ARCHIEF
INVESTIGATION S ON
THE TRAN SONIC FLOW AROUND
AEROFOILS
by B.M. SPEE F -,-Lab.
y.
Scheepsbouwkund
Technische Hogeschool
Dellt
Werner toont door middel van een numeriek experiment aan dat zich in de transsone strorning orn een profiel volgens Tomotika-Tamada een schokgolf ontwikkelt. Hij trekt hieruit ten on-rechte de conclusie dat de schokvrije transsone stroming orn een profiel instabiel is.
Werner, W.: Instabilität stossfreier transsonischer Profliströmungen. ZAMM. voI. 41, 1961.
De transsone stronling orn een quasi-elliptisch profiel kan worden berekend met behulp van differentie methoden. Er ontstaat dan geen schokgolf. De afwijkingen tussen de aldus berekende stroming en de potentiaal strorning komen overeen met de verschillen tussen theorie en experi-ment die veroorzaakt worden door onnauwkeurigheden in de vorm van het windtunnel model. Transonic aerodynamics.
AGARD CP no. 35.
De in dit proefschrift genoemde grafische methode ter bepaling van de ontwikkeling van een goiffront is een belangrijk hulprniddel bij bet onderzoek van instationaire transsone
stromiti-gen.
Tijdeman, H.. Berg, H.: Analysis of pressure distributions measured on a wing with oscillating control surface in two-dimensional high subsonic and transonic flow.
NLR-TR F.253. 1967.
De ontwikkeling van golffronten kan in de windtunnel langs optische weg worden bepaald. Uit bet resultaat kLinnen de grootheden van bet stationaire stromingsveld worden afgeleid. De in dit proefschrift genoemde invloed van de snelheidsgradiënt in de grenslaag op het gedrag van instationaire verstoringen kan in transsone stromingen aanleiding geven tot belangrijke schaaleffecten.
De weerstandsgroei die optreedt bij bet zogenaarnde "drag-rise" getal van Mach is bet gevoig van een toenarne van de verliezen in de schokgolf. Deze verliezen worden bepaald door de sterkte en de hoogte van de schokgolf.
A method of estimating drag-rise Mach number for two-dimensional aerofoil sections.
Royal Aer. Soc. T.D. Memo 6407, 1964.
De experimenten aan symmetrische quasì-elliptische profielen tonen aan dat de stroming in het supersoon gebied vrijwel onafhankelijk is van de condities verder stroomafwaarts. Dit betekent dat bij een schokvrij transsoon profiel met lift voor een afwijking in de circulatie kan worden gecorrigeerd met behuip van een kiep.
Een gloeidraad anemometer is ongevoelig voor geluidsgolven die zieh stroomopwaarts voort-planten in een hoog subsone stroming.
Morkovin, M. V.: Fluctuations and hot-wire ancmomctry in compressible flows. AGARDograph 24, 1956.
In een transsone windtunnel moet de overgang van meetplaats naardiffusorvloeiend. dat wil zeggen zonder sprong of onderbreking zijn. Alleen dan is het mogelijk de drukfluctuaties in de meetplaats op een aanvaardbaar niveau te brengen.
Het verdient aanbeveling in de opleiding aan technische hogescholen eco belangrijke plaats te geven aaii activiteiten die erop gericht zijn bij de ingenieur een grotere belangstelling te wekken voor sociale en politieke problemen.
INVESTIGATIONS ON THE TRANSONIC FLOW AROUND AEROFOILS
INVESTIGATIONS ON THE TRANSONIC
FLOW AROUND AEROFOILS
PROEFSCHRIFT
TER VERKRJJGING VAN DE (RAAD VAN DOCTOR
IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCI-LOOL TE DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M. VERHAGEN,
HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, TE VERDEDIGEN OP
WOENSDAG22OKTOBER 1969, DES NAMIDDAGS TE4 UUR
door
BERNARDUS MARIA SPEE
VLIEGTUIGBOUWKIJNDIG INGENIEURDIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR
Acknowledgement
T thank the Director of the Nationaal Lucht- en Ruimtevaartlaboratorium (National Aerospace Laboratory NLR) for his permission to publish investigations which are part of the research programme of the NLR in this form.
The work reported in chapter 6 has been performed under contract with the Nederlands Instituut voor vliegtuig-ontwikkeling (Netherlands Aircraft Development Board). I thank NIV for their support of this work, and for permission to publish some of the results in this form.
Contents Page Summary Symbols i I Introduction 3 I General 3
1.2 Outline of the investigation
2 General aspects of the transonic flow around
aerofoils 4
2.1 Transonic flow with shock waves 4
2.2 Shock-free transonic flow 5
3 The transonic controversy 6
3.1 General 6
3.2 The effect of viscosity 6
3.3 The non-existence of neighbouring flows 7
3.4 Unsteady aspects 8
3.5 The experiments of Pearcey 8
4 The behaviour of unsteady disturbances 9
4.1 Experimental results 9
4.2 Graphical experiments 12
4.3 The effect of the boundary layer 13 5 The stability of transonic flows 16
5.1 Introduction 16
5.2 The stability of quasi
one-dimensionaltransonic flows 16
5.2.1 General 16
5.2.2 Plane disturbances superposed on a
quasi one-dimensional flow 16
5.2.3 Instability of decelerating flow 19
5.2.4 Discussion 20
5.3 The stability of two-dimensional transonic
flows 20
5.3.1 General 20
5.3.2 The investigations of Kuo and Werner 20 5.3.3 The stability mechanism of transonic
potentiel flows 23
6 Experiments on quasi-elliptical aerofoil sections 25
6.1 General 25
6.2 Aerofoils 26
6.3 Test details 27
6.4 Results 28
6.4.1 General remarks 28
6.4.2 The design condition 28
6.4.3 Off-design conditions 31
6.4.4 Unsteady waves 33
6.5 Conclusions from the experiments 34
Summary
The question of the stability of steady compressible two-dimensional potential flows with embedded regions of supersonic flow adjacent to a body is examined. The behaviour of unsteady disturbances is investigated and an essential difference between the stabtlityoftransonic one-dimensional and two-dimensional flow is shown to exist. [lis shown that two-dimensional transonic potential flows are stable.
An experimental investigation on quasi-elliptical aerofoil sections is discussed. The results of this investigation suggest that the theore-tical potential flows can be approached arbitrarily close.
It is concluded that the use of potential flow theory in the transonic region is equally acceptable as anywhere else in aerodynamics.
