Supersonic wind tunnels - theory design and performance

l. b Q ' l hl ' l~ m v o o r ero-en hydrodynIm'C I

SUPERSONIC WIND TUNNELS - THEORY. DESIGN AND PERFORMANCE

BY

J. RUPTASH

..

ACKNQWLEDGEMENT

The auth or wis he s to express his sincere thanks to Dr-, Go N. Patterson, Di rec to r of the Ins t it u t e of Aerophysics , for his encou r age rn ent and discuss ions during the progress of this work• T'he author is als o glad to ex pre s s his thanks to Dr. 1.1. Glass for sugg esting im prove m e nts in the pr esentation, and to Dr. D.G. Gou ld for his valuable ass istanc e in che cking the contents of this review.

The auth or wish e s to express his fullest gr a titu de to the Institu te of Aero p hysics fo r the opportunity to underta ke this review during hi s course of gr a duate studies, and to the Aerona utic a l Library of th e National Re s e a rch Counc il (Canada) for the nece ssa r y papers and reports made available during the course of this wor-k,

The au th or is most grat e f u l to Miss No Fieldi ng and Mr s. H. Ha rtw i ck for their pat ien ce in typing this re view.

(ii). TABLE OF C ONTENTS NOTATION • • • • • • • • 1 I.

11.

111.

IV.

V.

VI. INTRODUCTION . •

SUPERSONIC WIND TUNNELS • 2. 1 General

2.2 Types of Supersonic Wind Tunnels

FUNDAMENTAL THEORETI CAL EQUATIONS -ONE- AND TWO-DIMENSIONAL SUPERSONIC FLOW 0 • • 0 • • • • • • • " • • • • e • • • • • •

3.1 Equation of State 3.2 The Energy Equation 3.3 Isentropic Flow Relations 3.4 Nor-marShock Wave

3Q 5 Oblique Shock Wave

3.6 Steady Two-Dimensional Flow - The Method of Characteristics.

FLOW IN A LAVAL NOZ;ZLE. . . .

TURBULENTBOUNDARYLAYERDEVELOPMENT IN COMPRESSIBLE FLOW • Q • Q • • • •

5. 1 Definitions

5.2 Turbulent Boundary Layer in Plane Compressible Flow .

5.3 Turbulent Boundary Layer Development along the Side (Plane) Wall of a Diver-gent Supersonic Channel.

TWO-DIMENSIONAL SUPERSONIC NOZZLE DESIGN 6.1 Graphical Methods - The Busemann

Method,

6.2 Semi-Graphical Me thods , 6.3 An aly tical Methods.

6.4 Nozzle Contours with Continuous Wall Curvature.

6.5 Nozzle Length.

6.6 Nozzle Contour Correction for Boundary Layer Effects • 4 5 18 55 64 88

.

.

.

.VII.

VIII.

(iii ).

SUPERSONIC DIFFUSERS . . • • • 7.1 Supersonic Flow Deceleration 7.2 Perfect or Ideal Diffusers

7.3 Minimum Diffuser Throat Area Required to Establish Supersonic Flow.

7.4 Variable Throat Diffusers and Flow Stability.

7.5 Fixed - Geometry Supersonic Diffus e r-sr -7.6 Variable Geometry Supersonic Diffusers 7. 7 Flow [n a Multip'le-iShock Wedge Type

Convergent - Divergent Supersonic Diff'us er-,

7.8 Diffuser Efficiency.

SUPERSONIC WIND TUNNEL PERFOR'MANCE RELATIONS • . • • • • • 0 • • • • • • • • • •

8.1 Test-Section Flow Parameters.

8.2 Minimum Pressure or Compression Ràtio Required to Maintain Supersonic Flow in the Test-Section.

8.3 Blast or Running Time Av ai lab le with an In t e r m i tt e nt Flow Supersonic Wind Tunnel with a "Vacuum Storage Drive" 8.4 Blast or Running Time Available with an

Intermittent Flow Supersonic Wind Tunnel with a "Pressure Storage Drive" 8.5 Power Requirements for Continuous Flow

Supersonic Wind TunnelOperation. 8.6 Compressor Intake Volume for Cont inuous

Flow Supersonic Wind Tunnels.

Pa ge 127 152 ·t -~

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REFERENCES 177

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NOT A T ION

speed of sound (rela tions 3.6 , 3.8).

•

A

A

(M)

Cp

Cv

'

e+-

J

c-E.

E.R .

r

(i~)

G-(i1)

HP.

h

Iè.

L

cro s s - s e c ti on a 1 area (4. 2, 4.7) .

area-Mach number re1ation (4.7, 6.5).

characteristic cell co o r din a te s (3.54. 3.55). specific heat at constant pressure (3. 3)~ specific heat at constant volume (3.3).

charaeteristies in the physica1 p1ane (3.39). .

inte r n a l energy per unit mass (3.2) . energy ratio (8. 3e) .

boundary 1a y e r functi on (5. 31). boundary 1a y er functi on (5. 3 3 ). ho rs e p ow er (8.37) .

no zz1e conto u r height (F i gu r e 5.8)•. th e r m a1 conduc tivity . length (8.10).

L

_o

diffuser 1ength (7•.9) .

M

Mach nurnbe r,

rn

. ' 1 1

P

J

P

J

P

P

r

mas s flow per unit tim e (4.5, 8.5) . mass of air (8.15, 8.16) . speed ind e x (3. 5 8 ).

boundary 1a y e r prof ile paramete r (5. 13). pres s ure.. .

power (8. 32 ).

boundary 1a y e r fun cti on s( 5, 25, 5.28, 5.40) .

fr

'_

-

1,C

e

,

Q)Q

Q,

t

l'

I

R

Re

r

v

V~

x

~$I.

1',

(xe -

x.)

~

,B

y

[,)/2

S-cS"

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%/7

3

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boundary la y e r functi ons (5 . 2 6. 5.29).

te st- s e ction dynamic pressure (8.4). velocity.

maximum possible or ultimate speed (3.31).

gas content (3.1).

Reyno lds nu m b e r (8. 10).

num b e r of wave r-eflec tions (6. 3) .

polar coor-d i n a te's in the physical plane , en tropy (3. 2 0) . tim e (8. 22. 8.28. 8.31). te m pe r atur e . velocity com p one nts. volume of storage ta n k (8 .15. 8.23, 8.29). volume inta k e (8 . 4 0). rec tàngular co o r dinate s . nozzle throat to exit.le ngth (6 . 2 6 ). half test -section height (F i gu r e 6.'12).

Mach angle (3 . 4 3 ).

ratio of spec ific heats (3 . 3) .

characteris tics in the hodograph plane (3.40). boundary la y e r th i c kne s s (Figure 5.2).

boundary la y e r displacemen t th i c kne s s (5. 1). diffuser effi ciency (7. 12, 7.16. 7.17).

com pr es s or efficien cy (8. 3 6).

flow deflection angle (3.28 ) .

.

'

r

.(3)

8-

_w obliq u e shock wave angle (3 . 2 8 ) .

1J

boundary lay er mom en tum thickne s s (5 . 2) .

..

.

mul tipl.ier-sfS,34)•

co e ffi cient of visc osity (8. 12).

r

/./ Prandtl -Meye r expans ion angle (3. 5 2).

f

dens ity.

rr

shear ing stres s at the wall (Fig u r e 5.3) . o

SUBSCRIPTS

*

sonicconditions; es g, T* te m p era t u r e.a t th e

nozzle thr-oat,

*2 con ditions at the diffu s e r or second throa t; e, g.

A*2 diffus e r th roat area.

1 no zzle exit or test-s ection conditions; e.

g

,

P,

tes t - sec tion stat i e pressure.

1 state ups tr eam of shock wave.

2 state downs tream of shock wave.

3 cond itions at diff u s er exit section .

Sim ila rly sub scripts a, b, e denote conditions at

points or in re gion s a, b, e.

..

f i o 01,02 , 03..• w final sta t e in th e stor a ge vessel. initial state in the sto rage ve ss el.

re s er v oi r, total or stagnation conditions ; e.

g,

T0 in let stagn a tion or total te m p e r atur e.

stagna tion conditions at points or in regions 1, 2, 3, ••0

e.g.

