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Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar boundary layers and shock waves with separation of flow. Koepcke, W. W. University of Minnesota, 1957. http://hdl.handle.net/10945/24741
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Page 1: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar

Calhoun: The NPS Institutional Archive

Theses and Dissertations Thesis Collection

1957-05

Interaction between laminar boundary layers and

shock waves with separation of flow.

Koepcke, W. W.

University of Minnesota, 1957.

http://hdl.handle.net/10945/24741

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Title

INTERACTION BETWEEN LAMINAR BOUNDARY LAYERS

AND. SHOCK WAVES WITH SEPARATION OF FLOW

A Thesis

SUBMITTED TO THE GRADUATE FACULTY

OF THE UNIVERSITY OF MINNESOTA

by

W. W. Koepcke, Lieutenant U. S. Navy

In Partial Fulfillment of the Requirements

for the Degree of

Master of Science in Aeronautical Engineering

May 1957

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ACKNOWLEDGMENTS

The author wishes to express his appreciation

to Dr. Rudolf Hermann, Professor of Aeronautical

Engineering, University of Minnesota, for his in-

terest, encouragement, and advice In the develop-

ment of the project; to Mr. Frederick Moynlhan,

principal engineer, Rosemount Aeronautical Labora-

tories, for his timely suggestions during the wind

tunnel operating period; to Mr. Miles Mock, raachin-

1st In the Department of Aeronautical Engineering,

who fabricated the working models; and to his wife

for her understanding and patience throughout the

entire period of the author's postgraduate study.

W. W. K.

35^3

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TABLE OF CONTENTS

1. Summary 1

2. Introduction 3

3. Table of Symbols 5

4. Equipment 7

5. Procedure 12

6. Presentation of Results 13

7. Discussion 16

7.1 Theoretical and Historical Background . 167.2 Preliminary Discussion of the

Experimental Results 197.3 Upstream Interaction Distance 207.4 Inflection Points on a Pressure Profile 247.5 Impulse and Step Shock Waves 277.6 Transition Within the Interaction Zone .297.7 Separation 367.8 Pressure at the Top of the Laminar

Foot 387.9 Shadow Photographs .40

8. Conclusions 43

References 45

Tables 48

Figures 57

Appendices

A. Calculations to find shock impingementpoint

B. Calculations to determine interactiondistance

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INTERACTION BETWEEN LAMINAR BOUNDARY LAYERS

AND SHOCK WAVES WITH SEPARATION OF PLOW

1. SUMMARY

Shock waves generated by a 5° and a 10° half

angle wedge located In the main stream of a Mach -— 3.0

test section were Impinged on the laminar boundary

layer of a flat plate causing separation of flow. The

flow was considered to be two dimensional with zero

heat transfer. Reynold's numbers, from 150,100 to

1,098,000, were produced by varying length along the

flat plate, and by changing stagnation pressure.

Analysis was mainly accomplished through the study

of static pressure profiles, supplemented by shadow

photographs.

Important results were:

(1) The ratio of the pressure at the separation

point, and the pressure Just upstream of the sharp

pressure rise denoting separation was nearly constant

regardless of magnitude of shock, and did not vary

with Reynold's number.

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(2) The ratio of the pressure at the top of

the laminar foot (region of nearly constant pressure

within the Interaction zone upstream of shock Im-

pingement point) and the pressure just upstream of

the Interaction region varied Inversely with Reynold's

number to the (.12) power.

(3) The ratio of the Interaction distance up-

stream to the boundary layer thickness varied In-

versely with Reynold's number to the (§) power.

As many as five Inflection points were found In

the pressure profile of a laminar boundary layer

acted upon by a shock wave. In general, step type

shock waves showed consistency while Impulse type

shock waves showed Inconsistency with a variation

of parameters. Evidence was presented showing dif-

ferent results according to whether Reynold's number

variation was obtained by changing length or stagna-

tion pressure.

The experimental study was carried out at the

Rosemount Aeronautical Laboratories, University of

Minnesota during the school year 1956-1957 in con-

junction with Lt. E. E. Irish, U. S. Navy, who in-

vestigated turbulent boundary layers utilizing approxi-

mately the same configuration.

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INTERACTION BETWEEN LAMINAR BOUNDARY LAYERS

AND SHOCK WAVES WITH SEPARATION OF FLOW

2. INTRODUCTION

This Investigation was conducted principally to

bring out features of the Interaction between shock

waves, strong enough to cause separation, and laminar

boundary layers on a flat plate. The analysis was

concentrated chiefly on the static pressure distribu-

tion on the flat plate supplemented by shadow photo-

graphs.

The equipment was designed so that the effects

of a pressure gradient caused by a shock wave would

be observed on a body In supersonic flow. The shock

wave was generated by means of wedges, of varying

half angles, suspended In the main stream of a super-

sonic tunnel. The wave Impinged on a flat plate below

the wedge. In tnis manner since the flat plate had

initially a zero pressure gradient along its upper

surface, the effects of the externally imposed shock

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wave on the flat plate's boundary layer could be

observed. The boundary l&yer was always laminar

upstream of the zone of Interaction. Surface

roughness of the flat plate was minimized, Mach

number was held nearly constant, and zero heat

transfer was assumed. The wedge was moved stream-

wise within the test section resulting in the shook

wave impinging at different locations on the flat

plate. Thus, the parameters varied were Reynold's

number, by varying both length and total pressure;

and shock strength. The flat plate and wedge com-

pletely spanned the test section thereby simulating

two-dimensional flow.

This experimental Investigation was carried out

at the Rosemount Aeronautical Laboratories, Depart-

ment of Aeronautical Engineering, University of

Minnesota during the school year 1956-1957 in con-

junction with Lt. E. E. Irish, U. S. Navy, who con-

ducted experiments of the same general nature with

turbulent boundary layers.

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3. TABLE OF SYMBOLS

d - corrected Interaction distance upstream

of the theoretical shock wave Impingement

point on the flat plate (d' corrected for

static hole diameter).

d 1 - measured Interaction distance upstream of

theoretical shock wave Impingement point

on the flat plate

h - diameter of static pressure taps In flat plate

I - theoretical shock wave Impingement point on

the flat plate

Mj - Mach number of flow at shock wave's Impingement

point on the flat plate

M - Mach number at the upstream beginning of inter-

action

P - total or stagnation pressure

p - static pressure within the Interaction zone

p - static pressure Just upstream of the Interaction

zone

ps - static pressure at the separation point

p«P- static pressure on top of the laminar foot,

denoted by second inflection point (T) In the

pressure profile curve

psla - pounds per square inch absolute

psig - pounds per square inch gage

Rj - Reynold's number at the shock wave's impinge-

ment point on the flet plate

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R - Reynold's number based on the distance from

the leading edge of the flat plate to the

point where the pressure begins to rise

(beginning of upstream interaction)

S - first inflection point in the pressure pro-

file curve denoting separation point

T - second inflection point. in the pressure pro-

file curve

Xj - distance from leading edge of flet plate to

shock wave's Impingement point on the flat

plate

X - distance from leading edge of the flat plate

to the upstream beginning of interaction

S r - boundary layer thickness at the point where

the shock wave impinged on the flat plate

calculated assuming no added thickening due

to the shock wave

JQ - boundary layer thickness at the upstream be-

ginning of interaction

Jq - displacement thickness of the boundary layer

at the upstream beginning of interaction

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4. EQUIPMENT

To study the interaction between the boundary

layer on a flat plate, and shock waves produced ex-

ternally outside the boundary layer, half angle

wedges were suspended in the main stream of a super-

sonic wind tunnel. Shock waves of varying strengths

generated by these wedges were projected onto the

boundary layer of a flat plate mounted within the

test section.

