approved - virginia tech · master of science in mechanical engineering approved: ... ( p1 - pe) c...

A STUDY OF TIP-LEAKAGE FLOW THROUGH ORIFICE INVESTIGATIONS

by

Gregory S. Henry

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State Cniversity

in partial fulfillment of the requirements for the degree of

Master of Science

in

Mechanical Engineering

APPROVED:

Dr. John Moore, Chairman

Dr. Hal L. .Moses Dr. Thomas E. Diller

OCTOBER, 1987

Blacksburg, Virginia

-I

A STUDY OF TIP-LEAKAGE FLOW THROUGH ORIFICE INVESTIGATIONS

by

Gregory S. Henry

Dr. John Moore, Chairman

Mechanical Engineering

(ABSTRACT)

Compressible fluid dynamics of flow through plain-faced long orifices was investigated

with the hope of gaining insight into the fluid dynamics of tip leakage flow. The

Reynolds number range investigated was greater than 104. \'leasurements were made of

the discharge coefficient as a function of back pressure ratio for a sharp-edged orifice

and long orifices with an 1/d from 1/2 to 8. The discharge coefficient measurements in-

dicate the mass flow rate in an orifice with an l,'d of approximately 2 is the largest and

the flow rate in a sharp-edged orifice is the smallest for pressure ratios greater than 0.27.

The mass flow rate in a sharp-edged orifice is largest for pressure ratios below 0.27.

To visualize the flow in a long orifice and model centerline pressure variation, a water

table study was performed. The results demonstrate that the flow separates from the

sharp corner at the orifice entrance, it accelerates to a maximum Mach number, and

then the pressure increases. For back pressures above 0.50, a pressure decrease follows

the initial pressure increase. If the maximum Mach number is supersonic, oblique

shocks will exist. At the higher back pressures that produce supersonic maximum Mach

numbers (0.50 < P8/P0 < 0.70), the oblique shocks ref1ect from the centerline as "Mach

reflectionsH and the flow is subsonic after the pressure increase. The maximum Mach

number for a back pressure ratio of 0.50 is approximately 1.5. At lower back pressure

ratios (P8/P0 < 0.50), the oblique shocks reflect from the centerline in a "regular" manner

and the flow remains supersonic on the centerline once supersonic speeds arc reached.

The flow in a long orifice is relatively constant for all back pressure ratios below ap-

proximately 0.30. The maximum Mach number for pressure ratios below 0.30 is ap-

proximately 1.8.

One-dimensional analyses were used to model the flow in long orifices with maximum

Mach numbers less than 1.3. Higher discharge coefficients of long orifices compared to

sharp-edged orifices are due to pressure rises in the orifices caused by mixing and shock

waves. These increases in the discharge coefficients are partly offset by friction and

boundary layer blockage. For maximum Mach numbers greater than 1.3, the flow in

long orifices is believed to become significantly two-dimensional because of supersonic

effects.

Ackno\vledgements

The author would like to recognize that this project was partially funded by Rolls-

Royce, Aero Division, England. Special thanks are given to the author's adviser, Dr.

John Moore, for the tremendous amount of time he spent on this project, and for the

patience he exhibited when working with the author. Thanks also go to Dr. Hal Moses

for the knowledge this author gained in his compressible fluid flow course and for the

insights he freely gave to questions raised during the research for this project. Dr.

Thomas Diller deserves thanks for serving on the author's advisory committee. Also, the

technicians in the mechanical engineering department's shop at Va. Tech are recognized

for the fine work they did constructing apparatus used in experiments performed for this

study.

Acknowledgements iv

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Sharp-edged orifice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Two-dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Cylindrical

Long orifices

Cylindrical

.......................................................... 17

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Shock wave behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Discharge coefficient measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

\Vater table study • . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

1-lydraulic analogy ....................................................... 61

Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 7

Table of Contents v

Water table results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

( l) Subsonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

(2) Subsonic/supersonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

(3) Supersonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

One-dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Basic calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7

Pressure decrease calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Friction ..................................................... 95

Blockage .................................................... 97

Water table pressure decrease .................................... 100

Summary of one-dimensional flow calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Maximum Mach number calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 03

Conclusions . . . . . . • . • . • . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Discharge coefficient measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 06

Water table study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 08

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Discharge coefficient equations and tenn errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Table of Contents vi

List of Illustrations

Figure I. Potential streamline pattern for incompressible flow through a slit. .... 6

Figure 2. Theoretical slit discharge coefficient. . .......................... 9

Figure 3. Comparison of calculated and measured slit discharge coefficients by Benson and Pool. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Figure 4. Comparison of calculated constant velocity lines and density fringes from interferogram (Benson and Pool). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Figure 5. Flow distribution in a slit for a back pressure ratio of 0.425 (Benson and Pool). . ................................................ 12

Figure 6. Flow distribution in a slit for a back pressure ratio of 0.123 (Benson and Pool). . ................................................ 13

Figure 7. Supersonic flow in vicinity of slit entrance (Guderley). . ........... 15

Figure 8. :vtaximum flow through a slit (Guderley). . ..................... 16

Figure 9. Supersonic slit flow downstream of "last" characteristic (Benson and Pool). . ................................................ 18

Figure 10. Comparison of slit and cylindrical sharp-edged orifice discharge coeffi-cients. . ................................................ 20

Figure 11. Incompressible long orifice discharge coefficients ( Lichtarowicz). 24

Figure 12. Pressure measurements for incompressible flow in long orifices (Lichtarowicz). . ......................................... 25

Figure 13. Shock waves at a wall. . ................................... 29

Figure 14. Sketch of discharge coefficient measurement system. . ............. 32

Figure 15. Photograph of discharge coefficient measurement system. . ......... 33

Figure 16. Sketch of orifice holder. . .................................. 34

List of Illustrations vii

Figure 17. Photograph of orifice holder. . .............................. 35

Figure 18. Photograph of orifices .................................... 38

Figure 19. Expected error in discharge coefficient measurements. . ............ 40

Figure 20. Sharp-edged orifice discharge coefficient vs pressure ratio (present ex-periment). . ............................................. 4 l

Figure 2 l. Discharge coefficient vs. pressure ratio, l/d = 1/2. . ............... 42

Figure 22. Discharge coefficient vs. pressure ratio, l/d = l. .................. 43

Figure 23. Discharge coefficient vs. pressure ratio, l:d = 2. . ................. 44

Figure 24. Discharge coefficient vs. pressure ratio, l,'d = 4. . ................. 45

Figure 25. Discharge coefficient vs. pressure ratio, l/d = 8. . ................. 46

Figure 26. Sharp-edged orifice flow factor vs. pressure ratio. . ............... 47

Figure 27. Flow factor vs. pressure ratio, l/d = l /2. . ...................... 48

Figure 28. Flow factor vs. pressure ratio, l/d = l. ......................... 49

Figure 29. Flow factor vs. pressure ratio, l/d = 2. . ........................ 50

Figure 30. Flow factor vs. pressure ratio, l/d = 4. . ........................ 51

Figure 3 l. Flow factor vs. pressure ratio, l/d = 8. ......................... 52

Figure 32. Dependence of discharge coefficient on orifice length for various pressure ratios. . ................................................ 57

Figure 33. Minimum length necessary to produce maximum discharge coefficient . 59

Figure 34. Surface wave velocity vs. wave length. . ....................... 63

Figure 35. Free surface water flow. . .................................. 64

Figure 36. Sketch of water table apparatus. . ............................ 68

Figure 37. Photograph of water table apparatus. . ........................ 69

Figure 38. Test section details. . ..................................... 70

Figure 39. Centerline pressure vs. distance from channel entrance. . ........... 73

Figure 40. Sketches of flow in channel for back pressure ratio (k = 1.4) of 0. 70. 74

Figure 41. Sketches of flow in channel for a back pressure ratio (k = 1.4) of 0.58. . 75

List of Illustrations viii

Figure 42. Sketches of now in channel for a back pressure ratio ( k = 1.4) of 0.50. . 76

Figure 43. Sketches of now in channel for a back pressure ratio (k= 1.4) of0.41. . 77

figure 44. Sketches of now in channel for a back pressure ratio (k = 1.4) of 0. 30. . 78

Figure 45. Sketches of now in channel for a back pressure ratio (k = 1.4) of 0.005. 79

Figure 46. Photograph of now in channel for a back pressure ratio (k = 1.4) of 0.005 (top vie\v). . ............................................ 80

Figure 47. Photograph of now in channel for a back pressure ratio (k = 1.4) of 0.005 (side view). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

figure 48. Method of calculating percent pressure changes. . ................ 88

Figure 49. Mixing analysis stations. . ................................. 89

Figure 50. Pressure increase vs. maximum Mach number. . ................. 92

Figure 51. Comparison of discharge coefficients calculated in analysis with measured values for l/d = 2. . ........................................ 94

Figure 52. Pressure decrease vs. maximum Mach number, l,'d= 2. . ........... 96

Figure 53. Pressure decrease vs. maximum Mach number, l,'d= 4. 101

Figure 54. Maximum Mach number vs pressure ratio. . ................... 105

List or Illustrations ix

List of Tables

Table l. Orifice dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Table 2. Liquid/gas analogous quantities ............................. 119

Table 3. Station l areas used in mixing analyses ........................ 120

Table 4. Velocity profile parameter n ................................ 121

List of Tables x

List of Sy1nhols

P - pressure

P 8 - back pressure

T - temperature

p - density

V - total velocity

u - velocity component in x-direction

v - velocity component in y-direction

w- streamline angle with respect to downward vertical

l - length

I"- the maximum duct length in Fanno-flow that does not result in choking

d - orifice diameter

dh - hydraulic diameter

P,. - wetted perimeter

List of Tables xi

D - supply tube diameter

A - area

w - slit or channel width

Red - Reynolds number calculated using orifice diameter

m- mass flow rate

ffi.0 - actual mass flow rate

rh,d .. 1 - ideal mass flow rate used in calculating discharge coefficient.

C d. h ffi · rl\ct 0 = isc arge coe 1c1ent, -.--

rn,d•••

Cc = contraction coefficient

(P - Pe) Cp = static pressure coefficient, ----

( P1 - Pe)

c - water wave velocity

A. - water wave length

h - water height

~h - change in water height, h., - h

he· water height that produces sonic flow

List of Tables xii

:vi - Mach number based upon local speed of sound

f - mean friction factor over duct length

Cp - constant pressure specific heat

Cv - constant volume specific heat

k - specific heat ratio, Cp/Cv

R - ideal gas constant

Vol - collection tank volume

ilt - time that a discharge coefficient test was run

g - acceleration due to gravity

subscripts

atm - atmospheric conditions

T2- final collection tank condition

T1 - initial collection tank condition

o - total condition

g - gage pressure reading

List or Tables xiii

Chapter 1

Introduction

The high cost of fuel and engine manufacturing has produced a large interest in the

turbomachine community to determine methods of increasing turbomachine efficiency

and prolonging engine life. Some of the research in recent years has been concerned with

studying turbine blade tip leakage. Blade tip-leakage is the escape of high energy fluid

from the pressure side of a blade along the casing or shroud to the suction side. The fluid

loses energy irreversibly through mixing and shocks. Researchers have reported that tip

leakage flow may contribute as much as 5% of the inefficiencies in a turbine.

Tip-leakage flow research areas have included the experimental testing of efficiencies

associated with different types of blade geometries ( eg. squealer, winglct, and grooved),

and the development of explanations for the fluid dynamics of this flow. Several

incompressible tip leakage flow studies have been been made but there do not appear to

have been any compressible flow studies. This paper is concerned with developing a

picture of the fluid dynamics of compressible flow in a tip gap.

Introduction

There is an interest in determining the flow physics of tip-leakage flo\\' not only because

of the inunediate penalties in efficiency associated with this flow, but also because it al-

lows for the determination of its heat transfer characteristics. At very high inlet tem-

peratures, heat transfer to blade tips may cause material failure which produces leading

edge rounding and results in increased flow area. If the damage due to heat transfer to

the blade becomes too great the blade is no longer useful. This means that tip-leakage

flow is not only important in determining present efficiencies but also the rate that the

efficiency is decreased to the point that the blade life is over.

Researchers studying tip-leakage flow have demonstrated similarities between this type

of flow and the much better understood orifice flow. The analogy between these two

types of flow is based upon the geometric similarities, and agreement of discharge coef-

ficient and pressure measurements. It is desirable to relate tip leakage flow to orifice

flow because the information on orifice flow provides a firm foundation of knov.:ledge

on which to build. The present research is rooted essentially in this tip-leakage/ orifice

flow analogy. In particular, the work included measurements of discharge coefficients

for cylindrical sharp-edged and long orifices, a water table study on a long rectangular

channel, and various one-dimensional flow analyses.

Introduction 2

Chapter 2

Literature revie'v

Interest in orifices dates back to the time of the Romans when they were used as a

method of regulating water flow to houses. Orifices have continued to be an important

method of metering flow, and because of this, most of the research on orifices has been

concerned with developing empirical correlations of their flow producing capabilities.

Theoretical studies have also been performed but these have been confined to the flow

in a two-dimensional slit. The discharge coefficient is a dimensionless flov.r rate param-

eter often used to present results. The discharge coefficient is defined as the actual flow

through an orifice divided by the isentropic one-dimensional flow rate that would occur

through an ideal converging nozzle with an exit area equal to the orifice area. In the-

oretical research, calculation of a discharge coefficient that compares well with a meas-

ured value is often a means of justifying results.

In this section, the flow from a large reservoir into square entranced sharp-edged and

long orifices with parallel sides is discussed. The sharp-edged orifice is of concern be-

Literature re-view 3

cause the fluid dynamics of flow through these orifices has been largely explained and

this provides a starting point for the explanation of long orifice flow. Long orifices have

a closer geometric similarity to the turbine blade tip. Square entrances and parallel sides

are easily reproduced and, therefore, eliminate parameters that would otherwise have to

be included (eg. radius of curvature, divergence angle). For these simplified conditions,

dimensional analysis indicates the important parameters in discharge coellicient calcu-

lations are Red (where d is a characteristic flow area dimension), the length to diameter

ratio, l/d, for long orifice measurements, and the back pressure ratio, P8/P0 , if

compressible flow is tested. Furthermore, the flow in orifices becomes largely inde-

pendent of Reynolds number for Red > 104. Orifice flow for Reynolds numbers in this

range was of concern in this study.