Po'S

stagnation or total press ure at th e diffuse r

exit section .

conditions at th e wall or surface.

BAR

Parameters with a ba r indic at e co r r e s p on din g va lues at

th e outer edge of the boundary la y e r; e.g.

Ni

Mach

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I. INTRODUCTION

The purpose of this review, which was started in January 1951, in conjunction with the author's work on the design of the U. T.1. A. 5- by 7- inch supersonic wind tunnel, is to present a treatment of some of the fundamental supersonic wind tunnel design and performance

problems. Although any graduate student can readily work his way through the many theoretical and experimental papers availab le, a demand does, however, seem to exist for a publication which presents a compilation of these fundamentals .

Unfortunately, due to the limited time ava.ilab le , many important details have been only briefly mentioned. However, the author feels that the selected topics which have been reviewed in detail are at

least among the important fundamentals of supersonic wind tunnel design and performance.

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Il. SUPERSONIC WIND TUNNELS

. The present section wiU be primarily devoted to a brief

out-line of some of the problems associated with s uper-s onic wirid tunnel de-sign and operation and a general de s cription of a numbe r of typical

in-,s t a lla ti on s . The following sections wil! be devoted to a dis cus s ion and .a n a lys is of some of the basic aerodynamic principles as s oc ia te d with

supersonic flows and supersonic wind tunne l design.

2. 1 General

Wind tunnels may be c las siffe d, in gene na l, according to îour main categories: fluid cycle, drive aystern..:test-section

arrange-..ment and test-section speed range. The Huid cycle may be a continuous open circuit, a continuous closed circuit, an intermittent open circuit

.or an in t e r m it t e n t closed circuit. The drive system may consist of a direct-drive blower, a pressure storage vessel, a vacuum storage vessel or the drive system may be based on the principle of in duc e d flow. The test-section may be completely enclosed, i.e., c los e d jet; consist of an open space with the air streaming from the nozzle exit to the diffuser entrance, i. e., open jet;or it may be half-open. Mor-eover-, the flow in the test-section of a supersonic wind tunnel may be e ithe r two- or three-dimensional depending on whether the nozzle is des igne d 50 that the air is expanded two- or thr-eee dirne ns ional.ly, The speed r-arige may be low

subsonic speed (0 to 400 mph), high subsonic speed (400 to 700 mph), transonic (t e s t- s e c t i on Mach number f'r-om ab out 0.8 to 1. 4), supersonic (Mach number from L 2 to about 5) and hypersonic (the term hypersonic in this case is used, loosely, to describe the Mach number range above about 5).

The conventional subsonic wind tunnel te s t-cs e ction suffers

from the limitation produced by the phenomenon of choking. When the

speed of sound is reached at th e smaUest cross-section or' the wind tunnel, the mass flow cannot be further increased. Thus a new feature (conve rgent-cdiver-gent nozzle) must be incorporated into the design of a test-section suitable for supersonic wind tunnel te s ting, By the use of a converging-diverging or de Laval type of nozzle, a large number of wind tunnels have now been constructed for testing mo de Is and basic flow studies at s upe r soni c speeds. Howe'ver-, the problems involved in the design methods and principles of operation of the s e tunnels are in certain respects quite different from those involved in the case of sub-s onic wind tunne Is.

As is weU known, the Mach number in the test-section or measuring chamber of a supersonic wind tunnel is dete rrn ined uniquely by the geometry of the convergent- diver-gent noz z le , i.e. by the ratio of the test-section to throat area. Moreover, in section (IV) it will be jshown that, for purely is entropie s upe r s onic flow over the entire

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ratio or pre s s ure diffe r ential is necessary to attain a given Mach nurnbe r , For all pressure rati os grea ter tha n this minimum the flo w within the

noz-zie re mains un change d, Furthermore. in a supersonic wind tunnel the

actual va lues of the pressure and dens ity upstream of the nozzle merely co n t ro l their magnitude in th e test section. Since a different·nozzle

con-tour is require d fo r each value of th e te st- s e c ti o n Mach numbe r , super-son ic wind tunn els are us ually designed to accomodate readily removable

nozzle blocks in order to provide test data at various Mach numbe r s, On

the ot he r han d. satis fac tory nozzles have been designed with adjustable

flexible walls in order to prov ide a continuous variation of the test-section Mac h num b e r over a limite d range. The essential features of typical

flex-ible no zzle s are descr ibed in references (18. 32, 66 and 78).

The r e are several genera l methods that may be used to increase

the pressure energy of th e air suffi ciently-' i.e•• to establish the r equir-ed

pr essure differen tial.vt o drive th e air through the test-section at supersonic speeds . Basically, all th e s e methods use a compressor but the method of

tra n s mitting th e energy may be effected in several different ways: (a) By

increasing th e pressure ahead of th e working-section. (b) reducing the pres s u re down s tream of the worki ng section - in this case the compressor

is used as a va cuum pump i ng sys te m to draw the air through the noz z le, (c) pa s s i ng th e compressed ai'r th r ough an injector slot aft of the

test-section thus producing th e required pressure ratio by the resulting high

sp e ed jet and (d) byin c r e a s i n g the pressure upstream of the

working-se ction and simu ltaneous ly decreasing th e pressure downstream as in a con tin uou s clos ed ci r c uit wind tunne l where th e compressor is both

in-cre asing and decreas ing th e pressure sim ulta n e ou s ly. Power Req u ireme nts. '

As th e design speed of the wind tunn e l s was in c r e a s e d a major

pr oblem had to be surmounted - the power a vaila b le, The flux of kinetic ene rgy per unit ar ea of th e wi nd tunne l te st - section varies as the cube

of the velo city fo r a given dens ity of th e airs tream. Since the power neces-sar y to drive a high speed wind tu nnel is lar g e. it is often advantageous to

store the require d ener-gy from a source of c ompar-attvely low power ánd

to disch a r g e it in to th e wind tunne l circ uit duri ng short testing periods. This is th e schem e used in the operation of inte r mitte n t or blowdown type win d tunnels. The powe r in s t alle d fo r intermitt e n t flow supersonic wind

tunnels is us ua lly about 10 per cent of th e required horsepower for

con-tin u ou s flow oper- a t fo n, On th e -o th erhand, since the power required de...,

creases in propo r tion to a decrea se in stagnation pr-es s ur-e, the power

requir e m ent s for a con tinuous flo w clos e d circuit wind tunnel may be re d u ced by operating at reduced pr-ess ur e s, in which case the Reynold's

number will he les s in proportion to th ep r e s s u r e.

Another method of reducîng th e high power required by.large con tinuous flow super sonic win dtunnels is th r o u gh the use of a gas with

a high mo lec u la r weight, 10 e.• a gas in whi ch the speed of sound at a give n temperat u r e is low e r than th at for air. For examp l.e, for a given

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te s t-section siz e, the powe r required to operate a wind tunnel at a Mach number of 2 with air is about seven tim es as hig h as th e power required to operate the tunn el with fr eon 113 [47J Howe ver, apart fr om th e high cost of some of the s e gase s, certain difficu lties may ar-Ise, The viscous

properties of some of the gases ar e, in general, not well known, More-over, it is advisa ble to us e a ga s with th e ratio of the sp e cific heats equal to th at for air ( r '" L 40) if an accurate comparison between tu nnel tests and flight data is to be obtai ned - in this regard a mixture of gases would

have to be use d,

Cooling

The energy su p plied to th e drive system in a wind tunnel

finally emerges as an inc r ea se of th e heat energy of the air stream in a closed cir c uit ty pe wind tunne l. Th is inc r e a s e in the temperature of th e

tunn el air str eam cont inu es until th e heat los s es balance the heat input .