The continuous flow supersonic wind tunnel

utilized was located at the Rosemount Aeronautical

Laboratories, facilities of the Department of Aero-

nautical Engineering, University of Minnesota, Air

was delivered to the stilling chamber of the tunnel

from a 1750 cubic foot pressure storage tank (240 pslg

capacity). The air was dried to -40° P dew point

before delivery from the compressor to the storage

tank. The compressor was three staged with a capacity

of 195 cubic feet per minute at 1500 psig. Downstream

of the tunnel was a vacuum system consisting of a 30

foot diameter sphere (14,000 cubic feet) plus five

cylindrical tanks (total volume 8,750 cubic feet).

The system could be evacuated to a pressure of £ in,

mercury, absolute. The test section was 1.75 in. wide

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8

by 1.9^ In. high with an asymmetric nozzle block

designed for Mach 3.05j (Fig. la). The upper por-

tion of the block was made of luclte and furnished

the curvature providing the expansion to the design

Mach number. This half was designed to compensate

for boundary layer growth. The lower part of the

block was straight, did not compensate for boundary

layer growth and was made of steel. The working

section had circular glass side walls approximately

3 In. In diameter, thereby permitting photography of

the flow.

The flat plate, 4£ in. long, completely spanned

the tunnel and was anchored to the lower tunnel wall

by means of a 0.715 In. high, 0.125 in. thick steel

pylon. The plate was made of stainless steel; was

hardened and had a commercial plating of chrome

0.0001 In. thick to reduce surface roughness. It

had a 5° half wedge angle at the leading edge and

was 0.125 in. thick at the maximum point. The lead-

ing edge thickness was 0.0010 in. There were five

static pressure holes staggered along the center line

of the flat plate. The most forward hole was O.725 In.

from the leading edge. Each succeeding hole was O.25O

In. further downstream, the fifth hole being 1.725 in.

from the leading edge. Each hole was 0.006 in. in dia-

meter. Leading from each hole beneath the plate was a

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copper tube 1/16 In. In diameter. The tubes ran along

the lower side of the plate downstream Into the dlf-

fuser section of the tunnel and then out to a mano-

meter board. The tubing was continuous from a solder

attachment at the bottom of each hole until Joining

the plastic tubing of the board. (See Figs, lb and

lc for a schematic drawing and photograph of the flat

plate).

The wedges which generated the shocks impinging

on the flat plate's boundary layer completely spanned

the tunnel and were suspended from the upper nozzle

block by means of a steel pylon 0.2 in. in height,

and 0.125 in. thick.

The pylon in turn fitted in a"T" slot cut into

the upper block, the slot being 0.125 in. wide run-

ning along the longitudinal center line of the upper

block. The pylon was attached to a lead screw which

extended downstream through a flange connecting the

test section and dlffuser. The lead screw (40 threads

per inch) was turned by means of a small ratchet

wrench. The wedge thus could be moved streamwise

within the test section, and was always parallel to

the upper nozzle block surface. Fig. Id is a photo-

graph showing the test section, the ratchet wrench

and lead screw mechanism, and the pylon which sup-

ported the wedge.

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Two wedges were used in the laminar flow runs

for generating the shocks. One had a 5° half angle

and the other a 10° half angle. Both were 1/8 In.

thick at the maximum point. The 5° wedge's leading

edge was .0006 in. thick, the 10° leading edge was

.0011 in. thick. See Fig. 2a for a photograph of

the two wedges. See Fig. 2b for a drawing of the

10° wedge imposing a shock wave onto the flat plate.

Total temperature within the stagnation chamber

was measured by means of thermocouple leads connected

to a potentiometer (manufactured by Leeds and North-

rup Co.). The thermocouple within the chamber was

shielded by a plastic covering.

Six static pressures, five from the holes on

the flat plate, (location shown on Fig. lb) and one

from a hole in the lower tunnel wall upstream of the

flat plate were measured within the test section.

The static pressure hole in the lower tunnel wall,

about one inch upstream of the leading edge of the

flat plate was only utilized in determining starting

of the tunnel , and as an aid in checking for choking

within the test section. These six holes or orifices

were connected to a multiple tube mercury manometer,

which also had tubes exposed to atmospheric pressure

for reference readings. The barometric pressure was

taken before each series of runs by means of a

standard brass mercury barometer. Parallel light

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shadow photographs were taken by means of a BH-6

power source and a mercury arc lamp with a columina-

ted light beam and with a flash of approximately

three microseconds.

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5. PROCEDURE

For the laminar flow runs, two stagnstlon pres-

sures of about 14.7 psla and 4-5 psla were used to

obtain a variation In Reynold's number from 150,100

to 1,098,000. With the lower pressure, Reynold's

number varied along the five holes on the flat plate

from 150,100 to 355,000. With the higher pressure,

the variation was from 456,000 to 1,098,000. Rey-

nold's number was held constant by varying total

pressure with each change In total temperature of

10° Fahrenheit.

For each series of runs (either low pressure or

high pressure) a static pressure reading of the six

orifices was taken with the shock generating wedge

In Its most downstream position. The wedge was then

advanced upstream In regulated Intervals by means of

a calibrated lead screw. The tunnel was not operated

In a continuous flow manner. The sequence was to ad-

vance the wedge, start the tunnel, shut off the mer-

cury manometer tubes by means of a clamping device

when pressures were steady, shut down the tunnel, read

the total temperature and the mercury column heights,

advance the wedge, and begin the sequence again. The

wedge was advanced forward generally until the most for-

ward hole on the flat plate had reached a pressure peak,

At this time, the series was completed.

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6. PRESENTATION OF RESULTS

Tables I to IV show important results obtained

from the pressure profiles of each tap hole. These

tables Include Interaction distances; pressure ratios

of separation, and at the top of the laminar foot;

Reynold's numbers. Tables V to IX show the static

pressure measurements in Inches of mercury absolute

for each hole on the flat plate for the 5° wedge,

45 psla runs with reference to the wedge's position

from its most downstream position in the test section.

Figs. 3 to 12 show the static pressure profiles for

Reynold's numbers varying from 150,100 to 1,098,000

as generated by the 5° wedge. Figs. 13 to 22 show

the same profiles and Reynold's number variation as

generated by the 10° wedge. Fig. 23 is a log-log

plot of -4* versus (R ) for the 5° and 10° wedge.

AFig. 2k is a log-log plot of -^f- versus (Rj) for the

ox

5° and 10° wedges. Fig. 25 is a log-log riot of £lPo

versus (Rq) for the 5° and 10° wedges. Fig. 26, 2?,

28, and 29 are a series of shadow photographs show-

ing the shock waves generated by the 5° a^cL 10°

wedges Impinging on the flat plate at various Reynold's

numbers. Fig. 30 shows Mach number distribution within

the test section at each hole with the wedges In the

most downstream (zero reference) position.

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The pressure profiles were plotted with £_Po

as the ordinate where pQ was the static pressure of

each hole when the wedge was In Its most downstream

position (wedge position = zero Inches In Tables V

to IX) In this manner, the profiles then show how

the Initial pressure at each hole was disturbed as

the wedge was advanced upstream. Each of the pres-

sure profiles presented In this report (Figs. >-22)

thus show the pressure at a single static hole as

the shock wave advanced from a position downstream

and passed over the hole, I.e., Fig. 3 shows the

pressure profile obtained as the 5° wedge, Imping-

ing a shock wave onto the flat plate's boundary

layer, was advanced from a position approximately

one inch downstream of the most upstream hole on

the flat plate until the shock wave passed over and

upstream of the hole. The abscissa of the pressure

profiles is always the wedge's position with refer-

ence to its most downstream (zero) station In inches.

