THE DESIGN AND CONSTRUCTION OF A FREE SURFACE

IATER TABLE FOR THE INVESTIGATION OF

COMPRESSIBLE—FLOW PEENOMENA

by

Robert William Eberhard

Thesis submitted to the Graduate Faculty ct the

Virginia Polytechnic Institute

in candidacy for the degree of

MASTER OF SCIENCE

im

MEGHANICAL ENGINEERING

APPROVED: APPROVED:

’W

„.irectoro ra ua e tu es par men

_,;..,.6.ä?;· ,„l.;.. ~· · .P=an o‘ =ng neer ng 3or/ ro essor// E

February, 1956

Blacksburg, Virginia

-2-

TABLE OF CONTENTS

Page

LIST OF FIGURES. . . •„.... . . . . . . 4

LIST OF DESIGN DRAWINGS. . . . . . . . . . . 5

LIST OF PLTES . . . . . . . . . . . . . . . 6

LIST OF TABLES . . . . . . . . . . . . . . . 6

LIST OF SYMBOLS. . . . . . . . . . . . . . . 7

I. INTRODUCTION . . . . . . . . . . . . . . . . 11

II. LITERATURE REVIEW. . . . . . . . . . . . . . 13

Theory. . . . . . . . . . . . . . . . . 14

Temperature Ratio. . ...... . 15

Density Ratio. . . . . . . . . . . 20

Pressure Ratio . . . . . . . . . . 25

Mach Nunber. . . . . . . . . . . . 26

Specific Heat Ratio. . . . . . . . 30

Validity of Analogy . . . . . . . . . . 33

Ratio of Specific Beats. . . . . . 33

'*ä°.?.ä.’䧑£.$‘£‘é‘„’€‘°f“?‘.......33Shock Waves. . . . . . . . . . . . 37

Vertical Mtious . . . . . . . . . 38

Surface Tension. . . . . . . . . . 40

Wave Propagation . . . . . . . . . 40

Methods of Experimentatioa. . . . . . . 42

Water Channel. . . . . . . . . . . 42Entrance Sections. .... . . . . 42

-3-.{J

Page

Test Sections. . . . . . . . . . 44

Circulating Systems. . . . . . . 46

Model Design . . . . . . . . . . 46

Depth-Survey Systems . . . . . . 47

Choice ot later Height . . . . . 47

III. DESIGN AND CONSTRUCTION. . . . . . . . . . 49

Entrance Sect1on• . . . . . . . . . . 49

Test Section. . . . . . . . • • • . . 59

Tanks . . . . . . . . . . . . . . . . 61

Circulating System. . . . . . . . . . 64

{Elevating System and Bottom Support , 67

Depth Survey System . . . . . . . . . 68

Model . . . . . . . . . . . . . . . . 80

Apparatus and Materials . . . . . . . 82

IV. OPEBATIONAL PBOCEDURL • • • • • • • • • • 84

V. DISCUBSION . . . . . . . . . . . . . • . . 87

Discussion of Design . . . . . . . . . 87

Representative Results . . . . . . . . 91

Method of Calculation . . . . . . 94

VI• SUMMARY. . . • • . . . . . . . . . . . . . 102

VII. RECOMMENDATIONS. . . . . . . .... . . . 103

VIII. BIBLIOGRAPHY . . . . . . . . . . . . . . . 105

Addenda . . . . . . . . . . . . . . . 107

IX• ACKNOWLEDGEMENT8 . . . . . . . . . . . . . 108

X. VITA . . . . . . . . . . . . . . . . . . . 110

XI• APPENDIX . . . . . . ........... 112

Materials for Construction. . . . . . 112

At

LIST OF FIGURES

Page

Figure 1g FIÜW Fi1am€¤t• • • • • • • • • • •

•Figure2, Infiuitesimal Fluid Prism, , , , , . 21

Figure 3. Infinitesimal Parallelepiped ofI I I I I I I I I I I I I

IFigure4, Pressure Ratio vs, Hach Number forIsentropic Flow, Showing EffectQf k I I I I I I I I I I I I I I

IFigure5, Pressure Ratio and Radius Ratio vs,Trning Angle for Prandtl-MeyerFlow, SÜOWÄIIQ Effüßt of k• • • • • 35

Figure 6, Pressure Ratio Across Gas Shock andSquare of Height Ratio AcrossHydraulie Jump, Both vs, meh

I I I I I I I I I I I I I

IFigure7, Propagation Speed vs, Wavelength forCombined Gravity and Capillary

I I I I I I I I I I I I I

IFigure8, Schematic Wave Pattern for SupersonicFlow Past Wedge. . .. . . . , , 43

Figure 9, Schematic Cross—Seetiona1 View ofChannel Showing Sluice Nozzle, , , 45

Figure 10, Schematic Diagram of Equipment , , , 66

Figure 11, Calibration Curve tor Orifiee, , , . 68

-5-

LIST OF DESIGN DRAIINGS

Page

Dwg, No, 1, Sluice Gate. , , , , , , , , , , , 55

Dwg, No, 2, Gate Guide , , , . , , , , , , , , 56

Dwg, No, 3, Details of Parts Sluice Gate . , , 57

Dwg, Ne, 4, Channel Frame. , , , , , , , , , , 60

Dwg, No, 5, Superstrueture . , , , , , , , , , 62

Dwg, Ne, 6, Pivot Shaft, , , , , , , , , , , , 69

Dwg, No, 7, Bottcm Support , , , , , , , , , . 70

Dwg, No, 8, Bottom Wedges. , „ , . , , , . . , 71

Dwg, No, 9, Wedge Screw. , , , „ ,,,,... 72

Dwg, Ne, 10, Top Wedges , , , , , , , , , „ , . 74

Dwg, Ne, ll, Adjustable Bearing . , , , , , , , 75

Dwg, No, 12, Details of PartsDepth Survey System, . , , , , , 77

„ Dwg, No, 13, Details of PartsDepth Survey System, , , , , , , 78

Dwg, No, 14, Model, , , , , , , , , , , , , , , 81

-3-

LIST OF PLATE

Page

Plate l, Water Table Assembly....... , . . 50

Plate 2, Views of Water Table Components . . . . 52

Plate 3, Flow of Water About A Fifteen-DegreeWedgeM0de1,,,••••,,,,,,,92

LIST 0 TABLES

Table I, Summary of Analogous Relationships. , . 32

Table II, Tests for Flow About A Fi£teen·DegreeWedge Model. . . . , . „ , „ . . . . 93

J}

LIST OF SYMBOLS

Sysbols Definition Units

a velocity of sound ' ft/sec

c velocity of propagation ofsurface waves in fluid ft/sec

Cp specific heat at constantpressure Btu/lb-°R

Cv specific heat at constantvolume Btu/1b—°B

FI flow work Btu

g acceleration due to gravity ft/secz

J h height cf water surface abovefloor of test section ft

ho stagnation depth of water ft

H specific enthalpy of gas Btu/lb

ä enthalpy of gas Btu

IE internal energy Btu

J Joule's constant (• 778) ft-lb/Btu

k ratio of specific heat atconstant pressure tospecific heat at constantvolume ·---

KE kinetic energy Btu

H Mach number ---—p static pressure lb/ftz

p° stagnation pressure lb/ft2

-3-

Symbols Definition Units

PE potential energy Btu

Q transferred heat Btu

B specific gas constant ft·lb/°B/lb

r radius ft

T :tatic absolute temperature °Rankine

Tb stagnation absolute temperature °Rankine

U specific internal energy Btu/lb

u X—component of velocity ft/sec

V velocity of flow ft/sec

V specific volume of substance fta/lb

v Y—component of velocity ft/sec

I work Btu

w Z—component of velocity ft/sec

z height offlow filament above floorof test section ft

ze etagnation depth of flow filament ft

Jß shock angle degrees

6 slope of water surface withhorizontal degrees

jo density slugs/fts

;\ wavelength of surface waves inliquid ft

g' surface tension of liquid poundals/ft

$91

Symbols Definition Units

A indicates a difference or achange of value —~-·

ß velocity potential in two-dimensional flow -—--

ßx partial derivative of ¢with respect

tg X•• • • • •

ßxx second partlal derivative of ¢with respect

2tg X

• Ög -

• • •

Ö1:

üy partial derivative of ¢with respect

b ¢tg y• ...57, • •

ßyy second partial derivative ot ¢with respect

2tg y • -Ö,-„-éé • • • •

Övßxy partial derivative of ¢

with respect

to x and y-

Ö zg ————ÖJÖY

Subscrlpts:

No subscript Any value of variable -·-—

O value at stagnation condi-tions • — • •

Symbols Definition Units

Subscrigts: (Coutinued)

1 value of variable atpßilltl -·-•••

2 value ot variable atpoint 2 -·•-

max maximum value ofvariable -

• • •

-11-

l. INTRODUCTION

The demand and active development of supersonic

aircraft has required engineers to become familiar with

the phenomena of supersonic flow. Experimental investiga-

tion of supersonic flows in a supersonic wind tunnel

results in limited observation techniques and extremely

high costs.

For approximately forty years, the hydraulic

analogy between water flow with a free surface and two-

· dimensional compressible gas flcw has been known. The

analogy lies between the gravity waves on the free

surface of a liquid and the pressure waves in a

compressible gas. The ”hydraulic jump" in water flow

is analogous to the shock wave in compressible gas flow.