For low spe e d wind tunn els this heat balance is realized at a reasonably

low temperature through surface cooling and the aid of än air exchange r , For high spe ed wi nd tunnels th e balance no lo n g e r occurs at a low temp-erature. Accordingly, additional cooling must be accomplished by one

of th e following methods: In c r e a sing the surface cooling by running

cold water over the tunne l exterior; inte ri o r cooling by water cooled turn i ng vanes; in stalli ng water cooled radiators; or by a continual re-placement of the hea ted tunnel air with cool air through a la r g e external

he a t excha nger,

Effects of Humidity

As th e air flows th r ou g h the noz zle of a supersonic wind tun-nel. th e relative humidity of the air rises ve r y rapidly due to th e la r g e decrease in sta tic te mpe ratu r e. For ex arnple, for a reservoir or stagn- /

ation temperatu re of about lOOoF th e re lative humidity in c r eas es approx-Im a t ely twenty tim es as the flo w is ac c ele rat e d to M ~ 1 at th e th r oat

[S. 69J . Th us, in order to re uce th e in t en s ity of the condens ation effects to a ne glÏ$ible am oun t, it is es se~tial to use highly dried air.

Experimentally L62, 69. 70J it has been sho wn tha t the use of ordinary

(atm os phe ric; air causes th e fo r mation of conden s ation sho c k s. The ex

-is tanc e of th e se sh ocks, whic h usually appear as oblique condensation

shocks downstream of the nozzle thr- oat, will affect the flo w uniformity and reduce the mean flow Mach num b e r, with a corresponding change in

th e other flow parame ters. These unde'sirable effects are caused by the

la r ge amount of heat relea sed in the condensation process .

It is weU known that, when air expands adiabatically, the

temperat ure and the vapor pres s ur e of th e water va por in the air decrease. Mor-eover, in this process the saturation pr-es sure drops ve r y rapidly and the air becomes saturat ed wh en th e satu r ati o n pressure is equal to th e vapor pressure. When the vap o r pres sure exceeds th e satu r at i on pressure

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th e water vapor tends to condense. In the case of atmospheric processes

(fo r m ation of f'o g, clouds) the variations of state are usually slow 50 that•

-as a r ule, supersaturation is not observed. Howeve r , in the f ow of moist

air th r ough a supersonic wind tunnel the variations of state are very rapid

and in this case the condensation originates in molecular collisions.

More-over. according to the deve lopment of the theory of condensation for rapid

pr-oces ses [62. 69. 70J condensation phenomena in the flow through a

..supersonic nozzle begin only with heavy s upe r satur-ation, i. e;, when the

supersaturation is ab out sixty fold and that, once it be gins, it will

con-tinue very rapidly with the consequent formation of condensation shocks.

Experimentally . condensation shocks begin at about 80 to 300 fold

super-saturation i.e, , where the ratio of the vapor pressure to the saturation

pressure is about 80 to 300 .[69J .

Although the effect of condensation in the flow through a nozzle

ca n be estimated [5. 62. 69 J J conventional practice in supersonic wind

tu nnel ope r atio n is to use dried air. It should be not e d, however-, that eve n in the case of very highly dried air, the condensation effects are not

co mpletely eliminated . In pr-acfice, the air is dried sufficiently to

re-duce the dis t u r b a n c e s caused by the condensation phenomena to a

neglig-ible am ount, For most test pur pos es , the dewpoint of the supply air must

~ be of the order of _gaF or lower (at atmospheric pressure) in order to

re-a;

U'" uc~~he ~gnitude of the condensation effects to a negligible amount· [62J.

, ........, f} IlQ~ .

~Q ~ ~

" ' ~ Several methods of drying adequate amounts of air for

super-I' ~,1 sonic wind tunneloperation have been suggested: (i) Chemical dr-ie r-s,

'Iil'~

i.e, J using a desiccant such as silicagel, activated alumina or sova bead

6·

[45J,wh ich remove the moisture from the air by adsorption. In

pr-ae-tice the dr yi ng beds may be installed in the tunnel circuit (see Figures

2. 1 an d 2.5) or externally in a secondary circuit (Fig u r e 2.4). After

the desiccan t has taken up a quantity of rn oîstur- e, the beds must be

re-a ctfvated, i.e., th i s moisture must be driven off by passing hot air over

th e beds. (ii) Compression. If saturated air at atmospheric pressure

and te m pe r a t u r e is compressed to say

n

atmospheres at atmospheric

temper ature then the fraction of the moisture condensed from the in iti a l

-ly sa turat ed air is given by

n;;

I This method of dr-ying the supply

air can thus be feasible with supersonic wind tunnels operated by a

com-pres sed air storage drive. If the compressed air is allowed to stand in

the ve s s el fo r a short whiIe , the condensed moisture can be drained off.

For example ;:if saturated air is compressed from 1 to 50 atm.ospheres

at atmospheric temperature then, when the air is released again to at-mospheric pressure and te mper-atur e , the relative humidity of the air

will be only 2 per cent. (iii ) Increasing th e tunnelstagnation

temper-atur e. By in c r e a sing the tunnel stagnation temperature to about 400°F 104°C

the condensa tion effects can be reduced to a negligible amount for Mach

num b e r s up to about 2 [62J . (iv) Refrigeration. i. e .• cooling the

sup-ply air to a te m p e r a t u r e of the order of the test-section tempera ture

thus condensin g out the moisture before the air is passed through the

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[62J tha t by in stalling an auxiliary or condensation nozzle upstream of the main nozzle and expandi ng th e mois t air to a supersonic velocity in this nozzle (MI

=

1.65), th e condensation shocks we re not observed in the main nozzle (MI

=

2.48).

Air Condensation - Hypersonic F low

Among th e major ae r o dy na mic problems connected with the operat ion of hyperson ic wi nd tunnels (MI> 5) is th at of air condensation sin c e, at very high Mach num b e r s, th e te m p e r atu r e in the test-section

may be in a range which would te n d to produce condensation of some of the airflow• In the case of water va p o r condensation in supersonic flows, it has been noted th a t condensa tion phenomena begin only with heavy

supersaturation . Howeve r-, in the case of air condensation the experi-mental evidence availab le [89J sh o w s th at th e condensation phenomena appear without any app r e cia ble sup e r s a t u r ati on. It has been suggested that th e absence of conside r a ble su pe r s a tu ration in th e case of air con-densation may be due to th e presence of fo r e i gn nuclei such as carbon

dioxide and ic e particles. .

It has been shown [52, 89, 90, 91J tha t for a constant stagnation supply pres sure th e te m p e r a t u r e of the supply air must be increased in order to attain high Mach nu m b e r s without air condensation. For example, fo r a stagnation su p ply pressure of 50 atmospheres the Mach number of eq u ilibrium condensation is about 4 for a stagnation

o 0

temperature of 80 F and about 10 for a temperature of 1340 F .

!'l.° C

7

2

₇

_°

_C

Another ma jo r pr oble m connected with th e operation of hyper-son ic wind tun nels is the very high pressure ratios r-equir-ed at high Mach numbers. For ex arnple, at a Mach number of 10 the required pressure ratio is about 75 0. However-, the Ma ch num b e r of equilibrium air con- .

densation decrea ses with in c rea sing stagnati on press ures fo r a constant

stagnation temperature . T'hu s, as the stagnation pres sure is in c r e a s e d to achieve higher and highe r Ma c h nu m b er-s, th e stagnation te m pe r a tu r e mayalso have to be inc reas ed to avoid air condensation . For exam ple ,

fo r a stagna tion te m peratur e 80°F th e Mach num b e r of equilibrium con-densat ion is about 5 fo r a stagnation sup ply pressure of 1 atmosphere and about 3.7 fo r a pres s u re of 100 atmospheres [91] .

2.2 Types of Supersonic Wind Tunnels

By selecting various combinations of the different fluid cy c le s , drive systems and test-section ar r a n g e m e n t s, many different types of supersonic wind tunn els may be constructed - the design and principle of operation being la r g ely dependent on th e power available, test-section

size required, te s t-s e ctio n Mach number to be attained and the type of testing to be undertaken.

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In gen er-al, supersonic wind tu nn els may be cla ss ifi e d accord-ing to two main typ e s: the continuous flow and the in t e r mitte nt flo w type and th en sub-classifie d according to the other variab les in the design and

principles of ope rati on. In the remaining part s of th i s section a brief

descr iption of a number of conventional supersonic wind tunnel

installa-tion s is presented in a genera l form. For a detailed description s e e,

for exam ple, reference s (18, 19, 20, 41. 48. 61. 63. 66, 78. 94).