This method had certain distinct advantages:

(1) To get as many data points as are shown on

the pressure profiles by impinging the shock at only

one point would have required a large number of static

pressure holes.

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(2) Some of the small pressure differences

measured probably would have been completely lost

because of the difference In pressure that could

be expected within the test section between holes.

However some error could be Introduced by re-

flection of tunnel waves from the wedge as It moved

forward. Also the closer one approached the lead-

ing edge of the flat plate, the more error one might

introduce because of the lack of constant character-

istics of a boundary layer In that region.

The series of shadow photographs (Figs. 26-29)

show the state of the boundary lsyer as the shock

wave was directly over each hole. It should be

noted that the pressure profiles do not necessarily

represent the exact conditions of the boundary layers

as shown by the shadow photographs, but if boundary

layer characteristics did not vary too widely over

the flat plate, then the profiles show a close approx-

imation of the actuel conditions.

Tables V to IX have been included to show the

data for the 5° wedge, 45 psia runs for each hole.

From these tables, Fig. 8 to 12 were constructed.

The remaining pressure profiles were constructed in

the same manner from similar data.

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7. DISCUSSION

7.1Theoretical and Historical Background

Boundary layer-shock wave Interaction affected

one of the most significant concepts in fluid flow,

Prandtl's theory of the boundary layer. Prandtl's

basic assumption was that viscous stresses In a low

viscosity fluid were small compared to other terms

In the momentum equation except in a relatively" thin

layer near solid boundaries. Here large velocity

gradients occurred and thus significant viscous

stresses existed. These viscous effects were Ig-

nored in calculations Involving the flow external

to the layers near the boundaries. This was an

outstanding simplification provided the boundary

layer flow did not appreciably affect the external

flow. For then the two regions of flow were calcu-

lated separately; the external flow as though it

were non-viscous and without heat conduction with

its boundary assumed to be the solid object In the

stream, and then the viscous effects at the wall

according to boundary layer theory. If the boundary

layers were thin compared to a dimension of the solid

body, then it was shown that the boundary layer had

only a second order effect on the exterior flow,

this effect principally being an outward displace-

ment of potential flow streamlines due to the dis-

placement thickness of the boundary layer.

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The principal assumptions of boundary layer

theory are that the rates of change of velocity

and temperature perpendicular to a bounding sur-

face are large compared to rates of change in a

direction parallel to the flow. In a shock wave

on the other hand the reverse is true. Pressure,

temperature, and velocity gradients perpendicular

to the wave are large compared with changes paral-

lel to the wave. When a shock wave occurs near a

solid body, so that there is an interaction between

the boundary layer and shock wave, the two are in

basic conflict.

Thus the simple boundary layer theory is broken

down, for the interaction between the boundary layer

and shock wave produces first order effects affect-

ing both the external and the boundary layer flow.

The shock wave imposes such a large pressure gradi-

ent upon the boundary layer that it is distorted.

It has been found that this distortion in turn

causes additional compression and expansion waves

to be generated from the boundary layer into the

external flow which changes the original shock pat-

tern. The Interaction between a shock wave and a

boundary layer leads to a flow pattern different

from one which is predicted by simple shock wave

theory and boundary layer theory separately.

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Attention was first drawn to the effects of

boundary layer, shock wave Interaction by Ferri

(Bef. 1) who during tests In a supersonic tunnel

observed boundary layer separation near the trail-

ing edge of a wing section at a point where a fav-

orable pressure gradient was expected. He stated,

"On the side of the wing on which there Is expan-

sion and which should therefore have a compression

shock at the trailing edge, there Is observed In

every case a phenomenon not predicted by theories,

namely that before reaching the trailing edge,

there Is a sudden pressure Increase, well brought

out In the photographs, by a shock wave which sep-

arates two regions of very different luminosity."

Although In most practical cases today, turbulent

flow prevails, the chances of extensive laminar

boundary layers seem to be growing as planes and

rockets fly to greater altitudes. With decreas-

ing density the Reynold's number decreases which

enhances the possibility of extensive laminar

boundary layers (Ref. 2).

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x^

7.2Preliminary Discussion of the Experimental Results

Two types of shock waves were generated In the

experiment, the 5° wedge creating a "step- type"

shock, and the 10° an "Impulse-type" shock wave.

There were marked differences In the two types.

See section "Impulse and Step Shock Waves"; also

Fig. 31 i for definition of the two types of shock

waves. The pressure profiles and results shown In

the tables for the 5° wedge generally agreed with

those In Eef . 3> that Interaction distances upstream

for Initially laminar boundary layers generated by a

wedge of about a 5° half angle are approximately 50

boundary layer thicknesses. The Ref. 3 runs were

conducted at Mach 1.5; the present report at Mach 3»0

Interaction distances for the 10° wedge reached 70

boundary layer thicknesses upstream. Peak static

pressures, In agreement with Ref. 3» were slightly

higher than theoretical for the 5° wedge and were

considerably lower than theoretical for the 10° wedge

Pressure profiles, with as many as five Inflection

points were found in certain Instances, confirming

results of Ref. 4. The quantity -7—* was found to

vary as R 4 for Reynold's numbers within the known

laminar flow range for the 5° wedge, while the 10°

results of the same quantities were inconsistent.

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The quantity -~ was found to vary as Rj for theo I

5° wedge over the known laminar flow range. The

quantity —7— for the 10° wedge varied Inconsistently

with Rj. Every pressure profile within the laminar

flow range showed separation. The pressure ratio £sPo

did not appear to vary with Reynold's number nor with

the strength or type of shock, Impulse or step. The

pressure ratio £t varied approximately with Ro~*

for both types of shocks. Shadow photographs showed

the point of actual shock impingement on the flat

plate to agree approximately with the theoretical

impingement point for the 5° and. 10° wedge. The

primary shock was, however, bent visibly as it

approached the flat plate, thus creating doubt as

to the reliability of the pressure profile to accu-

rately pin point the impingement. No doubt exists

however, in laminar flow as to the tremendous smear-

ing of the high pressure behind the shock wave up-

stream through the boundary layer.

7.3Upstream Interaction Distance

Tables I through IV show the interaction dis-

tance (d) upstream of the shock's impingement point

on the flat plate with respect to the displacement*

thickness (fQ at the point where interaction began

upstream. (This ratio was used extensively in Ref. 4).

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The point was always taken to be at the Initial

pressure rise in the pressure profile. The quan-

tity (d') was the actual distance measured upstream

at which pressure began rising and (h) was the dia-

meter of the static pressure hole. The diameter of

the hole (h) was subtracted from the measured in-

teraction distance (d') to get the true interaction

distance (d). In this manner, the effect of the

diameter of the hole in increasing the upstream

distance was cancelled. This method was taken

from Hef . 15, which showed good experimental re-

sults because of this correction. The displacement

thickness was calculated from the formula in Ref. 5 '

X * - 1.72 (1 + 0.277 M 2) £2* where

J s displacement thickness of the boundary

layer at the upstream beginning of

interaction

X a distance from leading edge of the flat

plate to the beginning of interaction

M a Mach number at the upstream beginning of

Interaction

The boundary layer thickness was calculated from the

formula In Ref. 6: #

(fx j>2 + 1.03 ( 2T-1) M-j-^j *i whereEjS

I s boundary layer thickness at the point

where the shock wave strikes the flat

plate

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Xj » distance from leading edge of flat plate

to shock impingement point on plate

My = Mach number at the shock impingement point

For MQ = Mj - 3

So* = 6,xn

h = 8,>9 ;

In Ref • 3i Llepmann discussed the interaction

distance (d) in terms of the boundary layer thickness

and Reynold^ number where the shock wave theoreti-

cally impinged on the flat plate. This interaction

definition is shown in Table I through IV in the

column —-r- • See Appendix B for sample calcula-oi

tlons with this method. The method of Ref. 4 sim-

plifies calculations considerably in contrast to

Ref. 3> since one needs to determine the Reynold's

number for each tap hole once, and it remains the

same. In the method of Ref. 3 however, one must

calculate the Reynold's number at each point the

shock wave strikes the boundary layer or flat plate.