Symbols
Al
BI
.11 . . . f5 k nl M Mcr Mdr M* n s P Po P Q qs
T u U Vw
constants, equation (5.29)- speed of sound
- chord length
- critical speed of sound - stagnation speed of sound - constants, equation (5.41)
- drag coefficient
propagation speed of wave - dimensionless slope of wave
value ofEat t=0
- frequency
functions, equation (5.26) - curvature of body at sonic point
characteristic length of body slope of body (at sonic point)
Mach number
free-stream Mach number critical Mach number - drag-rise Mach number
dimensionless velocity, M*=q/c
wind tunnel Mach number
- streamline coordinates - (static) pressure - stagnation pressure
u'+5c'
u'-5c'
- velocity vector magnitude cross section area
- time
- dimensionless time - velocity in x-direction - shock velocity - velocity in y-direction - rotation speed of wave
7 The stability of transonic flows with a shock wave 34
8 Conclusions 36
References 37
Summary in Dutch 38
-
general rectangular coordinates - dimensionless rectangular coordinates - coordinate normal to the chord - angle of incidence- constants, equation (5.27) - ratio of specific heats - thickness parameter
boundary layer thickness - slope of wave
- aerofoil parameters, chapter 6
- coordinates along characteristics flow direction
- wave length
constants, equations (5.41), (5.42) Mach angle
(y+ l)x
- shock displacement, chapter 7
- density
(diniensionless) time velocity potential
- velocity potential of steady motion
e(
n,s )
- conditions at the edge of the boundary layer
- equilibrium, steady flow
X y X Y ß y (5 { E E
t
17 o 't j.1 V ç pt
(- disturbance velocity potential
- velocity potential defined by =c1(X+J)
- velocity potential defined by 4' = c 1(1.:"
-
angle of wave front w - Prandtl-Meyer angleSubscripts
x,y differentiation with respect to the
coordi-X,Y nates
11»
w
- conditions al the wall
t
- differentiation with respect to time
t
-
conditions upstream of shock2 - conditions downstream of shock
Superscripts
- primed quantities denote departures from the equilibrium conditions
I Introduction
LI General
Research work on transonic aerodynamics has been going on for many years and reached a first peak some twenty years ago. At that time. the efforts of aeronau-tical studies were concentrated on the transonic range by the practical problem of "passing through the speed of sound".
The flow patterns in this range appeared to differ considerably from the flow patterns at low, subsonic speeds and from those at supersonic speeds. Normal shock waves and strong shock wave-boundary layer interaction effects were shown to be essential features of transonic flows. These new phenomena were, in fact, so difficult to understand that the term "sonic barrier" arose.
In this period, the main effort of research in transonic aerodynamics was on the experimental side. The
devel-opmenl of test sections with slotted walls made it
possible. for the first time, to obtain reliable results from wind tunnel experiments at transonic speeds.
These investigations produced sufficient general
information on transonic aerodynamics to achieve
sonic measure of confidence and success in applica-tions. The results enabled designers to develop super-sonic aircraft which had to pass through the transuper-sonic region. A number of fundamental problems, however, remained unsolved.
Following this first period of activity, little attention was paid to the transonic regime for many years. The reason for this was not entirely due to the absence of practical interest in transonic aerodynamics: most of the problems in this field appeared to be highly intractable mathematically and experimental investigations could not be supported by a firm theoretical basis.
In recent years, the interest in transonic flows has increased again as a consequence of the advent of a class of civil airplanes cruising at high subsonic speeds. There is an urgent need nowadays for detailed informa-tion on transonic flows, since for these airplanes the transonic regime is no longer a transient phase in flight. but determines the main aerodynaniic design condition. In view of the cruising efficiency of such aircraft, the significant increase in drag, connected with the appear-ance of normal shock waves terminating local super-sonic flow regions has become the central problem. The
attention of transonic research to-day
is therefore confined to a large extent to transonic flows in which the shock waves are weak, and more in particular to flows with a smooth, shock-free deceleration from supersonic to subsonic speeds.These shock-free flows have been the subject of much
controversy in the past. Theoretically, examples of
transonic potential flows could be constructed with shock-free recompression from a supersonic flow region
3
to subsonic speeds. Experimentally, however, only transonic flows involving shock waves were found, at least when a well developed supersonic region was present. lt was supposed therefore that these theoretical solutions were unstable in some sense and that a shock-free potential flow would collapse into the standard transonic flow pattern with a shock wave.
The arguments against the existence of shock-free transonic flows were generally accepted until Pearcey (ref. 1) evolved the concept of peaky pressure
distribu-tions, and showed that a for all practical purposes
shock-free flow around an aerofoil could experimentally be realized. This important result stressed the need for a better understanding of the essential physical
pheno-mena in tran sonic flows.
The possibilities for a mathematical treatment of transonic flow problems have been extended,
fortunate-ly, in the last years by the advent of large and fast
computers. The first opportunity of systematically con-fronting transonic potential theory and experiment for aerofoil flows occurred recently with the development of the theory of quasi-elliptical aerofoils by Nieuwland (refs. 2, 3). These potential flow solutions, describing the shock-free flow around a family of aerofoil sec-tions, formed the basis for the present investigations.
I .2 Outline of the investigation
The investigation presented is concerned with two-dimensional transonic flow around aerofoils. It consists
of two main parts. In the first part a detailed description of the behaviour of unsteady disturbances in transonic flows is given. Such disturbances were considered to play an essential role in the problem of the stability of shock-free transonic flows. In the second part the results of an experimental investigation. set up as a verification of the theoretical potential flow around quasi-elliptical aerofoil sections are discussed.
In chapter 2 an introduction is given to the general aspects of two-dimensional transonic flows. This chap-ter deals with the development of the transonic flow around aerofoils and describes the nature of a local supersonic zone.
In chapter 3 the different arguments advanced against the existence of shock-free flow are summarized and discussed.
A general description of the behaviour of unsteady disturbance waves is given in chapter 4. The results of some experimental observations are discussed, and the essential features of disturbances moving upstream are displayed by means of graphical experiments. Suggest-ions that these waves are unable to penetrate into a local supersonic region are shown to be incorrect.
A detailed analysis on the stability of transonic flows with respect to unsteady disturbances is given in chapter 5. A description of the analysis of Kantrowitz (ref. 4) for quasi one-dimensional flow, the flow in a channel,
is given. The results of this analysis show that in a
quasi one-dimensional flow deceleration through the speed of sound is unstable. Similar investigations for two-dimensional flows, given by Kuo (ref. 5) and
Wer-ner (ref. 6) are discussed. It is indicated that their
results are not conclusive with regard to the problem of stability of two-dimensional shock-free flows. A
simple analysis on the local behaviour
of acousticwaves moving upstream demonstrates
that two-dimensional transonic potential flows are stable.A description ofthe results of experiments on quasi-elliptical aerofoil sections is given in chapter 6. The results of these experiments show clearly the physical significance of transonic potential flow solutions.
Finally, some remarks on the stability of transonic flows with shock waves are given in chapter 7.
2 GeneraI aspects of the transonic flow around aerofoils 2. 1 Transo,iic flow with shock oaves
In order to introduce the problem it is convenient to consider the development of the flow pattern around an
M<1 Moe <Mc r Moe>] SUBCRITICAL FLOW SUPERCRITICAL FLOW M1
SUPERCRITICAL FLOW (WITH SEPARATION)
SUPERSONIC FLOW Fig. I Transonic flow pattern
aerofoil with increasing free-stream Mach number as observed in experiments and illustrated in figure 1.
If the free-stream Mach number, M, is low. the
flow around the aerofoil is everywhere subsonic. When the free-stream Mach number is increased, a certain critical value, Mer, occurs at which the local Mach
number, somewhere on the aerofoil contour, first
reaches unity, and a local supersonic region starts to
develop. 1f the values of Moe, are only slightly larger than
the critical value, the local supersonic Mach numbers are close to unity, and experimentally no shocks are observed. For values of M. appreciably higher than the critical value, however, the local supersonic region is terminated by a normal shock wave. This shock wave appears for the first time in the usual experiments when the maximum local Mach number is about M = 1.1. The position ofthe shock wave moves downstream with increasing free-stream Mach number, and the super-sonic region grows rapidly.