(a) Intermit te nt F low Supersonic Win d Tunne ls

In one of the sim ple st îor ms, th e in te r m it t e n t flow or blow-down supe rs onic wind tunnel is operated with a vacuum storage vessel

-th e air being drawn either fr om a dry air storage reservoir or from the atmosph e r e th r ough an air drying bed and then through acontraction se ction no zz le, te st -s e ctio n diffuser and into a la r g e evacuated vessel as sh own sc h e matic ally in Figure 2.1.

VACVUM STORAG-E Ve:SSEL D/rFusER. ,AIR DRyl!R TEST SECTION CONTRACT/ o N

$ECT/ON QUICK

ACTINe-CONTROL VALVI:

VA C()U!Vi

PUMPS_{, r}

FIGURE 2. 1

OPEN CIRCUIT. ''VACUUM STORAGE DRIVE".

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In the vacuum actuated int e r mittent flow sup e r s onic wind tunnel the pres-sure ratio available to drive th e air thr ou gh the cir c u it is a direct func -tion of the degree of th e va cuum .which can be attained in the evacuated ve s s el, With this arrang e m ent, i.eG, with th e flow fr om the atmosphere or fr om a dry air sto r a ge cha m b e r which is usually maintained at atmos-pheric pressure by means of a fle xible diaphragm [19, 41. 94J the up-st r eam up-stagnat ion or res e rv oir pre ssure, density and temperature will remain consta nt fo r the dur a tion of any tes t run.

Ariother variation of th e in te r mitte n t flow or blowdown wind tunnel is th at in which a larg e sto r a g e ves sel is filled with air at an a.p-propriately high pre s s ure . With this ar r a n ge m ent th e air is drawn from the atmosphere through a dryer an d st or e d in the high pressure ves se l as shown schematica lly in Figure 2.2 . From the point of view of opera -ting economy blowdown operation with a hig h pressure vessel would be <,

advantageous but this arrange ment ha s th e disadvan tage th a t th e stagn-ation pressure and te m p e r-atur e will change continuously during a test run. If an adjustable automat ic throttli ng valve is in s talle d between the high pressure st o ra ge vess el and th e working -section. th e test runs can be made at a cons ta nt upstream stagnation pressure . However, the

stagnation te mp eratur e wou ld still fall sinc e any thr ottlin g process can have no effec t on it and in cer t ain case s, e,g. , heat tr a n s fe r experiments, this may have a serious effe ct. In order to maintain a selected constant stagnation te mpera t u re in this arrang em ent an inlet heat exchanger with automatica lly variable rates of hea t exc han g e would be necessary.

Expe r-Lme n tal evidence ha s sh ow n that th e Reynold's numbe r may have an im p o rtant effect on th e flow at high Mach numbe r s, In this regard one adv a n tage of a pr essur e stor a ge dr ive wind tu n n el is that th e Reynold's numbe r can be varie d in dep en de ntly of th e te st- s e c tion Mach number and model size over ce rta in limits - by va r ying th e reservoir prèssure in th e sto ra g e vess el andjor th e thrott ling valve settin g. With the vacuum actuated in t er mi ttent flow wind tun nel th e upstream stagnation condit ions are con s tant (atm os ph eric) and thus th e Reyno ld's number can-not be varied in depende n tly of th e test se ction Mach number fo r a given model siz e,

Another ar r ange m ent may be pr ovide d by adding a vacuum receiver at th e end of th e diffuser of the la y o u t shown in Figure 2.2. Thus a wider operating Mach num b e r range and, for a given Mach num b e r and test-section, a lon ge r running tim e can be obtained with this combination, This arrangeme n t is pr ob a bly the most feasible method fo r obtaining the very high pressure ra tios req uire d at high Mach num be r-s , For example, to ac cele ra te th e at rstr-ea m to a Mac h num b e r of 10, the required pres-sure ratio is about 75 0' [92J •

(12) tllGH PR,ESSURE SUP P L Y

vessec

AUTOMATIC THROTTt.E VALVE j)IPFVSEIi!. NOZZLE TëST SECT/ON coNTR.ACT/ON SEC TI ON QUICt< ACrtAltT CONr~OL VALVE

/

d

1111;":' - il:-

"it-'

:

I:l

i

Ij

I

,

:1/1 111 'I I

I

H/&H p,eeSSUR.E SErrLIN6- CHAM8E~ 01L. FILTé~

t...==~I

I

;

1i~

.IÎr==='~it======::::::;;fi~

AIR./

aevee

COMP/?ESSOIif:.

1

FIGURE 2.2

OPEN CIRCUIT, "PRESSURE STORAGE DRIVE ", INT E R M IT T E N T FLOW SUPERSONIC WIND TUNNEL.

Th e primary advan tage s of an inte r mitte nt flo w supersonic wind tunne1 are its re latively lo w cos t, sim plicity an d low operating cost, It is

well known that it is economi c ally advan tag eou s to sto r e th e re quired energy

fr om a source of compa rat ively low pow e r and th e n to di scharge it in t o a

wind tunnel circuit during sh ort testin g periods. Anothe r primary advan

t-age in th e use of intermitten t flow wi nd tunn els operated by a high pressure

supply and/or a va c uu m receiver is that , for a given storage ve s s e l in s t a

ll-ation, sever al intermit te nt flow wind tu nn el circuits can be readily arranged

to use the sam e supply vess e l or vacuum receive r. However, since th e

run ning tim e of am intermittent flow wind tu n n e l with a given test-section

size and Ma ch num b e r is a fu nction of th e volume and initia l pressure of the

high pre ssure supply vessel and/or th e volume and degree of vacuum of th e

evacua ted rec eive r, a lo n g running tim e may re q ui r e excessively large

stor-age ve ss els . For -e x a rn ple, in the case of an interm it tent flow tu n n e l

•

'<.

....

(13)

in a tunnel with a test - s e ction area of 25 square fe e t, a va cuum stor a ge vessel of about 360,000 cubic feet would be required . Thus, inte r mit tent flow sup e r s onic win d tunnels are usually accepted as an installation con -fin ed to te st runs of short duratîon in which case the in s t r u m e ntation must be designed to meet this lim ita tion.

One dis a dv a n ta g e of the open circuit or non-return superson i c

wind tunn e l is that a la r ge quantity of air must be dried fo r each te st r-un, In order to reduce th e required capacity of the air drying equipmen t an int e r mitte n t flo w wind tunn e l can be designed with a return duet, i.e,

closed circ u it, [41, 63J so that th e air discharged fr om the vacuum pum p s

can be filte r ed and.returned to th e dry air storage vessel.

As has been previously no te d , the energy of a high pressure

supply vessel can be used to drive a wind tunnel in a number of ways of

which that usi ng the inj e c t or system and that using the direct di scharge of th e air th r o u gh th e working-section are the s irnp lest, The prin ciple of in duc ed flo w ha s been applied to both in t e r m itt e n t and con tinuous flow

su pe r sonic (and subsonic) wi nd tunnels. Moreover, there are tw o ty pes

of induc ed flow or indu c tion wi nd tunnels: Wind tunnels wh ich us e the in je ctor (o r eje ctor) system [18, 57, 59. 60J and wind tunne l s which us e the ex haus t of a gas tu r b i n e to in d u c e the flow through th e test-se c -tion [34. 57J 0

The principle of an indu c e d flow injector syst e m wind tunne l can be briefly described as follow s: Air at a high pressure is passed th rough a throttling valv e to an annu l ar high pressure chamber lo cated downstr e a m of th e test section as shown in Figures 20 3 and 2. 5. Th e air

th en es capes thr ough th e inje c t o r slo t ro und the periphery of the tunnel and for m s a high speed [e t, This je t, mixing with the tunnel air, cr eates a low pressure in an d aft of the tes t-section and thus induc es a flow through th e te st-se cti ono

A schematic diagram of a typica l open circ uit. ind uction ty pe. int e r mitte n t flow sup e rs onic wind tunne l is sh o wn in Figu re 20 3.