Both methods will show trends with Reynold's number

but the method of Ref. 3 seems to be preferable for

practical application since it gives variation with

the Reynold's number of the boundary layer where the

shock strikes, and not at a point where the shock is

felt upstream.

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*3

For a 5° wedge, the values of -~ were com-dl

parable to those found in Hef . 3 for a 4^° wedge,

and (d) was found to be approximately 50 boundary

layer thicknesses. It is to be noted that the ex-

perimental runs of Hef. 3 were conducted at Mach =

1.5, while in the present report, the runs were at

Mach = 3.0. For the 10° wedge, the distances were

increased up to 70 boundary layer thicknesses de-

pending on Reynold's number. Log-log plots of y-fd

°and -7- versus Reynold's number are shown in Fig. 23

and 24 and are discussed in the section "Transition

in the Interaction Zone".

The theoretical position of the shock wave

striking the flat plate was shown on each pressure

profile figure with the symbol (I). The pressure

peak always occurred downstream of the position (I).

In general, more of the total pressure rise occurred

. upstream of the position (I), in the case of the im-

pulse type shock than the step type shock wave. This

seemed reasonable for the total pressure rise of the

Impulse shock type had been blunted by the expansion

wave; consequently the position (I) would appear to

move downstream relative to the pressure profile.

This result also corresponded with Hef. 3 differen-

tiating impulse and step shock waves. One should

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note that It Is practically Impossible to pinpoint

the actual Impingement of the shock wave by observ-

ing the pressure profile. In the laminar case, the

smearing of the shock wave's pressure causes rapid

thickening and sometimes transition upstream. If

the transition was not too pronounced, It would not

be observable on the profile (see discussion on

"Inflection Points on a Fressure Profile") as an

inflection point, but would still Influence the

gradient enough to apparently cause the steep pres-

sure gradient near the shock wave to move further

upstream. At other times, transition could occur

far enough upstream of the primary shock to cause

an added inflection point in the profile, (see

Fig. 22). One thus could only pinpoint the actual

shock impingement as being located somewhere on the

profile where the steepest gradient occurred.

7.4Inflection Points on a Pressure Profile

In Hef . 4 the author stated that under certain

conditions, an initially laminar flow, acted upon by

a shock wave would show five inflection points within

the pressure profile. Beginning from the point where

a rise in pressure was first noted the points In suc-

cessive order would denote:

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(1) Separation

(2) Transition

(3) Thickening of the turbulent layer

(4) Shock impingement

(5) Reattachment of flow downstream of shock

The five inflection points would occur if tran-

sition were a considerable distance ahead of the shock.

Fig. 22 shows five inflection points where the pressure

profile of hole number five, 1.725 In. from the lead-

ing edge at a stagnation pressure of about 45 psia

and a Reynold's number of 1.098 (10°) has been

plotted to two longitudinal scales. Also on Fig. 22

Is a pressure profile of hole number one, 0.725 in.

from the leading edge of the flat plate at a total

pressure of about 101 psia and a Reynold's number

of 1.01 (10°). The profiles are very similar show-

ing a definite break in the curve at E_ ^ 1.9.Po

This inflection point is masked on a shortened long-

itudinal scale. The pressure profiles in Ref . 3 in

general did not show this break, probably due to the

extremely shortened scale. Since the profiles of

holes one and five at about the same Reynold's number

show the five inflection points, Ref. 4 appears to be

substantiated. The author (Ref. 4) explains the five

points in the following manner:

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1. There is first a foot where the boundary

layer Is laminar and the pressure falls off after

separation because the separated region becomes

thick.

2. When transition occurs, the boundary layer

although separated can withstand a larger pressure

gradient and the pressure rises steeply.

3. The dead air region (separated region)

becomes very thick further downstream so that the

pressure gradient must fall again since even the

turbulent friction forces cannot withstand a large

gradient.

4. When the shock strikes the boundary layer

the flow is deflected towards the wall, the separated

region becomes thinner, and the pressure gradient rises.

5. Reattachment of the boundary layer occurs

near the peak pressure position and the gradient falls

off once more as the pressure profile approaches Its

final downstream form.

In most Instances within the present report how-

ever, only three inflection points were observed,

presumably because the boundary layer did not thicken

enough between transition and shock for the pressure

gradient to be visibly affected or else transition

did not occur until shock impingement on the boundary

layer.

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7.5Impulse and Step Shock Waves

An Impulse shock wave Is defined to be of the

type where the generated shock wave Is produced by

a flow deflection angle closely followed by a gen-

erated expansion wave, produced by the same magni-

tude flow deflection angle. A step shock wave Is

defined to be of the type where the expansion wave

does not occur within the interaction distance

downstream of the impingement point of the shock

wave on the flat plate. See Fig. 31 , a sketch

showing the difference in the two types of shocks.

Within this report, the 5° wedge created an expan-

sion wave which theoretically impinged on the flat

plate 0.91 in. downstream of the shock wave. Since

interaction distances upstream of the shock wave

were approximately one inch, 5° wedges were assumed

to generate step waves. The 10° wedges however

generated expansion waves which theoretically im-

pinged on the flat plate 0.15 in » downstream of the

shock wave, thus generating Impulse type shock waves.

As shown on Figs. 3 to 12, the 5° wedge created pres-

sure peaks which were always greater than theoreti-

cal, and the 10° wedge created pressure peaks always

lower than theoretical (Figs. 13 to 22) throughout

the range of Reynold's numbers tested. Of particular

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£0

note is Fig. 17(b) which is Fig. 1? (a) plotted

to a different horizontal scale and extended be-

yond the pressure peak. It shows that the pres-

sure returned to a position below initial pres-

sure i.e., that there was an over expansion. The

results above were in agreement with those of

Ref. 3 and corroborated the general characteris-

tics of an impulse type wave. Step shock wave

pressure data did not show this over expansion.

Theoretical pressure peaks were defined to be those

stream pressures attained downstream of the shock

pattern of a given oblique shock wave if it struck

the flat plate and was regularly reflected in the

absence of a boundary layer. In the case of the

5° wedge with an initial Mach 3 flow:

p = 2.04^initial

In the case of the 10° wedge, with Mach 3 flow:

PfinaL_ . 3>93pinitlal

The above values were calculated by charts in Ref. 7.

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29

7.6Transition Within the Interaction Zone

An effort was made to create an entirely lam-

inar flow throughout the region of Interaction, up-

stream and downstream, even with flow separation.

In Ref. 4, this was only achieved (at Mach 2) with

a wedge of 6° and Reynold's numbers below 150,000.

Results of Ref. k showed that for completely lam-

inar flows the quantity —- graphed versus Ro

showed a positive slope. When transition occurred

d --within the zone of interaction, =—% varied as RQ

3

over a range of R from 2 (10-0 to 4 (10-O, I.e.,

the graphical slope was negative.