Eventually the shock becomes so strong and the
resultant adverse pressure gradient so large that the boundary layer separates. This happens, for a turbulent boundary layer, when the local Mach number upstream of the shock is about M= 1.25.
A further increase in free-stream Mach number
causes the position of the shock to move to the trailing
edge of the aerofoil. The trailing edge is reached for Moe close to unity. The flow over the surface is then
predominantly supersonic. except near the leading
edge. The flow pattern remains substantially the same for free-stream Mach numbers slightly greater than one, but a detached shock appears well ahead of the nose of the aerofoil. For higher values of Moe, the supersonic flow pattern with a detached bow wave and oblique trailing edge shocks is produced.
The behaviour of drag with increasing free-stream Mach number is given in figure 2. As long as the local
supersonic region is small and the maximum local
velocity only slightly above the sonic value, the shock wave is weak and the wave drag negligible. The drag coefficient differs little from its value at subcritical free-stream Mach numbers. At a certain value of Moe, the
Cd
Mr Fig. 2 Drag at transorsic speeds
Mr Mc, Mdr M M <1 /M>1 M ,>Mc r
r.,
local supersonic region usually grows quite suddenly. The shock wave increases in strength and its length, in the direction normal to the aerofoil, grows rapidly. The wave drag associated with the shock wave increases considerably and this results in a sharp rise in the drag curve. The free-stream Mach number at which this happens is called the drag-rise Mach number. Mdr.
The wave drag increases further with increasing free-stream Mach number while the total drag increases moreover by the effect of boundary layer separation. The drag coefficient reaches a maximum for values of
M close to unity and decreases again when M, is
allowed to increase further.
The development of the flow pattern is illLlstrated in figure 1 for a symmetrical aerofoil at zero incidence.
For a lifting aerofoil the flow pattern is, of course,
different for the upper and lower surface of the aerofoil. The whole picture is then complicated by the interac-tion through the trailing edge and the effect of separa-tion on the circulasepara-tion. However, the development of shock waves is essentially similar to what is shown in figure 1 for a symmetrical non-lifting aerofoil.
2.2 Shock-free transonic flow
For an aerofoil of arbitrary shape, the development of the flow with increasing free-stream Mach number as observed in experiments was described in the previous section. However, this type of transonic flow, with a local supersonic region terminated by a normal shock
wave, is not the only possible flow configuration.
Theoretically, two-dimensional compressible potential flows may become locally supersonic at sufficiently high free-stream Mach numbers. Such flows exhibit local supersonic flow regions together with a smooth, shock-free deceleration along the streamlines from the
supersonic zone to the subsonic region downstream. This observation suggests that for so far as viscous effects can be ignored, it should be possible to design specially shaped aerofoils which may have supersonic speed locally and yet be free of shock waves. This may
be of great practical importance since the drag is
expected to be small.
In a shock-free transonic flow, the supersonic region
EXPANSION WAVES M=1 "L M>] COMPRESSION WAVES
\
\
is attached to the surface and is embedded in the sub-sonic flow field around the aerofoil as shown in figure 3. The supersonic flow region is bounded on one side by the solid surface, the rest of the boundary is the sonic line, the M= I isobar.
The properties of the flow in a local supersonic region in steady, two-dimensional potential flow have been summarized by Moulden (ref. 7). The main points are: - waves incident on the sonic line must be expansion waves and those leaving the sonic line compression
waves.
- a compression wave must be a
compression wave along its whole length. The same applies to expansionwaves.
along the expansion waves, the velocity monotoni-cally decreases towards the sonic line. Along the compression waves the velocity monotoiiically increases towards the surface.
The condition that the waves running from the surfa-ce to the sonic line must be expansions and those run-ning from the sonic line to the surface compressions will be put into an analytical forni to be used in the investigation on the stability. Selecting the coordinate s along the streamline and n perpendicular to the stream-hne as shown in figure 4, the equations of conservation of mass and momentum, together with the irrotionality condition can be written in the form
(pq)5+pqO,, = O
2
q
qqO. = O
where p and p are the local pressure and density. q and (3 represent the velocity vector magnitude and direction. Assuming that the fluid is an ideal gas, with constant specific heats, and that each fluid element has the same entropy, it follows from the second equation in (2.1) that C2
/+l
*2 Cy1 = 2(y-1)
(2.2)i
J = constant ST R E AM L. IN E (2.1)FREE STREAM DIRECTION
Fig. 3 Local supersonic flow region Fig. 4 Coordinate system
with e denoting the local speed of sound and c' the
critical speed of sound (q=c=c*). Since the flow far away is assumed to be a uniform parallel flow with
con-stant entropy the expression (2.2) has the concon-stant
value not only along a streamline, but through the
entire flow (homentropic case).
Introducing the Mach number M and the
dimension-less velocity M* defined by
M* = = M c* y+1
=M[
2y1
1+
2 + 2 (2.3) 6supersonic region. Returning to (2.4) and 2.5) it then
follows easily that
(M2-1)
2 < 1 (2.6) q O.Since O is negative in the entire supersonic region envisaged, the contour of the aerofoil forming part of the boundary of the supersonic region must be convex. This indicates that if even a short segment ofthe aerofoil adjacent to the supersonic region is straight or concave, a shock wave may be formed as the result of the
coales-cence of compression waves originating from this
segment.
More generally, the properties of a local supersonic flow region indicate that a shock-free flow will exist only for specially shaped aerofoils and that a small change in aerofoil shape can result in the appearance of a shock wave.
3 The transonic controversy
3.1 . General
The shock-free transonic flow discussed in section 2.2 has been for a long time a controversial subject. Theo-retically, examples of transonic potential flows could be constructed with shock-free recompression from a supersonic flow region back to subsonic speeds. On the other hand, only transonic flows involving shock waves were found in experiments, at least when a well develop-ed supersonic region was present.
Several arguments have been advanced at various times to explain this discrepancy between theory and experiment. These arguments were based upon: the effects of viscosity, the non-existence of neighbouring flows and the instability with respect to unsteady distur-bances. The arguments will be summarized in the next sections.
3.2 The effect of viscosity
In the description of the shock-free transonic flow around aerofoils in section 2.2, the effects of viscosity
have been neglected. In a real flow, viscous effects are always present, but, in a shock-free flow, these effects may be thought of as being significant only in a thin boundary layer near the solid surfaces. This permits the flow field to be treated in two parts. the inviscid potential flow outside the boundary layer and the
vis-cous flow in the boundary layer itself.
The displacement effect of the boundary layer changes the effective contour of a profile. Since, as indicated in section2.2, a shock-free transonic flow exists only for
specially shaped aerofoils, this may lead, in a local supersonic region, to the appearance of a shock, des-troying the potential flow character of the flow field.
If this were the only effect of viscosity, it would ap-pear that the shock-free character of the flow could be where y is the ratio of specific heats, the equations (2.1)
can be brought in the form
M5q5
M*
q - pq2M2-1
M
q PnM*
q pq2 sIn the supersonic case the equations (2.1) admit two sets of real characteristics intersecting the streamline under the Mach angle p=sin 1(l/M). Standing in a point of
the streamline and looking downstream, the charac-teristics departing from the streamline to the left are called the left-running waves or -characteristics, and the characteristics departing from the streamline to the right are called the right-running waves or -charac-teristics, with
and i
selected as coordinates along the characteristics.Denoting partial differentiations by subscripts it fol-lows that
M' = M' cos p + M'sin p =
(2.4)
M*1
M1
=
iL°
VM2-1--j
Since along the -characteristic the velocity decreases when going from the surface to the sonic line the expres-sion (2.4) has to be negative.