A review of sharp-edged and long orifice flow will be presented in this section. Also,

because the water table study demonstrated that shock waves can exist in long orifices,

shock wave behavior will be discussed.

Sliarp-edged orifice

Two-dimensional

The sharp-edged orifice has been particularly well researched. The exact incompressible,

two-dimensional, potential flow solution for a slit with infinite walls was developed by

Kirchhoff( I) and Rayleigh (2) who assumed the fluid developed an axial uniform velocity

at infinity and that the boundaries of the jet were at the downstream pressure. The

Literature review 4

streamline pattern of this flow is presented in Fig. l. Although, the jet asymptotically

approaches an axial, uniform flow, it can be seen that it is very close to this state within

one slit width, w. The region of this minimum flow area is called the vena contracta.

Fluid contraction is due to the vertical momentum of the fluid moving along the up-

stream face of the slit causing the flow to separate at the sharp slit entrance. The

streamline that separates from the orifice face will be referred to as the separation or free

streamline. Rayleigh found that the contraction coefficient was 1t . 2 or approx1-

7t + mately 0.611. Benson and Pool(3) made experimental measurements of the discharge

coefficient of air through a two-dimensional slit and extension of their data down to a

pressure ratio of one (incompressible limit) gives a value of approximately 0.614. Benson

and Pool did not quote a Reynolds number for flow in this range, but the present author

calculated an approximate value of 1.5xl0', using twice the slit width as the character-

istic dimension. The good agreement of the experimental result and the theoretical cal-

culation indicates that the incompressible mass flow through a two-dimensional slit is

determined by isentropic jet contraction to a one-dimensional state.

Analytical compressible flow solutions have been developed with various degrees of ap-

proximation. Chaplygin( 4) developed an "exact" solution for the two dimensional,

isentropic flow from a slit with infinite walls for subsonic pressure ratios. (The reason

exact is in quotation marks is that, although Chaplygin solved exact equations, the sol-

ution involved finite equation approximations to infinite series). As boundary condi-

tions, Chaplygin assumed the flow developed a vena contracta and the jet boundaries

were at the downstream reservoir pressure. His solution reduces to Rayleigh's for a

pressure ratio of unity. It will be shown later that the discharge coefficients calculated

by Chaplygin compare well with experimental measurements of Benson and Pool.

Therefore, it can be concluded from the boundary condition assumptions made by

Literature review 5

\ separation (or free)

streamline

I

I slit centerline

Figure I. Potentilli streamline pattern for incompressible now through a slit.

Literature review 6

Chaplygin that compressible subsonic flow behaves similarly to incompressible flow in

that the mass flow is determined by isentropic fluid contraction to a one-dimensional

state corresponding to the downstream pressure. Chaplygin was unable to develop sol-

utions for pressure ratios below critical because his boundary condition assumptions

were no longer valid. Namely, it is impossible for a flow to reach subcritical pressure

ratios through contraction to a one-dimensional state. After discussing the studies that

developed an explanation of the supersonic fluid dynamics of sharp-edged orifice flow,

a description of this flow will be presented.

Frankl(5) was able to solve the two-dimensional isentropic equations for a special case

of supersonic flow in which the flow rate was constant for all pressure ratios below a

certain value. For gases with a specific heat ratio, k, of l.4 such as air, the limiting

pressure ratio is 0.0395 and Frankl calculated a discharge coefficient of 0.35.

Solutions of the fluid dynamics of two dimensional sharp-edged orifice flow over the full

range of pressure ratios were developed by Benson and Pool( 6 ). A combination of nu-

merical and graphical methods was used to solve the two-dimensional equations. Finite

difference forms of the stream-function equation were solved in the hodograph plane.

Other equations were used to transform the hodograph plane to the physical plane. For

subsonic flow, the equations are elliptic and were solved by relaxation techniques. The

equations for supersonic flow may be solved without iteration. Since the flow is

transonic for subcritical pressure ratios, a combination of relaxation and direct solution

techniques was employed in which conditions at the sonic line were matched. Small

shocks at low pressure ratios prevented the numerical technique from being used, so that

graphical methods were reverted to. Large shocks prevented any calculations from being

performed.

Literature review 7

Theoretical discharge coefficients calculated by Benson an<l Pool arc compared with

those calculated using Chaplygin's exact solution in Fig. 2. :'\oticc that Chaplygin's re-

sults compare well with Benson and Pool's down to a pressure ratio of 0.5. Included in

Fig. 2 are the two limiting exact values calculated by Rayleigh and Frankl.

Experimental measurements of the discharge coefficient, intcrfcrograms, and Schlicren

photographs were also made by Benson and Pool. The discharge coefficient measure-

ments for a pipe to orifice area ratio of 0.177 arc compared to the theoretical values in

Fig. 3. lnterfcrograms of the flow are compared with computed constant velocity lines

in Figs. 4(a), (b), and (c) for pressure ratios of 0.8616, 0.7085, and 0.5283, respectively.

Schlieren photographs arc compared to calculated flow distributions in Figs. 5 and 6 for

pressure ratios of 0.425 and 0.123. The calculated flow distribution for a pressure ratio

of 0.425 was developed by graphical techniques because there were not enough mesh

points to perform a numerical calculation. The good agreement of the calculated and

measured results indicates that Benson and Pool have modelled well the fluid dynamics

in two-dimensional sharp-edged orifices for the region of flow without large shocks.

Based on the results of the above calculations and measurements, a qualitative expla-

nation of the fluid dynamics of two-dimensional sharp-edged orifice flow is now pre-

sented. Subsonic flow behavior is similar to incompressible flow in that the flow

contracts isentropically to form a vena contracta. Compressible effects produce an in-

crease in the discharge coefficient by decreasing the radius of curvature of the separation

streamline and, therefore, decreasing the amount of jet contraction.

For pressure ratios below critical, the flow is transonic and everywhere two-dimensional

with a Mach wave system influencing the flow. The transonic flow region in the imme-

diate vicinity of the orifice entrance is described by Gudcrlcy(7) with reference to Fig.

Literature review 8

0 0

Literature review

0 Ga

0

)(

., I)

0 ... . ' .... -0 I)

0 0 0

c..;

0 ..... 0

.... :..0 ._, ~-.

0 C>

0 0

., 0

., .,

0 0

0 0 0 N

0 .., 0 ... 0 ., 0 C>

0 ..... 0 IO

0 0 0 0

,; .J = 0

.:! " 0

E

ll. " :>

-...

.. 0 0

u -

zz ~

;;-..

-<( ..

-~

-o

'-'

0 >-Z

II

... JO

:.;

<'3 fl.Ill

... Ji

~

<CZ Oi

... I:l&I

::::J ()CD

·= V

I ..,.

VI

.. ~

:> ...

... c..

~

""" N " .. :I

! 9

084

080

0 76

0 72

068

0 64

E:aperimenlal points ,., Computed po1n1s v

/ /

~··

09 08 07 06

Pressure ratio ( P. P0 )

04 Ol 02

Figure J. Comparison of calculated and measured slit discharge coefficients by Benson and Pool.

Literature review 10

(a)

constant velocity lines

t.__ _

(c )

(b)

-----1

---j constant velocity lines

constant velocity lines

Figure 4. Comparison of calcuhtcd constant velocity lines and density fringes from interferogram (Benson and Pool). Pa/ Po = (; 1 ) .~ 616, (b) .i(Jq 5, \\.: ).5283 .


s ~

g ;i D

~-£

----- constant velocity lines

graphical calculation

Schlieren photograph

'-. 5HOC"

Figure 5. Flow distribution in a slit for a back pressure ratio of 0.425 (Benson and Pool).


constant velocitv lines

numerical calculation

Schlieren photograph

Figure 6. Flow distribution in a slit for a back pressure ratio of 0.123 (Benson and Pool).


7. Streamlines and 'characteristic' curves in the physical and hodograph planes and the

"sonic line" are presented. The hodograph plane is one in which the axes are velocity

components with the abscissa representing horizontal velocities and the ordinate vertical

velocities. The angle w is the angle the flow makes with respect to the downward vertical

direction. Guderley explains that the flow at B is sonic and expands through Prandtl-

Mcyer waves to the downstream pressure which corresponds to point D in the

hodograph plane. Notice points B and D arc the same in the physical plane. The Mach

expansion waves emanating from B,D are reflected from the sonic line as compression

waves and these waves in turn are reflected from the free streamline back toward the

sonic line as expansion waves. The interactions of the waves with the sonic line allow the

downstream conditions to communicate with the upstream subsonic flow. The wave

reflected at E on the free streamline is the last wave that intersects the sonic line. This

characteristic or wave has been called a 'last' characteristic ( 8) because all characteristics

downstream of it have no effect upon the upstream flow.

As the pressure ratio moves further below critical, the sonic line moves closer to the

orifice entrance, the radius of curvature of the free streamline ED decreases, and point

E moves closer to D until they coincide as shown in Fig. 8. When this occurs, further

decreases in the downstream pressure have no influence upon the upstream flow because

the radius of curvature of that portion of the free streamline that influences the upstream

flow has become a minimum (zero). Under this maximum mass flow condition, the 'last'

characteristic in the hodograph plane intersects the free streamline at w = 45°. Frankl

exploited the symmetry in the hodograph to develop the maximum mass flow solution.

It is evident that the discharge coefficient for supersonic orifice flow is no longer deter-

mined by jct contraction to a one-dimensional condition but by the two-dimensional


f A

F

-A

physical plane

D

hodograph plane

Figure 7. Supersonic now in Yicinity of slit entrance (Guderley).

Literature review

C Hlast* characteristic

I~

F

physical plane

hodograph plane

Figure 8. Maximum now through a slit {Guderley).


shape of the sonic line and the amount the streamlines have contracted when they

intersect it.

The region downstream of the last characteristic is described by Benson and Pool with

reference to Fig. 9. These authors describe the flow in stages with stage l being that

described by Guderley. In stage 2, ECF, the velocity increases along CF to point F on

the free streamline. In stage 3, bounded by the characteristics connecting points CFG,

the velocity increases along the centerline to point G where the compression wave from

F reflects. Notice there is a line between point F and the centerline in which the velocity

is uniform (although not axial ). The velocity in stage 4 ( FG II) decreases from G in the

direction of the free streamline FH. In stage 5 (GHI) the velocity along the centerline

decrease from G to I and another line of constant velocity exists between the free

streamline and the centerline. Ideally, region 6 marks the beginning of a repetition of

stages 2 through 5, but the streamline pattern in this region is different from stage 2, and

a shock may occur.

From this discussion it is apparent that transonic flow is everywhere two-dimensional

and may be discontinuous if shocks exists.

Cylindrical

The solution to the axisymmetric flow through cylindrical orifices is very difficult (see

Leipmann(8) for a discussion). The author knows of no exact analytical calculation.

However, because of their wide practical application, there have been numerous meas-

urements of discharge coefficients.


B

B ·"i ( \

"last" characteristic

' '

-- . ·-·

c

.,,,

D

F, J

hodograph plane

const~nt velocity lines

SUOCt(

\

physical plane

Figure 9. Supersonic slit now downstream or .. last"' characteristic (Benson and Pool).

Literature reYiew 18

The experimental value of the discharge coefficient for incompressible flow with the

Reynolds number greater than 10' is 0.61 and has been obtained by numerous authors

(9, 10, 11 ). This value is approximately the same as the one obtained for two-dimensional

flow. BirkhoIT{ 12) mentions Trefftz, Southwell and Vaisey, and Rouse and Abul-Fetouh

as among some who have attempted theoretical calculations to confirm this value.

Perry(l 3) and Grace and Lapple(l4) performed compressible flow experiments on

sharp-edged orifices. Their results are compared with the two dimensional calculations

of Benson and Pool in Fig. 10. The results are very similar, differing by a maximum of

1.6% at the lowest pressure ratio. The agreement between the axisymmetrical and

two-dimensional values of discharge coefficient for both compressible and

incompressible flow suggest that the flow behavior for the two cases is similar.

Lollg 01·ijices

Cylindrical

Compared to the sharp-edged orifice, the long orifice has been much less extensively re-

searched. This author is unaware of any analytical calculations - two-dimensional or

axisymmetric- or detailed studies of the compressible fluid dynamics of this flow. How-

ever, because of either their use as flow metering devices for high pressure ratio flow,

or their similarity with other types of flow geometries, some experimental discharge co-

efficient measurements have been performed with long cylindrical orifices. Also, several


c: ;; .. II> c .. " .. " < ;;· :E

N Q

1. 00

0.9~

0.90

0.8~

0.80

C0 o. 7~

0.70

o.e~

o.eo

o.~~

o.~o

1 . 0 0.9 0.8 o.7 o.& o.~ o.• 0.3 0.2 Pressure ratio (P/P0 ) -.-.. PERRY (CYLIN8RICAL)

BENSON ANO POL (SLIT) ~ GRACE ANO LAPPLE

Fi&urc 10. Comparison or 11lit and cylindrical sharp-edged orifice di!>chargc cocffidcnls.

0. 1 0.0

(CYLINDRICAL)

authors have attempted qualitative explanations of the fluid dynamics of long orifice

flow.

Lichtarowicz( 15) compiles the results of many experimental incompressible long orifice

discharge coefficient studies. The range of Reynolds numbers covered was from 1 to

105 with length to diameter ratios up to 10. Through observations of orifices made with

perspex, Lichtarowicz also presents a qualitative explanation of the effect of Reynolds

number and l/d on the behavior of the fluid flow. The following observations are made

concerning the Reynolds number when the I;d is long enough to allow for reattachment:

1. Very low Reynolds number results in creeping flow in which the fluid remains at-

tached to the orifice through the full length.

2. Increasing the Reynolds number results in laminar separation at the duct entrance

followed by mixing and laminar reattachment. If the orifice is long enough, a laminar

velocity profile starts to develop. The flow recirculates in the separation region.