In the present investigation =—s versus RQ wascfo

graphed on a log-log plot over a range varying from

150,100 to 1,098,000 for the 5° wedge. To achieve

the Reynold's numbers from 150,100 to 355,000,

tunnel stagnation pressure over the five holes on

the flat plate was approximately 14.7 psla. For

the higher range, stagnation pressure was about

45 psla. The two series of runs are shown on Fig.

23. The lower Reynold's number series showed a

-Ivariation of -~ with R . The higher series

00fl a

showed a variation of * with R ~ 4, with the

o oslope showing a tendency to become less steep for

the Reynold's numbers above 776,000.

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Note on Fig. 23 that there Is a discontinuous

jump in the curves rather than a straight line con-

tinuation when the stagnation pressure was changed.

An explanation of the possible cause of this dis-

continuous Jump follows:

In Hef . 12, a complete resume' of the transi-

tion to turbulent boundary layers at supersonic

speeds, the author stated that transition extends

over a finite distance and depends on:

(a) Conditions of the test stream

(b) Conditions of the leading edge of the

test body

(c) Thermal conduction in the test body

Taking case (a) first, (Conditions of the test

stream), several experimenters have noted a system-

atic increase of transition Reynold's number with an

increase of stagnation pressure (Ref. 12) i.e., as

Reynold's number per length increased, so did the

Reynold's number of transition. The Increase was

proportional to the stagnation pressure increase,

since stagnation temperature was approximately the

same. If one speculates on the possible conse-

quences of the change of stagnation pressure and

Reynold's number per unit length, the following

effects present themselves:

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(1) urease of Ex yer ur.lt length lncreeses

the 1< af the laiinar part of the t ary layer.

This increases the :ctal extent of the effect of cls-

turbances In the test section ed ace r ness,

is tc reduce the Reynold's number of

r sition.

(2) Z-..e thickness of the boundary layer Is

l?rger In tne case of the lower Hj per unit length

, xfor any given leng I : = (constant) ^~* . There-

fore the boundary layer is more susceptible to out-

side disturbances. Zr.^s the lower Hx per unit length,

e sore chance that a ils.urbance : £ earlier

transition.

In the present experic n, when the stag-

nation pressure was c. i :r:~ 1-. 7 to ^5 psla,

for the saxe Reynold ' r, a nole nearer the

leadlr^, eige z: the flat plate would show charac-

teristic; M>re typical of a 1 boundary layer

than a hole further d: nstrean. Ine r.cle rest up-

stream would tr.us shew a greater —-—^ ratio, assux-

in* that laainar flows alwa: :w greater inter-

actionQ

» ratios than turbulent flows. This is

copre wisely what occurred In the experiment. Cn

7lg. 23, tne discontinuous Juxp occurred when Rey-

nold's number was changed froa ;,r c,DOO tc ~;:,:00

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JZ

by increasing total pressure and shifting from

hole 5» the furthest downstream, to hole 1, closest

to the leading edge.

Takln£ case (b), conditions at the leading edge

of the test body, it has been noted that there is a

systematic decrease of transition Reynold's number

with reduction in thickness of the leading edge of

a test body due to vibrations, (Hef. 12). Assuming

that vibration of the leading edge occurred, the

higher the stagnation pressure, the more vibrations

per second, and thus the possibility of earlier

transition.

In the present experiment however, when stag-

nation pressure was increased, for approximately

the same Reynold's number, the boundary layer acted

more like a laminar one than one undergoing transi-

tion. Therefore, vibrations were not the cause of

the discontinuous Jump in Pig. 23.

Taking case (c), thermal conditions in the test

body, it has been noted in many tests that a rise of

recovery temperature occurred near the leading edge

due to thermal conduction within the test body.

(Ref. 12). In the present experiment in order to

get desired strength, the nose of the flat plate

was built in the form of a wedge with the top sur-

face parallel to the stream. The wedge thus causes

a lower Mach number on Its surface and consequently

a higher temperature on the lower surface. Conduc-

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33

tlon in the nose wedge would then cause a higher

observed recovery temperature on the upper surface.

Since the temperature was higher on the nose of the

upper surface, this could lead to a greater degree

of instability of the boundary layer than if the

nose were cooler.

However in the present experiment, stagnation

temperature was very nearly the same regardless of

the stagnation pressure. Therefore, the effects of

possible thermal conduction within the test body

should have been the same regardless of pressure.

In evaluation of the three possible causes,

it would appear that case (a), conditions of the

test stream, was the most likely reason for the

discontinuous Jump in Fig. 23.

With regard to the tendency for the change of

slope at RQ of 776,000, previous investigators

(Hef. 13) utilizing the same wind tunnel as the

present report found that natural transition began

taking place at about R of 800,000. Likewise Fig.

12, a pressure profile at a Reynold's number of

1,098,000, did not show a separation point, whereas

the other profiles within the same series were con-

sistent in this aspect. The separated region may

have been too small to appear on the profile. How-

ever It would be reasonable to assume, that once the

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y*

flow entered the natural Reynold's number realm

of transition, that inconsistent results from

either wholly laminar or wholly turbulent flow

would result, and in this instance the flow did

not separate. Ref . 12 also stated that parameters

affecting flows which were in a natural transition

status were still very far from being understood.

Hence, the present report although showing the

pressure profiles for R of 936,000 and 1,098,000,

in general disregards the data since there is

Justified doubt as to their being stable laminar

flows.

In order to decide whether or not completely

laminar flow had been attained, the shadow photo-

graphs were scrutinized, the pressure profiles were

examined and references were checked. The photo-

graphs were not dlscernable enough In this respect.

If the pressure profiles showed five inflection

points, this would have indicated transition. In

the range of Reynold's numbers from 150,100 to

355 » 000, all of the profiles showed only three in-

flection points. In Ref. 4, and reiterated again

by the same author in Ref. 14, Fig. 7> completely

laminar flow was only accomplished below a Reynold's

number of 150,000 at Mach 2 with a 6 wedge. The

dslope of "T~¥ versus R was positive in that case.

o

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35

In the present investigation, the lowest Reynold's

d -^~number was 150,100, and -7—5 varied with R ** I.e.,

00slope was negative. It was therefore concluded

that pure laminar flow had not been achieved, and

that transition probably occurred at the point of

shock Impingement or Just downstream.

The quantity ~ versus R was plotted on/o

Fig. 23 for the 10 wedge also. For the lower Rey-

nold's numbers, the variation was in good agreement

with the 5° wedge results. In the upper R range,

the slope was considerably steeper. In Ref . 5» the

author stated that defective results were obtained

when the compression wave was too closely followed

by an expansion. This corresponds to the present

case for the 10° wedge, v/hlch generated an impulse

shock wave.

The quantity —£— , discussed in the "Interaction

Distance" section was plotted versus Rj on Fig. 24.

An arbitrary slope of (-§) was superimposed through

the data points. The 5° wedge results showed the

same break in the curve upon a shift of stagnation

pressure as noted earlier, and with a tendency to

vary away from the slope, above Rj of 776,000 al-

though not as pronounced as noted before. The 10°

wedge results were not as consistent; there was more

scatter, and a change in slope with a change of stag-

nation pressure. This again was in agreement with

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36

other results, that an Impulse type shock produces

Inconsistent results for many quantities compared to

a step type shock.

7.7Separation

Evidence of separation occurred In every pressure

profile for an Initially laminar flow except Fig. 12.

This discrepancy Is more fully discussed In the sec-

tion, "Transition Within the Interaction Zone". It

is sufficient to note here that Pig. 12 is a pressure

profile of a flow at a Reynold's number of 1,098,000.

At this Reynold's number, the flow probably was In a

state of natural transition and may not have been a

valid laminar profile.