Similarly for the ¡j-characteristic running from the sonic line to the surface one obtains
M = M cosp - M sin p =
(2.5)
=
[_o+
]
and this expression has to be positive, since the velocity increases with i when proceeding from the sonic line to the surface.
Since (2.4) is negative it follows that O has to be
negative for M' >0 (accelerating flow). In case M
<0 (decelerating flow) no conclusion can be drawn from (2.4) concerning O. However, in that case the positiveness of (2.5) shows that again O
has to be
readily re-established by changing the shape of the aerofoil in such a way as to take the boundary layer displacement thickness into account.
The separate treatment of potential flow outside and viscous flow inside the boundary layer, however, can only be justified, even in subsonic flow, as long as the boundary layer is thin. Liepmann (ref. 9) has pointed out that there may be, in supersonic flow, a kind of instability which leads to an excessive thickening of the boundary layer.
Consider a boundary layer in a supersonic flow, and imagine that at some point the thickness of this boun-dary layer increases due to a disturbance. A compres-sion wave is then generated as a result of the change of the effective contour. This wave produces a pressure rise in flow direction, which in turn produces a further thickening of the boundary layer. In this way there may exist a mechanism of self-excitation leading to a cata-strophic thickening of the boundary layer.
There is no experimental evidence which supports
these ideas. On flat plates for example this kind of
instability has not been observed.
lt has been suggested that the phenomenon may occur for a boundary layer close to separation. that is, in a strong adverse pressure gradient. Indeed, transonic potential flows can possess adverse pressure gradients which are sufficiently strong to induce boundary layer separation. However there is no evidence to support the idea that this phenomenon in transonic flow will be different in principle from its occurrence in subsonic flow, as Liepmann's hypothesis would suggest. There seems to be no special transonic problem, except when a shock wave is already present.
3.3 The non-existence of neighbouring flows
Another explanation why transonic shock-free flows were not observed was sought in the non-existence of
neighbouring flows.
This line of argument was initiated by Busemann
(ref.
il). Consider first the flow in a nozzle which
smoothly decelerates a parallel, supersonic flow through
the sonic velocity as shown in figure 5. Imagine that there
is somewhere a small error in the shape of the nozzle contour. The effects of such a small perturbation can be superposed on the original flow. For a small perturba-tion, the effects of the disturbance will be propagated as pressure waves along the Mach lines of the original
flow.
M>
MI
Fig. 5 Disturbance in a reversed lavai nozzle
7
If the disturbance wave is a compression, it will each time be reflected as a compression wave. The waves are inclined at the local Mach angle to the flow, and they
coincide near the throat. From small perturbation
theory for two-dimensional supersonic flow it follows that the pressure rise across the wave becomes infinite as íW- I. Linear perturbation theory does not apply, in fact, for M- I but nevertheless the argument makes plausible that a finite shock will form near the throat
and that, although there are many contours which
could decelerate the flow from supersonic to subsonic velocity without shocks, such contours are isolated in the sense that a neighbouring contour, being almost identical with the original one, does not give smooth
flow.
A similar phenomenon was expected to occur in the shock-free flow past an aerofoil. Consider a supersonic flow region adjacent to a profile and suppose that a small alteration is made somewhere in the contour. The disturbance is then propagated along a Mach wave which leads downstream to the sonic line. There it is reflected along another Mach wave and so on. until the downstream sonic point is reached as shown in figure 6.
Fig. 6 Disturbance in local supersonic region
Again, the perturbation waves become infinitely
numerous and the pressure increment becomes larger as M decreases. Thus there is a tendency for a shock to occur near the sonic point and hence for the potential flow to be destroyed.
There is one factor in two-dimensional flow which makes the argument less convincing than in a quasi one-dimensional flow. The whole flow, even upstream of the origin of the disturbance is affected through the sub-sonic flow outside. There exists, in principle, the pos-sibility that the original disturbance is cancelled to an extent sufficient to avoid irregularities at the down-stream sonic point. Nevertheless it seems reasonable to
suggest that profiles for smooth transonic flow are
isolated in the sense that the slightest alteration in the contour destroys the potential flow.
This suggestion has been confirmed mathematically by Morawetz (ref. 12). Morawetz proves that, given a transonic potential flow solution describing the flow around a profile. there exists no solution of the
equa-tions for plane compressible potential flow for the
boundary value problem with a perturbed aerofoil
shape and the free stream as boundary conditions.This means that, in contrast to the subsonic flow case, a transonic potential flow does not depend in a continu-ous fashion on the boundary data and that potential flow solutions should be regarded as isolated solutions. The significance of Morawetz' theorem has been discussed by Nieuwland (ref. 3). He stresses the point that the physical relevance of transonic shock-free flows is a problem that, in fact, can not be answered on the basis of potential theory. The existence of a potential flow solution does in no way indicate the possibility of a shock-free realization in a viscous fio. Alterna-tively, it would appear that a contour for which non-exis-tence of a potential flow solution could be proved, and which can be obtained from any admissible contour by
an arbitrarily small change, would not necessarily
generate a pronounced shock wave.
/
M =/
/
./ / SHOCK
8
flow with a small and weak shock wave. The argument does not only apply to the region near the sonic point on the surface, but for all points of the sonic line where
the flow decelerates from supersonic to subsonic
velocity.
This indicates that the shock wave has to be as high. normal to the surface, as the local supersonic region itself. The argument suggests that the potential flow will be destroyed completely and that a nearly isen-tropic flow, with negligible wave drag. is impossible as soon as a well developed supersonic region is present.
lt will be shown in chapter 4 that this idea is incor-rect. and that disturbance waves moving upstream can
penetrate the supersonic region. 3.5 The experiments oJPearcey
The arguments against the existence of shock-free transonic flow have been generally accepted for some time, until Pearcey (ref. l)evolved the concept of peaky
pressure distributions and showed that, in fact, a for all practical purposes shock-free transonic flow could experimentally be realized.
Pearcey showed that a valuable delay in drag-rise Mach number can be obtained for aerofoils whose low
speed distribution has a velocity peak close to the
leading edge. At Mach numbers above the critical
value, this peak develops to produce high suction over
the forward part of the aerofoil. and the associated
expansion waves produce. after reflection against the sonic line, a useful amount of isentropic compression in the supersonic flow downstream. This isentropic recompression red iices the strength of the shock wave and, therefore, the drag.
Pearcey worked this out by using a special property of the flow in a local supersonic region. the property of connected points. Consider the conditions along charac-teristics meeting at the sonic line, as shown in figure 8.