Application of the hydraulic analogy attained with

a free surface water table to the investigation of

compressible-flow phenomena has the advantages of

simplicity, rapid operation, and law costs. lt is net

intended that the free surface water table replace the

wind tunnel, but that it supplement the wind tunnel.

The water table may be advantageously used for

preliminary investigations, thereby making available

the wind tunnel for more complicated investigations.

-)g-

The primary objectives of this thesis project

were: (1) to present the existing theoretical analogy

between water flow with a free surface and twe·

dimensional compressible gas flow, (2) te design and

construct a free surface water table, and (3) to

demonstrate the operation of the free surface water

table by the investigation of flow about a basic model,

The water table was designed and constructed by

the author for use in the Hydraulics Laboratory of the

Mechanical Engineering Department at Virginia Polytechnic

Institute,

-13,

II• LITERATURE REVIEW

A comparison between the gravity wave patterns

(around a model) observed on a thin, rapidly flowing

sheet of water and shock wave patterns (around a

model) observed in supersonic wind tunnels is

qualitative evidence that an analogy exists between

the gravity waves on the free surface of a liquid and

the pressure waves in a compressible gas(19).

The origin of the hydraulic analogy seems to have

originated with Jouguet(°) who pointed out the analogy

for one-dimensional flow only, The hydraulic analogy

between water flow with a free surface and two·

dimensional compressible gas flow was presented on a

mathematical basis, and an apparatus for the

investigation of flow in a Laval nozzle was described

by Biabouchinsky(16), Later, he outlined the probable

usefulness of the hydraulie ana1ogy(17), The

application of the methods of gas dynamics to water

flows with a free surface was conclusively proven by

Preiswerk(15), Additional investigations to determine

the practical information that might be obtained by

application of the hydraulic analogy were stimulated

by the development of supersonic aircraft and the

-14-

great expense of supersonic wind tunnels(2'3'4'6'7'8'12*

13•14•18•l9), lt is notable, however, that useful

results of the hydraulic analogy can be obtained only

by a proper combination of experimental technique and

method of analysis of the experimental data(2°"21°),

The following sections of the literature review

present the general theory and validity of the analogy,

and outline the experimental techniques employed in

hydrodynamic studies,

Theory

The following theory development for the hydraulic

analogy, between water flow with a free surface and two-

dimensional compressible gas flow, was based mainly on

the work of Preiswerk(14),

ln the mathematical development of the theory, the

following assumptions are made:

1, The gas flow is isentropic,

2, The gas flow is steady and irrotational•

3, The water flow is frictionless,

4, The vertical acceleration of the water is

negligihle as compared with the acceleration

due to gravity,

5, The water flows over a smooth horizontal surface

bounded by vertical walls geometrically similar

-15-

te the walls bounding the corresponding

compressible gas flow,

6, The water flow is steady and irrotational,

By employing the above assumptions and applying the

energy and centinuity equations, the following relation-

ships for the hydraulie analogy may be developed:

1, The relationship between the gas temperature ratio

and the water depth ratio,

2, The relationship between the gas density ratio

and the water depth ratio,

3, The relationship between the gas pressure ratio

and the water depth ratio,

4, The relationship between the veloeity ef sound in

a gas and the veloeity ef prepagatien of gravita-

tional water waves,

5, The relationship between the Heh number and the

Freude number,

Temperature Ratio, The general energy equation for the

steady flow of gas may be expressed in the following form:

KE1 + PE1 + PII + IE1 + Q = KE2 + PE2 + FI2 + IEZ + I (1)

For an adiabatic process during which no work is done and no

change in potential energy occurs, the general energy

equation becomes

KE]. + Fwl + IEI‘ KE2 'F

•16•

or, fer one pound of substance,

2 — 2V nV v V..L.+....L}. CT • 2 + ++22 c·r2;.1 .1 +*1;;.1* *6****2

‘°’

However, g

n- 11+ BV (4)

Therefore,

..1;. + -

V;+ (5)

2gJK1

235Hz

Assuming stagnation conditions at point 0,

so -vz + xx (6)

E8]

By rearrunging terms,

V2·· 2gJ(H° · H) - 2gJCp(T° - T) (7)

Using cp in the units ot foot·pounds per pound•

degree Rankine,

vz- 2 (2 2) (6)gc?

o °

Assume the conditions of incompressible fluid

flow in a passage to be steady and frictionless,

Then, the energy equation may be written

(9)

„17•

or, for unit volume of substance, equation (9)

becomes

-6-;ä+)¤8Z·1+P1'—/E-;—;·+ßB¤2+p2 (10)

This form of the energy equation is recognized as

the Bernoulli equation,

The Bernoulli equation applies to a flow

filament (Figure 1) which passes through the

point y0, 20 of the initial cross section

x• 0,

P+/°V2

+ )¤8¤ **90+AVZ

+ pgz (11)T -2-2 <·

The pressure p on the surface of the water is constant

and equal to the atmospheric pressure, Since only

pressure differences are of physical significance in

the case of incompressible flows, this pressure may

he set equal to zero, In the case of water flowing

from an infinitely wide hasin, V0• 0 and the

curvature of the free surface at this point is equal

to zero, It is logical to choose this point x0, y0,

z0 as the reference point, The maximum water depth is

ho which is also the water depth at the reference

point,

-13-

X

>

-1:Q

1-ZL-1E<(;‘L1.

EO..11-

L11(ID*21-

.1N I

112

-é’

•1Q•

The Bernoulll equation for the above reference

paint is

p+.!i!!2;+,ogz-p„+pg¤° (12)

Therefore,

v2 • 2g(z° - z) + Eigg-:.£l (13)P

Under the assumption that the vertical acceleration of

the water is negligible cempared to the acceleration due

to gravity, the static pressure at a point of the field

of flow depends linearly on the vertical distance under

the free surface at that pesition

p„•,0g(h„ —z„) (14)

and

P 'p8(h-z) (15)

Substituting (14) and (15) in (13) results in

v2 • 2g(h° — h) • Zgzlh (16)

Therefore,

vmax = 'V Zgho (17)

Expressed in dimensionless form

V2

ABv.... ' Th (18)o

-20-

Using Ho in the units of foot·pounds per pound and

CP in the units of foot-pounds per pound-degree

Rankine, the maximum velocity of a gas is

Vmax•• • (19)

and

2V • AT (20)

"°"i|l°

These non-dimensional magnitudes of velocity are

equal if

Tb- .2.. (21)

o To

Density Ratio, Another analogous relationship

may be obtained by comparing the equation of

continuity for water flow with a free surface to

that of a gas flow,

Consider at location x, y a small fluid prism

with the dimensions dx, dy, and h (Figure 2), The

horizontal components of velocity are designated u and

v with w as the vertical component of the velocity V

in the direction of the X, Y, and Z axes respective1y•

Assuming that the vertical acceleration of the

water is negligible in comparison with the

acceleration of gravity, equation (15) may he used,

>—

KP

QS

7*

21

N

E

§

‘

3%

\3

;2

><

><

_;

Q

·{

ä

;Ä

Q3

[

-33-

Iritten in differential form

b anFg " /78;-*- (22)

and

6 an·;·P·y • ;¤s—;···y (23)

The above equations are independent of z, therefore,

the horizontal accelerations of all points along a

vertical also are independent of z, Con:equently,

for the entire depth, the horizontal velocity

components u and v are eonstant.

The continuity equation, based on the prineiple

of the conservation of mass, states that the mass of

fluid passing any section per unit time is constant•

The density of water being constant, the inflowing

volume per unit time must equal the outflowing fluid

volume•

The inflowing fluid volume • dqa

dqa • uhdy + vhdx (24)

The outflowing fluid volume • dqL

d• (buQL¤ + 1¥ädx)(h + 1¥xdx)dy

e Öv äh

By expanding, and neglecting infinitely small magnitudes,

the equation of continuity for steady water flow is

+• 0 (26)

-23-

The continuity equation for the most general

case of three dimensional gas flow will now be

derived, Consider an infinitesimal prallelepiped

of fluid with sides dx, dy, and dz (Figure 3),

Mass flow entering in left direetion • dme

due • p udydz (27)

Mass flow leaving the opposite face • dmL

dm" • (/0 u + )dydz (28)

Net mass flow in X-direction-

dmu

den ·= ,0 u)dxdydz (29)

Therefore, the total outflowing mass in the

X, Y, and Z directions is

ö a-5-Rip u) ·•· -;.i(pv) + Z(/ow) dxdydz (30)

Also, the net change in density per unit time is

)dxdydz (31)

By application of the prineiple of the conservation

of mass, there is obtained the equation of

continuity in three-dimensions•

Ö- +‘5(nu)+

ö(v)+ ö(w)•0 (32)t 3 x 5 y P E z IG

-24-

X

*1;

0

Q E*9

—'1-

$11„.

„1_ Og„ Q

I.1Lu9;0.L-Iaj.1Ex]0:

·E

I

N———# .1

x é>< -

1 xLu

1 xI;

1\

Z\ E

E

N.

QS ro‘¤u

lu

¤„ txDS21-

-25-

lf the motion is steady, 1.6,, no time variation

at a point, the equation of continuity is

ö( u) + Ö (pv) + ä (pw) -0 (33)5;:/O Sy Sz

Considering the case for two-dimensional steady flow,

the equation of continuity is

b( ö-E;-pu) + —ö**§(,¢V) • 0 (34)

Since the equations (26) and (34) have the same

form, another analogous relationship may be deter-

mined, That is, the density of the gas flow

eorresponds to the depth of the water, Expressing

this analogous relationship as a non-dimensional

magnitude:

P , n---

———- (35)Po hg

Pressure Ratio. Another analogous relation-

ship is obtained from the gas equation

p • /0gRT (36)

or

P.2....-

......T.'!‘ (37)po ·po 0

-26-

By substituting the values of _fL and T in terms/°o To

of .%;, the result is

E_ 11 2 (38)

Fo Bo

Mch Number. Through the combination of the

energy and continuity equations and the condition

for absence of vorticity, there is obtained the

differential equation for the velocity potential in

water.

