High speed unsteady separation about concave bodies: A physical explanation

van

KARMAN INSTITUTE

FOR FLUID DYNAMICS

TECHNICAL NOTE 123

HIGH SPEED UNSTEAVY SEPARATI0N ABOUT CONCAVE BOVIES

- A

PHYSICAL EXPLANATION

~

A.G. PANARAS

SEPTEMBER 1977

TECtiNiSf;t1C UNIVERSITEIT DELFT LUCHTVAART· Eli RUiMTEVAARTIECHNIEK

BIBUOTHEEK

Kluyverweg 1 - 2629 HS DELFT

7 JAN. 1988

~A~

-~Q~-

RHODE SAINT GENESE BELGIUM

VON KARMAN INSTITUTE FOR FLUID DYNAMICS

TECHNICAL NOTE 123

HIGH SPEEV UNSTEAVY SEPARATION ABOUT CONCAVE BOVIES.

- A

PHYSICAL EXPLANATI0N

-A. G. PANARAS

TABLE OF CONTENTS SUMMARY . . . . LIST OF SYMBOLS LIST OF FIGURES 1. INTRODUCTION . . . 2. GENERAL ANALYSIS .

2.1 The separated flow about spiked cones 2.2 Spiked hemispherically capped cylinders

and other bodies . . .

2.3 Free interaction . . . . 2.4 Chapman's reattachment criterton

3. PHYSICAL EXPLANATION OF THE INSTABILITIES

3.1 The proposed explanation of the pulsation mode 3.2 Explanation of the oscillation mode

4. VERIFICATION TESTS . . . . . 4.1 Test facility models . . . .

4.2 Spiked flat ended cylinders

4.2.1 Detection of the supersonic jet 4.2.2 Further evidence for the existence

of the supersonic jet . . . . 4.2.3 Effect of the position of the shock

intersection point . . . . 4.2.4 Oscillation mode of instability

4.2.5 Influence of the Re number - tempera tu re 4.3 Spiked cylinders with rounded shoulders

4.4 Concave conic bodies 5. CONCLUSIONS . . . . REFERENCES FIGURES PLATES i i i iv 1 4 4 6 6 9 13 13 17 23 23 24 24 26 27 28 29 31 32 33

.

35 39

ACKNOWLEDGEMENTS

The au thor wishes to acknowledge his debt to Professor J.F. Wendt, who suggested this research, for his support and constructive criticism.

Thanks are also due to Professor B.E. Richards for his help and encouragement in the initial phase of the research.

Finally, thanks are expressed to the personnel of the VKI High Speed Laboratory, especially to Mr R. Conniassell e , for his valuable help.

- i

-SUMMARY

A physical explanation is given of the observed large amplitude instabilities of the flow about some families of

concave bodies. By means of theoretical arguments and an inter-pretation of the experimental data, it is shown that the

"pulsation" mode is due to the effect which an annular supersonic jet, appearing at the shock intersection point of the

fore-shock and af ter-fore-shock, has on the separation bubble. Addition~ nally, it is shown that the "oscillation" mode, which appears experimentally when the separation point coincides with the

tip of the forebody, is most probably due to the non-satisfaction of the reattachment mechanism in a manner securing equilibrium of the pressure.

Cf cp d dl f k t M N P q Red Re_o R r Str T Tr u ., .-u -u y i i -LIST OF SYMBOES

skin friction coefficient pressure coefficient

diameter of concave body diameter of spike

frequency

length parameter, t/d length of the spike Mach number

pressure parameter of Nash pressure

dynamic pressure

Reynolds number based on body diameter Reynolds number based on running length radius of concave body

radial distance Strouha 1 number

temperature

recovery temperature velocity component

velocity along the dividing streamline velocity ratio u:: / u

e

gas constant (cp/cv)

two dimensional deflection angle; boundary layer thickness

afterbody inclination angle shock angle

cr d e o R t w 2 3 00 i i i -coefficient of viscosity

density, radius of curvature of shoulder separation angle

angle of the tip of the forebody

Subscripts

dead air region

conditions along the limit of the boundary layer beginning of shock-boundary layer interaction conditions at the reattachment point

total quantity wa 11

conditions downstream of reattachment conditions behind a strong shock

iv

-LIST OF FIGURES

1 Wood's diagram 2 Model body

3 Classification of instabilities 4 Steady axisymmetric separation 5 Free interaction plateau angle 6 Chapman-Korst reattachment model 7 Reattachment angle vs Mach number 8 Generation of supersonic jet

9 Similarity with Edney's IV shock formation 10 Henderson's diagram

11 Instability limits for various Mach numbers 12 Shoulder reattachment

13 Conic body calculations 14 Jet-impingement region

15 Pressure fluctuation vs spike length 16 Transition from pulsation to oscillation 17 Instab~lity limits for rounded cylinders 18 Initial dead air region

- 1

-1. INTRODUCTION

The phenomenon of flow separation from spikes ahead of blunt axially symmetric bodies has been extensively studied during the last two decades (Refs. 1 to 17). The interest has its origin in the effort to compromise two desired and non-similar characteristics of reentry bodies; that is, a large nose radius (low heat transfer) for the reentry phase, and a pointed nose (low drag) for the phase of motion through the atmosphere. The presence of a spike of controlled variable length protruding from the stagnation point of an axially sym-metric blunt body seems to provide at least a partial solution

(Ref. 2).

One characteristic of the flow around spiked bodies is an unsteadiness which occurs for certain combinations of the parameter K (ratio of spike length ~ to body diameter d) and the shape of the body, provided that the body alone causes a detached shock wave (Fig. 1, taken from Ref. 10).

The spiked body may be regarded as a degenerate form of what could be called a concave axisymmetric body (Fig. 2). When such a body was tested as a possible shape for space vehicles (Ref. 18), or appeared as a consequence of ablation on reentry nose cones (Ref. 19), it was observed experimentally that for some specific configurations the flow presented un-steady characteristics, similar to those of spiked bodies.

Similar large scale unsteady flows were also observed in two dimensional hypersonic separation induced by the deflec-tion of control surfaces (Ref. 20).