1f O is the local flow direction and w the
Prandtl--
SONIC LINEG+ wOR
a
-FLOW DIRECTION AT SONIC POINT
Fig. 8. Properties of reflected waves
SHOCKFREE b) STRONG SHOCK C) WEAK SHOCK
Fig. 7. Disturbed shock-free flow; possible flow pattern The uniqueness of two-dimensional transonic poten-tial flow suggests, that a completely shock-free flow will not be found experimentally, because there is always a certain inaccuracy involved in manufacturing theoreti-cal aerofoil shapes. There is, however, no reason to assume, as has been done often in the past. that the only alternative would be the complete collapse of the flow into a flow with a strong shock wave. lt does not exclude the possibility that, if the deviations from the theoretical shape are small enough. the flow is very close to the potential solution, that is. involves only a small and weak shock wave as shown in figure 7.
This is an important point. From a practical point of view, one is not in the first place interested in a comple-tely shock-free flow, but in a flow with negligible wave
drag.
3.4 Unsteady aspects
A third argument against the physical existence of shock-free transonic flows was based on the supposed
instability of these flows with respect to upstream
moving disturbance waves. Such waves are always generated by the boundary layer and the wake.
In its briefest outline, the argument was that these waves when superposed on a steady shock-free basic flow could move upstream as long as the local steady flow speed was subsonic, but as they entered the region of supersonic local flow they would necessarily come to a standstill and coalesce to form a steady shock wave. If this were true, that is, if a shock-free transonic flow is unstable with respect to unsteady disturbances, this excludes the possibility of an almost isentropic
Meyer angle, Ow will be constant along the left-run-ning characteristics and O+w along the right-runleft-run-ning characteristics. At the sonic line, w=0 and the value of the constant is equal to the local flow angle °L Thus. if along a characteristic originating at the surface at A.
Ow=O. the characteristic reflected from the sonic
line and striking the surface at B must be such that
II + w11 = = 01 (Ji1.
For a particular aerofoil. the values of O at the surface are known, and the values of w at the surface may be found from the experimental pressure distribution. It is then possible to draw curves for the values of Ow and 0+w at the surface as shown in figure 8. Points such as A and B. connected in the sense that a distur-bance introduced at A will affect the surface conditions at B. are found from the intercepts of horizontal lines with these curves. This niakes it possible to determine where the surface slope must be changed in order to
modify the local Macli number at a chosen point
downstream, and thus, by making successive changes. to determine the optimum geometry.
The curve for the surface slope bisects the vertical distance between the O - w and 0 + w curves. The rate of change of Mach number depends on the relative slopes of theO and O+W curves. lfthese curves are convergent,
the Mach number is decreasing due to the effect of
compression waves. Near the initial sonic point the curves are necessarily divergent since the Mach number there is increasing. The problem is now to find a distri-bution of surface slope producing a continuous con-vergence further downstream in order to reduce the
Macli number ahead of a possible shock wave.
A
_________ X
Fig. 9 Recompression by reflected waves
This may be achieved by using high values of dO/dx immediately downstream of the initial sonic point and then reducing dO/d.v as shown in figure 9, where the reduction is assumed to occur discontinuously at the point A. To the first order, the change of surface curva-ture at A has no effect on the slope of O+w upstream of the connected point B. Thus in the region AB the required convergence of the O and t9+w curves is ob-tained. If the shock is present at, or upstream of point
9
B, the Mach number ahead of the shock will be reduced by reflections from the sonic line of the relatively strong expansion originating at the surface in the region be-tween the sonic point and point A where dO/dx is large. Pearcey demonstrated that aerofoils with a peaky pressure distribution conform to the behaviour outlined above. He also succeeded in som.e cases, by making successive changes for successful sections of the peaky type, in realizing experimentally an almost isentropic flow with a well developed supersonic region.
These experimental results do not contradict the
argument of uniqueness of transonic potential flows since this argument. as indicated in section 3.3, does not exclude the possibility of flows with a small and weak
shock wave.
Pearcey's results, however, showed that the argument
that shock-free deceleration through the speed of
sound is unstable has no general validity. The results stressed once more the need for a better understanding of the essential physical phenomena in transonic flows. in particular with regard to their unsteady aspects. 4 The behaviour of unsteady disturbances
4. 1 E.vperimen tal results
It has been pointed out in section 3.4 that unsteady disturbance waves are expected to play an important role in transonic flows.
A good impression of the unsteady character of these flows can be obtained from schlieren and shadow photo-graphs made with a short dLlration spark exposure that arrests unsteady waves. Figures 10 and Il show such photographs together with those obtained with a con-tinuous light source. The spark photographs were taken using a flash duration of 12 microseconds.
The usteady waves on these photographs are moving upstream. Unsteady waves, if sufficiently weak, propa-gate with the local sonic speed, and if the velocity of the steady flow is close to the speed of sound. but subsonic, the absolute velocity of waves moving upstream issmall enough for optical detection. The absolute velocity of waves moving downstream is then roughly two times the velocity of sound. and these waves are therefore
not visible.
The pictures of figures 10 and 11 show that there is a considerable concentration of unsteady waves in the decelerating part of the flow. The waves are generated downstream of the supersonic region, apparently by the boundary layer and the wake. The structure of the unsteady flow as shown on these pictures, is too com-plicated for a detailed interpretation. It is impossible to follow one particular disturbance wave on its way up-stream, and this will be necessary to obtain some idea of the essential features of these waves.
In order to come to a better understanding of the behaviour of waves moving upstream in a transonic
a) Shadow-spark
b) Schlieren-spark
Fig.12 Wave generator
flow, a special wave generator has been developed (ref. 15). This wave generator consists of a two-dimensional
IO
M5 = 0:784
Fig. 10 Unsteady waves in transonic fiow shadowgraphs
M5= 0.810
Fig. 11 Unsteady waves in transonic fiow schlieren photographs.
Shadow-continuous
Schlieren-continuous
rectangular cavity connected with a resonance chamber as shown ¡n figure 12. The resonance chamber behaves like a closed organ pipe and the wave generator prod u-ces disturbanu-ces with one discrete frequency. These
disturbances are of much higher intensity than the
natural disturbances generated in the boundary layer and the wake of an aerofoil.
Wind tunnel experiments have been done with the wave generator set in the tail part of an aerofoil and
20 20
7.5 /
f
a) Schlierenpbotograph
MI.
IFrORIGINAL
WAVES WAVES REFLECTED AGAINST TUNNEL WALL
fr.
WAVES REFLECTED AGAINST MODELb) Constructed wave pattern (acoustic theory) Fig. 13 Wave generator in afiatplate withellipticnose and wedged
tail (Mt =0.59)
placed downstream of the test section. Some of the results of these tests are given in figures 13 and 14. The frequency of the disturbances was about 3000 cycles per second. lt is shown that in this way a relatively clean flow pattern is obtained which is much easier to interpret than the natural sound fields of figures 10 and Il.
The results of these experiments have drawn attention to several important points. In the first place it was found that upstream moving waves turn over when they propagate around an aerofoil, as shown in figure 14. This effect was disregarded in the stability argument of
section 3.4.
Another Interesting point is that the Riemann steepe-ning effect of finite disturbances is clearly shown in figure 13. The signal emitted by the wave generator is roughly sinusoidal, as can be concluded from the fact that the dark and bright bands on the photograph are of equal width and gradually fade into each other.Farther away from the origin, the sinusoidal character disap-pears and the compression is concentrated in a con-tinually narrowing band.