3. Further increase in the Reynolds number causes the separated flow to become

turbulent. The transition to turbulence probably occurs after the vena contracta and is

promoted by the destabilizing effects of the increasing pressure gradient. The turbulent

transition process results in reattachment occurring over·a noticeable length rather than

at a point (as in in laminar reattachment).

Decreasing the length to diameter ratio with the flow fully reattached and turbulent is

explained as having the following influence:

1. As the length of the orifice is reduced, the flow has less of a chance to mix and

reattach.


2. At some value of l/d < 1 the flow does not reattach and emerges as a jct. Recir-

culation of the fluid in the separation region still occurs 'with the result that fluid from

the downstream side of the orifice is drawn upstream ... : This implies that there are

pressure changes in the separation region and that the pressure along the separation

streamline is no longer at the downstream pressure as was the case with the sharp-edged

orifice. A pressure recovery still occurs along the orifice's bore after the vena contracta

due to partial mixing.

3. Reducing l/d further eventually results in the sharp-edged orifice \\'here no pressure

recovery is possible.

To sum up, Lichtarowicz explains the pressure drop across the orifice as consisting of

the pressure required to accelerate the fluid to the minimum pressure, the pressure re-

covery resulting from mixing and reattachment, and the pressure loss in friction.

Lichtarowicz predicts that when the l/d is not long enough to ensure reattachment,

hysteresis effects may occur with the fluid reattaching or not depending on whether the

now rate is increasing or decreasing.

Plots of discharge coefficient vs. Reynolds number were presented for lid = 1/2, 1, 2,

4, and 10. The C0 vs. Rea curve for l/d= 1/2, shows a peak and then a decrease to a

constant value Nleaving a small but noticeable 'irregularity' at somewhat higher

Reynolds numbers". This 'irregularity' persists in decreasing amounts with increasing I;d

until an l/d ratio of 2. Lichtarowicz claims this is 'almost certainly' the result of chang-

ing separation and reattachment with Reynolds number. For all long orifices, the dis-

charge coefficient becomes independent of Reynolds number for values greater than

approximately 2 x 104- This ultimate discharge coefficient is plotted as a function of l/d

in Fig. 11. Notice that the discharge coefficient increases rapidly with increasing l/d until


a maximum is reached at an l/d of about 2. Length to diameter ratios greater than 2

result in a gradually decreasing discharge coefficient. Apparently for Reynolds numbers

greater than 2xl0', an l/d= 2 is the limit for fully reattached, mixed Oow without pres-

sure decreases due to friction or boundary layer development.

Lichtarowicz also presents the result of measurements he made of the pressure distrib-

utions along the orifice wall for orifices with l/d ratios of 1/2, 1, and 4 for various

Reynolds numbers. A reproduction of these results in the form of pressure coefficient

vs. length is presented in Fig. 12. Lichtarowicz states that reattachment is indicated in

the 4 l/d orifice by the pressure decrease following the initial increase. On the other

hand, reattachment does not occur in the 1 l/d orifice bec.:ause the minimum pressure is

not as small as that in the 4 l/d orifice. Two different pressure distributions were re-

corded for the l /2 l/ d orifice. In one case, the pressure is everyv;here equal to the

downstream value, and in the other, the pressure coefficient is nearly constant at -.16

over the length of the orifice. Lichtarowicz made a calculation to determine if the in-

crease in discharge coefficient was due to this "underpressure". The calculation is made

assuming the flow forms a vena contracta at the minimum pressure with a contraction

coefficient equal to the sharp-edged orifice discharge coefficient. The Bernoulli equation

and the definition of discharge coefficient are used to calculate the actual flow rate,

The velocity is calculated from the Bernoulli equation,


r-;::;.· " .. ~ ~ .. " ... " <I!

~-

N ~

<>8 I

O·~

Co

0·1

0 6!> )

06

~

• • . ' •

•

I'-•

• • • •

I I

• ~-! . .. -

~ ---• • • r;--. •• ~ ~ r----: • r----.. • • ----.,.

• -------• JOHAN UN

• WE15BAC:14

• llOENHECKE

• JOA•SSEN ANO N[wfo,. .1/D •0·2224

• JOAISSEN ANO NEWTON d/D •0 l/OQ

• JOAIS:s.[N AlllO NEWTON a/tJ •0 4944

• sr1•ES AND P[NNINC.J()N

• ASIHMIN

• NAKAYAMA

• JAPPAAO ANO REIO

• JAMES

• SANDERSON

• - undcrpressurc calculati'ln I l ___ L. ____ 2 } 4 !> 6 1 8 9 '0

l,'<l

1-"igurc II. lncumprcniblc Ion& orifice di~char&c cocfficicnb (l.ichlaru"iu).

distance from entrance, x(in.) 0·1

-.. , ·.~.-.

:~~--c

I 1·5 2

J d•O·S02 in ... V//t ((( i I . ' -...; - -0·2 1 ~ P, -~ ·v = u

0 u c,-~ u ... -o 4 P,- I =' "' l/d ~. "' ':,) --·--- 3.ga !1100 ... c.

3·98 1&gc)0 -~ -·-- --·--- 0·994 6900 ~ -0·6 - -·- 0·994' 17200 V'l

I __ .,,_ __

0·498 ,3800 _....,_ 0·498 :e&oo " • ..._.J

-0·8

Figure 12. Prc..,sure measurements for incomprcssihlc flo" in loni; orifices (l.i..:htaro"i..:1).

Literature: re' icw 25

The ideal mass flow is calculated similarly, but this time the exit pressure is used to cal-

culate the velocity,

The discharge coefficient is now calculated,

or,

0.6lpA(l/p(P0 -Pmm)).S

pA( l/p(P 0 - Pc)).s

After canceling common terms, and adding and subtracting (P.)..s from the numerator

of the above equation, it becomes,

Pmm - Pe .5 = 0.61(1 - p - p ) .

o e

Finally, from the definition of pressure coefficient,

Using this equation, Lichtarowicz calculated a discharge coefficient for the l,'2 l/d orifice

to be 0.66. Similar calculations were performed by the present author for the orifices

with l/d = I, and 4. The results of these "underpressure" calculations have been in-

eluded in Fig. 11 to demonstrate that they compare well with the experimental values.

The agreement of these calculations with the measurements suggests incompressible flow


in a long orifice is similar to that of a sharp-edged orifice with the f1ow forming a vena

contracta at the minimum pressure. The higher discharge coefficient is accounted for

by the underpressure at the vena contracta.

Nakayama(l6), Grace and Lapple(l4), and Deckker and Chang(l 7) performed

compressible discharge coefficient experiments on long orifices. These studies were pri-

marily concerned with the determination of discharge coefficients as a function of pres-

sure ratio for various length to diameter ratios. Only qualitative attempts were made to

explain the f1ow physics. Particularly lacking was an account of the supersonic f1ow in

long orifices.

Deckker and Chang researched bevel backed orifices with diameters ranging from 2.4

to 13 mm and 2, l, and .5 values of l/ d. Sharp-edged orifices were also tested. In a pre-

liminary investigation, they found the Reynolds number had no inf1uence on the dis-

charge coefficient for all I/d's when it was greater than 104. This is approximately the

value obtained for incompressible f1ow. The Reynolds number for Deckker and Chang's

discharge coefficient measurements was greater than 10'. -:\akayama made measure-

ments on square backed orifices with diameters between .25 mm and 1.2 mm and l/d

ratios between 0.8 and 16.5. Nakayama suggests that some his measurements were made

for Reynolds numbers slightly less than 104. Grace and Lapple did experiments on an

orifice with an l/d of I and Reynolds number greater than 104. The presentation and

discussion of the compressible long orifice discharge coefficient results are deferred until

the results of the present study are presented.


Shock wave behavior

Because the results of the water table study by the present author demonstrate shocks

exist in long orifices if the back pressure is low enough and no information on supersonic

flow structure was contained in the long orifice literature, general shock wave behavior

is reviewed. Some of the pertinent information is presented here.

Leipmann et. al (18) present a discussion on the interaction of shock waves with sur-

faces. Two different types of shocks may exist, normal and oblique. Furthermore, ob-

lique shocks may reflect from surfaces as "regular" or \1ach reflections. A normal shock

is shown in Fig. 13(a). Regular reflections, Fig. 13(b), result if the upstream ~tach

number is large enough to allow the flow to negotiate the turning required by the inci-

dent oblique shock and reflection. Otherwise, a :Vlach reflection ( Fig. 13( c) ) will occur

in which the incident shock branches above the reflecting surface into a reflected wave

and a near normal shock that intersects the wall. The pressure downstream of a \1ach

reflection is the same but the velocities are different thus producing a "slip line" or

"vortex sheet" over which the velocity is discontinuous. Because a normal shock occurs

at the wall, at least part of the flow downstream of the shock is subsonic and, therefore,

can be influenced by downstream conditions.

The present study involved building on some of the ideas contained in the orifice litera-

ture through discharge coefficient and water table channel measurements, and simple

analyses to try to shed some light on compressible flow in tip gaps. The discharge co-

efficient measurements will be discussed next.

Literature reYiew 28

'11t .. !!!!/?!??!:~-l//l/1/ll//1111//;/12j2~,t1u,0j!'-,;11ZZ "011

(a) normal shock

lnc1dlflt Atfltcfld St.ec• Shoe•

wu::SZ:w;uJ;,i Woll Woll

(b)rcgular reflection ( c) \Lich rellcction

Figure 13. Shock waves at a wall.


Chapter 3

Discharge coefficient measurements

When the present study was undertaken the author was able to locate only a small

amount of information (Grace and Lapple's) on compressible flow through long

orifices. Therefore, a system was assembled to measure the discharge coefficient of air

through a sharp-edged orifice and long orifices with l,'d values of 1,'2, 1, 2, 4, and 8. The

diameter was a nominal 0.16 cm. All the long orifices had plane faces except the l ,' 2 l,' d

orifice which had a beveled back. The 1/2 l/d orifice had a beveled back because it was

intended to be the sharp-edged orifice; the discharge coefficient results, however, dem-

onstrated the length at the entrance was too large to allow it to be used for this purpose.

The Reynolds number for the measurements ranged between 2 x 104 to 8 x 104, and,

therefore, was believed to be uninf1uential. A description of the measurement system,

and a presentation and discussion of the results of the present study, and those of

Deckker and Chang, Grace and Lapple, and '.\:akayama, will be presented in this section.

Discharge coefficient measurements 30

Apparatus

A sketch of the discharge coefficient measurement system is presented in Fig. 14, and a

photograph in Fig. 15. The system was constructed with the aid of the description given

in the ASM E Flowmeter Computation Handbook ( 19). Compressed air with a nominal

pressure of 620 KPa was filtered and delivered to and from the orifice by two 12. 7 mm

ID ASTM A213 stainless steel tubes each of 30 cm length. A stainless steel tank with

a nominal volume of .295 m1 was used to collect the mass of air that Oowed during a test.

The type J-1 capsule shaped collection tank had a cylindrical mid-section with a diameter

and length of 60 m and spherical ends. The time it took to collect the air was measured

with a Cronus digital stop watch. A Fairchild regulator with a sensitivity of 3.2 mm of

water maintained the upstream pressure.

A sketch of the orifice holder is shown in Fig. 16 and a photograph in Fig. 17. The brass

holder was made by modifying an existing orifice holder in which holes were drilled to

accommodate the supply tubing and pressure taps were added. The supply tubing was

soldered to the holder to make it plane with the holder face. To test the orifices with

l/d = 4 and 8, two new securing nuts were manufactured from hexagonal brass stock.

0-ring seals were used to seal the orifice in place.

Pressure across the orifice was measured through . 79 mm pressure taps drilled at lo-

cations of 1-D upstream of the front of the orifice and 1/2-D downstream of the orifice

back. The pressure ratio across the the orifice was measured with a 0-690 kPa upstream

pressure transducer and a 0-395 kPa differential pressure transducer for tests with pres-

sure ratios above approximately 0.4. The transducers were manufactured by Statham.


compresse<l Jir

l

~mercury manometer

thermocouple

<l11Tercnttal pressure trJnsduccr

pressure transducer

bndge amp.

null amp.

thermometer

D oscilloscope

DV\.t

amp.

\..,-......:. ____ _,,...T'"""--L-------.1~:...s!!t~art1 stop valve

. ! I I' ~~pressure ratio

I alljustment valve

I

collecuon t3nk

lhcrmocouplc

Figure 14. Sl..c:tch of discharge cocffii:icnt mcuurement sy!'ltcm.


DV\.t

32

Discharge coefficient m

easurements

33

Flow

~

15.9 mm

Figure 16. Sketch of orifice holder.


nut for 4 l;J orifac

nut for S I,' d orifice

thermocouple __....

nut for S[; 1.J~ I.'~. I,::!

orifices

34

The upstream transducer was connected to a Vishay null amplifier which was connected

to a Keithley DVM. The differential transducer was connected to a Vishay bridge am-

plifier which fed a Tektronic oscilloscope. The pressure transducers were balanced for

pressure measurements. Below a pressure ratio of 0.4, the downstream pressure was

measured with an Ashcroft dial test gage because the differential transducer was over-

ranged.

The additional error introduced by using the dial gage was believed to be small because,

I) by definition of discharge coeflicient, measured pressure ratios below critical (0.5283

fork= 1.4) were not used in the discharge coefficient calculations, 2) supersonic pressure

ratios were used only to determine the pressure ratio at \\.·hich a measurement was made,

and 3) it will be seen from the results presented in this section, that the discharge coef-

ficient changes little for pressure ratios below 0.4.

Omega ungrounded, l.6 mm diameter, iron/constantan thermocouples were used for

temperature measurements. The temperature of the air flowing through the orifice was

measured 5-D downstream of the orifice back. The tank temperature was measured by

a 60 cm Jong thermocouple. Both thermocouples were connected to a battery operated

IOOx amplifier which fed a Keithley DV\1. An ice bath was used for a reference tem-

perature. The voltage indicated on the DVM was converted to temperature by linearly

interpolating between values on a l°C increment Omega table (20).

The pressure ratio across the orifice was adjusted with a valve downstream of the orifice.