The energy equation (16) may be written in the

form

u - no -lg (39)

26

and I

bu, _ 1 6es 2?Tz"'2>"‘°’

The horizontal velocity components u and v being

constant on a vertical and V being constant, as

shown by equation (16), the vertical component w is

also constant• In comparison with the components

u and v, w may be neglected since it disappears at

the bottom boundary of the f1ow•

Thenv2

-u2 + V2 (41)

-27-

Substituting in equation (40)

A11 -- 1 uöu + vöv (42)Ex Ex

and

äh 1 Ö¤+ äv· - .... (43)äv sEyFor

steady water flow, the equution of

continuity (26) may be written in the form

ou Bh bv bh15; + u-gi + big? + vis? ( )h • 0 44

Substituting equations (42) and (43) in

equatiou (44),

Bu u öu övlläx g 3x + v1§ä

öv v B ön -V + V — 0 (45+ jgg vjyy )

This may be written in the form

Ö1: u2A 2

[EV ab BY gh

- Ö1! dv uv • O (46)

A vertex is flow in which the streamlines

are concentric circles, In irrotational motion,

-23-

an infinitesimal particle moving along the streamline

does not rotate about its own axis, For the condition

of irrotational motion, or absence of vorticity,

öv ou (47)85* BY

For expediting the analytical investigation of

velocity fields, potential functions have been

devised, The velocity potential, ß, is some

function of x and y and is defined such that

6;: 6;;u • v-

(48)

The use of the potential function ¢ is limited to

irrotational motion,

Let

¢··

bgX 31:

gxx ' Q2;ox

ößgy --5-Y

6%g CYY Ԥ"'2'y

629¢XY·äxEy

Substitutins ¢(x,y) in equation (46)

2 2

¢ 1--2 + ß 1- ßy·2¤

axgy-0(49)xx s VV e5 xY‘§¤'

Equation (49) is the differential equation for the _

velocity potential of two·dimensional incompressible

flow, It may be shown that the equation for the

velocity potential of two—dime¤sional compressible

flow is

2 2·——- + • *’ •0 50¢ 1

uxß 1 .gZ2

zuxxa2 yy axy——:g

Since equations (49) and (50) have the same

form, another analogous relationship may be determined,

That is, 4/gh for the free surface liquid flow is

analogous to the velocity of sound, a, in gas f1ow•

The value q/gh is the basic surface wave velocity,

i, e., the velocity of propagation of long gravity

waves,

xpressing this analogous relationship as a non-

dimensional magnitude

¤ • ax (61)Z55; zsq

The Mach number, V , in the gas flowa

correspends to the Freude number, 7V;K_, in free

surface liquid flow, If equation (16) is substituted

in the expression for the Freude number, an equation

is obtained by which the corresponding Mach number

for gas flow may be easily computed, This equation is

1/2(2) (ll • ll)N · (52)

h

In gas flow, if the Mach number (M-

.§.) is

less than one, the flow is known as "subsonic"; and,

if the Mach number is greater than one, the flow is

known as "=upersonic". Similarly, in liquid flow, if

jvää. is less than one, the flow is known as "streamingflow"; and, if g is greater than one, the flow is

known as “shooting flow”•

Qgecific Heat Ratio. For isentrepic flow in a gas

1k-1.8. - .g,_ (53)

/°o o

-31-

and in the analogous relationships from

equations (21) and (35)

P , h .. 1-p° 'Q ··T; (54)

Then, we have the equation

1E'-TTTT - T (ss)

o [ilo]

Obviously, this condition is satisfied only for

Thus, it is shown that the flow of water is‘

quantitatively comparable with the flow of a gas

having a specific heat ratio, k, equal to 2,0,

In Table l, a summary of analogous relation-

ships is given, These analogous relationships

between water flow with a free surface and two-

dimensional oompressible gas flow, with a specific

heat ratio equal to 2,0, are based on the assumptions

listed on page 14•

TABLE I

Summary of Analogous Relationshigs

Two•Dimensiona1 CompressibleGas Flow, k

• 2,0 Hydraulic Analogue

Temperature ratio, T/Tb later-depth ratio, h/ho

Density ratio, füpg Iater—depth ratio, h/ho

Pressure ratio, p/pa Square of water- 2depth ratio, (h/ho)

Velocity of Have velocity, ·V ghsound, a • EB

V PHach number, V/a Froude number, V/M gh

Subsonic flow Streaming flow

Supersonio flow Shooting flow

Shock wave Eydraulic jump

-33-

Validit; of the Analog;

Tb evaluate the significance of data obtained

with a free surface water table, various factors

affecting the analogue are now to be reviewed•

Ratio of gpecific Beats. The analegy is not as

seriously restricted by the fact that k must equal 2.0

as might at first be expected. Changes in k do not

seriously affect subsenlc Mach numbers(13), Aceording

to various theeries, pressure distribution at moderate

subsonic speeds is independent ef the specific heat

ratio(18),

For supersonic flow, however, the effect of

specific heat ratio considerably influences reeults.

For this reason alone, lt is not considered feasible

that the free surface water table be used to obtain

accurate design data for direct application to super-

sonic air flows, Figures 4 and 5 illustrate errors

which are a result of the difference in specific heat

ratio.

Viscosit; and Thermal Conductivit;. For purposes

of the analogue, the fluid viscosity(18) has been

ignored in both the water flow and the gas flow; however,

the model, wall, and floor bonndary layers must be

considered in the analysis of experimental results,

..34-

LO"*‘l“'§%Mw

U°

\ANALOGUE)*1 0.u}IKK

IIILW°·°‘o¤ 2 6 4

Maca uumasa

aisuas 4. aasssoas anno vs. moa Nomssa roa isamaoaio now,saowms Eraser or a.

Shapiro, A. H.: Free Surface Water Table, "Physical Measurements inGas Dynamics and Combustion" (R. W. Landenburg, B. Lewis, R. N.Pease, and Ei. S. Taylor, Editors), 2, p. 3lL. Princeton UniversityPress, Princeton, New Jersey, l95A.

-35-

LO:R: L4

} L-‘“*"2·° '

l. ‘Ä-X Y ‘~ ‘$

\L\e L ki

P° k: 2.0 \z0.ng_—\cix_

_¤Ä'g„/r sl

/ \

4 \\x00.0 |0 20 30 40

0, DEGREES

HGURE 5. PRESSURE RAT•0 AND RADIUS RATIO vs. TURNING ANGLEFOR PRANOTL—MEYER FLOW, sH0w|NG EFFEcT OF k.

Shapiro, A. H.: Free Surface Water Table, "Physical Meaeurements inGas Dynamics and Combustion" (R. W. Landenburg, B. Lewis, R. N.Peese, am H- S- Taylw, Ediwrs), Q, p. 315. Princeton UniversityPress, Princeton, New Jersey, l95Z„

-36-

The model boundary layer causes flow separation

which results in pressure distributions that are

different from those produced if separation does not

occur, Since the wall boundary layer is some distance

from the model and is representative of only a small

percentage of the total mass flow, its effects are not

considered, The floor boundary layer, representing

an appreciable percentage of the entire flow, results

in the most serious effects, which observations have

indicated to be as follcwsz

1, how supersonic stream nnen numbers are attained,

2, Subsonic wall interference effects are reduced,

Any channel used for research should have as large

a Reynolds number as pe:sible because of the serious

effects of the boundary layer, As explained by

Matthews(9), "A channel permitting Reynolds numbers as

large as 3,000,000 at tunnel choking would have to be

approximately 10 feet wide if the water temperature

were 200 °F or 20 feet wide if the water temperature

were 100 °F '°°. At the suggested Reynolds number of

3,000,000, separation phenomena at the model should be

similar to comparable phenomena found in a 2—foot

diameter wind tunnel; whereas the 20-inch wide

demonstration channel is comparable to a wind tunnel

of 0,4 inch in diameter”•

-37-

lt is rather impractical due to low water speeds,

to duplicate the Reynolds number of high speed air

flow, Furthermore, aerodynamic heating effects in a

high speed compressible gas result in different

behavior for the boundary layer of the gas than for the

bcundary layer of low speed liquid flow(18),

Shock Waves, A discontinuity, known as a "plane

compression shoek"(1), may occur in air flowing at a

supersonic velocity such that the velocity suddenly

decreases to a subsonic value which satisfies the

conditions of flow, Through a shock, the velocity

suddenly decreases and the pressure suddenly increases,

An "oblique shock"(1) will occur when supersonic gas

flow is forced to change its direction due to an

obstruction•

The analogous discontinulty which may occur in

shooting water flow is known as the "hydraulic Jump",

The two types of Jump which may occur are: (1) the

right Jup, where shocting water is converted into

streaming flow; and (2) the slant or oblique Jump,

where, depending on the Reh number, the flow may be

either streaming or shooting after the jump,

When a hydraulic Jump occurs, part of the kinetic

energy of the water is dissipated as heat, Therefore,

the total head of the water flow decreases across the

-33-

jump, The heat generated by a shock in gas is not

lost, but is converted into thermal energy and the

total temperature and the total energy remain the

same across the shock, Consequently, when shocks

are present, the hydraulic analogy is not strictly

valid,I

Compared to the relations for the compreseion

shock of a gas with the specific heat ratio equal

to 2, the relations for the hydraulic jump are

different, Figure 6 "shows, for example, that, com—

pared with a gas having k• 1,4, the error(18) in

pressure ratio across a normal shock in water is

about twice as large as the error for a gas with

k •2 and is about 25 per cent between M ¤ 1,5 and

M ¤ 3,0, For weak shocks, both normal and oblique, the

change in entropy across the shock is small and the

analogue does not suffer appreciably",

Vertical Hotions, The assumption that the water

flow is two-dimensional limits the analogy to the

condition that the vertical components of velocity and

acceleration are negligibly small, To attain this

condition, the slope, 8, of the water surface with

the horizontal and the ratio of the depth of the

water, d, to the radius of curvature, R, must be small,

Matthews(9) states that, "When either 9 or d/R or both

-39..