Due to this steepening effect, both the first derivative and the second derivative of the density in the compres-sion phase increase. Since the schlieren and shadow effects depend on these quantities, the steepening of the waves is, in addition to the small absolute velocity, the reason why upstream moving disturbances are so clearly visible on short exposure spark pictures ofhigh subsonic and transonic flows.
11
a) Shadowgraph
REFLECTIONS OF ORIGINAL WAVES
p,
WAVES REFLECTED FROM MODEL REFLECTIONS TUNNEL WALLOF REFLECTED WAVES
b) Sketch of wave pattern
Fig. 14 Wave generator behind a model (Mt =0.67)
IRROR SOURCES
--I
II,
I N-
//
' / >I\
\\\7K
- - -
/ N / N / \ / \ V / TUNNEL WALL SOURCE POUTION / / APPARENT / SOURCE POSITION / / /__L______
ORIGINAL WAVES \ \ \ MIRROR SOURCESFig. 15 Construction of wave pattern (wave generator in fiat plate)
Figure 13 shows that, although the waves have a relatively large amplitude, the wave pattern can be described fairly accurately by acoustic theory.
A sound wave that is not plane may still be regarded
as plane in a small region if the amplitude and the
direction of propagation vary only slightly over dis-tances of the order of the wave length. If this condition holds, the sound is propagated along rays, these being lines such that the tangent to them, at any point has the
same direction as the propagation velocity. The study of the propagation of sound in such cases is called geome-trical acoustics. Geomegeome-trical acoustics corresponds to the limit of small wave length. i. - O and can be used to investigate the propagation of a wave front, the leading edge of an acoustic pulse.
The steepening effect makes the periodic waves be-have like separate pulses and when the steady flow is homogeneous, which is approximately the case for the flow around the flat plate model of figure 13. the wave pattern can easily be constructed. The wave fronts are circles as shown in figure 15. The distance between the centres of these circles is Me/f. where M and e are the Mach number and the speed of sound of the steady flow and f is the frequency. Figure 13 shows that the con-structed wave pattern is in close agreement with the optical observations.
These results have led to the graphical experiment to be discussed in the next section.
4.2 Graphical experiments
The complete time history of a wave front in a non-homogeneous flow can be constructed by geometrical acoustics. The relative propagation speed of an acous-tic disturbance in a steady flow field is equal to the local velocity of sound. This means that, for a given steady flow, the position and shape of a wave front at a time t can easily be obtained from its position at a time tt. The absolute propagation velocity follows from the local speed of sound and the magnitude and direction of the local veloctiy vector of the steady flow field as shown in figure 16.
In figures 17 and 18 the wave front of an acoustic pulse, generated at the trailing edge of an aerofoil is given for equal time intervals. As basic, steady flow
q
POSITION
AT TIME t
\
q
Fig. 16 Propagation of acoustic wave POSITION
AT TIME t-nt
12
solutions the flow around two quasi-ellipitical aerofoils have been taken. For these aerofoils not only the veloci-ty distribution along the surface is known, but also the
entire flow field.
AEROFOIL QI 1.0 775.1 3
Fig. 17 Propagation of unsteady waves
Disturbances generated in the subsonic region down-stream of a local supersonic flow region propagate up-stream. and due to the existence, in the basic steady flow field, of velocity gradients normal to the chord-wise direction, they move faster at some distance from the profile than close to the surface. The disturbance waves incline as they move forwards. The inclination depends on the value of the velocity gradients. Large gradients give a large inclination, small gradients a small inclina-tion. The waves reach the sonic line at a certain angle with the mean flow and for this reason they can move forwards into the local supersonic region. The waves keep moving as long as the component of the velocity vector of the steady flow normal to the wave front is smaller than the local velocity of sound. that is, if the wave angle with respect to the direction of the basic flow is smaller than the local characteristic angle. The waves then behave as in a subsonic flow.
The inclination of waves reaching the sonic line de-pends, except on the velocity gradients normal to the chord, on the position of the origin of the disturbance. 1f only disturbances generated in the boundary layer
and the wake are considered, the inclination at the
sonic line is larger for source positions more down-stream. However, this effect is generally not very large
AEROFOIL O 7.0 70.0
due to the fact that the inclination is mainly defined by the velocity gradients in the neighbourhood ofthe sonic line where the propagation speed is usually minimal.
The aerofoil given in ligure 17 is a symmetrical aero-foil with a peaky pressure distribution. The maximum local Mach number is Mmax= 1.18. The velocity gradi-ents normal to the chord are relatively large which may be concluded from the fact that the local supersonic region is low. Consequently, the disturbance waves generated at the trailing edge have a large inclination when they reach the supersonic region. The aerofoil given in figure 18, a doubly symmetrical, approximately
bi-convex section with cusped ends and Mmax= L08 has small velocity gradients norma! to the chord. The waves have a small inclination at the sonic line.
For the aerofoil of figure 17, the waves propagate rapidly and without deformation through the superson-ic region. Their propagation speed appears to be mini-mal in the subsonic region somewhat downstream of the sonic line. For the aerofoil of figure 18. the waves move slowly into the supersonic region. It appears that they concentrate in a narrow strip in the recompressive part of the supersonic region.
In fact, it has been found that the propagation velo-city becomes zero for the waves shown in figure 18, that is, they coincide with a right-running characteris-tic. lt should be noted, however, that the accuracy of the
constructed wave pattern is restricted, so that only
rough. qualitative conclusions can be drawn.
If, as indicated in figure 18, the lower part of the wave, adjacent to the surface, comes to a standstill, the upper part, which still propagates. turns over the top of the lower part and then moves rapidly upstream into the supersonic region. This pattern agrees qualitatively with that of unsteady waves in a flow with a shock, and it suggests that in this way a shock wave will be formed for this aerofoil. However, as has been noted, the ac-curacy of this graphical experiment is too small to make definite conclusions.
lt has already been indicated that the effect of the position of the source is not very large in general. For the aerofoil of figure 17 for instance, all disturbances generated in the boundary layer and the wake, even those from rather close behind the sonic line, move rapidly and without difficulty through the local super-sonic flow region. Only disturbances geiierated very close behind the sonic line behave as shown in figure 18. For the aerofoil of figure 18 all disturbances generated downstream of the supersonic region behave as shown
in this figure.
A similar construction can be made for the flow with a shock wave. This is shown in figure 19 which has been obtained from ref. 18. In this case, the velocity distribu-tion in the steady flow and the shock posidistribu-tion had to be estimated from experimental results.
Figure ¡9 gives a good impression of the wave
pat-13
tern when a shock is present in the steady flow. The lower part of the waves corne to a standstill at the shock wave. The upper part turns around the top of the shock wave and then propagates into the supersonic region in the same way as shown on figure 18.
ESTIMATEC SONIC LINE
SHOCK WAVE
___
M 0.875
Fig. 19 Propagation of unsteady waves in a flow with a shock
The figures 17 through 19 give a rather complete
description of the wave pattern in transonic flows
around aerofoils as obtained from short exposure flow pictures. These graphical experiments have assisted a
great deal in clarifying the unsteady behaviour of
transonic flows.
The main conclusion is that, in spite of former sug-gestions, acoustic disturbances can penetrate upstream
in a local supersonic region. Depending oil source
position and velocity gradients in the steady flow they move rapidly or slowly. lt is not possible in view of the accuracy of the graphical experiment to decide whether or not there are waves which come to a standstill in the local supersonic region. This question is the key in the problem of the stability of shock-free flows as will be shown in the next chapter.