The flow through this valve was choked to maintain a constant pressure ratio. The

collection tank was partially evacuated before the start of a measurement to ensure the

pressure ratio control valve remained choked during a test. A quick operating 3-way

Whitey ball valve was used to start and stop a measurement. The pressures across the


orifice stabilized to the operating values within l sec of the beginning of an experiment.

A standing mercury manometer measured the pressure in the tank. Tank measurements

were taken prior to and subsequent to a test only after the tank reached steady state as

determine by less than a l°C fluctuation in temperature. Air tightness was determined

by pressurizing the system upstream of the start/ stop valve to the line pressure ( 620

kPa) and downstream of this valve to approximately 250 kPa and applying soapy \\'ater

to joints.

Photographs of the orifices are presented in Fig. 18 \Vith their dimensions presented in

Table 1. The orifices were manufactured from 25 mm diameter brass stock. To ensure

the orifice inlet was sharp, the faces of the orifices were turned on a lathe and the up-

stream face was polished with a fine stone. The diameters were measured with a E. Leitz

and Wetzlar lOx traversing microscope. A lOx Bausch and Lomb lens was added to

enhance the magnification of the microscope. The lengths of the orifices were measured

with either a hand-held micrometer or a dial gage attached to the microscope head. The

lengths of the sharp-edged and 1/2 l/d orifices were measured \Vith the dial gage and

microscope because of their beveled backs.

An error analysis for the measured discharge coefficients was performed and the results

for the sharp-edged and 2 l/d orifices are presented in fig. 19. Because the discharge

coefficients measured with these orifices bracketed the values measured with the others,

these errors provide a band that the errors for the other orifices fall within. It can be

seen that the expected error is between 1.5 and 2% for pressure ratios between 0.9 and

0.3. The larger expected error of 3 % at the higher pressure ratios is due the large sen-

sitivity of the discharge coefficient to pressure ratio measurements. At the lower pres-

sures, a test could not be run as long because the pressure in the tank would unchoke



easurements

38

the pressure ratio adjustment valve. This resulted in a larger error in both the time and

the tank pressure change measurements. The equations used in the discharge coefficient

calculation, the calibration techniques, least count data, and expected errors associated

with each measurement are presented in Appendix A.

Experi111e11tal Results

Results of the sharp-edged orifice measurements by the present author and by Perry are

presented in Fig 20. Long orifice discharge coefficient measurements vs. pressure ratio

by the present author, Deckker and Chang, Nakayama, and Grace and Lapple are pre-

sented in Figs. 21 through 25, with results for similar l/d orifices appearing in the same

figure. In these plots, the sharp-edged orifice discharge coefficient measured by the

present author is used as a base line. Deckker and Chang's results were taken from some

of the data points lying on the curves these authors drew. :\akayama·s data was taken

from multi-curve plots in which the symbols were sometimes difficult to distinguish.

Grace and Lapple's data is for the same orifice with either end upstream. For this au-

thor's data, the "flow factor" as a function of pressure ratio is presented in Figs. 26

through 31. The ideal mass flow used in calculating the discharge coefficients are in-

cluded in the flow factor figures.

The following general observations are made about the discharge coefficient plots. With

the exception of the present author's data at the highest pressure ratios tested

{P8/P0 > 0.95), the discharge coefficients for all orifices generally increase with decreas-

ing pressure ratio. The present author's discharge coefficients at the highest pressure


c ~· ':Z'

E: (1Q

" n ~ 3 n ;;· a 3 " t: c ... " 3 " a ""

A =

5

4

3 -0 c' ...... .... 0 t: u

0 u 2

0

I x 0

t

1 • 0 0.8 o.e 0.7 o.e o.~ o.4 0.3 0.2 0. 1 o.o Pressure ratio ( l>/I\.)

I I I L/D•2 M M M SHARP-EDGED 1-'ieure 19. [).peeled error in discharge coefficient measurements.

c ~· :r .. .. ~

" n 0 ,, 3 n ;;· = .. 3 ,, !: c .. ,, 3 ,, a ..

~

1. 00

0.9!1

0.90

o.e~

o.eo

Coo. 7~

0.70

o.e~

o.eo

0-~~

o:~o ..... ~ ....... ~.....-~.....-~.....-~--~--~--~---~---~--~--~---~---~---~---~---~---~--~--~--, . 0 0.9 0.8 o.7 o.e o.~

Pressure ratio (P/P0 )

· a a a PERRY 11 N 11 HENRY

0.4 0.3 0.2

Figure 20. Sharp-edged orifice discharge cocffit:icnt n proi.ure ratio (prci.cnt experiment).

0. 1 o.o

c ;;;· n :r ID ...

:rQ n n 0 n 3 n ;;· :! 3 n !: c ... " 3 n :a ""

~ N

1. 00

0.95

0.90

o.e5

o.eo

Coo. 75

0.70

o.e5

0. 60 l ~ x

0.55

0.50 1. 0 0.9 0.8 o.7 o.e o.5 0.'4 0.3 0.2 0. 1 o.o


HENRY SHARP-EDGED OECKKER ANO CHANG L/D-0.50 • • •HENRY L/0-0.45

Figure 21. Di~chugc cocnicicnt u. prc~surc ratio, l/d = 1/2.

x

o.9 o.e o.7 o 6 o.~ 0.4 o.3 0.2 0.1 o.o Pressure ratio (P/P.,) SAKAYAMA L/0-.94 • • •HENRY L/0-1.0J

ECKKER • CHANO L60-1.o M M M HENRY SHARP-EocEo RACE • LAPPLE L/ _, ••• NAKAYAMA L/o-.ao

Figure 22. Dis1:harge coefficient n. pressure ratio, l/d = I.

c ~· 1. 00 :r e:

(IQ

" ... ~ 0.9~

3 ... ;;· :I .. 3 0.90 " !: c .. " 3 o.e~

" :I ... ~

o.eo

Co 0.7~

0.70

o.e~

o.eo

o.~~

o.~o ....--1. 0 o.9 o.e o.7 o.e o.~ o.4 o.3 0.2 0.1 o.o

Pressure ratio (P/I\) • • • I I I

NAKAYAUA L/0•2.01 a • •HENRY L/0•1.99 OECKKER • CHANG L/0•2.0 M M M HENRY S~ARP-EOGEO

t a.·igure 23. Discharge coefficient n. pressure ratio, l/d - 2.

0 .,

0 .,

0 .,

0 .,

0 GI

GI I)

I)

.... ....

C>

0 0

0 0

0 0

0 0

u


easurements

x 0 .,

0 C>

., .,

0 0

0

0 0 0 N

0 .., 0 • 0 -3

... ~~

c)-0

- :'3 .... CJ C> .... ~ 0

"' "' ~ .... :..

.... 0 I)

0 ~

0 0 ...

I) a. .., I a ' .. ~

u ~

>-::::-

a:: ,g,·

z -;

LI.I ..

J: u .. ~

I = u .. Cl.

.; .. c .!:! u E ... Q

lllQ

"' -11.1

u M

·O

... .. •O

.c

111.1 "' "'

01

a 'A

A

:: <

..; <J:

N

:Jiii u ...

< ~

>->-M

<It l.i:

~2

<1&.1 ZJ:

II

45

0 ..,

0 ..,

0 ..,

0 0

Gt Gt

., .,

" "

0 0

0 0

0 0

Q

u


euurements

)(

If) 0

.., C

) C

) If)

0 0

0

0 ., 0

0 0 0 N

0 ,, 0 • 0 .., 0 C)

0 " 0 ., 0 Gt

0 0 P'

Gt Gt

" I Q

~

' u

.J ~

::::;->-

.2 ct:

~

z ..

LI.I "

J: .. = "' "'

! " .. -

c.. 0

.; c..

... -

c c.. -

" 0

·v E

-"

:'3 Q

...

... C

) N

O

" ...

Doi

-ow ..

"' ..

"' ·O

..c:

u IOO

... ...

"' c..

111.1 c Q

I '\A

vi

.Jct: < M

<X " ..

:Jin :I

< ~

>->-~

<Ir :icz <ILi ZJ:

II 46

.. \ \ \ \ \ \ \ \,

..

x x

' ,, ' ,,

0 .. ,, ,,

Discharge coefficient measurem

ents ........ ........ ........ ........ ....___ .................. ---------...;; 0

0 -0 0

0 0 .. 0 N

0 II)

0 ci

..J -~

< .. w

u

a .. •

~

= 0

u .. "' -

.; 0

c.. >

:;:;--..

., ~

g u

o·S? .!

.... ~

~

0 ...

= 1..>

"'=' ...

u I>

~ >-

oa "'

It -:I

.,., "

0 u z

Q.

... ..

c.. w •

I: .r:: C

l)

..... I ..0 N

0

u .. ~ 1111 l.i:

'° 0 a. 0 0

47

' ' \ \ \ \ \ \ \ \ \ ' ' ,, ' ,,

0

........... ..........

~

0

........... ......... ............... ...... --- ----

x ----.._..._........__ _

_ -.a

__

__

_

.., 0

0 0

0 0

0

[

0~V ] lO

lnJ .~OJ.:J ;(".L

:))tµ

N

0 0


easurements

0 0

0 0 0 N

0 .., 0 • 0 -0

0.. - ,..

in :::. 0

o·:: :o:s ..... 0 .....

0 =' .,, .,,

0 cu ..... c..

" 0 ., 0 ~

0 0

.J <

,..i II.I -Q ~

::::-.i '; .. " .. = "' "' II.I ... C

l.

.; > in .. •

Q

u I

.! a

~

' Q

.J ~

>-,.:

ct: N

z

" Ill .. J: = !IA

I i.i:

48

0 0

0


easurements

0 0 .. 0 N

0 _, 0 " 0 ., 0 0 0 ,... 0 IO

0 a-0 0 .. -c..o -- :.. -0 .... '.'3 .... ~

.... :3 IJ) .,, QJ

....... -

~

.J <

"I:! ...

:::::-0

.ri -.. .. u .. = .. Ill

u ... a. ..; .... ... Q

<:i .!

0 ~

Q

.. ~

I 0 ~

' .....

.J u ... =

~

Cle

It ~

z l&.I I:

I 49

' \ \ \ \ \ \ \ \ \ \',...__ ,,

N

0 ..

.......... ............


easurements '"'""- ..._ __ ---

--~-...;;:::

"' 0 ---N

0 0

0 0

0 0 0 N

0 "' 0 • 0 -0

" ., ~

·-o-0

-~

... G

u ... :::>

0 "' "' u ... i:..

"' 0 I)

0 ()

0 0

"'4 .J

I <

~

.., ::::--

Q

.i .. .. u .. :I

"' "' ~ .. ~ ..; ... .. Q

'C .!

() ~

() 0 ~

I d.

Q

' M

.J u .. :I

>-tot

It ~

z .., J: I so

0 0 .. 0 N

0 .., 0 'it

0 -Cl

:.. .,, :: -0 0 -~

...

0 u ... ~

0 .,, .,, u ... :.. ,... 0 ., 0 ca 0

_________________ __..... __________ ~;:::~=tJ.o

0 0

.. ..

0 0

0 ~

0 0

[

0 ~ V

] JO

lJ'l?j .\\Of.:J

!:"(a .1 :J)lµ

..


easurements

.., 0 0

0

0 0

..,; ..I

I "Cf <

::::::-... .i Q

.. .. " .. ::s ... ... " .. Cl.

.; ... .. g <.I .!

., ~

ca ..: L.r.

.., r = '"I a:

u z .. ::s ...

M

x ;.::

t SI

c ~· ;r e ~ ,, n :i 3 n ;;· a 3 ,, e: c: .. ,, 3 ,, a "'

UI N

1 . s

1 . 2

1 • 1

1. 0

0.9

r--i ,_ o. a

0

!- .... 0 c....-u <o.7 :s L........'.___Jo • e

t.--0 ..... u ~ o.& ~ 0 c 0.4

0.3

0.2

0. 1

// //

/ I

I

I I I I /• I I I I I

I I

___ ..,,.,,,...-~..---------,,,''/.// --------------------------·

o.o ·~~ ........ ~-.-~-.-~-.-~-.-~.......-~.....-~.....-~........-~........-~--~---~---~---~.....-~--~.....-~--~--~--1 • 0 0.9 o.e o.7 o.e o.& 0.4


HENRY L/0•7.99 ----- I DEAL

1-'igurc 31. Flow factor vs. prcuurc ratio, 1/d "". 8.

0.3 0.2 0. 1 o.o

ratios are consistently higher (on the average by about 3%) than the values measured

at the next highest pressure ratio. The cause of these larger values is unknown. All the

orifices Deckker and Chang tested have consistently higher discharge coefficients than

the values measured by the other authors. This is believed to be the result of the beveled

backs on their orifices allowing for larger pressure recovery.

The sharp-edged discharge coefficient measurements of the present author (Fig. 20) are

less than 2% different from the values measured by Perry. This indicates that the dis-

charge coefficient measurement system used in the present experiment was operating

properly. It can be seen from the flow factor plot (Fig. 26) that the mass flow through

the sharp-edged orifice continues to increase for the full range of pressure ratios that

measurements were made.

Deckker and Chang's results for the 1/2 l/d orifice (Fig. 21) are peculiar in that the au-

thors drew two branches of the curve through the results. The maximum difference in

discharge coefficient between the two branches is 10%. The authors claim the branching

in the data is due to reattachment hysteresis of the type speculated about by

Lichtarowicz; in this. the upper branch of the curve is the result of the flow remaining

attached when the pressure is increased (mass flow decreased) to values that do not

otherwise ensure reattachment. Deckker and Chang predicted that hysteresis may only

occur in orifices with an l/d of 0.50 and. later in a letter in "Nature", Deckker (21)

claimed that hysteresis may only occur in orifices with beveled backs. The necessity of

these particulars is not explained.

None of the other authors' results indicate the existence of hysteresis. This might be

attributed to the difference in measuring technique and orifice size, or to the fact that

measurements were made with plane backed orifices. Although the present author's 1/2


l/d orifice had a beveled back, it would have been impossible to detect hysteresis because

the measuring method resulted in the mass flow always increasing for test start up.