The efficiency of in d u c e d flow wind tunnels can be de s c rib ed in te r m s of th e mass ratio. i.e•• the ratio of the mass flow through the te s t - s e ction to the mass flow through the inj e cto r slot. In th e s e terms, th e la r ge r the mass ratio the long e r will be the blast tim e of an in duce d flow tunnel operated by a compressed air storage drive. Although a pr es sur-e of only one atmosphere (ga u ge) in the ejector box is sufficient to produce s onic speed at th e injector sl.o ts, it is advantageous to use higher pressures sin c e, with higher pressures and hence den s ities, th e momentum of the injected air is in c r- e a s e d,

( 14) ." HIG-H PRESSUR,E .sUPPLY VESSEL • COMPRESSOR

f

AIR. DRyER CONTRACT/ON SEC TI ON TEST $6C710N ~Hl<aTTLING­ VALVE SLOT ZONE HIGH PRESSUR.I;= EJECTOR 80X FIGURE 2.3

OPEN CIRCUIT, INDUCTION TYPE,

IN T E R MIT T E N T FLOW SUPERSONIC WIND TUNNEL.

The air injector can be designed to work over a wide range of

inducing mass flows and pressures. However, to obtain maximum

opera-ting effi cie ncy at any given Mach number, the ratio of the in j e c t o r slot area

to th e te s t - s e ction area must be matched to the blowing pressure. Usually

the in j ecto r is designed to give M

=

1 at the slot. Another advantage with

open cir c u it in du c e d flow wind tunnels is that the air injector can be design-ed to operate with a fluid other than air. Small scale induction type super-soni c wind tunnels have been designed using steam as the dr-Ivlng.fluid [60J.

The ratio of the mass of air flowing through the test-section

(in du c ed mass flow) to the mass of air flowing through the injector s lots

(inducin g mass flow) is about 3 for a Mach number of about 1. 4 with the

slot to test-section area ratio of 0.08 [18, 59J . T'hus , for a Mach

num-ber of 1. 4 a given test-section and quantity of high pressure air. the

run-ning tim e with an induced flow drive will be about 3 times as long as that

•

•

'f

(15)

For Mach num b er s below about L 5 th e inj e c t o r system is: in gene r- al,

more efficien t than the dire c t di sch arge methode Howeve r, at higher

Mach num b e r s only th e dire c t discharge method is in general use since it has not been pos sible to obtain Mach numbers above about 2 with single

sta g e injecto r syst e ms [18, 59J .

If the comp r ess o r andjor vacuum pump capacity is la r g e enough the in termi t tent flow wi nd tunn els. as shown schematica lly in

'Figu r e s 2.1. 2.2 and 2. 3. may be operated as a continuous flow win d tunnel but, in order to retain the relative cost advantage, the compone n ts

are usually designed fo r inte r mittent flow of short duration. However-, since a common high press u re supply andj or vacuum receiver can be used

to drive several separate win d tunnels it is often advantageous to design

the additiona l circuits with a smaller test-section, thus providing an ins tall

-ation with long er running tim e s.

eb) Con tin uous Flow Supersonic Wind Tunnels

A continuou s flo w wind tu nne l may be either a closed cir c uit

(r etu rn) or an open cir c uit (non- retur n) ty p e. A ty pica l closed circuit,

direct drive, continuou s flow supersonic wi nd tu nn e l is shown schema ti -cally in Figure 2. 4. The major components comprising a wind tunnel

of this type are: The motor driven compressor. cooler, contraction

se ctton, nozzIe, test.-aection, diffus e r, return duet and the drying system

which may be located eithe r inside or outside the main circuit. The com

p-ress or mus t su p ply the pre ssure diffe r e ntial ne c e s s a r y to dri ve th e air

around the circuit . Although multi- s tage axial flow blowers or compr ess ors

are the mos t comm on ty pe in use for obtaining la r ge pressure ratio s and ma s s fl.ows, the ra dial or cen trifugal ty pe may be used when th e pow er

re q uirements ar e not as la rge.

Since a lar ge quantity of he a t is ad d e d to th e air strea m eac h

time it pa s ses through the comp r es so r, a cooling un it must be added to the circuit of a supers on ic wind tunnel in order to keep th e stagnation

te m per atur e at a cons tant value, Although the cooling unit may be lo c

-ated eith er dow nst rea m or upstream of th e working-sect ion it is usually

loc a t ed upstr-e arn, i.e•• betwe en th e compressor and working-s e ction as shown in Figure 2.4. For a giv e n amount of heat exchange th e coole r ma y be several tim es sm alle r fo r the upstream loc a ti on than fo r th e down

-stream locatton, If the.c oole r is lo c ate d downstream of th e working-s ec tion

the test-sec tion waUs wi .Il be at a high te m p e r a t u r e which is undesirable .

Moreover, the velocity distrib ution downs t r e a m of the compressor is ,

in gene r-al, no t uniform hence a cooler loc ate d upstream of the working

-section ma y serve simultane ou sly as a honeycomb to im p r ov e th e velo

(16)

•

TURNING-VA NES TEST SECTIO'" DIFFUSER NOZZLE

I

CONTRACTION SECTION

I

--COOLER. RETUR.N

Dver

COMP~ESSOR.

]1

G-EAR BOX AIR ORYER FIGURE 2.4

CLOSED CIRCUIT. DIRECT DRIVE. CONTINUOUS FLOW

SUPERSONIC WIND TUNNEL.

As was stated pr-e viously , th e power requirements for a

con-tinu ous flow clos e d cir-cuit wind tunnel may be reduced by operating at reduced pressures - in this case. in addi tion to the main components

shown in Figure 2.4. a vacuum pumping system must be included to

re-du ce th e pressure in th e entire circuit . The continuous flow closed

circuit wi nd tunnel therefore has the advantage that within certain limits

the Reyno ld's num b e r can be varied independently of the Mach number.

Moreover• with a closed circuit only a small quantity of air must be dr-ie d, other gases may be used and the tunnel may be operated at re-duced pressures adjusted to correspond to high altitude flight.

With an open circuit (n on - r e t u r n ). direct drive, continuous

flow wind tunnel the air is taken in at atmospheric conditions and pump-ed th r o u gh an air dryer before passing through the contraction s e ction, no z zle, test- s ectton, diffuser and out to the atmosphere again. Because of th e la r g e power requirements and the quantity of air which must beo

'"

..

(17)

dried con tinuously , this ar rang e m ent is not a ve r y common one - the closed cir c uit (re turn) ty p e is usually used when continuous operation

with a direct dri ve is de sir ed. An open cir c uit tu n n e l is usually used

in testing devices with realo r simulated combustion processes [78J •

A

ty pic al clo s e d circuit (r etu r n). indu c ti on type, continuous

flow supersonic wind tunnel is shown schematically in Figure 2.5. The

arrangemen t is similar to th at shown in Figure 2.3 except that a return

duet has been adde d fo r con tinuous flo w c losed circuit operation and an

exit valve th r ough wh ich th e excess air is drawn out.

FIG URE 2.5

CLOSED CIRCUIT (RET UR N). INDUCTION TYPE.

CONTINUOUS FLOW SUPERSONIC WIND TUNNEL.

Some adva n tages and dis a dva nta g e s of in du c t i o n ty p e wind

tunnels have already bee n pointed ou t. In addition a disadvantage of the

con tinuous flow induction type wind tunn el is th a t even at low Mach nu m b e r s

the effi cie ncy is low er than that of a conti nuous flow closed circuit wind tunnel with

a

dir e ct driv e. Moreover , it is no t convenien t to operate in

-duced flow wi nd tunn els at sta gnat ion pressures below atmospheric pressure

which restric ts the test- s e ction size for a give n power. The main advantage

in using an induction drive fo r a con tin uous flow win d tunnel is th a t th e

com-pressor and tunne l chara cteri sti cs ne ed no t be closely matched but this

(18)

III~ FUNDAMENTAL THEORETICAL EQUATIONS

-ONE- AND TWO-DIMENSIONAL STEADY SUPERSONIC FLOW.