In Ref . 8, Fig. 2, separation was defined to be

the point where the first inflection point occurred

in the pressure curve. In Ref. 9 the separation

region was defined to be the position on the profile

curve where the pressure was nearly constant. Both

definitions are In agreement, the former more precise

as to the onset of separation. In Ref. 4, evidence

was presented to show that separation of laminar flow

did not occur if the half angle wedge generating the

shock was smaller than 2°. In the non-separated cases,

the pressure profile only showed one Inflection point

at about the position of the shock impingement. In

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37

the present report, the wedges utilized had half

angles of 5° and 10° , the pressure profiles within

the laminar range had at least three Inflection

points - the flows were separated.

Theories have been advanced as to the ratio value

between pg ,pressure at the separation point, and p ,

stream pressure Just upstream of the sharp pressure

rise denoting separation. In Ref. 10, for laminar

flow, the author predicted, at Mach 3 and R of

2.5(10^), that £s would equal 1.18. Ref. 8 stated

that the ratio value depended on the external flow

Mach number and state of the boundary layer, either

laminar or turbulent. For undisturbed flows definite-

ly within the laminar range of Reynold's numbers, the

pressure ratio causing separation at Mach 3 would be

approximately 1.14.

In Ref. 5 i the point of separation was measured

experimentally with a half Pltot tube with values of

—— = 1.14 at Mach 3» although the value was quail-pofled with the statement that the exact separation

point was very difficult to detect. The ratio did

not vary over the range of Reynold's numbers tested,

2 (10^) to 4 (105 ).

In the present report, testing with both the 5°

and 10° wedges in a Reynold's number range from

150,100 to 776,000, results were in good agreement

with the above ratio of Ref. 5. Taking the first

point of Inflection on the pressure profiles as ps ,

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38

the following resulted:

1.08 - Hi - 1.17 with a meanPo

value of 1.12. There was no consistent variation with

Reynold's number. However, selecting the exact point

of inflection on the curves was somewhat nebulous,

and a consistent variation with Reynold's number

may not have been perceived. In Ref . 10, theory

predicted a variation of -Ji with R©- *, and in Ref. 11,°

*the prediction was a variation with R

" s. Ref. 10

advanced the following theory toward —IL: "When a

sufficiently strong oblique shock wave Is incident

upon the boundary layer on a flat plate, it causes

both the pressure to increase and the boundary layer

to thicken and separate upstream of it. The thicken-

ing of the boundary layer generates a band of compres-

sion waves that determine the pressure distribution

acting on the boundary layer upstream of the shock,

and this pressure distribution in turn governs the

rate of thickening of the boundary layer. The two

processes must adjust themselves to be in equilib-

rium so that the pressure distribution upstream of a

certain point where a shock impinges on a boundary

layer would presumably remain unaltered If the shock

were increased in strength and simultaneously moved

downstream to some new point such that the separa-

tion point did not move. Hence the ratio of pressure

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39

at separation to the undisturbed free stream pres-

sure should be a function of Kach number and Rey-

nold's number only, Independent of shock strength."

The present report appears to substantiate In

Pspart the above theory, for =r- did not appear to vary

^oregardless of strength nor type of shock, Impulse or

step, while Interaction distance upstream changed de-

pending on shock strength and Reynold's number. How-

Psever, a consistent variation of ^ with Reynold's

^onumber was not observed.

7.8Pressure at the Top of the Laminar Foot

In Ref. 8, the author defined the second inflec-

tion point on the pressure profile as the top of the

laminar foot. The laminar foot is defined as the

region of relatively small pressure gradient between

the shock wave's impingement point and the most up-

stream interaction point. This laminar foot is the

outstanding feature distinguishing laminar from tur-

bulent pressure profiles. In Ref. 5, a graph of -Z,Po

pressure at the top of the laminar foot (second point

of inflection) over pressure of the undisturbed

stream was shown versus Reynold's number. In log-log

coordinates, £?_ varied with R "** for a range of RoPo ~

from 100,000 to 400,000. On Fig. 24, ^£ versus R

for the present report is shown with the results of

Ref. 5« The variation with R agrees well with Ref. 5,

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40

except for R above 776,000 for the 10° wedge.

Again, for flows which In the absence of the shock

are known to be laminar, correlation of results Is9

good.

For the 5° wedge, ~£ varied approximately with

-.11 o PtH as In Ref. 5. For the 10 wedge, —= varied

i*°

with R" ,iD from 150,100 to 776,000. Apparently

the variation with Ro was consistent regardless of

type of shock wave, Impulse or step. In this case

also, when stagnation pressure was changed, there

was a discontinuous Jump In the curves, as In the

discussion section on "Transition Within the Inter-

action Zone".

7.9Shadow Photographs

Figs. 26 to 29 are shadow photographs of the

conditions existing within the test section during

the experimental runs. The pictures were not In-

tended to be utilized for their quantitative value,

but were Initially used to check flow configuration

because of choking difficulties. They did show the

characteristic thickening of the boundary layer due

to the interaction, and certain aspects pointed out

below.

For the 5° wedge runs, Figs. 26 and 27, the photo-

graphs show that the actual shock impinged close to

the theoretical position on the flat plate. The pic-

tures were taken at the wedge position when the theo-

retical impingement point was over each hole. In

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41

looking at the photographs one can observe the tub-

ing leading from each hole under the flat plate.

Each hole Is almost directly vertical to the up-

stream edge of the tubing. The tubing was braced

with a solder backing and thus looks much thicker

than Its 1/16 In. actual diameter.

The 10° wedge runs, Pigs. 28 and 29, show that

the primary shock was deviated (bent) much more than

the 5° wedge shocks. This could have been due to:

(1) More Intense compression shocks from the

relatively thicker boundary layer upstream.

(2) Expansion fan effect off the rear of the

wedge, which caused the 10 shock to be originally

defined as an impulse type.

One would believe however that the compression

shocks emanating from the region of separated bound-

ary layer were the main factor because the primary

shock wave showed a sharp kink when it deviated,

rather than a gradual bend, typical of expansion

wave reaction.

The photographs showed evidence of a shock wave

at the leading edge of the flat plate. This leading

edge compression shock in the case of the 5° wedge

configuration (Fig. 25(e)) was mild, as shown by

static pressure measurements. The Mach number im-

mediately forward of the flat plate was 2.99, while

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42

on the flat plate, the average Mach number was about

2.95 (Fig. 30). Assuming the leading edge shock wave

caused the drop in Mach number, the intensity would

have been that caused by a flow deviation of less

than one degree. However since pressure profiles

were graphed always as the ratio of the static pres-

sure for each hole, when the wedge was in its most

rearward position, to that pressure as the wedge was

advanced, the leading edge disturbance was essentially

cancelled out.

The bending of the shock wave would have affeoted

the apparent interaction distance upstream, because

this distance was always measured relative to the

theoretical impingement point of a clean shock wave

generated off the leading edge of the wedge. The

Ps

it is primarily dependent on equilibrium between the

ratio «=• would not be affected by the bending because

undisturbed flow and thickening of the boundary layer,

and not on the position of the shock wave.

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43

8. CONCLUSIONS

A step shock wave generated by a 5° wedge and

an impulse shock wave generated by a 10° wedge located

in the main stream of a Mach <—' 3.0 test section were

impinged on the laminar boundary layer of a flat

plate causing separation of flow. The flow was two-

dimensional; Reynold's number varied from 150,100 to

1,098,000; zero heat transfer wes assumed. Impor-

tant results were:

-. (1) -^~ varied with Rj"* f0r the 5° wedgeo I

d -^ n(2) —* varied with RQ

4 for the y wedge0q

(3) £s = (1.12 1 .05) regardless of Rpo

(4) _T varied approximately with R~

Pofor the 5° and 10° wedges

Although both types of pressure profiles showed

laminar characteristics, the profile of the 5° wedge

peaked at a value slightly higher than theoretical,

while the profile of the 10° wedge peaked considerably

lower than the theoretical value. In general, the

step type shock wave was consistent, the Impulse shock

wave inconsistent, in parameter variations.