Also peculiar about the 1/2 l/d measurements of Deckker and Chang is that this orifice

has the largest maximum discharge coefficient ( C0 = 0. 9 ) of all the orifices tested. This

discharge coefficient is about 5% larger than the maximum theoretical discharge coeffi-

cient for a sharp-edged orifice. Deckker and Chang's data for the lower and upper

branches of the curve arc, respectively, on the average 8% and 12<% higher than the

discharge coefficients measured by the present author. The cause of this difference is

unknown.

The flow factor plot of the present author's data (Fig. 27) show that a constant mass

flow resulted in the 1/2 l/d orifice for back pressure ratios belov .. · approximately 0.45.

The discharge coefficients measured by the present author for this orifice increase from

a value of 0.66 at a pressure ratio of 0.97 to a value of 0.82 for pressure ratios below

0.45.

The discharge coefficient measured by the different authors on the orifice with an l/d = l

(Fig. 22) shows a spread as great as 6% at pressure ratios above 0.6 and, excluding the

data of Deckker and Chang, shows nearly no difference for pressure ratios above 0.6.

The present author's data for pressure ratios greater than 0.6 is generally lower than the

data of the other authors. Discharge coefficient measurements by ~akayama on a 0.8

l/d orifice have been included in this plot to demonstrate that the shape of the curve

through the present author's data is reasonable. A straight line was drawn through

Grace and Lapple's data because the spread was as large as 4%. The discharge coeffi-

cients of Deckker and Chang become progressively larger than the values measured by


the other authors as the pressure ratio decreases with the difference becoming about

4.5% for pressure ratios below 0.50.

The mass flow plot of Fig. 28 indicates that the mass flow rate in the present authors I

l/d orifice was constant for pressure ratios below 0.60. The present author's discharge

coefficient measurements had a minimum value of approximately 0. 77 at a pressure ratio

of 0.81 and a maximum value of0.82 for pressure ratios below 0.60.

The 2 l/d orifice measurements (Fig. 23) by the present author and Nakayama compare

well over the complete pressure ratio range tested. The 2 l/d data by Deckker and

Chang is consistently higher by about 3.5%. Figure 29 indicates the mass flow in the

present author's orifice with an l/d = 2 was constant for back pressure ratios below 0.6.

The discharge coefficient measurements by the present author had a minimum value of

approximately 0.80 at a pressure ratio of0.91 and a maximum value of0.84 at a pressure

ratio of 0.98. A discharge coefficient of approximately 0.83 was measured for all back

pressure ratios below approximately 0.6.

The discharge coefficient data for the 4 l/d orifice (Fig. 24) shows that the values meas-

ured by the present author are smaller than those measured by i\akayama by a near

constant value of 2%. This is a reversal of the results of the discharge coefficient meas-

urements of orifices with l/d < 4, in which Nakayama's values were larger than the

present author's. The mass flow factor plot of the present author's data (Fig. 30) shows

constant flow for pressure ratios below 0.55. The discharge coefficient measured by the

present author had a minimum value of 0.78 at a pressure ratio of0.90 and a maximum

value of 0.82 for pressure ratios below 0.55.


The measurements for the l/d= 8 orifice (Fig. 25) shows the discharge coefficients

measured by Nakayama are consistently lower than the values measured by the present

author with a difference as great as 5% occurring at the higher pressure ratios. The

mass flow factor plot (Fig. 31) demonstrates that the flow rate in the 8 l/d orifice used

in the present study became constant for pressure ratios below approximately 0.55. The

present author's discharge coefficient for the 8 l/d orifice has a minimum value of 0. 79

at a pressure ratio of 0.89 and a maximum value of approximately 0.82 for pressure ra-

tios below 0.55.

To compare the variation in discharge coefficient with l/d, Fig. 32 is presented in which

the discharge coefficients measured by the present author are plotted as a function of!,'d

for various back pressure ratios. The discharge coefficients for this plot were obtained

by linear interpolation between data points, or, for pressure ratios of I and 0, by ex-

tending the curve drawn through the data.

With the exception of the values for the sharp-edged orifice at low pressure ratios

(Paf P0 < 0.3), the discharge coefficients of the 2 l/d orifice represent maxima that the

discharge coefficients of orifices with an l/d less than 2 approach as the pressure ratio is

decreased. As the l/d of the orifice increases from 2, the discharge coefficient decrease

slightly.

The increasing discharge coefficient with decreasing pressure ratio for the orifices with

an l/d less than 2 could be the result of the radius of curvature of the separation

streamline decreasing as the pressure ratio is decreased allowing for larger amounts of

mixing and reattachment in a smaller distance. A plot was made of the minimum l/d

producing the maximum discharge coefficient as a function of pressure ratio and this is

presented in Fig. 33. It is believed that this plot shows the minimum i/d that is necessary


0 ~-':I" e: ~

~ 3 n ;;· ~ .. 3 " r: c ... " 3 " ~ i:

~

Co

1. 00

0.9&

0.90

a o.e&

o.eo --- , ·-:---::::~ - ----- . -·----~----·---·-=·-·---·-·:--== ,,,..7L.........-= ~--::::·-..~.:.:::::::-·---·-·-·-· . . ·--~----~,;- ·----.;: . .:.::::::.:_--;-.-==-· .

0.7&

0.70

o.eo

o.eo

0 • ~~ ..,, u u u u u u ; u u I o a o o a u u u u I o u e ; u o u ; u ; a c u a a c u : ; o a u u u u o a I u u : u o c a u ; I u u a u o u a e u ; u u u u u u u u u I

0 1 2 3 4 & e 7 8 l/d

a---- 0 b ............ 0 . 1 l: -------- 0 . 2 d ----- 0. 3

e--- o.4 f __ ._ o. !;-. g 0.6 h ·-·-·-·· 0 . 7 . . O.B J ·--·- 0. 9 k •

•·igure 32. Dependence of discharge c~ffident 011 orifice length for nrious pressure ratios.

for the flow to fully "develop" reattachment capabilities (not in the sense of developing

a boundary layer velocity profile) for a given back pressure.

Discharge coefficient measurements S8

.. 0

p/J wnw~u~

Di.~hargc coefficient m

easurements

0 0 N

0 ... 0 IO

0 G

0 0 .. -0

.. -- .. --0 ·-- :u ... u ... ~ en en u ... Q.,

~

.!! ... E a: u " M .. • .I: u .. ~ e :::I e ·= e " u :::I

1 .. a. s >.. .. • .. .. " u " = .I: D. =

.!:! e :::I e :5 :;;; ...; ~

" .. :::I .. ;.:

59

Chapter 4

\Vater table study

A water table study was performed so that the fluid flow in long orifices could be visu-

alized and the centerline pressure distribution modelled. The water table was partic-

ularly useful in obtaining the shock wave structure in the orifice. The water table was

used rather than an optical system (eg.Schlieren, interferometry) and pressure probe

traversing system, because the water table was available whereas an optical/pressure

traverse measurement system would have to have been constructed. Therefore by using

the water table time and money was saved. However, since the results rely on the hy-

draulic analogy, this water study is viewed as preliminary to an optical/pressure traverse

study with air. A brief discussion of the analogy that relates water flow to compressible

flow will presented next.

Water table study 60

Hyd,.aulic a11alogy

A brief review of the hydraulic analogy is presented here. Readers interested in obtain-

ing more information should consult papers by Johnson (22) or Preiswerk(23) from

which this information was taken.

Analogies are methods of relating physical properties of different subjects to one an-

other. Common analogies are those that relate mass transfer or electrical flow to heat

transfer. Another analogy is the hydraulic analogy in which incompressible fluid surface

wave motion is related to compressible fluid flow. The correct situation for the analogy

occurs if water is allowed to flow over a horizontal surface with vertical side walls under

the effect of gravity.

Analogous to the sonic velocity in a gas is the Hsmall height" gravity surface wave ve-

locity. The velocity of a gravity surface wave is

c = [~~ tanh [2~hJl5

If the water is very deep, h > > I. so that

inh h d h [ inh ] h I d h 1 · · -X- approac es cJJ an tan -X- approac es , an t e wave ve octty 1s

c = ( ~~ ).s which is a function of wave length alone. On the other hand, if the water

. 21th [ 2rrh J 27th 1s shallow, A. > > h so that -X- approaches 0 and tanh ---,:- approaches -X-

and the wave velocity is c = (gh)..s , which is a function of depth alone. It is this second

wave velocity that is analogous to the sonic velocity in a gas.

W atcr table study 61

Water surface tension allows for the water to vibrate as a membrane and thus produce

another type of surface wave - capillary waves. Capillary waves are most influential in

shallow water. Therefore, a compromise in the height selection must be made so that the

surface wave velocity becomes independent of wave length and is unaffected by capillary

waves. A plot of the combined surface wave velocity as a function of wave length is

presented in Fig. 34 for various heights. Johnson states that a good height for high ve-

locity experiments is between 1/4 and 1/2 of an inch. In the present experiment, the

height of the water in the test section was between 0.3 and 0.8 in or equivalently between

7.6 and 20.3 mm. The upstream reservoir height was maintained at approximately 22.

mm.

The property analogies are derived from equations for isentropic flow in each type of

fluid motion. For instance, equations will now be derived that relate water surface

height to the compressible fluid properties temperature, density, and pressure. These

equations will be derived with reference to Fig. 35. It is assumed that the fluid flows out

of an infinitely large basin (subscript o) so that the velocity is zero. The Bernoulli

equation is written for a stream tube extending from the basin to a general location

downstream,

= p + .£..y2 + pgz 2

Solving for the velocity term, the following is obtained,

y2 = 2g(zo -z) + 2 <Po - P) p

It is assumed that the vertical acceleration of the flow is negligible compared to the ac-

celeration produced by gravity so that the static pressure is the hydraulic pressure ,


2.0 h(in) -u u

"' -c:: 1.5 -c.i ~ 0.5 -·u 0 u 1.0 > u > 0.2 <IS ~

0.50 0.1

5 10 wave length, A(in)

Figure 34. Surface wave velocity vs. wave length.


x.,, Yo• la

Figure 35. Free surface water now.

Water table study

z

y x~~~~~~~~~..::...x

(into page)

p

x,y,z h

64

and

P = pg(h - z) .

Inserting these values into the above equation for the velocity term gives,

V2 = 2g(h0 - h) = 2g.1hih0 •

This equation tells us that the velocity at any x,y location is independent of z and,

therefore, is constant over the entire depth. The maximum velocity the flow can have

is V max = (2gh,y, and the velocity ratio may be written,

For an ideal gas the following equation for the velocity ratio may be derived from the

energy equation

These equations show that the velocity ratios for water and gas flows are equal if

The relation between water height and density is found by relating the differential forms

of the continuity equations. For water the continuity equation can be written

o(hu) + o(hv) = ox oy 0 .

The compressible two-dimensional continuity equation is


o(pu) + ox o(pv)

oy = 0 .

It can be seen that these two equations are the same if p = h.

Remembering the isentropic ideal gas relation between density and temperature,

Pf Po = (T/To) 11(k - t> ,

the equivalent water table specific heat ratio may be found. Since the hydraulic analogy

requires p/p0 = T/T0 , it can be seen that k must equal 2. Finally, the ideal gas equation

gives the following relation between water depth and pressure:

The equation for the "Mach" number in terms of the water height is found from the ratio

of the water velocity to the water wave velocity,

or,

From this equation it can be shown that a height ratio, h/ho, of 2/3 produces "sonic"

flow. A summary of the analogous quantities is presented in Table 2.


If the flow becomes supersonic, shocks may occur and, because shocks are non-

isentropic, the above equations do not strictly apply. However, Preiswcrk points out

that for a wide range of conditions (small velocities, and/or \\'eak oblique shocks), the

shock loss is small and the analogy "of the two types of flow is still satisfied to a first

approximation."

Appa1·at11s

The equipment used in the water table study is presented in a sketch in Fig. 36 and in a

photograph in Fig. 37. The water table, which was constructed at Va. Tech prior to the

present study, generally consisted of two large .22 ml tanks, a .91 m x .61 m plexiglas test

section, and a traversing micrometer. For leveling, the table could be rotated by a

wedge/screw system and the height of each leg could be adjusted. Water was circulated

with a pump between the two tanks from left to right. The pump was fitted with a

rotameter which allowed for flow monitoring and reproduction. A throttle valve and a

plate downstream of the test section were used to regulate the upstream and differential

heights of the water in the test section.

A channel with a length to width ratio, l/w, of 3.65 and a width w = 30.5 nun was con-

structed from plexiglas; the orientation of the channel on the water table is shown in

Fig. 38.

A micrometer with a 0.0254 nun least count capable of traversing both longitudinally

and latitudinally was used to measure the water heights, the evenness of the test section


Flow~

L 45.7cm ... ,."1.------91.4<.:m------61.0cm_j

1 · lra ,·ersmg :rn<.:romctcr

T .... c u ,..... t'""I ,....

I I

i E ~

_L~~~-----~~~__L level

adjuc:m1ent~ wheel

throttle valve ~

Figure 36. Sketch of water table apparatus.

Water table study

Flowl

~ -rotameter

68

Water table study

69

test section

! eycomb hon

SC re en

I

\

\ I'\

60 cm

l

test section

\

Figure 38. Test section details.

Water table study

~[ ::i:

top view

channel

l I I

side view

~l =- 111 mm j wat er height ...:L ..f--adj ustment

plate T w= 30.5 mm

wing nut ~channel ~

I

70

bottom, and the level of the table. Longitudinal and latitudinal distances were measured

with 1.6 mm and 2.5 mm least-count rulers, respectively. The bottom was found to be

even within 0.25 mm. The table was leveled by filling with water and checking its height

at the four corners and on the center line at the exit of the channel. The maximum dif-

fcrence in height of the water at "level" was no greater than 0.15 mm.

A hydraulic diameter for the channel used in the water table study was calculated as

follows. The hydraulic diameter is defined as four times the flow area divided by the

wetted perimeter, dh = ~\ . The flow area for the channel is, A= wh, and the wetted w

perimeter is, P., = 2h + w. The hydraulic diameter is then,

h = 4v .. ·h c 2h + w

The height used in the calculation was the approximate height that produced sonic flow.