Finally it should be noted that although the waves can move upstream through the supersonic region, the
component of the propagation velocity along the
streamline is always in downstream direction. The
acoustic rays bend downstream in the supersonic region as shown in figure 17. When a wave front comes to a standstill, the propagation speed isin the direction of the characteristic. The acoustic rays then locally coincide with the wave front.
4.3 The effect oft/ic boundart' layer
The effect of wave reflection against the aerofoil surface has not been considered in the previous section. Reflections are expected to appear since, due to the turn-over effect, the waves are not normal to the sur-face. Schlieren and shadow photographs show that there are no noticeable reflections as long as the angle between the waves and the normal to the surface is
small.
It will be shown that this may be due to the presence of the boundary layer. Some insight in the phenomenon can be obtained by taking into account the velocity
gradient in the boundary layer with viscous effects
omitted. IMìc Mc Mc
///////// //
(lM)c EDGE OF BOUNDARY LAYER l.-Mc WAVE FRONT AT tO WALLWAVE FRONT AT tAL
'/7
WAVE FRONT AT t=2A L
Fig. 20 Effect of boundary layer on wave propagation The boundary layer effect will be illustrated for waves of small wave length () -+0) using geometrical acoustics.
Consider a plane wave front moving upstream and
being normal to the flow direction as shown in figure 20. The velocity and velocity of sound of the flow out-side the boundary layer are assumed to be constant. This means that the part of the wave front outside the boundary layer will remain plane and normal to the stream direction. This part of the wave front moves
with a velocity (1 M)c. In the boundary Jayer, the
Mach number decreases from M at the edge to zero at the wall and if the speed of sound is assumed to be constant, the velocity of the wave front increases from (1 - M)c at the edge to c at the wall. After a short time the wave front in the boundary layer has turned and since the propagation velocity c is normal to the wave front, the front will be lifted-off from the wall.
This description of the effect of the boundary layer suggests that the disturbance will in effect not reach the surface. Actually, the wave front can not end abruptly. There will be a spreading effect corresponding to the phenomenon of diffraction at a straight edge, and a relatively weak sìgnal will reach the wall.
14
Periodic waves witlì a wave length which is small compared with the thickness of the boundary layer, that
is A/ic J, are expected to behave as shown in figure 20.
Waves with a large wave length, A/ö» I will not be affected by the boundary layer, at least not in this sense. For wave lengths which are of the same order as the boundary layer thickness the effect may be expected to be somewhere between these extremes.
Waves moving upstream will always be affected by the boundary layer at high subsonic speed since their wave length then becomes small. The wave length of waves moving upstream is given by
c(lM)
f
(4.1)where f is the frequency, c the speed of sound and M the Mach number of the steady flow. The expression (4.1) shows that the wave length decreases with increas-ing Mach number and becomes zero at M= I.
WALL
lt
Fig. 22 Turned wave
ORIGINAL SIGNAL FUNDAMENTAL FREQUENCY OF TRIANGULAR WAVE TRIANGULAR WAVE
Fig. 21 Triangular wave form (due to steepening effect)
Another reason why the boundary layer effect will, in general, be important at high subsonic Mach numbers is due to the steepening effect of finite disturbances. Consider a source emitting a sinusoidal signal. At some distance this signal has obtained a triangular form as shown in figure 21. The amplitude of the fundamental frequency of this triangular wave is only 2/ times the total amplitude, that is, 2/70 times the amplitude of the
original sinusoidal signal. An important part of the
acoustic energy is shifted to higher frequencies, that is, smaller wave lengths. It is likely, in view of the preceed-ing discussion, that the effect of the boundary layer will then become larger. Since the distance in which the steepening process takes place is, for plane waves mov-ing upstream. proportional to (1 - M). the boundary
VELOCITY
F RO FIL E
(IMIc
layer effect can be expected to be important at a short
distance from the source for high subsonic Mach
numbers.
So far it has been assumed that the waves are normal to the flow direction. In general. however, they will make an angle with respect to the flow as shown in figure 22. The wave front turns again due to the velocity gradient in the boundary layer, but now at the same time points from the original wave front in the free
stream propagate into the boundary layer.
If the angle i/e between the wave front in the free stream and the flow direction decreases, thispartof the wave front will penetrate further into the boundary layer, and a certain value of i/e occurs at which the wave front just reaches the wall. This value of i,Li will be called the reflection angle. For smaller values of ithe wave front is reflected against the wall.
In order to use these simple arguments in experiments the following calculations to determine this reflection angle have been performed.
Fig. 23 Rotation of wave in boundary layer
When the velocity of sound is assumed to be constant and the velocity in the boundary layer approximated by
a linear function of the coordinate y, normal to the
wall:
where u,,6 is the velocity at the wall, the variation of i/i with time in the boundary layer is (figure 23)
= c1 sin2 i (4.3)
Integration gives
i/i = arctg(c2 - c1 t) 1
(4.4)
where arctg I/c2 is the value of il' at the time t=0. Thus, for the linear velocity profile. /i is independent of the
coordinates x and y; a straight wave front remains
straight.
15
The velocity of the wave front in y direction is dy
dt
The y coordi nate of a point of the wave front is given by
y =
/(c,c1t)2+l+c3
(4.6)ii/ (r=0)=ii/ is equal to the reflection angle, ifthe point of the wave which is at the edge of the boundary layer at t=0 will just reach the wall. The value of the constant of integration e3 for this point follows from
6)
= - Vc+1+c3
- C
ci (4.7)
The point just reaches the wall if dy/dt=0 for y=O. It follows from (4.4) and (4.5) that dy/dt=0 at t=c2/c1 and when this is substituted in equation (4.6) for v=0, it follows that ('3= c/c1. The wave angle is then given by
Vc+1-1 =
C (4.8)and since e1 =(l u/u) Mc/ô, where u
is the velocity at the edge of the boundary layer, the reflection waveangle is given by
iiie=arctg[{(i
-
__)M+1}_l]
(4.9)This reflection angle is given in figure 24 for different values of uw/ue as a function of Mach number. When the wave angle t/i is smaller than the reflection angle, every disturbance, even ifA*0 will be measured at the wall. When Li, becomes larger than the reflection angle an increasing part of a signal. to start with the highest frequencies will be obscured by the boundary layer
effect.
Figure 24 shows that when u/u is taken as 0.8, which gives a reasonably good approximation for the mean velocity gradient in the boundary layer. neglecting a very thin part near the surface, high frequeiicy waves moving upstream will not be reflected, at high subsonic Mach numbers, as long as the angle between the waves
= ccosçl'
(4.5)LI = U+C1y (4.2)
06 I.0
and the normal to the surface is smaller than about 30 degrees. Experimentally it is found that visible reflec-tions occur when the angle between the waves and the
normal to the surface becomes of the order of 15 to 20 degrees
this can reasonably be correlated with the
effect of the velocity gradient in the boundary layer. discussed in this section.
The reflected waves tend to turn over in the flow
out-side the boundary layer for the same reason as the
original ones do, arid therefore they are concave. This is, in many cases, hardly visible because the reflection usually starts at a point where the waves are running rapidly.
5 The stability of transonic flows
5.1 Introduction
in this chapter the stability of transonic flows without shock waves will be examined.