HYDRAULICJUMPia

NORMAL SHOCK

> Ä an 6AsVa k = 2.0

ä L k = ·-4<¤:<6 ßX

'¤ 2 6Mx

neun; 6. pmsssuns mmo Acnoss GAS snocx Anm scums orHEIGHT mmo Acnoss Hv¤aAu¤.¤c Jump, sem v6.

man uumasn.

Shapiro, A. H.: Free Surface Water Table, "Physical Measurements inGas Dynamics and C0mbustion" (R. W. Landenburg, B. Lewis, R. N.Pwße, ¤¤d H~ S- Taylw, Ediwrs), 2, p. 316. Pr1¤c6c6¤ UniversityPress, Princeton, New Jersey, l95l..,

-49-

become large, for example, near stagnation points on

sharp-nose airfoils, near pressure peaks, and within

=hocks, the vertical acceleration is an appreciable part

of the acceleration of gravity, and hence may seriously

affect the results• The effect of appreciable vertical

accelerations is tc give an absolute value of the slope

of the water surface different from the absolute value

required to represent correct compressible flow condi-

tions"•

Surface Tension, Disturbances in the water surface

produce surface tension waves, called capillary waves,

which have small wavelengths and large propagation

speeds•

The capillary waves are not a part of the hydraulic

analogy and are a serious handicap in making depth‘

measurements and in interpreting flow photographs•

Iave Propagation• The velocity of propagation of

surface water waves is given by Shapiro(18) as

Q2 * + tlllhliZTIpA A

Figure 7 “shows graphically the form of the

relation between propagation speed and wavelength and

indicates also the shape of the curves when either

gravity or capillarity alone is act1ng"•

-41..

OZ,

oeE 9 .r- 1;**< Sh

0. N GRAVITY PLUSCAPILLARITY //

a Z

\ //GRAV|TY ALONEL5 \\ A ///

\ /Q ÄuJ \\ CAPILLARITY ALONEw / \\\\

/ \\—~..

/

WAVELENGTH, A

FIGURE 7, PROPAGATION SPEED VS. WAVELENGTH FOR COMBINEDGRAVITY AND CAPILLARY WAVES.

Shapiro, A. H.: Free Surface Water Table, "Physical Measurements inGas Dynamics and Combustion" (H. W. Landenburg, B. Lewis, R. N.Peaee, and S. Taylor, Editors), 2, p. 317. Princeton UniversityPress, Princeton, New Jersey, l95L,

-42-

Figure 8 "shows the nature of the flow pattern

past a thin wedge according to these con=iderations,

Although the analogue requires a sudden change of

direction across a single wave, the actual flow

consists of a gradual turn through a series of waves“,

Methods of Experimentation

Water Channel, Matthews(9) cxplains that a

vertical-return type channel is considered more

satisfactory than a horizontal-return type because the

water surface is undisturbed by turning vanes and space

requirements are less, However, a horizontal—return

type is probably more satisfactory for large installa-

tions•

Entrance Sections, Either a vertical entrance

section or the horizontal entrance section may be used

with either type of channel(9), Since the vertical

entrance section and the test section have the same

width, the flow of water is constricted by a Variation

in the depth of the channel, The entrance floor to the

test section should have a Very gradual final approach,

so that waves in the test section will be eliminated

and a uniform Velocity distribution will be maintained

throughout the water depth,

-43-

// ,/ / //

/ / // // //1.

1 /1 1 V/4 o

1 1111 1\\1 1 1 \1%*\ \ \ \\\ \,

\ \ \ \ \\ \1\\

FIGURE 8. SCHEMATIC WAVE PATTERN FOR SUPERSONIC FLOW PASTWEDGE. A REPRESENTS WAVE OF MINIMUM PROPAGATIONspass. 6 papnasams CAPILLARY °°HEAD°° mpptas, maPROPAGATION SPEED OF WHICH EQUALS THE SPEED

OF FLOW.

Shepiro, A. H.: Free Surface Water Table, "Physical Measurements inGas Dynamics and Combustion" (R. W. Landenburg, B. Lewis, B. N.Pease, end H. 5. Taylor, Editors), 2, p. 318. Princeton UniversityPress, Princeton, New Jersey, l95L,

-44-

The horizontal entrance section, having a floor

in the same plane as the test section, necessitates a

wide settling basin that requires greater space,

material, and expense, Iatthews(°) explains that an

entrance section of this type presents difficulties

as a result of secondary flow at the vertical walls,

Hwever, with either of these entrance sections,

it is difficult to avoid unwanted waves and to obtain

a uniform flow, 8hapiro(18) states, ”This difficulty

is eliminated through the use of a sluice-type

nozzle", (Figure 9), "which, experience has shown,

produces a test section flow with a very smooth

surface, Furthermre the sluice nozzle pßrmits the

Ich number to be varied merely by adjusting the

height of the water upstream of the sluice gate,

whereas a given Laval nozsle is restricted to a single

Hach number, The sluice gate is simple to construct and

may be designed easily for variable test-section depth”,

Test Sections, It is imperative that the pressure

of the liquid at any point depend only on the height of

the free surface at that point, Therefore, the floor

over which the water flows must be in a horizontal

plane and the walls through which the water flows must

be perpendioular to the water, A velocity gradient is

produced along the channel, similar to the gradient in

-45-

/

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QQQ Moos:.Q Y'? wATsR

or

9, scHsMAT:c cnoss-sscr:o~A:. v:sw os c:~:A~~s:.,SHOWING sLu:cs Mozz:.s.

Shapiro, A. H.: Free Surface Water Table, "Physical Measurements inGas Dynamics and Combustion" (R. W. Landenburg, B. Lewis, R. N.Pease, and H. S. Taylor, Editors), 2, p. 3].1. Princeton UniversityPress, Princeton, New Jersey, 1951+.

-45-

a wind tunnel, as a result of the development of the

boundary layer. This effect can be compensated by

designing the floor so that it may be adjusted to

make the effective instead of the physical floor

plane and horizontal,

A convenient floor for the test section is

obtained by using plate glass because it makes a very

smooth surface, permits the transmission of light, and

is suitable for many photographie techniques,

Cireulating Systems, According to Matthewslg),

“The pump used for the circulation of the water should

be designed for a pressure increase of two to five

times the total depth of the water plus the losses in

the antiturbulence screen, The maximum volume of flow

must be equal to the amount required to ehoke the

tunnel, A marine propellor is considered the best

suited for this purpose, as it can pump a large volume

of water at a low pressure and with a minimum increase

in the turbulenee level of the water",

Model Design, Two dimensional models are required

which have their axis perpendicular to the water

surface, Shapiro(18) suggests that the model may be

constructed with static pressure holes below the water

surface to avoid the height·d1storting effects of

-47-

surface tension near the model surface, The models

should be made of a material that is not affected by

water,

Depth—Survey Systems, For quantitative studies a

system is required which permits depth surveys(9) at

all points in the test section, A very satisfactory

sy¤tem may be designed that uses a depth gage with a

needle probe mounted on cross rails so that it can be

moved in any of three rectangular coordinate directions,

”The tolerances(9) in design and construction of the

cross-rail system should permit measuring the depth

of the water to one part in a thousand,“ The indicating

system of the depth gage should have a range from the

floor of the channel to the total depth of the water,

The tip of the probe should be non·corrosive,

Choice of Water Height, Photographs made during

experimentation by Orlin, Linder, and Bitterly(13)

indicated that air flow and water flow most nearly

correspond when water depths are between 0,75 inch and

1,0 inch, However, a shallow water depth is

recommended if capillary waves are to be eliminated

at high stream Ich numbers,

According to Shapiro(18), "To obtain the best

approximation to the analcgue, it is necessary (1) to

minimize the effects of capillary waves, and also

-43-

(2) to make the radius of curvature of the free

surface large compared with the water depth so that

vertical velocities and accelerations will be

neg1igib1e•

"Item (2) may best be managed by making the

model under test large compared with the water depth,

"For item (1), two courses are open, First, a

water height cf about 0.25 inch may be used; ,,, ,

_ for this height the capillary waves behave most

nearly like gravity waves. Second, depths of a foot

or more may be used (requiring prcporticnately larger

channels and models as noted in item (2)), with the

aim of making the gravity waves so large that

capillary effects are secondary•"