Both the general theory of one- and two-dimensional isentropic

flows and the theoretical re lations which describe the transition across shock fr on ts have been developed duringthe past years and provide methods

of calcu lation in the applied field of supersonics. Some of the well known

results of th e s e theoretical developments will be applied to the analysis of the flow ch a r a c t e ris ti c s in a supersonic wind tunnel.

In the present section we will be primarily concerned with steady is e n t r o pic flow and the flow relations across shock waves. Steady

flow is defined as a motion in which the flow velo c i .ty , pressure, density

and specific entropy remain unchanged with respect to time at each spatial

point. Ise ntr o pic flow is defined as a reversible adiabatic flow. Steady

flow in which there is no exchange of heat across the boundaries is called steady adiabatic flow. Moreover , if the flow is a so reversible, i. e., if

th e r e is no dissipation of mean motion energy into heat, then it is a rever-sib le adia batic flow and is called is e nt r o pic .

For subsonic flo w s the assumption of isentropic motion is, in

genera l, valid except in regions such as the boundary layer where viscous

for ce s predominate. For supersonic flows the assumption of isentropic motion is vali d for regions outside the boundary layer when expansion

oe-cu r-s, However, when compression occurs in a supersonic flow shock

wa ve s are usu ally present and the tr a n s ition through shock waves is

non-Isentr-opt c , There is an in c r e a s e in the entropy (according t,o the second

law of th erm o dy n a mi c s) in crossing the shock wave in the flow direction.

In th e flow region upstream of th e shock wave th e entropy is constant

thr ou gh ou t the region. Moreover, if the shock wave is straight, th e

en-tr opy is also co n sta nt th r o u g h o ut th e downstream region. If the shock wa ve is curve d, as in the case of a detached sh o c k, the change in the entropy ac ross th e shock is different fr om point to point along th e shock

and th e flow in the downstream region is non-isentropic [a n d hence

rot-ational).

Although it is assumed that the reader is fa mili a r with the

equations governing th e"fl ow of a compressible nonviscous f'lui.d, the basic equations which rela t e the flow parameters to the Mach number

wi ll be revie w e d . For their derivation see, for exarnple , references

(1, 2, 4, 5, 7, 8 or 10).

I

r

(1 9) 3. 1 Equa tîon of State

In th e development of supersonic flo w theory the fluid is usual-ly assumed to be a perfe ct or ide al gas with constant specific heats. More -over, the effects of vis c o sity and th e r m a l conductivity are usually neglect -ed.

The equation of state of a perfect gas is given by

f=i

R T

and the in t e r n al energy

E

per unit mass is defined by the relation

(3. 1)

(3.2) Moreover, fo r a perfect gas th e rela tion between the specific heats is given by

R

=

Cp -

C"

=

t I -/) Cv

3.2 The Energy Equation - Bernoulli's Equation.

(3.3)

The energy equation fo r ste a dy adiaba tic flows for a perfect or id e a l gas is gi ve n by

cpT-ri'j-Z

₌

CpTo

=

constant (3.4 )

r(

?

+

i.

rg.-Z

.::1.

.

R.

constant (3.5 )

HOf

2-

-

"(-I

_t:

=

a,z

},2

Cl:-or

--

_7-1 -f

-

.,

_-/

_-

₌

constant (3.6 )

The above relations hold fo r steady adiabatic flows whether th e y are is e nt r o pi c or not, Therefo re, from equation (3.4), it follows that th e stagnation te m p e r atu r e

7c;

mus t have th e same value thr-oughout an adiaba tic flo w.and is not affected by th e presence of friction or shock fron t s .

3. 3 Isentropic F low Relations

The law for is e n t r o pic changes of state is expressed by the relation:

"t

(2 0)

The speed of sound, q, , is given by

(3.8)

Dividing the energy equation (3.5) by 0,2 and using relations

(3 .7 and 3.8), th e static pressure - Mach number relation between any

tw o points a... ,

6

in an isentropic flow region becomes

(3. 9)

hence, th e relation between the static pressure } at any point and the reservoir or stagnation pressure

I;

becomes

-y

Ii.

(

'Y-/

~?=/

-

_,b

=

/+-M

_z

(3.10)

and the co r r e s p on di ng rela tions for the density, temperature and speed of sound are

(3. 11)

(3.12)

(3. 13)

= constant

Bernoulli"5 equation for isentropic compressible flow becomes

"/-1

1-

Z

+

i

fa(

?)

:y _ r(

h

2

r

r-r

t: {

I;,

-

1-1

fa

The is e n t r opic flow parameters

00

»

~)

7I-t

_o and

Q.

/a

o as fun ctions of th e Mach num b e r can be readily obtained from tab-ulate d andjor graphical values as given in references (3, 4, 7. 95 or 96). The variation of the s e is e n t r opi c flow ratios with in c r e a s i n g Mach nurrr-be r is show n graphically in F'Igur-e 3. 1.

I

.,

..

3. 4 Normal Shock Wave

In th e th e o r e ti c a l development of the relations which describe the varia tion of th e flow parameters across a shock wave the fluid is usually assumed to be a perfect gas and the effects of viscosity and ther-mal conductivity are usually neglected. Since the flow is assumed to be.

a ste ady adiabatic flow. the tot a l enthalpy or heat content and the ratio of th e sta g n ation pressure to stagnation density ,

elr:, ,

(s ee equation

3. 5) and henc e th e stagna tion temperature

7:

remain constant through-out th e flow .

\;

I

"

,, I

,

\\

-:

,,,,

'\

_~

_:

_'\

,

,

_,

~\ ,

,

,\\

_\

,

\

,

\ I

\"

I \

,

I \

,

,

,

\ \

-,

\

\

_'\

I \

,

I _{\ ~}

,

\ \.

,

1

\\

~

I

,

,

_r-,

,

,

:

\

\

\ " T I

'\,

"~

,

"

"

\ A.

,

, -

... \

\A

'~

I

1\

\. \. I

"

"

i'-,

.

~"'

,~

"

----,

-,

,

_"

,

.

"

I I'... " _"

"-,

_<,

<,

--

'-J r--, I

I---

--

--

---

-

--"

.

,

.

,

r 1.0 0. 8 0. 6 0. 4 0.2

o

o

1.0 (21) 2.0 . 3. 0 4.0

_/VI

5.0 FIGURE 3.1 ISENT R OP IC FL OW RELAT IONS ( Y :;: 1.40)

(22)

Ac ross a shock wave the air is compressed in the direction of

the flow - th e sta tie pressure , de nsity, temperature and the speed of sound

inc r ea se acro s s th e shock fr on t. The speed across a shock wave is de- .

cr ease d - to a subson ic speed in the case of a normal shock. In order to

find the chang e s in the flow parameters when the fluid passes across a shock wa ve, it is necessary to consider the conservation of mass, mom-entum and en ergy across the shock. These conditions show that the com -pression of the flui d in passing through a shock wave is not an is entropie process.

,

,

M,

>/

I;

---J!l-

ft

-r,

"

,

FIGURE 3.2 FLOW PARAMETERS ACROSS A NORMAL SHOCK.

The relation between the Mach num b e r before,

M;

,

and after,

M

_z

'

th e no r m al shock is

7-/

M2.

Z /

+

-2 '

M.

_.z

=

_{ï'M'I. _ ~}

I Zo

(3. 14)

The sta tie pressur e, density and te m pe r atu r e ra ti o across a normal shock can be expres se d as a fun c tion of th e initi a l Mach num be r

Z1

z

= - M

ï'of/ ' 7-/ ï'+/ (3 . 15)

=

("I-tlJM/"

2.