A classification of the observed type of instabilities for axisymmetric flows was made by Kabelitz (Ref. 15) who used the words "pulsation", "os c illation", and IIvibration". The cor-responding shock and shear layer configurations are shown in figure 3.

- 2 ..

IIPulsationll

(Fig. 3a) is the fundamental instability observed by Mair (Ref. 2) and Maull (Ref. 8) and manifests it-self as a periodic change of the entire flow field, as the conical separation region inflates like a balloon and expands

radially. IIPulsationll

has been observed around spiked flat cylinders and large angle spiked con es for a certain range of the spike length parameter, k.

When the spike length parameter k exceeds a certain value which depends on the Mach number, Reynolds number and the temperature of the body, the instability undergoes a quali-tative change to wh at Kabelitz calls l1os c illationl1 (Fig. 3b); i.e., the conical shock is attached to the spike tip and the accompanying shear layer oscillates laterally, its shape changes periodically from concave to irregular convex (Ref. 7).

Another kind of instability, observed on con es of a slightly greater angle than the detachment angle and of

consi-derable spike length, is th at of I1vibration l1 (Fig. 30). In this

case, a double shock system exists which oscillates laterally

(Ref. 15).

At this time no satisfactory physical explanation of the phenomena described above has been given. As Abbott et al. state in Ref. 27, at present the general consensus is that the

flow instabilities are related to changes in the effective

geometry of the body due to growth of the separation bubble. The bubble is thought to grow to some critical state at which time further growth cannot be sustained, and the bubble suffers a collapse due to outflow near the shoulder. The exact mechanism of bubble growth and collapse is presently a matter of specula-tion.

The objective of this work is to propose a rational explanation for the above mentioned instabilities. By means of theoretical arguments and an interpretation of the experimental data, we show that the pulsation mode is due to the effect which an annular supersonic jet, appearing at the shock intersection

p 3

-point of the fore-shock and af ter-shock, has on the separation bubble. Additionnally, it is shown that the oscillation mode, which appears experimentally when the separation point coin-cides with the tip of the forebody (bodies with short forebody), is most probably due to the non-satisfaction of the reattachment mechanism in a manner securing equilibrium of the pressure.

~ 4

-2. GENERAL ANALYSIS

For the information of the reader, the experimentally observed behaviour of the flow about some simple families of concave bodies will be summarized in this chapter. Additionally, Chapman's separation and reattachment analysis will be reviewed briefly, so that the proposed explanation of the oscillation mode will be better understood.

2.1 The separated flow about spiked cones

The separated flow past a spiked cone may be divided into a number of distinct flow patterns which depend on both the free stream conditions and the spike length and cone angle. A photographic study of the separated flow over spiked cones at a Mach numb~r of 10 and Reynolds number of 0.5.105 _{was made} by Wood (Ref. 10), while a similar study of flow geometry at a Reynolds number of 0.2.106 _{and Mach numbers of 10 and 15 was} made by Holden (Ref. 12) to deduce the effect of Mach and

Reynolds numbers. An important result of Wood's study was the discovery of five different types of flow. These occur in regions which were defined in terms of spike length and cone angle, as shown by figure 1. The regions are labelled A to E as shown. They are best described in the work of Wood:

"For small cone angles the flow does not separate from the spike. The conical body generates a shock wave, whose angle is the same -as if the spike were not present (type A). As the cone angle

increases beyond the boundary between regions A and B, a small region of separation appears. The flow separates fromfue spike and reattaches on the conical face of the body.

The size of the separated flow region increases with increasing cone angle, until the reattachment point reaches the shoulder of the body on the rather ill-defined boundary shown as a shaded area in figure 1. As can be seen flows of type C with shoulder reattachment exist for a wide range of model configurations. Type D is the unsteady flow first observed by Mair (Ref. 2). It occurs only on bodies with semi-angles greater

~ 5

-than the conical detachment angle. The cone angle has a conside-rable effect upon the amplitude of the oscillation. For angles only slightly greater than the conical detachment angle, the oscillation appears as a ripple in the flow pattern, instead of the violent expansion and collapse of the dead-air region which occurs on highly blunted bodies.

The last type of flow (type E), is relatively unimpor-tant. It is not different from the flow without a spike, and it occurs when the spike length is insufficient to penetrate the detached shock wave.

An interesting feature of figure 1 is that all of the boundaries are centered upon the conical detachment angle. Fully attached flows of the A type were never found for co ne angles

~greater than this value. Conversely, unsteady flow (type D) was

~ever found unless the body alone would have generated a detached

shock wave".

Wood discovered that the boundary between regions B and C was influenced by the temperature of the body. The size of region C increases considerably at the expense of region B as the temperatuee of the body increases and approaches the adia-batic recovery temperature.

All bodies used by Wood in the general tests had sharp shoulders. It was found that a marked effect on the unsteady boundary between regions C and D occurred when the shoulder was characterized by a small radius.

The unsteady flow type D mentioned in the above des-cription of Wood's diagram is what Kabelitz has cal led pulsation. The oscillation mode is not mentioned in the work of Wood, nor

in some other basic works, such as Maull (Ref. 8) and Holden (Ref. 12). A description of this mode first appears in the ex-perimental study by Bogdonoff and Vas (Ref. 7). Their schlieren photos of the flow about spiked flat-ended cylinders showed that for the

ra~ge

of the normalized spike length, 1.5 <

~

< 3, the

""' 6

-shock seemed to be attached to the tip of the spike while the downstream sections oscillated, appearing alternately convex and concave.

2.2 Spiked hemispherically capped cylinders and other bodies

The separated flow which is established about spiked hemispherically-capped cylinders has been found to be rather steady for all spike lengths (Refs. 2, 4, 5, 6, 7, 8, 13).

Neither pulsation nor oscillation of the flow has been observed. The reported tests cover a Mach number range from 1.6 to 14. We note here that a small unsteadiness of the flow consisting of the rapid osritl lation of the separation point up and down the spike, has been observed at low Mach numbers (1.6 to 3) by Hunt (Ref. 5) and Album (Ref. 9).