-49-

Ill, DESIGN AND CONSTBUCTIGN

The primary element of the free surface water

table is a smooth horizontal surface bounded by

vertical walls geometrically similar to the walls

bounding the corresponding compressible gas flow,

The other necessary elements are a frame to support

the smooth horizontal surface; the pump, piping, and

tank: to supply the water; and the measuring apparatus,

A vertical-return type channel was selected

because, with this type channel, the water surface

is not disturbed by turning vanes and space require-

ments are less, For versatility of operation, the free

surface water table was designed to operate with water

flow analogous to gas flow at all desired Mach numbers

between 0,40 and 7,00, In addition, it was designed to

permit a rapid change of models and stream velocities

for both qualitative and quantitative studies,

The assembly of the water table is shown in

Plate 1, and the water table components are shown in

Plate 2•

Entrance Section

A vertical entrance section was selected because

less space, material, and expense are required than

-51-

Legend

for

Plate 1. Assembly of Water Table

1. Inlet Tank2. Return Tank3. Test Section4. Superstructure5. Channel Frame6. Bottom Support7. Sluice Gate Assembly8. Deptb Survey System9. Stagnation Depth Micrometer

10. Elevating System11. Pivot Shaft12. Suotion Line13. Discharge Line14. Tbrottling Valve15. Orifiee Manometer

-53-

Legend

for

Plate 2, Views of Water Table Components

(A), Side View of Water Table1, Sluice Gate Assembly2, Depth Survey System3, Stagnation Depth Micrometer4, Model

(B), Sluice Gate Assembly1, Sluice Gate2, Gate Guide3, Gate Frame4, Slider Bearing5, Bearing Shaft6, Nut7, Adjusting Screw8, Mierometer9, Micrometer Bracket

(C), Depth Survey System1, Side Rail2, Side Track3, Side Slider Bearing4, Side Ubeel Elder5, Cross Rail6, Cross Track7, Cros= Slider Bearing8, Cross Wheel Holder9, Micrometer Bracket

10, Micrometer11, Depth Probe

(D), Stagnation Depth Micrometer1, Micrometer Nut2, Micrometer Screw3, Vertical Scale4, Circular Scale

-54-

with the horizontal entrance section, To avoid

unwanted waves, to obtain uniform flow, and to permit

variable test section water depth, a sluice-type gate

was selected, Tb provide for operation of the water

table at all analogous Nach numbers between 0,40 and‘

7,00, the sluice gate and its component parts were

designed to permit water depths rangin from 0-1 inch

above the glass floor,

The gate, made of lucite, was designed and

machined ae shown in Drawing Number 1, A frame of

1 x 1 x 1/8-inch angle was fabricated and the gate was

attached with number 10-24-H0-2 flat head sorews to

this frame, The flat head screws were countersunk

in the gate to provide a smooth surface, To obtain

a water seal at the sides of the gate, lucite guides

were designed and machined as shown in Drawing Number 2,

These guides were attached with nuber 10-24-NC-2 flat

head ¤crews to 1 x 1 x 1/8-inch angles that were

fastened, so as to be adjustable, to the sides of the

inlet tank with number 10-24-NC-2 round head :crews,

Two slider bearings, Drawing Number 3, and two

bearing shafts, Drawing Number 3, were used to ensure

correct alignment and smooth operation of the sluice

gate for all settings, The bearing shafts were

polished for a sliding fit with the slider bearings,

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By fastening the bearing shaft= at both ends with

1/2-13-NC-2 nuts, the slider bearings and bearing

shafts were assembled to the gate frame, The slider

bearings were attached with nuber 8-32-NC-2 screws

to a l·1/2 x 1-1/2 x 1/4-inch angle of the inlet tank,

A cast iron nut and steel adjusting screw were

designed and machined as shown in Drawing Number 3,

The nut was attached with four number 8-32-NC-2 screws

to the gate frame, A bracket, for a micrometer gear

unit, was constructed and attached with two

1/4-20-NC-2 bolts to the top of the inlet tank, and

the micrometer gear unit was fastened with four

number 10-24-NC-2 screws to this bracket, After the

adjusting screw had been assembled with the nut, the

micrometer gear unit shaft and adjusting screw were

joined with a cotter pin, The micrometer gear unit had

two circular aluminum scales, The 0 - 2 inch circular

scale was graduated in 0,025 inches, and the smaller

circular scale was graduated in twenty-five parts, By

observing both circular scales, sluice gate adjustments

to the nearest 0,001 inches could be made,

The sluice gate assembly is shown in Plate 2(B),

and the sluice gate location on the water table is

shown in Plate l and Plate 2(A),

-59-

Test Section

For the test section, a glass floor was selected

for two reasons• First, glass has a very smooth surface

and is sufficiently rigid to maintain a horizontal

plane, Second, glass permits the transmission of

light and is suitable for many photographie techniques•

The size selected for the glas: floor was 1/2 x 20 x 36-

inches, A thiekness of l/2—ineh was selected to provide

adequate strength to support a mdel plus a maximum

water depth of one inch, A width of 20-inches was

selected for convenience of operation of the depth

survey system, and a length of 36-inehes was selected

to insure uniform water flow at the model location in

the test section,

The glass was seated in elastic glazing eompound

on top of the channel frame, Drawing Number 4, which

was machined on its top surface, The purpose of the

elastic glazing compound was to provide a smooth

uniform seat for the glass over the entire bearing

surface, Although the channel frame was of more than

adequate strength to support the test section, diffi-

eulties would have been encountered in the machining

of an angle frame, In addition, the 4-inch x 5,4 pounds

per foot channel was selected for the channel frame to

provide a wider bearing surface for the glass than

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would have been provided with a channel of smaller

size, The channel frame was attached with four

l/2·l3·NC·2 holts to the top legs of the superstructure

shown in Drawing Number 5, The vertical walls of the

channel were made of two 1/4 x 5-l/2 x 38-inch pieces

of plexiglass that were fastened to the channel frame

with number l0·24—NC-2 screws spaced approximately

three inches between centers,

The minute space between the plexiglass walls

and the sides of the glass floor and between the tank

edges and the ends of the glass floor was filled with

elastic glazing compound to ensure watertightnesr,

IEEE.

At each end of the channel frame, a tank was welded

to the superstructure, The inlet tank collects the water

from the pump before it flows through the test section

and the return tank collects the water from the test

section before the water returns to the pump, Both

tanks were fahricated of l6·gauge sheet steel braced at

all corners with l x 1 x 1/8-inch angle,

The dimensions of the inlet tank are 18—inches

long x 24—i¤ches wide x 29-inches deep with a capacity

of approximately 7,0 cubic feet, ln addition to being

welded to the superstructure, the inlet tank was

-63-

fastened with 1/4-20-NC-2 bolts and nuts to the end

of the channel frame. To obtain a uniform connection,

a 1/4 x 2 x 22-inch steel plate wa: placed between

the 16-gauge steel of the tank and the 1/4·inch bolt

heads•

To fulfill the requirements of the analogy, it is

necessary tc maintain stagnation water in the inlet

tank before the water enters the test section, For

this purpose, an antiturbulence device was placed,

inside the inlet tank and parallel to the bottom of

the inlet tank, approximately sight inches below the

test section entrance• The antiturbuleuce device

consists of several layers of cellulose acetate between

two wire screens, This assembly was held securely in

place between two 1 x l x 1/8-1nch angle frames that

were fastened to one another with twelve 1/4-20-NC-2

bolts, The heads of the 1/4-inch bolts were tack-

welded to the bottom angle frame which was tack•welded

to the inside of the inlet tank, The top angle frame

was made detachable so that the assembly might be

easily removed if necessary.

A 1-1/2-inch pipe, four inches long, was welded

to the outside of the inlet tank, near the bottom of

the side adjacent to the channel frame, to accemmodate

the diseharge pipe1ine„ A 1/4-1nch gate valve was

-64-

attached to the bottom of the inlet tank to facilitate

the emptying of the tank,

The dimensions of the return tank are 24-inches

long x 24—inches wide x 24•inches deep with a capacity,

up to the glass floor, of approximately 6,8 cubic feet,

In addition to being welded to the superstructure, the

return tank was fastened with 1/4-20-NC—2 bolts and nnte

to the end of the channel frame, To obtain a uniform

connection, a 1/4 x 2 x 22—inch steel plate was placed

between the 16-gauge =teel of the return tank and the

1/4·inch bolt heads, A 1-1/2-inch pipe, four inches long,

was welded to the outside of the bottom of the return

tank, Attached to this 1-1/2-inch pipe was a l•1/2-inch

standard galvanized pipe tee to which was Joined the

suction pipe line and a 1•1/2-inch globe valve, The

purpose of the globe valve was to facilitate the removal

of air from the suction line and to facilitate the

emptying of the return tank,

The relative location of the inlet and return tanks

is shown in Plate 1 and Plate 2(A),

Circulating System

For this water table, the water is se1f—contained

and recirculating, The performance curves for an

existing fifteen horsepower centrifugal pump indicated

-65-

that its capacity was sufficient to supply the quantity

of water required, The piping and connections between

the centrifugal pump and the inlet and return tanks

are shown schematically in Figure 10,

Both the discharge and the suction pipelines were

constructed of two-inch galvanized pipe except for the

l-1/2-inch galvanized pipe of the discharge line which

enters the inlet tank and the l-l/2-inch galvanized

pipe of the suction line which leaves the return tank,

On the suction line next to the pump a two-inch gate

valve was installed to allow unobstructed flow when

the valve is opened, Adjacent to the inlet tank on the

discharge line, a l-l/2-inch globe valve with a plug-

type disc was installed to permit regulation of the

flow, A l/8-inch bleed-eff line with a 1/8-inch globe

valve was installed at the highest point of the suction

line to facilitate the removal of air from the suction

line, To absorb any vibration that might occur in the

pipelines, a 1-1/2-inch inside diameter flexible

rubber hose, fifteen inches long, was installed with

hose clamps in both the suction and discharge lines at

a location adjacent to the water table,

To measure water flow, a 1,188-inch diameter flat

plate orifice was installed in the 1-1/2·inch discharge

line leading to the inlet tank, The orifice and pressure

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taps were located and installed in accordance with