+

(-I-I)

/I1,Z

(3. 16) (3. 17)

(23)

Mor-eover, the vetocity and density may be re1ate d by the equation of con t inuity hence, io r a constan t ar ea

r:

-~

Z

+

('I-I)

/1,2-('1+1)

1'1/

(3. 18)

The los s in pres s ure energy across a shock wave is usually re pres ented by the decrease in the stagnation pressure. The ra tio of the

stagnati on press u res,

Po2.~

across a nor m a 1 shock can be readily

ob-rol

tained from th e known ra t ios of the sta g nation to statie pressure on each si de of the shock (s e e equation 3. 10) and the ratio of the statie pressure across th e shock. th us

(3. 19)

'y

It should be note d th at, according to th e above the or etical

r-elations , the statie pressur e ratio acros s a no r m al shock wave becomes

infi n ite as th e initia l flow Ma c h numbe r becomes inf irri te, i ,e. ,

Fip,

~

co

as ~ -

co

How ever, the density ratio approac hes a limit as

M,

be c om es in fi n ite; for air ,

~t:

---

b as

t1, -

00 •

The num e rica1relat ion between the flow pa r a m eters in front

and behi nd a nor m a1shock wave as a function of the initia1 Ma c h num b er

can be obtai ned from tabulated and/or graphica1va lue s as gi ve n in re

fer-enees (3. 4, 7. 95 or 96) ? The varia tion of thes e relations with in

-cre asing Mach number is sho wn graphically in Figu re 3. 3.

The increase in the entropy (

$'2. -

S,

)

ac ros s a shock wave

is given by

...

_(3.20)

3.5 Obliq ue Sho ck Wave

Acr os s an obliq ue shock wave the air is compressed in the dir ec tio n of the fbw and in addition the flow is deflecte d in a dire c tion

suc h as to decreas e th e acute angle betwe en the streamline and the shoc k

wave, as shown in Fig ur e 3.4, iaee. the flow is deflected towa rds the

1.

0

0.8 0.6

0

.4

0. 2 (24) ""

--

~

..

~\'\

... \ "

'r--...

~\\

,

-,

"

_"

,

_1',

~\

-,

,

,

"

~

"" -,

,

1'02-,

_"

\

1\,

<, "

"

',/t

"

,

",

-,

" ... '"

1\\

... ,

"',

-"

,

1'-

.

<::-

1':-: _,_

'""

--~

\

.",

....

r-;

1-- _ _

---

_---

r

-r-,

..

....

-,

...

_"'

, Ti

i'... ....

_{fi _}

_u«

"

_"

,...

-

_7i

r-,

...,~ ... f i - -

-

-r-,

_ lt.,

",

-

~,

- r----.

_---

...

-

-

----"

-r-;

h

< ,

~

_~

-..,

1----

I

-T

"

o

1.

0

1.5

2

.5

FIGURE 3. 3 3.0

3

.5

4.

0

RELATION BETWEEN FLOW QUANTIT IES

ACROSS A"NORMAL SHOCK WAVE.

(

Y

=

1.

40

)

I

(2 5)

The flow paramete rs across an oblique shock wave are usua ly

expressed as a function of th e initia l Mach number

M,

and the shock wave

angle

e

_w (Figu r e 3.4) as gi ve n by the following expressions

....

~I

OBLIQlJE .sHOCK

FIGURE 3.4

FLOW PARAMETERS ACROSS AN OBLIQUE

SHOCK WAVE.

M

Z

=

:2

+

(7-/)

!VI,2.

+

z

2 Y

11,2.

.sin2ew -('I-I)

~

2-( 1 . , 2 = -

M.

SII7

e

I;

Y+I I 4J 2.

M,2.

cos2.

e

_w -I-I "'1+1 (3.21) (3.22) ~

(-t+/)

M,z

s/n

2

Bw

(3.23)

f,

-

2

+

(1-/)

M,z

s/n

2 é14AJ ,

_..

7i

At:

_(3.24)

-

-I:~

/,

1z,

('''oS

Bw

₍₃0 2 5 )

-

=:

'J;

Cos

(fi

_{w -}

e)

(2 6)

=

/

(3.26 )

(3.27)

Moreover , th e relationship between the initia1 Mach number

M,

and the sho ck an g1e

ft

is given by _ S' 2Ll -~n Ow 5/'1

e"-J

5/

n

ti-Cos

(e

w -

8-)

(3.28)

The sh o c k ang1e is plotted against the flow deflection ang1e for various Mach numbers in Figure 3. 5.

The nu m e ric al relation between the flow parameters across an obliq ue sho ck wa v e can be readily obtained from the tabu1ated andj or gr aph ical va lues as give n in references (1. 3. 7. 8).

The inc r e a s e in the entropy across an oblique shock wave is given by relation (3 .2 0) .

From th e above r e

a.

t

ion s it is apparent that th e flow para -meters across an oblique shock can be com p1ete1y determ i ned as a fun

-ct ion of the two para m ete r s ~ and

e

w However, th e flow para-meters can also be speci f ied as a function of any other two parameter s .

If

UZ

is dete r mine d as a fu n ction of UI and I.I..

_z

by using th e equa

-tions ex p ressi ng th e cons e rvation of mass, momenturn and energy then

the following rela tion is obtained

z

Z 2 (::) ==

(-U.I -

u,t.\

z

U,

:2.. -

Q+

2-'*

't* ) Y-rl U, - U,

~'Z.

+- Q+ (3.29 ) J '4

wh er e

a..+

is the critic a1 speed of sound, i.e. , th e speed at which the

fre e stream sp e e d is equa1 to th e Iocal speed of sound, and from re

a-tion (3 . 6) is give n by

2. Z

a-'l:

=

V

~

II

!.

M'l.

V

~

[...---'

,

M,

-:

. /

e..,

~

V

V &

-

- -

-/ +~

/

/ /

~

...

~

~

V

/

~

/

/

/

~\ \ \

V

V

_[7

\

/

/

_...-

V--

h

\

V

_/

_V

_~

L-/ \ \

/

/

/" \

V

_V

₁

_/

/

V

.

_~' o 40 30 20 10

o

10 20 30 (27) 40 50 70o FIGUR E 3.5

OBLIQUE SHOCK AN G LE ~ VE R SUS DEFLECTION

ANG LE

e-

FOR VARIOUS MACH NUMBERS.

(28)

If

LLya...

is considered fixed then

U2.k..

and

lrYct..-.

are

1.Lz.J ();.I

gi ve n byequation (3.29). A plot of Ja,.,. versus %./0... for a given

value of

LL1/a..;.

is a strophoid - the "Folium of Descartes

'~

as shown in

Figure 3.6, and is known as the shock polar - fîrst introduced by

Buse-ma nn [2J • The shock po.la r has the following properties: (i) The

inte r s ectio n of the shock p olar with any ray fr om the origin represents

the value s of

1.L2./a..f<

and

LT'2.,/a....,..

for the flow deflection angle,

e

gi ven by th e in clina ti on of the ray with the IJ.2./o.,.fr -axi.s , In general

", " WAVE " " / , / , ///""-MACH

"

WEAK

"

SHOCK o~::::"1--~~---=':':~!---+---==-==~~-...---

__

FIGURE 3.6 SHOCK POLAR

th e r e are th r e e inte r s e ctio n s

A , B

,

and C . For the point C

Mz.

>

MI

which represents a los s of entropy hence is ruled out on

phys i cal gr-ou nds , Intersections

A

and

8

represent two

(2 9)

3.2 2 and 3.2 3) is associate d with the lts rong " sh o c k. corresponding to point

A

•

than with the "we ak" sh o c k corresponding to point

B

.

Ex-per-imenta l.ly, it has been found th at th e conditions behind an oblique

shock fr ont are usuaUy thos e give n by inte r s e ction

B

(iii) Any in t e r -section within th e unit circle predi cts sub s onic flow behind the oblique shock. Since th e unit circle does not pass th r o u gh the point denoting the maximum flow deflec tion, fo r flow deflec tions near th e maximum value the "weak"fa mily of oblique sh o c ks mayalso produce a tr a n sition to a subsonic sp e e d. (iv) A Iine drawn th r o ugh th e origin and normal to th e

lin e th r ough th e points

(I.l

_Y

a..'1 }

0)

and

B

•

the appropriate point of

in te r s e ction on the shock pola r (s ee Figure 3.4). represents th e oblique shock fr ont. (v) The inte r s ections of th e shock polar with th e UYa."lt"

- axis correspond to Ltl

u%~

=

1 (n o r m al shock) and

u-Ya...

=

u..Ya..;.