Maull (Ref. 8) has conducted tests at Mach 6.8 using spiked flat cylinders with rounded shoulders. He found that all bodies whose shoulder radius was less than half the diameter of the cylindrical portions of the body exhibited pulsation for certain ranges of spike length.

We also performed tests with s~ked cylinders having rounded shoulders; the objectives were to determine if the oscil-lation mode occurs on such bodies and to study the influence of the initial position of the shock discontinuity point on the behaviour of the flow. The latter point will be discussed in section 4.2.2. A significant observation was the suppression of the oscillation mode, even for very small values of the shoulder radius. More specifically, the flow abruptly changes from the pulsating mode into a steady mode above a critical value of the spike length.

2.3 Free interaction

It is a matter of experimental observation and of ma-thematical verification that once a boundary layer has been

7

-separated by some disturbance (step, shock), the flow field in the region near separation is dominated by the interaction be-tween the boundary layer and the external supersonic flow (Ref. 28).

Although the location of the point of separation is a function of the location and strength of the disturbance, the pressure distribution throughout the region removed from the immediate neighborhood of the disturbance, is giverned by local interaction and is, in this sense, independent of the disturbing mechanism.

This phenomenon is known as "free interaction" and it has been the basis of similarity analyses for the prediction of the pressure distribution in the separation region, as well as the basis of analytical methods for the solution of the boundary layer equations describing laminar and turbulent separation.

Chapman, Kuehn and Larson (Ref. 28), using the Prandtl-Meyer relation and the boundary layer equation at the wall have shown for the two dimensional case that :

(M~-2)1/4 ( 1 )

~ere the subscript 0 designates conditions at the beginning of

interaction, that is, at the downstream-most point upstream of which the pressure is sensibly the same as in the inviscid flow.

Erdos and Pallone (Ref. 30) developed and exploited the analysis both for laminar and turbulent flow, while Lewis, Kubota and Lees (Ref. 35) have correlated the pressure distri-bution measured on a plate upstream of a 10° ramp at M = 6.06

00

with the Chapman similarity variables (corrected for hypersonic flows by taking the next term in the pressure coefficient); they confirmed the ftee interaction concept.

8

-A further development of the semiempirical "méthodes globales"has been made by Carrière, Sirieix and Solignac who generalized the theory of Chapman, analyzing supersonic non uniform axisymmetric flows (Ref. 36). The non uniformity of the flow is characterized by the existence of a longitudinal con-stant pressure gradient. Their analysis is applicable on both laminar and turbulent flows and it confirms the work of Chapman et al.

Additionally, we mention that for spiked cones Wood (Ref. 10) has shown that the boundary layer equations for a thin spike, could be reduced to the two dimensional form if the

thickness of the wall sublayer, in which the adverse pressure gradient acts, was assumed to be small in relation to the spike radius. His analysis led to the relation :

( 2)

where Hand nare empirical functions of the Mach number only. For the case of the laminar axisymmetric separation about a cylindrical forebody, such as a spike, we have estimated the order of magnitude of the angle formed between the wall and the shear layer connecting the separatirn and reattachment region (angle of in Fig. 4). For this we have substituted, in equation 1, which according to the analysis of Carrière et al (Ref. 36) applies also to the axisymmetric separation for small values of the boundary layer thickness relative to the radius of the forebody, the skin friction law of Rubesin-Johnson (Ref. 29). The constant of proportionality is taken to be 1.4 from the Erdos-Pallone (Ref. 30) correlation. Equation 1 becomes

2.09

( 3 )

Index 0 again designates conditions at the beginning of inter-action. If the forebody is conical or curved, the Mangler (Ref.3!)

- 9

-relations have to be used for the transformation of the axisym-metric forebody into a flat plate.

The free interaction angle af corresponding to cp is taken from standard conical flow tables, e.g., NACA TR 1135. Application of equation 3 for various Mach and Reynolds numbers has given the results which are shown in Fig. 5. We note that tbe order of magnitude of the free interaction angle af is about 10°. The values of Rea used in Fig. 4 represent actual values observed in experimental works in flows about spiked bodies. For example, in the experiments of Wood (Ref. 10), the Rea for the separation to occur at the junction of the conical tip and the cylindrical part of the spike is equal to 1.1.104 , while in our tests the range of the corresponding Reynolds number is 3.104 _{to 2.105 .}

2.4 Chapman's reattachment criterion

Chapman et al. (Ref. 28) have established a simple relation for the reattachment of a laminar shear layer on a flat two dimensional surface, assuming the nature of the reat-tachment to be inviscid.

The basis of the calculation was a laminar mixing layer theory (Ref. 37), according to which a uniform stream of velocity u_e' Mach number Me and pressure Pe mixes with a dead air region (of pressure Pd

=

Pe) having large dimensions compared to the thickness of the mixing layer. The initial boundary layer thick-ness has been assumed to be zero, so the velocity profiles at different streamwise stations are similar; hence, the velocity ratio uX/u e (Fig. 6) along the dividing streamline does not change with Reynolds number and distance from separation.

Further, Chapman has shown that this velocity ratio depends only slightly on Mach number and on the temperature-viscosity relation. A reasonable approximation is the value U = uX/u

- 10

-If the initial boundary layer thickness is non-zero, it has beenbhown by Cooke (Ref. 38) that the value Lï::

=

0.587 is taken asymptotically for small values of the initial boundary layer thickness.

Of course, the value of u~ affects the numerical result of a specific problem, but the reattachment relation has been deri ved for genera 1 Lï::.

Chapman et al. state th at in the establishment of the dead air pressure the essential mechanism is considered to be a balance between mass flow scavenged from the dead air region by the mixing layer and mass flow reversed back into the dead air region by the pressure rise through the reattachment zone. For steady flow the dividing streamline at the separation point as calculated from the mixing layer theory must also be a dividing streamline at the reattachment. If this were not the case, air would be either continually removed from or rejected into the dead air region, and the scavenged mass flux would not balance the reversed mass flux.