the ABME research publication on fluid meters(5) and

were connected to a fifteen-inch U-tube mercury

manometer, The orifice was calibrated by weighing

the amount of water flowing during a measured interval

of time, The orifice calibration curve is shown in

Figure ll,

Elevating System and Bottom Support

In order that compensation might be made for the

frictional effects produced by the water flow, an

elevating system was designed, One end of the super-

structure was supported on a pivot shaft and the

other end was supported on wedges,

The pivot shaft, shown in Drawing Number 6, was

welded across the bottom of the superstructure under

the center of the return tank, Each end of the pivot

shaft was supported in a pillow block bearing that was

fastened to the bottom support, Drawing Number 7, with

1/2-13-NC-2 bolts,

At the other end of the bottom support, the

bottom of the bottom wedges, Drawing Number 8, rested

on the 2 x 2 x 22-inch steel bar of the bottom support,

The bottom wedges were connected with the wedge screw

shown in Drawing Number 9, The top wedges, designed

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and machined as shown in Drawing Nuber 10, rested

on top of the bottom wedges, The adjustable bearings,

shown in Drawing Number ll, were screwed into 1-8-NC—2

nuts that were welded to the bottom of the superstructure

so that the adjustable hearings were in position to

rest in the V-grooves on top of the top wedges, The

bottom wedges were adjusted to be equi-distant from

the longitudinal center line of the water table, The

wedge screw was locked in position with l/2-13-NC-2

lock nuts on each end of the wedge screw at the out-

side of the channel legs of the bottom support, A six-

inch diameter handwheel was located at uh! end of the

wedge screw to facilitate operation of the elevating

system,

This system of wedges permit= elevating the water

channel while maintaining the test section level in

the direction transverse to the water flow, This

design also ensures a very fine adjustment with smooth

and uniform operation,

A l-l/2-inch pipe, four inches long, was welded

to the inside bottom of each channel leg of the bottom

support, These pipes serve as holders for the

1-8-NC-2 x six-inch long bolts with nuts that are

used for adjusting the water table to a level

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Degth Survey System

For quantitative :tudies a depth survey system

was designed to permit depth surveys at all point:

in the test section, A depth mierometer, graduated

in thousandths of an inch from 0 — 1 inches, with a

1/8—inch needle point brass probe was mounted on a

set of cross rail: to permit movement in any of

three rectangular coordinate directions,

The cross wheel holder and the side wheel holder

were designed and machined as shown in Drawing

Number 12, The wheels and bearing shafts were

designed and machined as shown in Drawing Number 13,

After the wheels were pressed on Fafnir Ball Bearings,

Number F-4, the wheels, bearings, and bearing shafts

were assembled with the wheel holders so that the

wheels turned freely in the slots of the wheel holders•

The bearing shafts were peened on the ends to fasten

them in the wheel holders•

The side slider bearing and the cross slider

bearing were designed and machined as shown in Drawing

Number 12, The side rails, cross raile, side tracks,

and cross tracks were designed and machined as shown

in Drawing Number 13, The rails were polished for a

sliding fit with the slider bearings,

-79-

The entire depth survey system was mounted on

four 1-1/2 x 1-1/2 x 1/4-inch angles with l/4-20-NC-2

bolts and nuts• These angles were fastened with

1/4-20-NC-2 screws in tapped holes of the super-

structure, A solid foundation for the depth survey

system was obtained by supporting it in this mnner,

The parts for the depth survey system were

assembled as shown in Plate 2(C), and the location of

the depth survey system on the water table is shown

in Plate 1 and Plate 2(A),

A stagnation depth micrometer was designed and

constructed to measure the stagnation depth of the

water, in the inlet tank, above the floor of the

test section at entrance,

A twelve-inch long brass screw with a needle

point and a l x 1 x 3-inch bras= nut, both with

3/8-20-N-3 threads, were made, The nut contained

a 0 - 8 inch vertical scale graduated in tenths of an

inch, On top of the brass screw was attached a

circular scale that was divided into one hundred parts,

By reading both the vertical scale and the circular

scale, measurements to the nearest 0,0005 inches

could be made, The stagnation depth micrometer

assembly was attached with a 3/8-16-NC-2 bolt to a

1-1/2 x 1-1/2 x 1/8-inch clip angle that could be

-39-

adjusted in a bracket that was welded to the top of

the inlet tank,

The assembly view of the stagnation depth

micrometer is shown in Plate 2(D), and the lecation

of it on the water table is shown in Plate 1 and

Plate 2(A)•

@

The requirements for the model are that it be

two-dimensional with its axis perpendicular to the

water surface•

The model used for the performance tests was a

two-dimensional fi£teen—degree wedge, It was made of

seven pieces of 1/4·inch thick plexiglass eemented

together with CD Cement and machined to the dimensions

as =hown in Drawing Number 14, The top of the model

was painted with black enamel for photographie

purposes, Plexiglass was used because it is ea=y to

machine and is practically unaffected by water,

A view of the model located in the test section

of the water table is shown in Plate 2(A)•

*81-

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Apparatus and Materials

The apparatus and materials selected for this

project are listed below, A detailed list of the

materials selected for the construction of the free

surface water table is included in the appendix,

Pump, Centrifugal, 15HP, No, 725354, size 2 Cl OS.

Manufactured by Worthington Pup and Machinery Corp.,

Iorthington Iorks, Hrrison, N, J, Used to circulate

water in the system,

ßgjgg, Inductien, Model No, 40020, Type KT 954 4 15

1800, Form C, Three Phase, 60 cy., 220 v., 36 amp.,

1755 rpm @ full load, Ho, 4241449, 15HP, Continuous

40 °C, Manufaotured by General Electric 00,,

Schenectady, N. Y, Used to drive the eentrifugal pump,

Compensator, Starting, CR 1034-KIB, Cat,

No, 2019014-63, 5-15 HP, 220 v., 60 ey., Three Phase,

Manufactured by General Electric Co., Schenectady,

H, Y, Used for starting the induction motor,

gage, Discharge Pressure, 30 psi Vacuum to 60 psi

gage, Manufactured by Crosby Steam Gage and Valve Co.,

Boston, Mass, Used to measure discharge pressure of

centrifugal pump,

Qggg, Suction Pressure, 30 psi Vacuum to 15 psi

gage, Manufactured by Crosby Steam Gage and Valve Co,,

-33-

Boston, Mass. Used to meaeure suction pressure of

eentrifugal pump.

Glazing Compound, Worth Knife Grade, Natural,

three lbs, Manufaetured by the W, Bingham C.,

Bingham, Ohio, Used to seat the glass channel floor

and to insure watertightness,

Cement, CD No, 125, one pint, Manufactured by

Chemical Development Corp., Danvers, Mass. Used as

an adhesive to make the plexiglass fifteen—degree

wedge model,

Areskap 50, Sample No, 9-N·246, one lb, Manu-

factured by Monsanto Chemical Co,, Rubber Service

Laboratories Division, Akron, Ohio, Used as a wetting

agent to reduce surface tension in the water,

Water Table, Free Surface, Refer to the appendix

for a list of materials for construction, Used to

investigate two—dimensional compressible flow phenomena

by application of the hydraulic analogy,

-34-

IV. OPEBATIONAL PROCEDURE

With the free surface water table, water flow at

an analogous Mach number may be obtained by varying

both the stagnation depth of water in the inlet tank

and the height of the water surface above the floor

of the test section, The stagnation depth of water in

the inlet tank may be controlled by adjustment of the

throttling valve, The height of the water surface

above the floor of the test section may be controlled

by adjustment of the sluice gate,

The wave patterns of the water flow, around a

given model, may be conveniently photographed by

locating the camera directly above the test section,

For this location of the camera, a light source may

be placed under and to one side of the test section•

To diffuse the light, tracing paper may be used, The

tracing paper should be placed adjacent to the glass

floor of the test section and between the glass floor

of the test section and the light source,

The following procedure is recommended for

operation of the free surface water table.

