/ 5$

/

U4-..-V '- I

P' ACTUAL '.1ALL

-.

I

-F!9.

f --

c.h"

o

5)oe.

-fvQ

Rfid

(40)

40

30

20

0

0

%O

20

30

40

50

60

70

1NCIPNT At4GLE cC

Ft9 IZ

Re41ectiort of WeK Shocks trt

Jter

rd Ar

- .1

fh=.

WATER

0.7

0

THEORETICAL

0 EXPERIMENTAL

-CORECTON

£

p

4

---b-0

/

REGULAR

REFL.C,TIOM

/

/

I.

MAC4

FL E CT1 ON

/

/

,ç_LIMrTIWG cc

E.XPERIME.NrAL POINTS

FROM RE.FE.R.t-4CE. 16

/

AIR

REGULAR

REFLEC.1TION

V

MACH

REFLGTION

0

'ID

,/,000

EXPRIMEN7AL POI1JTS

FROM REFE.RNCE..J4

/

/

(41)

-'z0

Jo

Ui

J

I-L) LU

-2O

LI

220

z

Lii

2

Co

2

-10

-20

0

REGULAR

RE-FLaC7T-

ION

EXPER1ME.NTAL POIWT5

FF(Or"l REFERENC,E 14

10

20

30

40

50

0

70

60

INCIDENT ANGLE c*

AIR

Pi/Pz

0.5 ?

1.4

I

REFLECTiON

F. 13

Refc.t.ion o-

Medium

5hocics in. water

rtd Air

(h1/h)2=

WATER

0.5

0

09

/

-

T-4EORETICAL

o EXPERIMENTAL

4'-

CORRECTION

/

/

/

--.-0

/

.0

41ON

IHLIMmNG

FROM

EXPER%ME.NVAL

FE.FERENCE

PoUTS

16

/

40

30

(42)

20

I0

0

Lii

F-Ui

-j30

U-ui

2

1O

I-z

hi

0

2

-10

-20

0

10

20

0

40

GO

IP4C1DET ANGLE. CC

t4 Refe.ction o-f Strop,

hoc.Vs

rt VJate

70

nd

pp-WATE,R

9

0

/

p/I

/

/1

1/.

/

7

REFLECTION

_//

/

'LMtTING O

RE. FLECTIO N

I S

:

TNEORE.TIC,AL

c0RRc-rOW

0

'

I

EXPE.R1MENTAL P0NT5

FROM REFaRE.NCE..

I G

/

Pi/Pz

AIR

0.3,

1.4

o0

0

REFLECTION

1

REGULAR

p

RFLE.CT10N

-I

I

I

I

I

I

Po%NrS I

F0M REFEPEC.E. t4

/

Cytaty

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Słowa kluczowe: środki smarowe, ciecze eksploatacyjne, lepkość strukturalna, niskotemperaturowy moment obrotowy, mieszalność środków smarowych, indeks żelowania, odporność

14 Jak wyżej pokazano, liberalizm niemiecki odbiegł w swym rozwoju znacznie od pierwotnych koncepcji liberalnych, jednak nigdy, mimo polityki kulturkampfu nasta­ wionej

T eodorow icz-F Iellm an: „S zw edzkie przekłady Pana Tadeusza”. Stanisław em Falkow skim

Wysłuchanie publiczne może dotyczyć projektu ustawy (przeprowadza się je opierając się na przepisach Regulaminu Sejmu) bądź projektu rozporządzenia (zgod- nie z art.

leży w inny jeszcze sposób; nie tylko nowo ogłoszonemi na­ bytkami, ale także kilkoma pozycjami zaczerpniętemi z dawnych czasopism, przeoczonemi tam przez