In order for a particle flowing along a streamline within the mixing layer to be able to overcome the pressure rise

through the reattachment zone and to pass downstream, its total pressure Pt must be greater than the terminal static pressure P2 at the end of the reattachment zone. In figure 6, particle (a) passes downstream in this manner. Particle (b), however, has a low velocity with a corresponding low total pressure and is reversed because its total pressure is less than P2'

The dead air pressure is determined by requiring that that total pressure along the dividing streamline, as it ap-proaches the reattachment zone, be equal to the terminal static pressure P2' Thus the flow is divided into two regions : a viscous layer wherein the pressure is assumed to be constant, and a re-attachment zone wherein the compression is assumed to be such that little total pressure is lost along the dividing streamline.

11

-Taking into consideration the Buseman isoenergetic integral of the energy equation, Chapman et al. found that :

y :y:T P2

~

1 +

cl

2 M' e

~

_{( 4 )} -

=

Pd - 2

cl

2 + (l-u ) _{2 Me}

Obviously, in the physical description no distinction is made concerning the nature of the flow, so equation 4 should be applicable to either laminar or turbulent separation.

Nash (Ref. 39),supported by physical arguments (im-possibility of reattachment taking place at a point of zero pressure gradient) and by experimental data, corrected the Chapman analysis by assuming that reattachment takes place

oc

a point of strong positive pressure gradient and that a substan-tial residual pressure recovery is achieved downstream of the position of zero shear stress (see Fig. 6).

For the quantitative expression of the criterion, Nash introduced the parameter

( 5 )

the value of which has to be determined from experimental data. For lanlinar flow, Cooke (Ref. 38) proposed, on the basis of experimental results, the value N

=

0.5. Merritt (Ref. 34) found that the value N = 0.62 is the weighted value for

laminar two dimensional flow. Finally, compiling Horton's data (Ref. 40, we estimated that the mean value for reattachment on the surface of conical bodies is N ~ 0.10.

12

-Chapman (Ref. 32) has extended the reattachment analysis to axisymmetric separated flows by application of Mangler's trans-formation (Ref. 31). According to this transtrans-formation, pressure relations such as equation 4 remain unchanged while length quan-tities are subjected to changes, if two dimensional results are applied to axisymmetric flows.

Assuming the existence of an oblique shock at the reattachment point, we have estimated, for the various values of N mentioned above, the angle of the reattaching shear layer relative to the surface of reattachment, as a function of the Mach number Me (Fig. 7). The effect of the range of the values of the reattachment angle 0 on the features of a separated

13

-3. PHYSICAL EXPLANATION OF THE INSTABILIT~ES

3.1 The proposed explanation of the pulsation mode In this section we shall show that the pulsation mode of instability is due to the existence of a mechanism external to the separation bubble. The model body of the analysis is shown in Fig. 2. In the text the nose of the body which becomes a

simple spike in its degenerate form will be called the "forebody", while the main body will be called the "afterbody". The shock

envelope shown in the figure is that appearing at a certain point in time during the pulsation mode. The point where the foreshock meets the aftershock (point A) is called the "shock discontinuity pOint" or "triple shock point".

In order that the hypothesis can be more clearly explained, we will consider the development of the flow field, about a

geo-metry similar to that in Fig. 2, as it might seem if the flow was impulsively started at a supersonic Mach number. The

selec-tion of the impulsive start of the flow is not fortuitous. The flow conditions prevailing about a concave body, at the starting phase of each cycle of pulsation, resemble an impulsive flow, in which the shock envelope retains a position corresponding to an inviscid flow field. This impulsive nature of the flow field is due to the violent radial expansion of the separation bubble, which occurs during the final phase of each cycle of pulsation, as we will demonstrate with experimental data.

According to the experimental evidence (Refs. 10, 15, 19), the flow about a concave body may be attached or separated and steady or unsteady. The critical geometrical parameter for th-e 0 cc ure n ce 0 f s e par a t ion i s mos t pro b a b 1 Y t hes hap e 0 f t h e compression arc ,bc (Fig. 2), but so far no quantitative criteria have been established. In this analysis we assume that the geo-metry of the body is such that separation occurs.

Furthermore, we assume that the mean inclination angle

.. 14 ..

the conical detachment angle corresponding to M~. This condition,

according to the experimental evidence (Ref. 10), is necessary for the flow about a concave body to be unsteady.

A final point to clarify before we proceed with the analysis is the extent of the separation bubble. It can be shown

(paragr. 3.3) that if a steady separated flow is established about our model body, th en the separation region covers all the concave part of the body; the shear layer reattaches on the shoulder of the body (Fig. 4) and never on the conical part of the afterbody.

At the early development of the impulsively started flow about the model body, the field is inviscid. Thus, with the first flow pulse, the weak foreshock SA (Fig. 2) and the strong aftershock AB appear.

If the flow is quasi steady, a shock discontinuity point A, a weak shock, Ar, and a shear layer, Af, must appear.

The next phase is the development of the boundary layer along the forebody and its separation at the compression arc bc. The separation of the boundary layer and the formation of the separation bubble take place with a finite speed. Lighthill

(Ref. 33) has estimated that the disturbances in a separated boundary layer propagate upstream with a speed which is

some-what less than the local velocity of sound. Thus it is reasonable to suppose that the speed of the growth of the separation bubble, up to the point of filling the concave part of the model body, is about the same order as the local sonic speed.

The growth of the separation bubble has the following effect on the further development of the flow: as the distur-bances from the inflating surface of the bubble travel along the Mach lines at a speed compatible with the speed of the inflating surface itself, the shock system does not feel in time the suc-cessive positions of the everchanging shape of the IIbodyli and we may reach a condition where the shock system remains almost

- 15 ~

motionless while the shear layer, which envelopes the separation bubble, approaches it.

At this point we note that the free stream air con-tained in the streamtube F (Fig. 8) passes through the foreshock and is channeled between it and the inflating surface DA of the separation bubble. This channeled supersonic air is directed, in the vicinity of the shock discontinuity point, towards the body and impinges on its surface.