Before starting the pump, all valves in the

system, Figure 10, are closed except the suction gate

-85-

valve from the return tank, the tap water globe valve,

and the bearing sea1_valves. The bearing seal valves

are opened to prevent leakage of air at the pump

bearings, The tap water globe valve is opened to

prime the pump and, in conjunction with the suction

gate valve, to fill the return tank to within three

inches below the channel. To force all the air from

the suction line, the bleed-off globe valve is first

opened until a small amount of water is forced from

the bleed-off line. Second, the pipe opening in the

return tank is partially plugged and the return tank

drain valve is opened slightly until the suction line

is devoid of air. After the return tank is filled as

specified, the switch is thrown to start the pump•

When the pump has maintained an operating discharge

pressure of 50 pounds per square inch, gage, as

indicated by the discharge pressure gage, the tap

water globe valve is closed and the discharge globe

valve is gradually opened. The inlet throttling valve

is gradually opened until the desired stagnation head

is maintained in the inlet tank. After the b1eed·off

cocks are opened to remove the air from the manometer

lines, they are closed and the manometer globe valves

are opened to permit the registering of the orifice

pressure differential on the U—tube manometer• A

U-86-

wetting agent is then added to the water in the return

tank to reduce the surface tension of the water,

thereby reducing the effects of capi11arity•

When the stagnation head is maintained constant

by adjustment of the inlet throttling valve, the

following readings should be recorded:

W 1, Pressure differeutial across orifice

2, Stagnation depth of water

3, Depths of water in test section

' For shutting·dow¤ the system, the discharge

globe valve is closed before the switch for the pump

is moved to the off position, Next, in the order

mentioned, the suction gate valve, the inlet

throttling valve, and the manemeter globe valves

are closed,

-37-

V, DISCUSSION

To evaluate the design, it will now be analyzed

on the basis of the performance of the water table

and the water table components,

Discussion of Design

Although the free surface water table was designed

to operate with water flow analogous to gas flow at all

desired Mach number: between 0,40 and 7,00, its per-

formance was not satisfactory over this entire range

of Mach numbers,

The entrance section produced water waves which

originated at the corners and grooves of the sluice

gate guides and at the end of the glass floor at

entrance, The water waves that originated at the

corners and grooves of the sluice gate guides formed

diagonally across the glass floor of the test section

and were especially noticeable,

In an attempt to eliminate these unwanted waves,

the sluice gate was adjusted to a gate opening of

0,250 inches; and the grooves in the sluice gate guides,

below the sluice gate, were filled with elastic glazing

compound which was smoothed to make the sides as nearly

·88·

uniform and vertical as possible, By the further

application of elastic glazing compound, all =harp

corners were eliminated to provide a rounded entrance,

Also, a wetting agent was added to the water in the

return tank to reduce the surface tension of the

water, The application of the elastic glazing compound

and the addition of the wetting agent eliminated the

unwanted waves at analogous Mach numbers between 0,40

and 4,00, Beyond the analogous Mach number of 4,00,

performance was unsatisfactory because the water waves

which originated at the entrance section became very

pronounced,

The glass floor of the test section was thoroughly

checked with a level, Although it was level in the

transverse direction, a slight curvature existed in

the longitudinal direction, The curvature seemed to

be located at approximately one foot from the return

tank, and the glass surface sloped very gradually dawn-

ward from that location to the return tank, At

numerous points on the glass surface measurements were

made with the depth survey system, and it was determined

that the approximate total variation of the glass surface

was 0,005 inches, Since the curvature was incorporated

in the glass when it was made, the Variation in the

glass surface could not be corrected,

.3;-

The fifteen-horsepower centrifugal pump

adequately supplied the quantity of water required

for operation of the water table, Due to the length

of the overhead suction line, some difficulty was

experienced in removing all of the air from the

circulating system when the pump was started initially,

By means of the throttling valve, a plug·type disc

globe valve, fine regulation of the water flow was

obtained, For example, at an analogous Mach number

equal to 1,00, the hydraulic jump of the water could

be moved back and forth in the channel or maintained

at a desired location by adjustment of the throttling

valve, The flexible hose, in both the suction and

discharge lines, absorbed all apparent vibration that

might have been transmitted by the connecting pipelines

to the water channel,

Very satisfactory performance was obtained with

the elevating system; however, no attempt was made to

determine the optimu tilt of the water channel to

compensate for the frictional effects produced by the

water flow, Furthermore, no scale was designed to

indicate the amount of tilt of the water channel,

The performance of the depth survey system and

the stagnation depth micrometer was very satisfactory,

On account of the formation of capillary waves, the

-99-

depth probe reading at the water surface was valid

only at the instant the probe touched the water, The

use of the depth survey probe is invalid for measure-

ments of depth at the surface of models and at fluid

boundaries because of the capillary rise. Usually the

effect of capillarity is negligible at all locations

beyond 1/4-inch from an exposed surface; therefore,

at these locations, the depth probe measurements are

of much value, The graduations of the 0 - l inch depth

micrometer were numbered in the reverse order of that

for the most convenient reading, thereby making it

necessary to subtract each reading from 1,000,

The bottom support provided a solid foundation for

the other parts of the water table, The superstructure

provided a rigid frame for the support of the channel

frame, the depth survey system, and the inlet and

return tank:. For the requirements of the water channel,

the capacities of the inlet and return tanks were

adequate. If the return tank was filled to within three

inches below the channel, the depth of the water in

the return tank during operation was approximately ten

inches, This design feature prevented any leakage of

air into the suction line from the return tank,

The following section of the discussion includes

' the results of representative tests and the method

-91,

of caleulation employed to obtain these representative

results,

Regresentative Results

To obtain representative results of its performance,

the free surface water table was operated at water flows

analogous to both subsonic flow and supersonic flow of

a two-dimensional compressible gas,

Three tests were made using the fifteen·degree

wedge model, shown in Drawing Number 14, with the test

section maintained horizontal, For each test, data was

recorded as specified on page 86 of the operational

procedure; and photographs, shown in Plate 3, were

made of the water flow about the test model, In addi-

tion, the photograph shown in Plate 3(B) was made to

il1u:trate a hydraulic jump in the water flow (preceding

the model) at an analogous Heh number equal to 1,00,

The data and results of the tests are tabulated in

Table II, The results for Test 1 are in close agreement

with the values calculated for the aualogous gas flow,

The greatest variation oceurs in the corresponding

values for the pressure ratio, A photograph of the

water flow at an analogous Hach number equal to 0,99

is shown in Plate 3(A), For Test 2 and Test 3, the

results are in reasonable agreement with the values

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calculated for the analogous gas flow, In each case,

the greatest variation occurs between the corresponding

values for the pressure ratio, and between the

corresponding values for the Mach number after the

shock wave, In Plate 3(C) is shown a photograph of

the water flow, for Test 2, at an analogous Hach number

equal to 2,29, In Plate 3(D) is shown a photograph of

the water flow, for Test 3, at an analogous Mach

number equal to 3,33, In both Plate 3(C) and

Plate 3(D), it is seen that the actual flow consists

of a gradual turn through a series of waves to form

the shock wave,

The results of the representative tests and a

study of the photographs of Plate 3, indicated that

the water table functioned properly at analogous

Mach numbers between 0,40 and 4,00,

Method of Calculation, The method of calculation

employed to obtain the calculated values, for the

water flow, tabulated in Table II was the application

of the hydraulic analogy as explained in the theory

development of the literature review,

The method of calculation employed to obtain

the calculated values, for the analogous gas flow

tabulated in Table II for Test Number 1, was the

application of equations from the work of Keenan and

-95-

Kaye(10). To explain this method of calculation, the

sample calculations for Test Number 1 are given be1ow•

(1) Determination of Mach Number. The

values for M1 and M2 were assumed to be

identical to those calculated, from the data

of the water flow, by equation (52), page 30,

(2) Determination of Density Ratio,

k-1er - 1 1 T11?/01

1where:}91

-density of gas before shock,

slugs/ftg

/02-

density of gas after shock,slugs/ft3

k • specific heat ratio, (- 2.0)

ul ¤ Mach number before shock

M2 • Mach number after shock

2-1 2P2 1 +2/01

1ig

¤/°1

-96-

(3) Determination of Temperature Ratio•

K-]. 2

E2 ·1 + --2 M1

T1 1 + Erin3 2

where:

T1-

absolute temperature of gas beforeshock, °R

T2 • absolute temperature of gas aftershock, °R

k-

specific heat ratio, (• 2,0)

I M1 - Mach number before shockM2 - Mach number after shock

2-1 212 11TTTTT

1 1 T I-TIT]

*1*J- -· 0,66T1

(4) Determination of Pressure Ratio.

k/1-kK-1 2

pl 1-kk—1 2

E "2where:

pl • pressure of gas before shock, 1b/ftz

D2 = pressure of gas after shock, lb/ftz

k• specific heat ratio,

(•2.0)

M1 -Hach number before shock

M2 • Mach number after shock

21 + ?:.L.<1.-129p2 3——·— ° ·—·—·······——····——*‘27—*‘pl 2 1-22-1

Q+p

T2 «- 0771

The method of calculation employed to obtain

the calculated values, for the analogous gas flow

tabulated in Table ll for Test Number 2 and Test

Number 3, was the application of equations from the

work of Keenan and Kaye(1°•11), To explain this

method of calculatlon, the sample calculntions for

Test Number 2 are given helew•

(5) Determination of Shock Angle, By

using a protractor the shock augles were measured

from the photographs, Plate 3, of the water flow

about the fifteen-degree wedge mode1•

(6) Determinatlon of Density Ratio,

/02 _X

-tlll ß

P1 tan(F- w S

where:

}°1 • density of gas before shock,slugs/fta

•density of gas after shock/02

slugs/fts '

Ä9 • shock angle, degrees

dl-

half-angle of wedge, degrees

P2 ,, um 32°/01 tan(32¤ - 7.ä¤S

fig • 1,37P1

(7) Determination of Pressure Ratio,

P2 , 6X - 1P1 6 • X

where:

P• pressure of gas before shock,1 lb/ftz

P • pressure of gas after shock,2 lb/ftz

X • tan ßta¤(,8-w)

ß•• shock angle, degrees

w-

half-·ax1g1e of wedge, degrees

P2 _(6) (1,37) - 1

pl 6 — 1.37

P2

-100-

(8) Determinetiou of Mech Number Before Shock,

1/2

where:

M1 • Mach number before shock

X ,, ten ÄtI„11(ß

•• LU )

Ä •• bock eagle, degrees

w-

ha1f··ang1e of wedge, degrees

1/23 ,, cosec 32°1 6 - 1,37

M1 •- 2.30

(9) Determination of Mech Number After Shock,

1/2M2 cosec (,8-60)

where:

M2 • Mach number after shock

x

_tan ,8

taz1(ß -60)

Ä • shock eagle, degrees

cu • half-eagle of wedge, degrees

1/2°

scB e 7.II2 cosec (32 )

M2 • 2,00

-101-

(10) Determiuatiou of Temperature Ratio.