(M a c h wave). (vi) For a given value of

u'/o..'fç

th e r e is a maximum defle c tion angle,

0ma.x

•

th r o u gh which th e flo w may be turned. If the deflec t ion angle of a bo dy intr o duc e d int o th e flo w at a given value of

u"/a.*

is grea te r tha n

eml-

_X • a detached sh o c k wave wi U form in

th e flow ahead of the body. In this case th e flow downs tream of th e shock is non-isentropic and the ab ove analysis cannot be applied. The variation of th e maximum angle of flow deflection ew>~~ fo r which th e shock is stiU atta ched is plotte d again st the Mach number MI in Figure 3.7 . Mor-eover-, it sh o uld be note d that fo r a given Mach number th e maximurn

angle of a cone for which th e sh oc k is sti ll attached is la r g e r than the co r r es p on ding wedge angle as sh own in Figure 3.7 . (vii) There are two

limitin g cases fo r the shoc k polar, For

bi

':=

tl*

the shock polar loo p

shrinks to a point an d fo r

1;

=

q-

the sh o c k polar loop reduces to a

"'"

'

r

crr-cle , where . ~ is th e ma ximum possib le • ultimate or limitin g sp e e d that ca n be attained fo r given stagnation con dition s and is defi ned by relation (3.5) for th e con dition

I'

=

0 • i.e••

(3. 3 1)

\~

A

Theoretically . for the limiting case },

=

1

th e region ups tream of th e oblique shock wave cor r esponds to a zone of cavitation with

I:

-=

0 •

t: ;-

0

Mor~o

ver.

for

1,

'=

i

.

the maximum deflec tion angle

&m~x is 45.5 fo r Y = L 40. Thus, the maximum pos sible, ultimat e or limiting flow deflec tion angle acros s and oblique shock wave s approa ches 45. 50 fo r ']I'

=

1.40 as the initial Mach number

----

t--V

~

V

, /

~

cP

~

-~

/

V

,

V"

->

~

/

_~~~

~v

1/

_/

V

-)

/

/

V

"'1,

M~

I

/

V

_~

1

_V

_~

f/

o 50 40 30 20 10

o

1 1.5 2 (30)

2.5

FIGURE 3.7 3 3.5

M

I 4

MAXIMUM WEDGE AND CONE HALF ANGLES FOR AN

ATTACHED SHOCK.

( ""

=

1. 40)

(31)

Reflec tion of Shoc k Waves from a Rigid Wa:'l

Assumi ng pe rfect flui d flow, the simp lest oblique shock

re-fle ctio n fr om a s olid bou ndary or wall is known as a regular r-ef'lectton,

Consider an incid e n t shock wave ongina ting at 0 , say the vertex of a wedge, in a uni form airstream at Ma ch number

M,

as shown in Figure

3. 8. Acros s the incide nt shock the flow speed is reduced

(M

Zo

<:..

M,)

and the flow dire ctio n in the re gio n downstream of the inci d e n t shock is

inclin e d at an angle -& to the initial str eam dire c t ion . T'hus, the flow direction in region (Z ) is no t parallel to th e wall

ABC

50 that at the

point

8

the flow mus t be deflec t ed ba c k through th e angle

9

in order

to make the flo w paralle l to the wall in th e region downstream of

8

.

A

sh oèk wave, the refle cted shock, originat e s at

B

such that the flow deflec tion

8-

acro s s th e reflected sh ock is the same as that across the

incid ent shock but op p osite in dire ction. The speed across the reflected shock is re duced 50 that MI

>

/11/2,.

>

M

_{3 •} A INCIDENT SHOCK

MI

(I) (2. ) 8

c

RëFLEcrEo SHoCK (3) FIGUR E 3. 8

THE ü R ETICAL RE GUL AR REFLECTION OF A SHOCK WAVE

FR O M A PL ANE WALL.

Since th e defle ction angle

e

acr os s the incid e n t and reflected

shocks is the same and since the Ma ch number across the incide n t shock

is reduced, i.e. ,

M

_z

<

M,

'

th e refle c t e d shock will fo r m a greater

ac u t e angle with th e stream direction , i.e., &W2,

>

9w,

.

For

re g ula r reflec tio n at th e wall, th e angle of incid e nce is le s s than the angle

of reflec tion. Moreover, even though the direc tion of the airstream in region (3) is parallel to th e initial stream direction the flo w parameters are no long er the same, Since th e defle c tion angle is kn o w n , the flow para

-meters across eac h sh o ck can be dete rmined by using th e shock pola r or

th e ta b ula t e d andjor graphic al value s gi ven in refe rences ( 1, 3, 7 or 8)•

(32)

It has been shown (s e e Figures 3. 6 and 3.7) that for a given Mach num b e r there is a maximum deflection angle, em~

.

thr-ough which the flow can be deflected across an oblique shock wave. Corres-pondingly, there is a limit beyond which a regular ref ection cannot occur-, If th e flow deflection angle

e

is greater than the maximum deflection an gle éTm~><

o

.

co r r e s p on din g to Mach number

fV/

_z

'

regular reflec-tion cannot occur, In this case the reflection appears as shown in Figure

3.9 an d is usually known as a Mach reflection . In the case of a Mach ,

NORMAL

.sH'oc,l<-(I)

INCIDENT Sf/DeK.

(4)

---,--

_0'"

CONTACT ae: VELOClrY

(3) DISCONTllJulry sute,cAce (2) sI/oeI::.

..

, .' FIGURE 3.9. MACH REFLECTION.

reflection th e inciden t shock branches, at some point from the wal l, in t o a reflecte d sh ock and a "nearly" normal shock. i.e•• a curved shock that is normal at the wall, The pressure ratios

~/p,

and

P3/

~,

are the sarn e, However, since the entropy change across the normal shock is not th e same as that across the two oblique shocks. a slipstream or con ta c t dis c ontinuity surface across which tbere-Is a change in de nsity, temperatur e and velocity must exist [2J

It sh ould be no t e d, however, that in practice both the

regu-la r reflection and the Mach reflection configurat ions may be considerably

modi fie d by the boundary la y e r along the wall [16, 17. 58] . For exarnpl.e,

a phenomenon freque ntly observed near the surface is an apparent branch-in g or bifu rcatio n of the shock.

.1

'

.

(33)

3" 6 Steady Two-Dimensional Is e ntr o pic Flow - The Method of Characteristics

It is well known that the non-linearity of the differential equa-tion s for compressible fluid flow s (s ee, fo r exarnp le , equation (3.33) in which the dependent variab les u,) lT and their derivatives

~~:l

; ' : . , - • • , appear as products) makes them difficult to solve exactly for given bound-ary conditions, since a general me~hod fo r the treatment of non-linear

partial differential equations has not, as yet, been developed. However,

for fin ding solutions of the equations of motion of a compressible fluid three methods of attack have been developed: (i) The Small Perturbation Method - the equations of motion can be lin e a ri z e d by assuming certain parameters, e. g •• flow deflection, to be small so th a t the non-linear terms can be neglected; (ii) The Hodograph Method - a few exact solutions have been obtained by transform ing th e exact non- linear equations into exact

linea r equations - th e non-linea r equa tions describing the flow in the physical

( 'X,

IJ-

)

plane are tr ans for m e d int o lin e a r equations describing the flow in

the hodograph (

u-)

cr ) plane and ; (iii) The Method of Characteristics - in the special ca s e where the speed in th e flow field is everywhere supersonic the equations of motion assume such a for m that th e y may be solved by a procedure called th e method of cha r a c t e ristic s . Since the conventional methods of designing supersonic no z zle s (s e e section VI ) usually

in-volv e some fo r m of th e method of characteristics , an outline of the method

will be given in this sectton, Moreover, although this review is Iimite d to

two-dimens ional flows, it sh o uld be note d th at th e method of characteristics

has also been developed fo r th e case of non- s tatio n a r y one-dimensional

flow s and st ea dy th r e e-dim en s ional flows with axial symmetry (s ee, for examp le, reference 2, 4 or 10).

The differe ntial equations of motion defini ng steady, two

-dimensional, is entropic [a n d hence irrotationa l), compress ible flow of a perfect (n on-vis cous) fluid are given by

=. 0

which expresses th e irrotati onal character of the flow, and

(3.32)

+

((/'l._o..?)

dtr"

=

0

';;)1

(3.33) .