When Reynold's numbers were greater than 776,000,

results were erratic, thus showing the beginning of

natural transition. All flows, known to be laminar,

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44

separated under the influence of the shock waves

generated by the 5° and 10° wedges. There were no

wholly laminar flows throughout the Interaction re-

gion. Five inflection points were noted in certain

pressure profiles denoting transition upstream of

shock wave Impingement on the boundary layer.

Evidence was presented in Figs. 23, 24, and 25

in the form of a discontinuous break in log-log curve

plots showing that results were affected, depending

on how a variation with Reynold's number was accom-

plished, by a change in length or a change in stag-

nation pressure.

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^5

REFERENCES

1. Ferri, A.: Experimental Results with Aerofoils

in the High Speed Tunnel at Guldonla, NACA

TM 9^6 (19^0).

2. Young, D. D.: Boundary Layers and Skin Friction

in High Speed Flow, Aeronautical Quarterly 1,

page 137, (19^9).

3. Llepman, H. W., A. Roshko, S. Dhawan:v

On the

Reflection of Shock Waves from a Boundary Layer,

NACA TN 233^ (195D.

4. Holder, D. W., H. H. Pearcey, G. E. Gadd: The

Interaction between Shock Waves and Boundary

Layers, Aeronautical Research Council No.

16,526 Current Paper No. 180 (195*0.

5. Gadd, G. E., D. W. Holder, J. D. Regan: An

Experimental Investigation of the Interaction

Between Shock Waves and Boundary Layers, Royal

Society of London Proceedings, 195^, Serial A

226 page 227.

6. Kuethe, A. M., J. D. Schetzer: Foundations of

Aerodynamics, Wiley and Sons, 1950 $page 301.

7. Dailey, C. L. , J. C. Wood: Computation Curves

for Compressible Fluid Problems, J. Wiley and

Sons, 19^9.

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46

8, Gadu, 0. E., D. W. Holder: Boundary Layer Sepa-

ration in Two Dimensional Supersonic Flow, Aero-

nautical Research Council, Current Paper No. 270

(1956).

9. Shapiro, A. H. : The Dynamics and Thermodynamics

of Compressible Fluid Flow, Volume II, pages

1141-1142, The Ronald Press Company, New York,

(1953).

10. Gadd, G. E.: Interactions between Wholly Lam-

inar or Wholly Turbulent Boundary Layers and

Shock Waves Strong Enough to Cause Separation,

Journal of the Aeronautical Sciences, November,

1953.

11. Donaldson, C, R. H. Lange: Study of the Pres-

sure Rise Across Shock Waves Required to Separ-

ate Laminar and Turbulent Boundary Layers,

NACA TN 2770 (1952).

12. Probstein, R. F., C. C. Lin: A study of Transi-

tion to Turbulence of Laminar Boundary Layers at

Supersonic Speeds. Institute of the Aeronautical

Sciences Preprint No. 596, January 195&.

13. Bradfield, W. S., D. 0. DeCoursln, C. B. Blumer:

Effect of Leading Edge Bluntness on Momentum Loss,

Journal of the Aeronautical Sciences, June 1954.

14. Holder, D. W.: The Interaction Between Shock Waves

and Boundary Layers. Institute of the Aeronautical

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^7

Sciences, Preprint No. 550, June 1955.

15. Moynihan, F.: Normal Shock - Boundsry Layer

Interaction Studies on Cones at Mach Number 1.5»

University of Minnesota Rosemount Aeronautical

Laboratories Research Report 136, October 195^.

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48

TABLE I

TABULATED RESULTS

5° WEDGE, TOTAL PRESSURE 14.7 PSIA

Hole 1 2 3 4 5

X (inches) 0.725 0.975 1.225 l.*75 1.725

R (10-5) 1.50 2.01 2.52 3.04 3.55

tf '(lO^)(ifiches)

11.24 13.08 14.65 16.04 17.37

d • ( Inches

)

1.053 1.053 1.025 0.953 0.853

d ( Inches

)

1.047 1.047 1.019 0.9^7 0.847

d

Jo* 93.0 80.1 69.6 59.0 48.8

Ps

Po1.12 1.17 1.11 1.11 1.12

Pt

Po1.24 1.22 1.18 1.18 1.18

d

tfl

40.1 37.5 3^.7 30.9 27.0

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TABLE II

TABULATED RESULTS

5° WEDGE, TOTAL PRESSURE 45 PSIA

49

Hole i 2 3 4 5

X ( Inches

)

0.725 0.975 1.225 1.^75 1.725

r do"5)

4.56 6.18 7.76 9.36 10.98

/ *(io3 )

( Inches

)

6.50 7.55 8.35 9.15 9.92

d ' ( Inches

)

0.653 0.603 0.553 0.553 0.553

d (Inches) 0.647 0.597 0.5^7 0.5^7 0.5^7

d99.5 79.0 65.5 59.8 55.0

Ps

Po1.11 1.13 1.14 1.10 —

PT

p71.22 1.20 1.17 1.13 —

d

7i49.4 42.5 36.6 34.2 32.5

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50

TABLE III

TABULATED RESULTS

10° WEDGE, TOTAL PRESSURE ~ 14.7 PSIA

Hole 1 2 3 4 5

X (Inches) 0.725 0.975 1.225 l.*75 1.725

Ho (lO*-5 ) 1.50 2.01 2.52 3.04 3.55

r * do3) 11.24 13. 08 14.65 16.04 17.37

(Inches)

d f (Inches) 1.377 1.277 1.227 1.152 0.902

d ( Inches

)

1.371 1.271 1.221 1.146 0.896

d122. 97.2 83.4 71.4 61.6

Ps

Po1.16 1.12 1.11 1.11 1.09

Po*

1.30 1.24 1.18 1.18 1.10

d48.1 43.2 39.8 36.1 28.2

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51

TABLE IV

TABULATED RESULTS

10° WEDGE, TOTAL PRESSURE ~ 45 PSIA

Hole 1 2 3 4 5

X ( inches

)

0.725 0.975 1.225 1.475 1.725

R (10-5) 4.56 6.18 7.76 9.36 10.98

4*(103) 6.50 7.55 Q-35 9.15 9.92

( inches

)

d' (inches) 1.027 0.877 0.677 0.677 0.577

d (inches) 1.021 0.871 0.671 0.671 0.571

d157. 115. 80.

3

73.3 57.6

Is.