IT-]. 2T2 _ 1 + ::2: M1T" 1 1 1;-11 ——-—-2 **2

where:

T • absolute temperature of gas1 before shock, °R

T2-

absolute temperature of gasafter shock, °R

k • specific heat ratio, (• 2.0)

H1 • Mach number before shock

M2 • Mach nuber after shock

1 2 1 Q aäl 2*[—£—;] •T2 .„..._.„...........,.,.,._.__._ä_*1 1 1 [1;%-¤·ä1

·1·...2. -· 1.26T1

-102-

VI, SUMMARY

The purpose of this thesis project was to design

and construct a free surface water table for the

investigation of compressible-flow phenomena,

Although the design range of operation of the

water table was to include all desired analogous Mach

numbers between 0,40 and 7,00, the range of operation

was limited by the formation of unwanted waves at the

entrance section, Iith the application of elastic

glazing compound to provide a rounded entrance and with

the addition of a wetting agent to reduce the surface

tension of the water, the water table performed satis-

factorily at analogous lach numbers between 0,40 and

4,00, The test section depth could not be easily

varied because it was necessary to fill the grooves of

the gate guides, with elastic glazing compound, for each

gate opening, Except for the entrance section, very

satisfactory performance was attained with all the

components of the water table,

Satisfactory performance of the water table was

verified by the flow photographs and the results of the

representative tests,

•l03-

Vl1• BECOMMENDATIGHS

The following recomendations are made to

facilitate the use of the free surface water table

for the investigation of compressible-flow phenomena:

1, To avoid unwanted waves in the test section

and to ensure a uniform velocity distribution through-

out the water depth, the entrance to the test section

should have a very gradual final approach, For this

purpose, it is recommended that the grooves of the gate

guides and the vees of the sluice gate he eliminated,

Also, the vertical walls of the test section should be

extended into the inlet tank and should be attached to

a gradually rounded approach to the glass floor, thereby

eliminating all sharp corners and providing the equiva—

lent to a bell—mouth entrance,

2, For determining the amount of tilt of the water

channel, it is suggested that a protracter and indieator

be designed and conetructed for attachment on the channel

leg of the bottom support at the wedge assembly, The

protractor should be in the shape of an arc with a

radius equal to the distance from the center line of

the pivot shaft to the center line of the adjustable

bearings,

-104-

3, For greater accuracy and ccnvenience in making

field measurements for quantitative studies, it is

suggested that an electronic relay he incorporated

with the depth survey system, The electronic relay

would indicate the contact of the prebe with the

water surface•

4, Due to the capillary rise of water at the

surface of a model and at a wall, the depth probe

method of measuement is unsuitable at these locations,

Since pressure orifices are very suitable for this

type of measurement, it is suggested that small

pressure erifices be installed for determining water

depth measurements at the surface of a model and at

the wa1ls•

-105-”

VIII, BIBLIOGRAPHY

1, Bailey, N, P,: Abrupt Energy Transformaticns inFlowing Gases, ASHE Paper No, 46-A-71 (1946),

2, Bruman, J, R,: Application of the Water ChannelCompressible Gas Analogy, North Amer, AviationRept, NA-47-87 (1947),

3, Donaldson, C, DuP, and R, D, Sullivan: The Effect ofWall Friction on the Strength of Shock Waves inTubes and Hydraulic Jumps in Channels, NACATN 1942 (1949),

4, Einstein, H, A, and E, H, Baird: Progress Report ofthe Analogy between Surface Shock Waves onLdquids and Shocks in Compressible Gases,Calif, Inst, Technol, Hydrodynamics Lab,Rept, N-54 (1947),

5, Fluid Meters, Their Theory and Application, Part 1,ASM Research Publication, 1931,

6, Ippen, A, T, and D, R, F, Harlemanz QuantitativeStudies of Supersonic Flow Problems by HydraulicAnalogy, ”Hydrodynam1cs in Modern Technology"PP• 155-157, Mass, Inst, of Technol, Hydro-dynamics Lab,, Cambridge, Mass,, 1951,

7, Jacobs, D, H,: An Investigation of Water ChannelPhotography, North Amer, Aviation Rept, AL-129,(1947),

8, Johnson, R, H,, W, R, Nial, and N, C, Witbeck: WaterAnalogy to Twc-Dimensional Air Flow, Gen, Elec,nept. 55218 (1947).

9, Jouguet, E,: Quelques Problems d'HydrodynamiqueGenerale, Journal de Nathematiques Pures etAppliques, (8), Q, PD. 1-63 (1920),

10, Keenan, J, H,, and J, Kaye: "Gas Tables", p, 209,John Wiley and Sons, Inc,, New York, N, Y,, 1948,

11, ibid, p, 213,

-106-

12, Matthews, C. W.: The Design, Operation, and U=e: ofthe Water Channel as An Instrument for theInvestigation of Compressible-Flow Phenomena,NACA TN 2008 (1950). „

13, Matthews, C, W. and R, H, Wright: Investigations ofFlow Conditions and the Nature of the Wall-Constriction Effect Near and At Choking by Meansof the Hydraulic Analogy, NACA RM No, LSFI7(1646).

14, Orlin, W, J., N. J, Linder, and J, C, Bitterly:Application of the Analogy Between later Flowwith a Free Surface and Two-DimensionalCompressible Gas Flow, NACA Rep, 875 (1947),

15, Preiswerk, E.: Application of the Methods of GasDynamics to Water Flows with Free Surface,Part 1, Flows with No Energy Dissipation,NACA TM 934 (1940). Part II, Flows withMomentum Discontinuities (Hydraulic Jumps),NACA TM 935 (1940).

16, Riabouchinsky, D.: Mecanique des Fluides, ComptesRendes t, 195, No, 22, pp, 998-999 (1932),

17, z Mecanique des Fluides, ComptesHenäes, t, 199, No, 14, pp, 632-634 (1934),

18, Shapiro, A, H,: An Appraisal of the HydraulicAnalogue to Gas Dynamics, Mass, Inst, Technol,Meteor Rept. 34 (1949).

19, z Free Surface Water Table,"Päysical Eeasurements in Gas Dynamics andCombustion” (R, W, Landenburg, B, Lewis,R, N, Pease, and H, S, Taylor, Editors), 9,PP. 309-321, Princeton University Press,Princeton, New Jersey, 1954,

-107-

Addenda

20a, Harleman, D, B, F, and A, T, Ippen: The Range ofApplication of the Hydraulic Analogy inTransonic and Supersonic Aerodynamics,"Jubile Scientifique de H, DimitriRiabouchinsky, Memoires Sur La MecaniqueDes Fluides, offerts par Ses Amis, SesCollegues Et Ses Anciens Eleves, Le 8 Mai1954", pp, 91-112, Paris, Service deDocumentation et d'In£ormation Technique de1'Aeronautique, 1954,

2la, Laitone, E, V,: Developments in Gas Dynamics bythe Hydraulic Analogy, "Jubile Scientifiquede M, Dimitri Riabouchinsky, Hemoires Sur LaMecanique Des Pluides, offerts par Ses Amis,Ses Collegues Et Ses Anciens Eleves, Le 8 Mai1954”, pp, 203-217, Paris, Service deDocumentation et d'Information Technique de1'Aeronautique, 1954,

-108-

IX, ACKNOLEDGEMENTS

The author wishes to express his appreciation to

Professor J, B, Jones, Head of the Mechanical Engineering

Department, who approved department aid to supply the

necessary equipment and materials,

To the members of his thesis committee,

Associate Profe¤sor S, L, Wood, Major Professor,

P Associate Professor C, I, Long, and Professor J, P,

Mahaney, all of the Mechanical Engineering Department,

the author wishes to express bis sincere appreciation

for the suggestions, cooperation, and constructive

criticism rendered, Special recognition is due

Associate Professor H, L, Ibod for his animation and

invaluable aid,

The author wishes to express his gratitude to all

those who contributed pertinent information, and to

Professor A, E, Bock, of the United States Naval Academy,

for his courtesy, suggestions, and cooperation, Thanks

are extended to Assistant Professor H, N, Jones and

Assistant Professor B, E, Hedgepeth, both of the

Mechanical Engineering Department, for their coopera-

tion,

Grateful acknowledgement is due Mr, R, D, Tate, Jr,,

Laboratory Technician, for supplying the necessary tools

-109-A

and apparatus ter the project, For assistance during

the construction ot the project, the author wishes to

express his appreciation to Messrs, G, E, Bodell and

E, M, Henderson, both ot the Industrial Engineering

Department Machine Shop, and to Messrs, F, H, Grisson,

· H, M, Smith, and F, B, Fisher, all ot the Mechanical

Engineering Department Machine Shep,

-112-

XI• APPENDIX

Materials for Construction

-113-

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