The direction of the channeled supersonic air is defined by the pressure forces prevailing in the adjacent low speed

regions. As we have already mentioned, a third oblique shock Ar has to appear (Fig. 2) for the equilibrium of the flow around the shock discontinuity point. This shock, in the case of a non separated flo~ extends to the surface of the body and prevents the high pressure air, existing behind the strong shock AB, from flowing into the low pressure region behind the oblique foreshock SA. But in the case examined here the growing separation bubble causes the continuous shortening of the length of the weak shock Ar and its consequent detachment from the surface of the body. In this way the annular supersonic stream which impinges on the surface of the body is bounded between the low pressure region of the separation bubble and the high pressure field which exists behind the strong shock AB (Fig. 8).

This difference in pressure will act to turn the annular supersonic stream in the direction of the compression arc (bc) of the body. Thus, a certain fraction of the air contained in the streamtube (F) is fed into the separation bubble. This air is characterized by a high pressure, which is due to the appearance of astrong detached shock at the region of impingement of the annular supersonic stream.

To summarize, we have so far shown that the existence upstream of the afterbody of an intersection point between a weak conical shock and a strong shock may result in the appearance of an annular supersonic stream which envelopes the inflating

sepa 16 sepa

-ration bubble and feeds it with high pressure air. But as the volume available for the expansion of the separation bubble which is filled with high pressure air is limited, the foreshock itself will be pushed outwards and will eventually cover the concave part of the model body.

If the pressure in the separation region is much higher than the pressure of the surrounding free stream, a radial pres-sure imbalance exists and a radial expansion will occur. On the contrary, if the radial pressure imbalance is small, no reason for a radial expansion exists and the excess air in the separa-tion region may expand downstream at the shoulder of the body

(illustrated photographs will be presented in paragraph 4.2.1.1). According to the experimental evidence, when the condi-tions for the occurrence of the radial expansion exist, the fore-shock inflates abruptly and takes a hemispherical shape. However, as this hemispherical strong shock is due to the expansion of the high pressure air contained in the separation bubble and not to the existence of a proper blunt body in the flow, it cannot stand there attached to the tip of the forebody, but it has to approach the surface of the afterbody. Obviously, as the strong shock moves towards the afterbody, the forebody is exposed to the free stream and as aresult the foreshock must reappear. We no te that at this phase of the flow development no separation bubble exists; hence, simultaneously with the appearance of the conical foreshock the growth of the boundary layer along the surface of the forebody and its subsequent separation at the compression arc (bc) will start.

It is clear that the reestablishment of the initial condttions of the impulsively started flow results in the repe-tition of the appearance at the shock discontinuity point of the annular supersonic stream, the filling of the separation

bubble with high pressure air, etc. We may say that the pulsation mode of instability has been established.

17

-The dependence of the explosive inflation of the sepa-ration bubble on the appearance of an annular supersonic stream, at the vicinity of the shock discontinuity point, is a basic requirement of the above analysis. This supersonic stream acts as a supersonic jet at the region of its impingement and the flow at the vicinity of the shock discontinuity point may be viewed as the unsteady counterpart of the steady type IV shock formation

(Fig. 9) observed by Edney (Ref. 21).

A fundamental difference is that the Edney IV shock formation appears only when the flow behind the embedded shock Ar is supersonic and this condition is not present if Moo is low. This can be demonstrated by reference to Fig. 10 from Henderson

(Ref. 22) in which the regions of subsonic or supersonic flow behind the shock (Ar) are shown as a function of the Mand the ₀₀ strength of the foreshock (SA). However, in the unsteady flow case the supersonic jet is visible even for Moo ~ 2 (see photo-graphs in the study of Mair, Ref. 2).

3.2 Explanation of the oscillation mode

The rapid addition of high pressure air in the growing separation bubble and its consequent radial expansion are the peculiar characteristics of the pulsation mode. However, the geometry of the body may be such that during the hypothetical impulsive start of the flow the inflating separation bubble may cover the afterbody before a considerable amount of high pressure air is trapped within the bubble. The question then arises : what kind of flow will be established in this case? The alternatives are two :a steady separated flow or an unsteady flow of the oscil-lation mode. The dominant parameter which determines the type of flow to be established for a specific concave body is the shape of the shoulder of the afterbody. Experimental evidence and a theoretical justification for this conclusion will be presented later in this paragraph.

The experimental information presented in paragraph 2.1 and 2.2 support the view that the oscillation mode appears when :

18

-a. the mean inclination angle of the afterbody is greater than the detachment angle;

b. the length of the forebody is less than a critical value; c. the shoulder of the afterbody is sharp (small or zero radius of curvature).

The above special geometric characteristics have a significant effect on the development of the separated flow: if the flow were steady, both the separation and reattachment point would be fixed (tip separation and shoulder reattachment). Neither the free interaction formula, nor Chapmanls reattachment relation can provide information on the extent of the separation region. The shear layer would begin at the tip of the forebody and it would reattach on the sharp edge of the afterbody.

To prove the above statement, we start by examlnlng the reattachment of the shear layer. The fact that the reattach-ment mechanism cannot occur with face reattachreattach-ment, when the mean angle of the afterbody is large, is evident from the ex-perimental diagram of Wood (Fig. 1). This conclusion also follows from Chapman's reattachment relation. Indeed, we see in Fig. 7 th at whatever is the value of the empirical parameter N, the values of the reattachment angle 0, for steady flow, are small, the maximum value being 20°. The reattachment point therefore must move outwards until the shoulder of the body is reached.

Assuming the establishment of a shoulder reattachment we proceed to study the effect of the length of the forebody on the separation mechanism. For small values of the radius of cur-vature of the shoulder, the reattachment point is fixed; thus, the position of the separation point on the surface of the fore-body is defined only by the free interaction relation. No coupling between the separation and reattachment mechanism is required. The geometric inclination 0g of the shear layer (Fig. 4) has simply to be equal to the value of the free interaction angle af corresponding to the pressure coefficient cp (equation 3). In practice an iteration is required in order to find the separaticr point on the forebody for a given reattachment point. However,

19

-if the length of the spike is short, it is possible th at an equi-librium point will not exist, i.e., of will be less than 0g' Hence, the separation point must coincide with the tip of the forebody.