Po1.14 1.10 1.10 1.08 1.13

p«r

Po"1.22 1.14 1.12 1.14 1.18

d68.6 57.3 43.9 40.9 33.7

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52

TABLE V

P_Po

VALUES FOR 5° WEDGE, P - 46.4 psla

HOLE 1

Wedge Position(Inches)

p p(Inches mercury) Po

00000 2.57 1.001.625 2.57 1.001.675 2.57 1.001.725 2.60 1.011.775 2.65 1.031.825 2.72 1.061.875 2,85 1.111.925 2.96 1.151.975 3.03 1.182.025 3.08 1.202.075 3.11 1.212.125 3.13 1.222.175 3.13 1.222.225 3.13 1.222.275 3.19 1.242.325 3.24 1.262.375 3.44 1.342.425 4.22 1.642.475 5.40 2.102.525 6.09 2.372.575 6.20 2.412.625 6.15 2.392.675 6.04 2.35

pQ *» 2.57 (Inches mercury)

M. 3.00

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TABLE VI

~ VALUES FOR 5° WEDGE, P - 46.4 psia

HOLE 2

53

Wedge Position r P(inches) (inches mercury)

*S

00000 2.74 1.001.425 2.74 1.001.^75 2.74 1.001.525 2.77 1.011.575 2.79 1.021.625 2.96 1.081.675 3.09 1.131.725 3.18 1.161.775 3.23 1.181.825 3.29 1.201.875 3.29 1.201.925 3.31 1.211.975 3.34 1.242.025 3.^5 1.262.075 3.70 1.352.125 3.92 1.432.175 4.90 1.792.225 5.59 2.042.275 5.86 2.142.325 5.94 2.172.375 6.00 2.192.425 6.00 2.192.475 6.00 2.192.525 6.11 2.232.575 6.32 2.312.625 6.41 2.3^

Po 2.7^ (inches mercury)

Mo = 2.96

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54

TABLE VII

g_ VALUES FOR 5° WEDGE, PPo

HOLE 3

46.4 psia

Wedge Position(Inches)

000001.2751.3251.3751.4251.4751.5251.5751.6251.6751.7251.7751.8251.8751.9251.9752.0252.075

(Inches mercury)

2.662.662.662.692.772. 853.033.113.273.624.074.575.185.746.046.096.045.98

P_Po

1.001.001.001.011.041.071.141.171.231.361.531.721.952.012.272.292.272.25

p = 2.66 (inches mercury)

M, 2.98

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55

Po

TABLE VIII

VALUES FOR 5° WEDGE, p

HOLE 4

46.4 psla

Wedge Position P P_(inches) (Inches mercury) Po

00000 2.83 1.000.975 2.83 1.001.025 2.83 1.001.075 2.83 1.001.125 2.89 1.021.175 2.97 1.051.225 3.11 1.101.275 3.17 1.121.325 3.20 1.131.375 3.28 1.161.425 3.48 1.231.^75 3.79 1.341.525 ^.53 1.601.575 5.04 1.78I.625 5.41 1.911.675 5.63 1.991.725 5.83 2.061.775 5.89 2.081.825 5.89 2.081.875 5.83 2.061.925 5.77 2.04

p 2.83 (Inches mercury)

M, 2.94

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TABLE IX

JL VALUES FOR 5° WEDGE, P = 46.4 psiaPo

HOLE 5

56

Wedge Position P P(inches) (Inches mercury) Po

00000 2.70 1.000.775 2.70 1.000.825 2.70 1.000.875 2.7^ 1.010.925 2.78 1.030.975 2.89 1.071.025 3.0S

3.241.13

1.075 1.201.125 3.51 1.301.175 4.02 1.491.225 4.48 1.661.275 4.84 1.791.325 5.10 1.891.375 5.32 1.971.425 5.35 1.981.^75 5.^0 2.001.525 5.^3 2.011.575 5.51 2.041.625 5*65 2.091.675 5.75 2.131.725 5.79 2.151.775 5.83 2.161.825 5. 81 2.151.875 5.72 2.121.925 5.61 2.08

Po - 2 -70

M, 2.97

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57

Fig. 1A

ASYMMETRIC NOZZLE BLOCK

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58

-J

vb

Lu

S U")

1

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59

o

a. 1o o

Oh

E-"

3Eh

&Ho

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60

Fig. ID

RATCHET WRENCH AND

LEAD SCREW MECHANISM

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61

C5 SMU, O

Q

O

2

8

H

IA

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62

X

— O

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63

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65

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Page 173: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar
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82

Page 175: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar
Page 176: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar

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Page 177: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar
Page 178: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar

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Page 179: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar
Page 180: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar

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Page 181: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar
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36

(a) Rx - 150,100 (b) Bj « 201,000

(c) Hi - 252,000 (d) Ri = 315,000

(e) Bj = 355.000

Pig. 26

SHADOW PHOTOGRAPHS

5° WEDGE, P - 14.7 psla

Rx« 150,100 to 355,000

Page 183: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar
Page 184: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar

87

(a) Hi » 776,000 (b) Bi * 936,000

(c) Hj » 1,098,000

Pig. 27

SHADOW PHOTOGRAPHS

5° WEDGE P = 45 psla

Hi » 776,000 to 1,098,000

Page 185: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar
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88

(a) Rj = 150,100 (b) Rj = 201,000

(c) Rx - 252,000 (d) Rj = 304,000

(e) Ri = 355,000

Fig. 28

SHADOW PHOTOGRAPHS

10° WEDGE P « 14.7 psla

Ri = 150,100 to 355,000

Page 187: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar
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89

(a) Rj - 475 r500 (b) Rx * 63^,000

(c) Rx= 793,000 (d) ax - 951 1 000

(e) Rj 1,098,000

Pig. 29

.SHADOW PHOTOGRAPHS

10° WEDGE F =« 45 psle.

Bi - 475,50c to -. >;<- >00

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Page 190: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar

90

Page 191: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar
Page 192: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar

91

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Page 193: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar
Page 194: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar

APPENDIX A

Calculations to find position of shock Impingement

on flat plate.

1. When the wedge was In Its most downstream posi-

tion, Its leading edge was found to be 1.033 In.

downstream of the leading edge of the flat plate.

2. Distance vertically between the leading edge of

a wedge and the flat plate = .878 In.

3. At Mach 3.0, for 10° deviation of flow:

wave angle 2?. 4°

tan 27. 4° = .518

4. The shock struck the flat plate 1.694" downstream

of the nose of the wedge.

5. Adding (1) and (3) gave 2.727". Hence, the shock

struck the flat plate 2.727" downstream of the

leading edge of the flat plate.

6. To find the distance, the wedge must move forward

to place the shock on the fifth hole downstream,

since the fifth hole was I.725" from the leading

edge:

2.727• 1.7251.002"

Hence the wedge had to be moved forward 1.002" to

strike the fifth hole. The wedge would be moved

.25" forward each time to strike succeeding holes

upstream.

Page 195: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar
Page 196: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar

7. For the 5° wedge, at Mach 3,

wave angle = 23°

tan 23° - .^24

x « A§|g = 2.070-

8. Henoe the shock struck the flat plate 3.103"

downstream of the leading edge, and to strike

the fifth hole the wedge would be moved for-

ward 1.378". For each succeeding hole forward,

the wedge would be moved forward .25"

Page 197: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar
Page 198: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar

APPENDIX B

Sample calculations to determine the Interaction

distance upstream by the method of Hef • 3«

Hole 1, 5° wedge, 14.7 psla

Xn = .725" Adding: 1.047211

d = 1.047" 1.772"

Therefore shock impinged at 1.772" from leading

edge, when interaction affected hole 1.

For Mach 3,

= 8.9 —t (See section on "UpstreamR * Interaction Distance")x

x = 1.772"

Rx = 3^4,000 (based on total pressure of14.7 psla and 70° P)

/, = 6.9 [1.772) 8.0 (1.772) 261(36. ^(lO^) * " (6.04X102) " 2 - 61 (1° )

fx' J# C10*) - 40.1

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Page 200: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar
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Page 202: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar

ThesisK727

SE

Koepcke »

Interaction betweenlaminar boundary layersand shock waves withseparation of flow.

18 59 IHTERLIB,azjLa

ThesisK727 Koepcke

Interaction between laminarboundary layers and shock waveswith separation of flow.

Page 203: Interaction between laminar boundary layers and shock ... · Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1957-05 Interaction between laminar

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