With an approximate upstream reservoir height of 21.6 mm, the height that produced

sonic flow was,

hc=ho[ :0 l = 21.6 mm (2/3) = 14.4 mm

The hydraulic diameter diameter based upon this critical water height is then,

4whc 4( 30.5)( 14.4) dh = =------

2hc + w 2(14.4) + 30.5

or,

dh = 29.6 mm

The length to hydraulic diameter ratio for the channel is l/dh = 111.3/29.6 = 3.76.


A Reynolds number for the channel based upon the hydraulic diameter calculated above

and a critical velocity of 0.30 m/s was calculated to be Redh = l.1x1 O'. This Reynolds

number is close to the range of values in the discharge coefficient measurements

(2 - 8xl0').

fVater table res11lts

For various back pressure ratios (height ratios squared, see Table 2), the centerline to

upstream pressure ratios were measured, the top and side views of the flow drawn, and

the streamline behavior and separation locations sketched. Streamlines were visualized

through the injection of ink. The pressure measurements are presented in Fig. 39. Three

scales are presented in this figure. The pressure ratio scale for k = 1.4 was obtained from

isentropic relations using the Mach number. Sketches of the flow are presented in Figs.

40 through 45, with streamline sketches in Figs. 40, 42, and 44. In the top view sketches,

dashed lines are meant to represent general flow boundaries and solid lines shock waves.

Photographs of the flow for the lowest back pressure ratio measured are presented in

Figs. 46 and 47.

Observations related to the pressure measurements and the flow visualization will now

be presented. Regimes of flow behavior will be discussed corresponding to ( 1) subsonic

flow throughout the channel, (2) fluid acceleration to supersonic conditions with subse-

quent deceleration to subsonic flow, and (3) fluid acceleration to supersonic flow. For

each regime, approximate 'b·•(;k pressure ratio boundaries are presented.


entrance 1.0 o.o exit

1.0

a o.•

o.e b 0.5

0.7 0.8

(h/h.)' c d o.e

\I c

0.5 o.e 1.0 --------.1--

0.4 g (P. P,),.,,

h o . .) 0.4

0.2 , .:i

0.2

0. I 2.0

o.o -----Ai> -..--~..;;;....~--...-~----~-...--~-..----...... --~ ....... ~--...-~----~-..--~-..-----.-~--~o.o -.) -2 -1 2 .) 4 :I • 7 II 10

dut:incc from channel entrance (:\.'w)

J b Cd C fg h I J k. (h,. h.): • U.'Jf, I.I.Sil 0.65 0.62 0 . .!.I 0.50 0 . .1~ 0 J.I 0.2.1 \).13 0.01 (I'., r,J • I) 'J':' o.ss o. ~; 11. ~n u 62 o.58 0.50 0 . .11 o. jO o.~.i 0.005

Figure 39. Centerline pressure vs. distance from channel entrance.


--I ( I

-- -- -.

- ,_., - --top view

Flow~

side view

Figure 40. Sketches of flow in channel for back prCMure ratio (k • 1.4) of 0.70.

Water table study

--,

----

74

', ~---..... -- ........... \/ ll J\ -- / "' r-~~~~~~""'"-~~~~~~~~~~~~~~--,./ --

top view

Flow c=>

side view

Figure 41. Sketches of now in channel for a back. pressure ratio (k'"' I.~) of 0.58.

75 Water table study

--- -..., ' ------~--------~----' I /I _ _.,,,. --

top view

side view

Figure 42. Sketches or now in channel ror a back pressure ratio (k • 1.4) or O.SO.


top view

Flow~

side view

Figure 43. Sketches of now in channel for a back pressure r:uio (k .. I.~) of O.~l.

Water table study

' ' I I

/

77

top view

Flow c::::::::>

side view

Ficure 44. Sketches of now in channel for a hack pres.o;ure r:itio (k • 1.4) of O.JO.


top view

Flow c:::::>

side view

Figure 45. Sketches of now in channel for a back prCS!>Ure ratio (k ~ 1.4) of 0.005.

Water table study

/

/ /

/

79

c. 3 -... . -

Water table study

80

-~

·= .. ~ ~

:§, ~

c: ~

~ -~

::3:: 6

0 -

-;.;....

:-:::

0 ~ iii ~ ~

=-...:::: ;.J

.2 ~

"'" 2 ~

~ ;: u ·= ~ 'c -§_ ~

~

~ r-.: 'o:f'

~

~

Water table study

81

( 1) Subsonic

The pressure ratio range for complete subsonic flow throughout the channel is

0.62 < (h8/h,,)2 < I, or 0. 70 < P 8/P 0 < I. The flow visualization presented in fig. 40 is

for the back pressure ratio that corresponds to the subsonic flow limit, curve d in fig.

39. It can be seen that no wave structure was evident. The contraction of the stream-

lines towards the channel centerline at the orifice entrance indicates separation.

Boundary layer development is indicated downstream of the pressure increase by a con-

vergence of the streamlines.

The pressure measurements for the higher back pressures in this range (curves a and b

in Fig. 39) are generally smooth. At the highest back pressure ratio measured (curve a),

the maximum Mach number is 0.28 and is located about one width from the channel

entrance. The pressure levels off after increasing until about two channel widths. As the

back pressure is decreased, the pressure changes occur over a smaller distance. Rcf1ect-

ing on sharp-edged orifice flow behavior, this may be the result of the vena contracta

moving towards the orifice entrance. At the lower back pressure ratios (curves c and

d), the pressure increase is followed by a pressure decrease. The maximum Mach num-

ber for the lowest back pressure ratio in this regime, P8/P0 = 0.70, is 0.97.


( 2) Subso11ic / superso11ic

The pressure ratio range for this flow regime lS

0.42 < (h8/l\,f < 0.62, or 0.50 < P8/P0 < 0.70. Oblique shocks emanating from near

the channel entrance were observed in this flow regime (Figs. 41 and 42). The oblique

shocks are seen to reflect from the centerline as Mach reflections with a resulting normal

shock occurring at the channel centerline.

The shocks were, at first, a surprise because the flow was expected to be similar to that

in a sharp-edged orifice. However, upon considering the difference in boundary condi-

tions for the two types of flow, the cause of the oblique shocks seemed to be explained.

It can be seen from the streamline sketched in Fig. 42 that the streamlines contract and

start to diverge toward the channel wall. These walls require the streamline direction to

become axial. Since the flow is supersonic, the necessary turning occurs through an

oblique shock.

A back pressure ratio of 0.50 (k = 1.4) seems to be the limiting value for which a \iach

reflection occurs at the centerline. The maximum Mach number measured for this back

pressure ratio (curve g) was 1.5. The subsequent static pressure rise corresponds quite

closely to that of a normal shock and is in the range that would produce boundary layer

separation.


( 3) Supersollic

Supersonic flow occurred throughout the channel for (h8/h,,)2 < 0.42, or P8/P0 < 0.50.

The sketches (Figs. 43, 44,and 45) show that the oblique shocks emanating from the

channel entrance reflect from the centerline as regular reflections and subsequently as

Mach reflections from the channel walls. Just upstream of the shock reflection at the

channel wall, a second oblique shock is seen to emanate. The pressure measurements

demonstrate that the maximum Mach number for all back pressure ratios in this range

is approximately 1.8. For back pressure ratios below 0.30 (k = 1.4), the pressure meas-

urements and wave structure in the channel are nearly the same. The location of the

maximum Mach number is about I w from the channel entrance and the initial pressure

increase is complete at about 2 w.


Chapter 5

Analysis

Attempts were made to try to explain the discharge coefficient and water table pressure

measurements making use of fundamental compressible fluid dynamic equations of ideal

gas flow. Results of one-dimensional mixing and friction analyses and maximum Mach

number predictions are discussed. One-dimensional long orifice flow behavior is sug-

gested by I) the subsonic fluid dynamics of sharp-edged orifices in which the flow con-

tracts to a near uniform, axial velocity, and 2) the agreement of measured incompressible

long orifice discharge coefficients and those calculated from Lichtarowicz's pressure

measurements assuming the flow was uniform at the minimum pressure (see literature

review).

It was found, however, that the one-dimensional analysis explained the flow in long

orifices over a limited range of back pressure ratios. This is not surprising since l)

supersonic sharp-edged orifice flow, unlike subsonic sharp-edged orifice flow, does not

contract to a one-dimensional velocity, and 2) it was found in the water table study that

Analysis 85

oblique shocks, which are indicative of two-dimensional flow, exist in long orifices if the

back pressure is low enough. The inability to explain the flow under these conditions

with one-dimensional analyses was interpreted to mean that the flow has become sig-

nificantly two-dimensional due to supersonic effects.

011e-dbne11sional analysis

The one-dimensional analyses explained here consists of a basic uniform mixing calcu-

lation and modifications. The modifications involved adding friction or boundary layer

blockage effects to the initial calculation to try to explain the discharge coefficient in

orifices with l/d = 2. The 2 l/d orifice was chosen for comparison because Lichtarowicz' s

pressure measurements indicated that this length orifice was large enough to allow the

flow to reattach for all pressure ratios, and the discharge coefficient measurements for

this orifice were generally the largest.

The calculated discharge coefficients are compared with approximate values determined

from 2 l/d orifice measurements of Nakayama and the present author. (Deckker and

Chang's measurements were not considered in determining the approximate values be-

cause the beveled backs on their orifices seemed to significantly affect their perform-

ance). The approximate values are:

C 0 = .80 for .9 < Pa/Po < 1

C 0 = .81 for .7 < Pa/P0 < .9

Analysis 86

C0 = .82 for .6 < P8/P0 < .7

C0 = .83 for P8/P 0 < .6

The calculated pressures are compared with the centerline pressures measured in the

water table channel in percent pressure change plots versus maximum \fach number.

Percent pressure increase and decrease were calculated as shown in Fig. 48. This figure

gives a qualitative description of the pressure changes assumed to occur in the analyses.

The pressures from the water table experiments were converted to values for a gas with

a specific heat ratio of 1.4 through the Mach number.

Basic calculation

In the initial analysis the flow was assumed to expand isentropically to a uniform, axial

maximum Mach number, and then mix to a uniform axial velocity at the exit. The flow

was considered adiabatic throughout. This situation is shown in Fig. 49. At station I

the pressure is assumed to act uniformly across the full duct width. With these as-

sumptions the equations governing the flow are the continuity equation,

the momentum equation,

and the energy equation,

Analysis 87

p

uniform mixing

\ friction

.-pl-~--=-- -\- ---..

% pressure increase = 100[ ~: - l]

% pressure decrease = too[ 1 - ~: ]

Figure 48. \lcthod of calculating percent pressure changes.

Analysis 88

.... >

... 0

M

~ >-.. u

Analysis

.. e .. >

1"'-~ I

·~ c u u

.;. .§ iii 'ii "' 'iii .... ! ; ~

. :(

-f"': 0:. "' " .. = lllD ;.:

89

The mass flow was calculated from the following equation,

With an assumed area and conditions at station I, a quadratic equation for the velocity

at station 2 was found by combining the above equations. The following result was

obtained,

where,

and

.5 -b + [b2 -4ac]

V2 = --------2a

R a=---2Cp

With the velocity at station 2 calculated, properties at station 2 were determined from

the governing equations.

The discharge coefficient is calculated from the ratio of the actual flow rate to the

isentropic flow rate that would occur with a pressure ratio P2/P 0 u C0 =

Analysis 90

The area at station I was assumed to be the area at station 2 times I) the sharp-edged

orifice discharge coefficient at the pressure ratio at station I, A1 = Co.sE(P1/P01 ), for

subsonic maximum Mach numbers, or 2) the product of the sharp-edged orifice dis-

charge coefficient and the isentropic area ratio at the maximum \1ach number,

A1 = Co.sE ~· (P1/P0 ), for supersonic maximum \1ach numbers. Implied in the station 1

area assumption is that the flow through a long orifice is equal to that through a

sharp-edged orifice exposed to the conditions at station I - the location of the maximum

Mach number. This is noted because the maximum \1ach number calculations pre-

sented at the end of this section are based upon this assumption. The mixing calcu-

lations were performed until the area at the maximum Mach number was equal to the

full area. The areas used in the calculations are presented in Table 3.

The pressure increases calculated from the mixing analysis arc compared with the pres-

sure increases measured on the water table channel centerline in Fig. 50. It can be seen

that the measured and calculated pressure increases compare well up to a maximum

Mach number of approximately 1.3 which corresponds to a mixed pressure ratio of 0.68.

The pressure increase calculated at the largest maximum \1ach number, \1m.,. = 1.51, is

the one in which the area at station 1 is the full area and, therefore, corresponds to the

pressure increase produced by a normal shock.

Included in the pressure increase plot is the static pressure increase that would result if

a normal shock occurred at the maximum Mach number (dotted line). The pressure

increase calculated in the mixing analysis and measured on the water table for supersonic

maximum Mach numbers up to 1.5 is larger than produced by a normal shock alone.

This is the result of the pressure rise of the normal shock being enhanced by a pressure

rise due to mixing. The existence of a normal shock in this maximum \1ach number

Analysis 91

;;;.. :I !. ~

"" ;·

-Q N

" • . .. I x v

Ill

"' ~ Ill I: 0 z -Ill I: :>

"' "' la.I I: A. I

2.-0

220

200

180

180

1.-0

120

100

80

eo

.-o

20

0 o.o 0.2 0.4 o.e o.a

I I

I I

I I

I I

I I

I I

I I

I I

1 . 0 1 . 2 1. 4

MAXIMUM MACH NUMBER 1 • e

• • • ONE-DIMENSIONAL MIXING ANALYSIS ----- NORMAL SHOCK M " " WATER TABLE DATA Figure SO. Prc3t!>Urc incrc1uc n. maximum Mach number.

1 . 8 2.0 2.2

range is supported by the water table flow visualization (Figs. 41 and 42) in which the

oblique shock emanating near the orifice entrance reflects from the centerline as a Mach

reflection with a normal shock occurring at the centerline.

The water table pressure increases for maximum Ylach numbers greater than 1.5 are

generally less than those produced by a normal shock which is consistent with the ob-

servation that the oblique shock reflects from the centerline in a regular fashion in this

range of maximum Mach numbers (Figs. 43 through 45).