An examination of the experimental data concerning the unsteady flow about spiked cylinders reveals that the range of the spike for which unsteadiness is observed is such that tDe flow, if it were steady, would be of the tip separation type. This is evident in Fig. 11, where the minimum spike length for the establishment of a free interaction separation is compared with the experimental values of the minimum spike length for the stabilization of the flow, for various Mach numbers. Additional evidence is provided in plate 3a where the Mach 6 laminar flow about a spiked cylinder with spike length equal to the critical value for the stabilization of the flow is shown. It is observed that in this case separation starts from the conical surface of the spike tip.

The determination of the dead air pressure (Pd) solely by the free stream conditions and the inclination of the conical surface which envelopes the wncave part of the body is one signi-ficant result of the combined tip separation and shoulder reat-tachment. Neither the free interaction equation nor the reattach-ment relation participate in the determination of the flow con-ditions in the separation region. This observation which applies to bodies with a shoulder of sufficiently small radius has been confirmed by a sensitivity analysis of the reattachment relation (eq. 4). We have estimated that :

d

a) for a value of the radius of curvature p =

2

the de ad air pressure is affected by the position of the reattachment point on the curved surface of the shoulder;

b) for values p

~

{ the dead air pressure does not depend on

whether the shear layer reattaches tangentially or with a reat-tachment angle determined by Chapman's reatreat-tachment criterion.

Thus, if the radius of the shoulder is small, it is not the value of the downstream pressure which dictates the posi-tion of the reattachment point and the dead air pressure; the

20

-dominant factor is the latter pressure which is defined by other parameters. The positton of the reattachment point must ~ defined in such a way that neither air from the de ad air r~on will escape downstream (reattachment point very near the edge of the body), nor air from the external part of the shear layer will turn in-wards .at the reattachment region (reattachment point far from the edge of the body). The achievement of equilibrium is assisted by the fact that in the shoulder reattachment, though the dividing streamline must always meet the face of the body, the outer part of the shear layer passes outside the shoulder (Fig. 12). By this mechanism, the deflection of the external stream and the associated reattachment pressure rise, as Wood (Ref. 10) first observed, can be considerablY less than the values required to turn the flow parallel to the face of the body.

However, we know from experience that if the reattachment is of the shoulder type, the separation is not always steady.

Furthermore, the observed instability (oscillation) is charac-terized by a periodic inflation of the lower part of the conical separation region (plate 2d). This is an unquestionable indication that though in the shoulder reattachment the reattachment pressure rise is lower than the one existing if all the shear layer meets the face of the body, still it is higher than the stagnation pressure along the dividing streamline.

An examination of the shoulder reattachment model (Fig. 12) reveals that the deflection of the external stream depends on the radius of curvature of the shoulder and on the velocity gradient of the outer part of the shear layer. A greater deflec-tion (for similar shear layers) occurs wh en the shoulder is sharp.

If the shoulder is rounded the outer part of the shear layer is subjected to a smaller deflection and an equilibrium of the pressure may be reached. Obviously, the sealing of the proper radius of curvature, for the stabilization of the flow, is the thickness of the external part of the shear layer and not the diameter of the body. This explains the fact that a flow which otherwise would be unsteady is stabilized by a very small

21

-rounding of the shoulder (numerical details in paragraph 4.4). As far as the velocity gradient of the outer part of the shear layer is concerned we can see,from Fig. 12, that a lower value of the gradient means a lower velocity of the layers adjacent to the dividing streamline and consequently a lower reattachment pressure rise. The velocity gradient depends, ac-cording to Chapman's analysis, on the length of the shear layer. More specifically, a longer shear layer has a lower velocity gradient (~~), because the thickness of the shear layer grows parabolically with increasing length, while its velocity profile remains similar at different streamwise stations. The dependence of the velocity gradient on the length of the ~ear layer gives an explanation to the experimental observation that the oscilla-tion is established when the length of the shear layer (or of the forebody) is small and it ceases whèn this length increases.

The significance of the thickness of the shear layer is evident in plates 3a-b. The flow conditions for the photos were the same; only the spike length was different. We observe that while in the case of the longer spike a part of the shear layer passes outside the shoulder, in the case of the short spike almost all the shear layer meets the surface of the shoulder. With a further small reduction of the spike the flow was observed to oscillate.

A special test which we have performed shows that the factor which affects the establishment of the oscillation mode is the length of the shear layer (compared to the reference length of the body) and not the establishment of the tip separation.

The model for this special test was the conic body of Fig.13. The flow conditions were Moo = 6, Rea = 2·104/cm. Application of

the fr.ee interaction procedure gave as separation point that for which Xs/a

=

0.29 (the symbols are defined in Fig. 13). Thus, the

length of the forebody is much greater than the one required for the occurrence of tip separation and, consequently, if the dominant factor for the development of the flow was the tip separation,

22

-the flow was not steady, but unsteady of -the oscillation mode (plates 3f-g).

To conclude, the sharpness of the shoulder of the afterbody in conjunction with high values of the velocity gra-dient of the shear layer seem to be necessary conditions for the occurrence of the oscillation mode of instability. Both these parameters affect the stability of the flow adversely, by

in-creasing the reattachment pressure to a level greater than the total pressure along the dividing streamline. Thus, the external part of the shear layer turns inwards and feeds the separation bubble with high pressure air, instead of flowing downstream. This high pressure air is in imbalance with the environ~ent

and an inflation of the bubble occurs. The excess air escapes downstream and the shear layer takes again the initial position. In this way, the instability cycle is repeated.

23

-4. VERIFICATION TESTS

4.1 Test facility models

A verification program was conducted in the von ~rman

Institute hypersonic tunnel H-3. The tunnel is of the blowdown type with an axisymmetric contoured Mach 6 nozzle giving a free jet of 12 cm diameter, followed by an adjustable diffuser. A supersonic ejector is used to provide the necessary suction, downstream of the diffuser.

The tunnel is supplied with air at 40 bar and the air is heated with a pebble-bed heater to stagnation temperatures of approximately 200°C. More data for the H-3 are contained in Ref. 24.