The term stability, defined as the quality of being immune to small disturbances, has frequently more mathematical than physical significance. Stability stu-dies in fluid dynamics are in general related to hypo-thetical flow models, and it is not always clear that the real flow will behave in the same way as the simplified mathematical model. However, the stability of flows can, in many cases, only be studied theoretically; an
experimental approach to the stability of flows
isdifficult because unstable flows do not usually persist for a sufficiently long time to be measured.
In the mathematical treatment of the
stabilityproblem, it is usually supposed that the flow equations have a steady solution. An initial value problem is then considered, with flow variables slightly different from those in the steady state solution. 1f the unsteady solu-tion approaches the steady state solusolu-tion as time goes to infinity, the motion is stable, if the solution diverges with time, the motion is unstable. Thus, mathematically the problem is to solve, for a given steady flow, the equations of continuity, momentum and energy. subject
to a set of initial and boundary conditions.
A more physical description of the stability problem is the following. Suppose that the original flow exists, at least for a short time. At some instant, a disturbance is generated somewhere in the flow field. The develop-ment of the flow can then be followed, taking into ac-count the propagation of the disturbance and its growth or decay. If the original steady motion is stable, the disturbance will die out, if it is unstable, the disturbance will keep growing with time. There may also be cases where a disturbance approaches a steady state, result-ing in what is known as neutral stability.
The problem of the stability of two-dimensional
shock-free transonic flows is too complex to ask at present
for a strict mathematical treatment. lt will be shown,
16
however, that the question of the stability of transonic flows can for practical purposes be answered without obtaining a complete solution of the unsteady flow
equations or tracing the history of a disturbance in
detail. Only certain features of the behaviour of distur-bances are required.
5.2 The stability of quasi one-dimensional transonic flows
5.2.1. General
The stability of quasi one-dimensional transonic
flow, the flow in a channel with slowly varying cross-section, has been studied by Kantrowitz (ref. 4). His
analysis describes in a simple and physically clear
fashion the essential features of disturbances in quasi one-dimensional transonic flows. Since the results of
Kantrowitz will be used in the investigation of the
stability for two-dimensional transonic flow it is suit-able to give a summary of his analysis.
Kantrowitz made his analysis to show why channel flows involving shock-free deceleration through the speed of sound are unstable. The analysis is divided in two parts. In the first part. Riemann's theory of the propagation of finite amplitude disturbances in a homo-geneous medium is extended to the case where upstream moving pulses are superposed on a decelerating transo-¡lic channel flow. Shock waves are formed as the pulses propagate, and if the shocks are assumed to be weak, the pulses approach a steady state in which a part of the decelerating flow is converted into an accelerating flow. The flow has then a kind of neutral stability.
In the second part the assumption of weak shocks is dropped and the motion of shockwaves is considered more accurately. It is found then that expansion pulses are consumed by the shock motion and that compres-Sion pulses inevitably grow. The conclusion is that
smooth deceleration through the speed of sound is
¡instable with regard to compression pulses coming from the rear of the channel.
A similar anlysis can be made for the case where a shock wave forms a part of the original channel flow. The shock position is found to be stable in accelerating flow and unstable in decelerating flow. This will be discussed in chapter 7.
5.2.2. Plane disturbances superposed on a quasi
one-dimensionalfiow
Consider the unsteady flow field in a straight channel of slowly varying cross-section. Transverse velocity com-ponents will be small and if effects of viscosity and boundary layers are omitted it is reasonable to assume
that variations in the flow parameters over a
cross-section are small also and hence can be neglected. The resulting flow is termed quasi one-dimensional. The
equations of motion and continuity for the isentropic
flow of a perfect gas then become
au (5.1) at ¿x
0
+ ¿3(puS) axwhere S is the cross-section area.
The equation of continuity may be rewritten in the form
3 log p
+ ua log p..
++u
au d log S- 0
(5.3)et cx 3x dx
Multiplication of equation (5.3) by ±c, addition of equation (5.1) and further taking y= 1.4 leads then to
the equations
-(u±5c)+(u±c)---(u±5c)±uc
d logS0 (5.4)
The flow to be studied is a steady flow (indicated with subscript e) upon which unsteady perturbations
(denoted with dashes) are superposed. The steady flow satisfies the steady equations
d logS
-dx
u,'
''dx
and
d(5c,) du, dx
-
edxwhich M=u,/c,, the equilibrium flow Mach nLlnlber. In the perturbed flow, we have
UU,+U
= e + 8(u'+5c') du x-
+ (u'±c')(1R:M)
C+
ax'dx
2lu c
± (Me 1)( \ U. u' ,\ du, + M,+cJ
/dx
=0
aP / 3P+2Q\ ap+ u+c, +
) ox+
+(1-
M)(3P+2Q)duC
5dx+
+(M-1)
P+QPQ\dUe
20u,
+
Thle +iO)
dxaQ I 2P+3Q)5Q
+
- + (u,c, +
X at , 5 a (5.2) (5.5)and substitution of (5.5) into equation (5.4) leads to + (u
- ç +
3Q + dUe = 0 (5.9)Ct
Sjcx
5 dxx
at
-(5.6)
Denoting with P and Q the quantities P=u'+5c' and Q=u'-5 e', the equations (5.6) may be written
l7 ¡2P+3Q\ dUe
+(1+M,)
(M2l)(PQ
+ P+Q +P_Q\du,
-0(58)
20u, 2M,10 J dx
-The quantity P is essentially transported with the velocity u,+c, that is downstream, while Q, owing to the appearance of u,c may be moving upstream if the steady flow is subsonic. The appearance of both P and Q in the two equations indicates that the motion of downstream moving perturbations (P) is affected by the presence of upstream moving perturbations (Q) and conversely. In particular if initially only a
Q-perturba-tion is present it must be expected that also P-perturba-tions will be created (aP/at 0). Kantrowitz has shown, however, that for a pulse, a finite disturbed region
boun-ded on both sides by undisturbed flow, moving up-stream in the transonic region. P can be neglected in comparison with Q in equation (5.8) if there are no downstream moving perturbations to begin with. This
implies essentially that P will be assumed negligible and
hence that everywhere u'= Sc' and Q=2u'=lOc'. The propagation of a Q pulse can then be deduced
from the equation (5.8) alone. The pulse can be
visualiz-ed in the ux plane as a little mountain (expansion pulse) or a valley (compression pulse) with one section
where u' is increasing with x, called the expansion phase and one section where u' is decreasing with .v. called the compression phase.
Equation (5.8) can be simplified further since in the transonic regime Me' I, and the fourth term will be neglected entirely owing to the small factor M,2 1. Omitting P and putting I + M,=2, equation (5.8) then
takes the form
For values of M, close to i this may be rewritten, using
d due
-s--- (5e,) = M,
d'
a
i
/3Q
ut ax +
_c)]
= 0 (5.10)Integrating this equation with respect to x over the
range ato x, gives
fX1Qdx+[Q(
+ ue_Ce)]::=O (5.11)
If the integration is taken over the whole of a finite pulse. the second term of equation (5.11) vanishes and the pulse area JQd.v remains constant to the order of
accuracy of equation (5.9).
= 0 (5.7) Kantrowitz has shown that, as long as only upstream moving disturbances are present, this rule, conserva-tion of pulse area, is not affected by the presence of
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