The results of discharge coefficients calculated in the mixing analysis are presented in

Fig. 51. The values calculated from the mixing analysis arc represented by squares. The

approximate measured discharge coefficient is represented by a dashed line. The dis-

charge coefficients calculated from the mixing analysis generally increases with decreas-

ing pressure ratio from 0.83 at a pressure ratio of 1 to 0.89 at a pressure ratio of 0.67.

On the average the calculated discharge coefficients are about 5% larger than the values

measured for the 2 l/d orifice.

Pressure decrease calculations

To try to further explain the discharge coefficients for the orifice with an l/d = 2, a pres-

sure decrease was assumed to occur after the pressure increase produced by mixing and

the normal shock. The reason a pressure decrease results in a smaller discharge coeffi-

cient is that the ideal mass flow used in calculating the discharge coefficient increases

with decreasing pressure ratio (see flow factor plots in discharge coefficient measurement

chapter for ideal mass flow curve). A calculation was made to determine the "necessary"

pressure decrease after the flow mixes so that the discharge coefficient of the 2 l.'d orifice

Analysis 93

l l ' I \ l I ' I ' c

I ' I I I I ' I I \ I I ' I I I I I I I

0 in

0 .,

0 .,

0 a.

a. .,

., ,..

0 0

0 0

0 Q

u

Analysis

0 .,

0 .,

0 ,..

., .,

., IO

0 0

0 0

0

0 0 0 N

0 ~

0 • 0 i)~

c.. - 00.. -.~ .... ~

., .... .u

o~

"' "' C) .... c..

,... 0 ., 0 a. 0 0 ...

N u

.,, ~ ~

.. .,,

12 >-

"' .J

u ::I

< ii

z >

< ~

u ; ()

"' .. z

u e x

..t::.

:J .i "' ·~

N

.J ....

IN<

-.; c

OIZ

.. '°°

.: .J

,_ ~

.Jt/) u

• •Z ,;

ozw ::I

IUO:J

u 'i

11:--u

:>~a

"' tn(J I = .. <

-Ill ·u

l&.IQ:Z E

~o

~ u

IH u lie

.. .. ..t::. u .. :-a ... = c 0

.!!! .. .. Cl.

e = 1,,,,1

...: V'I

u .. ::I llll ~

94

was produced. The calculations were performed until it was no longer possible to correct

the discharge coefficients calculated in the mixing analysis to the desired values through

a pressure decrease.

The necessary percent pressure decrease is presented in Fig. 52 and is represented by

squares. It can be seen that at a maximum Mach number of 1.3, a pressure decrease

of approximately 12.5% is required to produce the discharge coefficient for the 2 l/d

orifice. Remember, for this maximum Mach number the pressure ratio after mixing was

0.68. If a 12.5% percent pressure decrease were to occur after mixing from a maximum

Mach number of 1.3, the back pressure ratio would be 0.60. Note that this back pres-

sure ratio is the largest value that resulted in a constant mass flow rate for the 2 l/d

orifice. This suggests that flow in a long orifice over the full range of pressure ratios that

the mass flow changes may be explained by one-dimensional mixing followed by a

pressure decrease.

To try to explain possible causes of the pressure decrease, friction and blockage calcu-

lations were performed.

Friction

Frictional affects were examined through one-dimensional, constant area, compressible,

adiabatic (Fanno flow) calculations. Property variations in Fanno flow have been 4 fl' tabulated(24) in terms of non-dimensional maximum lengths, -d-, which do not

" produce choked flow. The friction factor, f, is a mean value over the duct length that a

calculation is being performed. For given upstream conditions at 2, the conditions at

another location, B, can be determined as follows,

Analysis 95

N

N I

N

0 ' .J

0 <> z

N

x :::E .,

a: Id ~

I&. <

ro >-a: < II)

r-4 II) "

• Id

~

0 ::::-

a: Id

.: la.I z .a: m

e

:J ! ::I ~

c:: N

z ~

;j

.. J:

-.e 0

s <

::I

0 :J

s :J

·~

~

e :::E

.; ;,.

x "

., ...

< .. "

:::E ..

0 "' " ~ " .. ::I

ro " " "

0 :::

N .... ll'l

• e a

" .. 0

' ::I N

.J ~

z N

0

0 ~

0 It 0

I&.

0 I 0

If) 0

If) 0

If) 0

0 •

,., ,.,

N

N

( •• ~ -

)4

) 3SV3~030 ::a~nss3~d •

Analysis

96

where 128 is the duct length between locations 2 and B.

In the calculations, the conditions at l were assumed to be those that resulted from the

mixing calculation. The friction factor was found from the Moody diagram for a smooth

pipe and Reynolds numbers in the range in which the discharge coefficient and water

table measurements were performed (l - 8x 104). The friction factor under these condi-

tions was found to be nearly constant at a value of 0.005. Friction was assumed to act

over 2 l/dh.

The results of the decrease in pressure caused by friction have been included in Fig. 52.

Also, discharge coefficients were calculated and these results have been included in Fig.

51. It can be seen from these figures that the pressure decrease produced by friction is

not large enough to correct the discharge coefficients calculated in the mixing analysis

to the values for the 2 l/d orifice. This suggests that boundary layer blockage may con-

tribute to the pressure decrease.

Blockage

A rectangular cross-section blockage calculation was performed in which the flow was

assumed, as in the initial mixing calculation, to expand to a one-dimensional axial con-

dition at station l, but this time was assumed to mix to a l/n'h velocity profile,

Analysis

[ Jl'n V2= V2max _Y_ , ·

w/2

97

A representation of this situation has been included in fig. 49. The density at station 2

was assumed to be constant across the full width. The equations used in this calculation

were the same as in the initial mixing calculation but this time velocities at station 2 were

integrated over the half width. The continuity equation becomes

w/2 m = P1V1A1 = P2J V2 dy

0

The momentum equation is now

w:2 2 P1A2 + mV1 = P2A2 + p2J V2 dy

0

The velocity component in the energy equation was mass averaged,

v~ [ p J w12 v; T0 = -- + T1 = -- J -.- dy + T2 2Cp 2Cp 0 m

Once again, these equations were combined and a quadratic equation developed for the

velocity at station 2. The quadratic equation in terms of the maximum velocity at sta-

tion 2 is

where,

and,

AnaJysis

.5 -b + [b2 - 4ac]

V2max = --------2a

R[t+n] [ n J a = 2Cp 3 + n - 2 + n

98

c = -RT0 •

Values of n equal to co correspond to uniform flow at station 2 and, in the limit, the

equations reduce to those used in the uniform mixing analysis.

Values of the profile shape parameter, n, were chosen to produce the discharge coeffi-

cients for the 2 l/d orifice. Calculations were performed until it was no longer possible

to produce the desired discharge coefficients. The necessary values of n arc presented

in Table 4 together with the corresponding percent flow blockage. The flow blockage

was calculated by determining the flow deficit that resulted from a velocity profile ex-

isting at station 2 rather than the uniform maximum velocity,

or,

w/2 Blockage (V2maxP) = pJ (V2max - V2)dy

0

100 W/2[ V ] percent Blockage = --J 1 - V 2 dy w/2 0 2max

For the velocity profile assumed at station 2, the equation for the now blockage be-

comes,

percent Blockage = 100[ 1 ~ n J

It can be seen from Table 4 that for maximum \1ach numbers up to 1.05, the velocity

profile parameter is between 8 and 12 and the flow factor is between approximately 8

and 11 %. These values are close to those which approximate a fully developed turbu-

lent velocity profile in which n = 7 and the percent flow blockage is 13 % . For maximum

Analysis 99

~ach numbers above l.05, a larger amount of blockage is required ( 18%) which might

in the physical situation be caused by boundary layer separation.

fVater table pressure decrease

The pressure decrease measured on the water table is presented in Fig. 53. For com-

parison, the necessary pressure decreases after mixing calculated to produce the ap-

proximate discharge coefficients of the 4 I/ d orifice have been included in the figure. The

necessary pressure decrease for the 4 l/d orifice discharge coefficient measurements is

used for comparison because this orifice had a length to diameter diameter ratio that

was closest to the length to hydraulic diameter ratio for the water table channel,

l/dh = 3. 76. The approximate 4 l/d orifice discharge coefficients used in the necessary

pressure decrease calculation were,

C0 = .79 for .9 < P8/P0 < 1

C0 = .80 for .7 < P8/P0 < .9

C0 = .81 for .55 < P8 /P0 < .7

C0 = .82 for P8 /P 0 < .55

A friction pressure decrease calculation was also performed assuming friction acted over

4 l/d and the results have been included in Fig. 53.

It can be seen that the necessary pressure decrease for the 4 l/d orifice compares rea-

sonably well with the centerline pressure decrease measured on the water table. Both the

shapes of the curves and the actual values are in good agreement. Once again, the

Analysis 100

> :::s p

-40 ~

~·

3!>

,.... .. 30

~

I ::!£ v

w 2!> Ill < w ~ 0 20 w 0

w k ~ Ill

1 !> Ill w k 0.

10 • !>

0

~

o.o 0.2 0.4

WATER TABLE FRICTION,L/0-4

o.e

J•igure 53.

0.8 1. 0 1 . 2 1. 4 1. e 1. 8 2.0 2.2 MAXIMUM MACH NUMBER

8 8 8 NECESSARY AFTER MIXING,L/0-4

Pres~ure decrease n. maximum l\lach number, l/d = 4.

pressure decrease caused by friction is not great enough to account for the necessary

pressure decrease.

At a Ylach number of 1.3, the pressure decrease necessary to produce the discharge co-

efficient measured with the 4 l/d orifice is about 22%. This pressure decrease applied to

the pressure ratio that resulted from the mixing analysis for this maximum Mach number

(PJP01 = 0.68) results in a back pressure ratio of 0.53. As was the case with the neces-

sary pressure decrease calculations for the 2 l/d orifice, this pressure ratio is approxi-

mately equal to the largest back pressure ratio that produces constant mass flow in the

the 4 l/d orifice, P8/P0 = 0.55.

Summary of one-dimensional flow calculations

The results of the one-dimensional calculations indicate that for maximum \1ach num-

bers up to approximately l.3 the initial pressure increase measured on the water table

is explained by a simple one-dimensional expansion and mixing supplemented by shocks.

The pressure ratio after mixing for a maximum Mach number of l.3 is approximately

0.68. The discharge coefficients of orifices with an l/d = 2 and 4 can be explained by a

pressure decrease due to friction and blockage following the initial pressure increase.

For a maximum Mach number of 1.3 pressure decreases after mixing of approximately

12.5% and 22% produce the discharge coefficients of the 2 l/d and 4 l/d orifices, re-

spectively. If these pressure decreases are applied to the mixed pressure ratio of 0.68,

one-dimensional mixing followed by a pressure decrease could explain the flow in long

orifices for the complete range of back pressure ratios in which the flow rate changes.

Analysis 102

Friction alone does not adequately account for the pressure decreases that are necessary

to produce the discharge coefficients measured for orifices with an l/d= 2 and 4. If ve-

locity profile development causes the necessary pressure decrease in the 2 !,id orifice, the

flow blockage for maximum Mach numbers up to 1.05 is between 8 and 12'% and for

maximum Mach numbers greater than l.05 is 18%.

The limited range of maximum '.\.1ach numbers over which the one-dimensional analysis

applies suggests that the flow in long orifices becomes significantly two-dimensional due

to supersonic effects for maximum Mach numbers greater than 1.3.

l'Vlaximum Mach number calculation

The final calculation to be discussed was one in which a maximum :'vtach number was

predicted as a function of back pressure ratio. It was noted in the one-dimensional cal-

culations that this was implied in the station l area assumption.

The expected maximum Mach number/back pressure ratio relation was determined as

follows. For a long orifice at a given back pressure ratio, P8/P0 , the mass flow rate is

given by

Following the one-dimensional flow analysis, the mass flow rate is also assumed to be

given by the sharp-edged orifice discharge coefficient at conditions corresponding to the

maximum Mach number. Thus

Analysis 103

Equating these two expressions and using the measured discharge coefficients for the

long and sharp-edged orifices, the maximum Mach number can be found for any back

pressure ratio.

The results of this procedure are shown as squares in Fig. 54 for the l/d = 2 orifice.

For comparison, Fig. 54 also shows the maximum \fach numbers measured on the

centerline of the water table channel and the Mach numbers for isentropic flo\t: corre-

sponding to the back pressure. The predicted and the measured Mach numbers are in

reasonable agreement over the whole range of back pressure ratios, and they show quite

a different trend than the isentropic Mach numbers. For pressure ratios above approx-

imately 0.2, the maximum Mach numbers predicted and measured are higher than the

isentropic values. For pressure ratios below about 0.4, the predicted and measured

maximum Mach numbers remain relatively constant at about l.6 and l.8 respectively,

whereas the isentropic Mach numbers increase rapidly. Figure 54 may be useful for

predicting maximum Mach numbers in tip leakage flow in transonic turbines, for exam-

ple.

Analysis 104

> ~ ~ -< "' r;;·

Q CA

. 0 z l: 0 < 2

2 :::> :I -x < :1

2.G

2 ...

2.2

2.0

1 • 8

1 • e

1 ...

, . 2

, . 0

o.e

o.e

0 ...

0.2

0.0 1 • 0

.................... · ,..········

o.e

WATER TABLE EXPECTED

_,..,.·"" ... ~·· ...

,, .. ····· ... .. ~····

.. .. ... -···· .-···"'

. . .. ,.. .--..

o.e o.• 0.2 back pressure ratio

I

,/

•••••••••••• ISENTROPIC RELATION

l<-igurc S4. :\luimum l\lach number n prcs~urc ratio.

o.o

Chapter 6

Conclusions

This study was concerned with explaining the fluid dynamics of flow through plain-faced

long orifices. The Reynolds number range considered was Red > 104.