The models studied were :

a. flat ended cylinders of diameter d

=

46 mm and d

=

30 mm, equipped with spikes of variable tip semi-cone angle (at

=

7.5° to 90°) and of diameter dl = 3 mm;

b. spiked flat ended cylinders of d

=

30 mm with rounded shoulders of radius

%

=

0.125 and

%

=

0.25. The spikes had a diameter dl

=

3 mm and were either flat ended or has a semi cone angle at

=

7 . 5 ° ;

c. concave conic bodies (description in § 4.4).

The Reynolds number based on the diameter of the bodies Red' was varied from 3.105 _{to 9.105 • The frequency of the shock}

induced instabilities was measured only for the flat ended spiked cylinders of d = 46 mm and the value f = 3 KHz represents the mean value. All the tests were performed at zero incidence.

The flat cylinders were equipped with a flush mounted Kistler 601A piezo-electric transducer, the output of which was fed to a Kistler charge amplifier; type 568Y1 and the signal was shown on an oscilloscope, type Tektronix 502A.

24

-A schlieren system was used for the optical observation. For the photographs, a constant light source or a spark source, giving sparks of about 0.2 ~sec, were ~ed.

4.2 Spiked flat ended cylinders

In this section we shall follow a series of schlieren photographs (plate 1) which give the main points of a typical

pulsation mode of instability, in order to illustrate the proposed physical explanation. The large cylinder (d = 46 mm) was used as

a model for these experiments. The spike length parameter is equal to K

=

1.4 and the frequency of the instability, which was meas-ured from the variation of the pressure, is equal to f

=

3.0

KHz.

The corresponding Strouhal number, based on the diameter of the cylinder, is :

Str

=

~.d

=

0.15

00

The time between the photog~~s on plate 1 is not equal.

A good description of the flow development with time is included in Ref. 17.

In plate la the flow field is shown at what we have termed the beginning of the cycle of instability. The foreshock emanates from the tip of the spike and its shape is not affected by the presence of the separation bubble. The supersonic jet has been formed and, as it impinges on the surface of the body, a part of it is directed inwards and fills the separation region, while another flo~s outwards and expands downstr.eam at the shoulder of the cylinder. The strong aftershock is nearly normal at the

shock intersection point and is gradually turned downstream.

In plate 1b, the separation point has reached the shoul-der of the spike, the inflated separation bubble nearly touches the foreshock, the supersonic jet is more visible and the shock intersection point has moved a little outwards.

25

-In plate Ic, the inflating region enveloped by the foreshock has taken an irregular cylindrical form with a normal

forward part, and the shock intersec~ion point has moved out of

the projection of the bady. With a further movement of the shock intersection point, the high pressure inflated region communicates with the downstream low pressure region and the expansion begins. The lack of equilibrium with the environment results in the in-stantaneous radial expansion of the high pressure air and the foreshock takes a nearly hemispherical shape, while the af ter-shock is pushed downstream.

The expansion results in the decrease of pressure behind the spherical shock and so the reestablishment of the flow begins

from the tip of the spike. The front of the spherical shock is

pushed downstream and the conical shock appears. In plates Id-e two events of this reestablishment are shown.

In plate Id, the shock front has started to approach the surface of the body, leaving the spike exposed to the free stream; hence, the conical shock appears. The non existence of a separation bubble is the most significant feature of this photograph. Thus, the shock envelope at the end of the reestablishment phase takes a shape similar to the one which it would have if the flow had started impulsively. This observation is the key for the explana-tion of the mechanism of the pulsaexplana-tion.

In plate Ie, the front is nearer to the surface of the body and the air behind it expands downstream at the shoulder of the body. The triangular form at the base of the spike is the separation bubble which has started to be formed. In plate lf, the spherical front widens, the new strong shock appears and the separation point has moved up to the cylindrical part of the spike. The channeling of the flow between the foreshock and the separation bubble starts. This channeling leads to the appearance

of the supersonic jet, which in turn leads to the repetition of

I

26

-An interesting feature of the photographs is the observed explosive character of the filling of the separation region. In both plates lb and Ic, it is difficult to distinguish between the separation surface and the shock. There is the impression of a coalescence of the two fronts. This illustrates the lack of sufficient time for the establishment of communication between the inflating separation bubble and the accompanying shock along the characteristic lines.

4.2.2 Further evidence for the existence

Af ter the optical detection of the supersonic jet, which appears at the pulsation mode of instability, we performed sublimation tests and in plate 2a, the surface of the cylinder

is shown af ter a test under conditions similar to those of plate 1. The correspondence of the dark outer ring to the surface projection of the supersonic jet in plate lb can easily be veri-fied. The inner dark ring, most probably corresponds to a second-ary peak within the inward flowing stream (region KL, Fig. 14, taken from Ref. 21).

A similar sublimation te~ had been previously performed by Loll (Ref. 16). He attributed the dark ring to the reattachment of the separated shear layer or to be impingement of the shear layer created by the intersection of the two shocks.

The dynamic characteristics of the pulsating flow were also detected by the pressure measurements. The measured amplitude of the pressure fluctuation at the face of the afterbody at

i

=

0.25 versus the spike length is shown in Fig. 15. The flow conditions were M~ = 6, Pt = 20 ata. We observe that the am~li­

tude of the fluctuation is independent of spike length for the pulsation mode and equal to óp

=

1 atm. If we normalize this value by the pressure P3 behind the strong shock AB, we find :

~

=

2.0 P3

27

-This simply shows that the fluctuation of the pressure in the separation region is greater than the pressure behind the strong shock and consequently, the air which fills the dead air region cannot have come from the adjacent subsonic region, but actually comes through the conical shock wave and is compressed at the impingement region. In Fig. 15 we notice a marked change in the magnitude of the pressure fluctuation at the critical point

(K

=

1.4). In the case of oscillation the pressure fluctuation at the critical point is :

%;

= 0.56

and it confirms the argument of mass addition from the reattach-ment region and not from the annular supersonic jet.

4.2.3 lffe~t_oi !h~ ~o~i!i~n_oi the

~h~c~ in!e!s~c!i~n_p~i~t

The fundamental element of our analysis is the influence of the shock discontinuity point on the behaviour of the flow.

For spiked bodies, the initial position of the conical shock is affected by the shape of the spike tip and the diameter of the spike.