Disclia1·ge coefficiellt n1easure111e11ts

Discharge coefficient measurements of air flow through sharp-edged orifices and long

orifices with values of l/d between 1/2 and 8 demonstrate the discharge coefficient in-

creases with decreasing pressure ratio. The mass flow in orifices with an l/d of approxi-

mately 2 is largest and the mass flow in a sharp-edged orifice is smallest for pressure

ratios above 0.27. For pressure ratios below 0.27, the flow rate in the sharp-edged orifice

is largest. The mass flow through orifices with an l/d less than 2, increases with de-

creasing back pressure ratio until it is approximately equal to that in the 2 l/d orifice.

Conclusions 106

The mass flow in 4 and 8 l/d orifices is only slightly smaller than that in the 2 l/d orifice.

The discharge coefficient for orifices with l/d = 2 is approximately 0.80 for pressure ratios

above 0.90, and 0.83 for pressure ratios below 0.60.

The mass flow is relatively constant in the orifices with an l/d = l and 2 for back pres-

sure ratios below approximately 0.60, in the l/d = 4 and 8 orifices for back pressure ra-

tios below 0.55, and in the l/d = l /2 orifices for back pressure ratios below 0.45.

The sharp-edged orifice discharge coefficient increases from 0.61 at a pressure ratio near

one to a value of 0.82 at a pressure ratio of approximately 0.27. C nlike the flow in long

orifices, the mass flow in the sharp-edged orifice did not become constant in the pressure

ratio range tested. Theoretically, the mass flow in a sharp-edged orifice continues to

increase until a pressure ratio of 0.0395 is reached at which point the discharge coeffi-

cient is 0.85.

JV at er table study

A water table study of the flow in a channel was performed to gain an understanding

of flow behavior in a long orifice. The results indicate that the flow in a long orifice

separates from the sharp corner at the orifice entrance, it accelerates to a maximum

Mach number, and then the pressure increases. The flow reaches the maximum Mach

number within one diameter from the orifice entrance and the pressure increase is com-

plete by two diameters from the orifice entrance. The pressure decreases after the initial

pressure increase for back pressure ratios above 0.50.

Conclusions 107

If the maximum Mach number is supersonic oblique shocks will exist. For the higher

back pressures that produce supersonic maximum ~ach numbers, 0.50 < Ps < 0.70, Po

the oblique shocks reflect as "Mach reflections" and the pressure increase is large enough

to produce subsonic flow. The maximum Mach number can become as large as 1.5 for

a back pressure ratio of 0.50.

Back pressure ratios below approximately 0.50 result in the oblique shocks reflecting as

"regular reflections" and the flow on the centerline remaining supersonic throughout the

orifice once supersonic speeds are reached. For back pressure ratios below approxi-

mately 0.30, the wave structure and the centerline pressure variation is constant and the

maximum Mach number is 1.8.

Analysis

One-dimensional analyses were made to try to account for the orifice discharge coeffi-

cient measurements and the centerline pressure variations suggested by the water table

measurements. The basis for the analyses was the assumption that the How entering the

orifice accelerated isentropically to a vena contracta with an area ratio given by the

discharge coefficient of a sharp-edged orifice. For maximum :\tach numbers less than

or equal to 1.0, the flow then mixed to fill the orifice. For maximum :Vlach numbers

greater than 1.0, the flow further expanded isentropically from the sonic "throat" and

then mixed one-dimensionally. The pressure of the mixed-out flow was the back pres-

sure for calculating the orifice pressure ratio, and the minimum pressure in the orifice

Conclusions 108

(just before mixing) gave the pressure ratio for the sharp-edged orifice discharge coeffi-

cient.

These analyses worked well for maximum Mach numbers less than 1.3. They showed

that the higher discharge coefficients of long orifices compared to sharp-edged orifices

are due to pressure rises in the orifices caused by mixing and shock waves. These in-

creases m discharge coefficients are partly offset by friction and boundary layer

blockage.

For maximum Mach numbers greater than approximately 1.3, the orifice discharge co-

efficients and centerline pressure variations could not be explained by simple one-

dimensional analyses. This suggests that the flow in long orifices then becomes

significantly two-dimensional due to supersonic effects and oblique shock waves.

Conclusions 109

Bibliography

1. Kirchhoff, G., "Zur Theoreie freier Flussigkeitsstrahlen," Journal fur \1athematik Bd. Lxx.Heft 4, Berlin, 1868.

2. Rayleigh, J.W.S., "Notes on Hydrodynamics," 1876, Scientific Papers, Vol. I, pp. 287-334, Dover Publications, 1964.

3. Benson, R. S. and Pool, D. E., "The Compressible Flow Discharge Coefficients for a Two-Dimensional Slit," Int. J. \1ech. Sci., Vol. 7, pp. 337-353, 1965.

4. Chaplygin, S. A., "Gas Jets," Sci. Annals, Moscow Cniversitv Phys-\fath publica-tion No. 21, (1904). [Translation: J\:ACA TM: 1063, (194...t)].

5. Frankl, F. I., "The Flow of a Supersonic Jet from a Vessel with Plane Walls," Pok!. Akad. Nauk., SSSR, 58, 381 (1947).

6. Benson, R. S. and Pool, D. E., "Compressible Flow Through A Two-Dimensional Slit," Int. J. \1ech. Sci., Vol. 7, pp. 315-336, 1965.

7. Guderley, K. G., "The Discharge from a Vessel," The Theorv of Transonic Flow, pp. 124-127, Pergamon Press, Oxford, 1962.

8. Leipmann, H. W., "Gaskinetics and Gasdynamics of Orifice Flow," Journal of Fluid Mech., Vol. IO, pp. 65-79, 1960.

9. Sanderson, E. W., "Flow Through a Long Orifice," B. Sc. Thesis, :\'ottingham Cni-versi ty, 1961.

10. Johansen, F. C., Aeronautical Research Council, R and \1 !\'o. 1252, London, 1929.

11. Spikes, R. H. and Pennington, G. A., "Discharge Coefficients of Small Submerged Orifices,' Proc. Instn. Mech. Engrs., 173, p. 661, London, 1959.

12. Birkhoff, G., Jets, Wakes, and Cavities, p. 230, Academic Press Inc., 1957.

Bibliography 110

13. Perry, J. A., Jr., "Critical Flow Through Sharp-Edged Orifices," Transactions of the ASME, p. 757, October 1949.

14. Grace, H. P. and Lapple, C. E., "Discharge Coefficients of Small-Diameter Orifices and Flow Nozzles," Transactions of the ASME, p. 639, July 1951.

15. Lichtarowicz, A., et al., "Discharge Coefficients for Incompressible ]'\:on-Cavitating Flow Through Long Orifices," Journal Mechanical Engineering Science, p. 210, Vol. 7, No. 2, 1965.

16. Nakayama, Y., "Action of the fluid in the Air Micrometer (2nd Report, Charac-teristics of Small-Diameter Nozzle and Orifice, No. 2, In the Case of Compressibility Being Considered)," Bulletin of Japan Societv of Mech. Engrs., p. 516, Vol. 4, 1961.

17. Deckker, B. E. L. and Chang, Y. F., "An Investigation of Steady Compressible flow through Thick Orifices," Proc. lnstn. Mech. Engrs., Vol. 180, pt. 3J, p. 312, 1965-66.

18. Leipmann, H. W., et al., "On Reflection of Shock Waves from Boundary Layers," ~ACA TN 2334, 1951.

19. Flowmeter Computation Handbook, ASME, 7'ew York, 1961.

20. Temperature Measurement Handbook Tables, OMEGA Engineering, Inc., 1981.

21. Deckker, B. E. L., "Hysteresis in the Flow through an Orifice," Letter, ".'\ature, p. 904, Vol. 214, :\fay 27, 1967.

22. Johnson, R. 1-1., "The Hydraulic Analogy and Its Cse with Time Varying flows," G.E. Report No. 64-RL-(3755C), August 1964.

23. Preiswerk, E., "Application of the Methods of Gas Dynamics to Water Flows with Free Surface. Part I. Flows with no Energy Dissipation," :'\iACA T\l 934, 1940.

24. Saad, M. A., Compressible Fluid Flow, Table A4, Prentice-Hall, Inc., >:cw Jersey, 1985.

25. Kline, S. J., and McClintock, F. A., "Describing L'ncertainties in Single-Sample Ex-periments," Mechanical Engineering, Vol. 75, 0:0.l, pp. 3-8, Jan. 1953.

Bibliography 111

Appendix A

Discharge coefficient equations and term errors

The error in the discharge coefficient measurement was calculated using the square

error method (25). The equations used in the discharge coefficient calculation were:

= (P +P ~d2[ 2 k [1 -[ (P + Patm) ]¥]]·5[ P + Patm ]+

og atm 4 RT0 k - I (P0 g + Patm) Pog +Palm

Discharge coefficient equations and term errors 112

Partial derivatives were taken with respect to each measured term to evaluate their in-

fluence. These values were multiplied by their individual errors and the sum of the

squares was taken to evaluate the overall error,

A list of each measured item's error is presented below. The least count and calibration

technique have been included where applicable.

item error

I.Differential mercury manometer .2 inHg or 0.677 kPa

with a 0.1 in. least count.

2. Upstream pressure transducer . l psi or 0.689 kPa

calibrated with a deadweight

tester. This transducer was connected

to a null amplifier which in turn

was connected to a

10-l mV least

count DV:vt.

3. Orifice diameter measured with 2 in or 5.08x 10- 3 mm 10000

a traversing microscope with

0.0001 in. least count micrometer.

Each diameter was measured

at 8 locations around the orifice


and a mean taken.

4. Differential pressure transducer

calibrated with mercury manometer

and deadweight tester. The mercury

manometer was used for calibration

for differential pressures up to 68kPa.

5. Omega thermocouples.

The thermocouple was connected

to an amplifier which was connected to

a 10- 1mV least count DVM.

The amplifier amplification factor was

found by submerging a thermocouple

in a hypsometer and ratioing the

DVM output voltage to the input

value specified in the Omega tables.

6. Stopwatch with 1/ 100 of a

second least count.

7. Tank volume measured by filling

with water and weighing.

8. Mercury barometer with 0.01 in.

least count.

Discharge coefficient equations and term errors

.1 inHg or 0.339 kPa

1 K

1 sec

1 lbm of water or 4.4xl0' m- 3

. l inHg or 0.339 kPa

114

Although the lengths of the orifices and the downstream dial pressure measurements

were not used in the discharge coefficient calculations, they were used to distinguish the

measurements for the different orifices. The length of orifices with an l/d greater than

1/2 were measured with a hand held micrometer with a least count of 2.5xI0- 4mm. The

beveled backs on the sharp-edged and 1/2 l/d orifices prevented this from being used to

measure their lengths. Instead a dial gage with a 0.0254 mm least count attached to the

microscope head was used. The length was measured by taking the difference in dial

gage readings when the microscope was focused on the location where the cylindrical

and beveled sections intersect and the paper on which the orifice lay. By either meas-

uring technique four length measurements were taken each at different locations around

the orifice hole and a mean was taken to determine the length. The downstream dial

pressure gage had a 7 kPa least count and was calibrated with a dead weight tester.

The possible errors in the discharge coefficient measurements have been previously pre-

sented in Fig.19 as a function of back pressure ratio. The errors presented arc the ones

calculated for the sharp-edged and 2 1/d orifice measurements. Since the discharge co-

efficients of these orifices bracketed the values obtained with the other orifices, the errors

presented in the figure represent an error band for measurements with all orifices. It can

be seen that the errors in the measurements are expected to be between 1.5 and 3.5%.

However, the error for pressure ratios between 0.90 and 0.30 is nearly constant at a value

of 2%. Larger errors were predicted for pressure ratios above 0.90 because of the sensi-

tivity in the discharge coefficient calculation to the pressure ratio measurement. For

pressure ratios below 0.3, to maintain choked flow across the pressure ratio adjustment

valve, it was necessary that the final pressure in the collection tank be smaller than for

higher pressure ratio measurements. This resulted in the experiment being run for

smaller amounts of time and smaller pressure changes occurring in the collection tank.


Therefore, the predicted error is larger for pressure ratios below 0.3 because of larger

errors in the time measurement and in the tank pressure measurements.

The error for the present discharge coefficient measurements is comparable to the values

others authors quoted for their discharge coefficient measurements. Grace and Lapple

estimated that their measurements had possible errors between 1 and 3% and Deckker

and Chang estimated an error of 1.6%.


Tables

Appendix B

Tables

117

Tables

Ill c: 0 ·~

c: "' e :.; "' u I: ·: 0

"':)

---c: ~

-:: -CJ '.J t:: ·;:::: 0

0 "T

I')

'° 0 0 0 -0 0 v ... -VJ

;_ 17

72

77

UJ

;r.

a-. V'1 ..,. 0 ..,. I')

'° 0 0 0 ,._, ---0 N

---

~

,._, ~

:;-.. :=;

--r-0

N

,._, '° '° 0

0 0

0

,._, '° ,._,

•n -...:;

,..., 0 -0

0

-N

00

r-=--,._,

,._, ,._, '° 0 0 ~

-I')

N

0 "T 0

""> v I

r-:x: =--r-0

0

r'

~ '° ----""· ---00

~~ ~

1 (0

77

77

) ~~

t "-I~

118

Table 2. Liquid/gas analogous quantities

liquid flow gas flow

surface wave velocity, c = (gh)..s speed of sound, a = (kRT)..s

water depth ratio, h/h0 temperature ratio, T/T0

water depth ratio, h/h0 density ratio, p: Po

water depth ratio squared, (h/h0 ) 1 pressure ratio, P/ P0

[ (11a - h) r Mach number, 2 h \fach number, V/a

Tables 119

Table 3. Station I areas used in mixing analyses

\1mu A, Al

0.05 0.61

0.39 0.62

0.57 0.64

0.73 0.67

0.89 0.71

1 0.74

1.05 0.76

1.22 0.84

1.30 0.87

1.43 0.94

1.51 1.00

Tables 120

Table 4. Velocity profile parameter n

'.Vf max n % blockage

0.05 10.5 8.7

0.39 12 7.7

0.57 12 7.7

0.73 9.5 9.5

0.89 9 10.0

1 8 11.1

1.05 9 10.0

1.22 4.7 18.2

1.3 4.5 18.2

Tables 121

The vita has been removed from the scanned document

approved - virginia tech · master of science in mechanical engineering approved: ... ( p1 - pe) c...

Documents