We used spikes with tips of conical semi-angle at

=

7.5°, 15°, 30°, 45°, 60°, 90°, and of constant diameter

dl 1

(cr

=

12)

and testing step by step we measured the spike length for which the pulsation turns into oscillation. The results are plotted in Fig. 16 and confirm our hypothesis of the importance of the initial position of the shock intersection point in

determining the type of the flow.

Also important is the experimentally observed nearly constant initial value (plate la) of the conical shock angle,

- 28

-during a pulsation cycle, far from the tip of the spike. For spike lengths K

=

1.0 ~ 1.6 (range of pulsation) and variable spike tip angle, this shock angle was found to be equal to 14° ~ 16°, with a mean value of e = 15°. This mean value is a little smaller than

the given one for hemispherical cylinders at M

=

6.06 (Ref. 25).

The effect of the volume of the separation region on the amplitude of the oscillation was studied by varying the length of the spike. When fue separation volume is small, the amplitude is great (photograph d, plate 2). If the volume is increased, the amplitude is reduced and the shape changes periodically from concave to convex {photogra~hs are contain~d in Ref. 16).

Interesting is the fact that af ter the critical length for the suppression of the oscillation is reached, an unsteadiness of the laminar flow is observed locally at the reattachment region. The unsteadiness consists of the periodic oscillation of the

reattaching part of the shear layer and of the accompanying shock wave. The amplitude of the oscillation is not great and it appears even for spike lengths securing the establishment of a free inter-action type of separation (photographs b-c, plate 3). This local unsteadiness is also observed in the flow about spiked cylinders having a rounded shoulder of small radius (plates 3d-e). It is obvious that the forcing mechanism of this unsteadiness is again the non extstence of pressure equilibrium at the reattachment region, but the pressure differences in this case are not strong enough to induce an oscillation of the whole shear layer.

Another parameter which affects the behaviour of the oscillating flow is the foreshock. If this shock envelopes the afterbody, the amplitude of the oscillation is smaller and the flow stabilizes for a shorter spike length. Thus, while for a

conically tipped spike of angle at

=

7.5°, the flow becomes steady 1

for a spike length equal to d

=

2, for a flat ended spike the 1 >

29

-A sublimation test was also performed during an oscilla-tion mode. In plate 2b, the surface of the cylinder is shown af ter a test under conditions similar to those of plates 2c-d. It is observed that the surface of the cylinder is dark only at the shoulder. Thus, heat peaks appear only at the region where the oscillating shear layer reattaches periodically.

If the separation flow is transitional, the air is more energetic and it possibly may influence the development of the flow. More specifically, the range of spike length for the occur-rence of the pulsation mode of instability may be restricted.

We tested the influence of the transitional character of the flow by covering the tip of the spike with sand or metallic particles. This caused a deformation of the initially conical

tip and for this reason we also tested spikes with a hemispheri-cally rounded nose. Results

a) both the rounded and the rough spikes eliminated the pulsating character of instability;

b) roughness was more effective in reducing the pressure fluctua-tion. Specifically, the amplitude of the pressure fluctuation for the rough model is reduced to 20% of the value without roughness, while for the rounded nose models the reduction was to a value of approximately 35% of the value without the rounded nose. In both cases, the spike length was equal to K

=

1.1;

c) the frequency of the instability was not affected by the pre-sence of the roughness or by.the rounding of the spike tip; d) roughness elements in the dead air region did not affect the pressure fluctuation;

e) smooth conical-tipped spikes with annular grooves at the tip were ineffective in influencing the pulsation. This creates some questions about the cause of the flow change in the case of rough spikes. Perhaps the deformation (rounding) of the spike tip may cause the improvement because of the movement outwards of the shock intersection point.

30

-But a study of the photographs shows that in the case of roughness the limits of the shock waves are two straight lines slightly curved, while the flow about the rounded nose spike is similar to the one observed in the case of oscillation, and it indicates the small resistance of the laminar shear layer.

Also Stainback, using a 5° half-cone at Moo

=

8, observed that the (attached) flow became easily transitional when a single row of spheres was used, while the annular grooves and screw

threads were completely ineffective over the Reynolds number range of his tests.

Finally, Kenworthy and Richards (Ref. 19), testing concave conic bodies at the VKI Longshot tunnel at M

=

20 ob-served that for the rough nose the pulsating flow obob-served at the peak pressure (Reynolds number) was transformed first into an oscillating flow as the pressure decreased and then, as the pressure diminished further, it turned again into a pulsating flow (in the Longshot the flow conditions decay monotonically during 10 to 20 milliseconds running times).

This observed Reynolds number influence is in agreement with the flow description given in Chapter 3. If the flow is

turbulent/transitional, the incoming supersonic stream meets resistance and depending on the initial position of the shock intersection point relative to the shoulder, the downstream

expansion may begin before the trapped air in the conical separa-tion region has forced a radial expansion.

The temperature of the bodyaffects the flow field in the opposite way, because of the lower density of the air near the surface. That was checked by using the spike with at

=

7.5°. For Tw/T_r = 1, K for transition was equal to 1.6, but for

TwLT r = 0.90, K was reduced to 1.4.

Experimentally we found that the transition from pulsa-tion to oscillapulsa-tion occurs, for M = 6, when the extrapolation

31

-of the conical foreshock (initial position) intersects a point on the cylinder whose radial distance from the center of the body is

r

~

=

0.85 ~ 0.70

The smaller values correspond to the case Tw/T r = 1. The above values may be dependent on Mach number.

4.3 Spiked cylinders with rounded shoulders

Maull (Ref. 8) has presented data for this family of bodies. We performed a series of tests, on one hand bacause Maull did not mention the oscillation mode of instability, and on the other hand, to study the influence of the initial position of the shock discontinuity point.

The fact that Maull did not report the oscillation mode was favorable for our analysis, but ambiguity arose about the

reason for this omission, as in the same report, he includes tests at Mach 6.8 with spiked flat ended cylinders for which oscilla-tion has been observed to occur.

The results of our tests were completely in agreement with the predicted behaviour of the flow about those bodies

(chapter 3); i.e. :

a) as long as the initial shock intersection point lies above the flat part of the forebody, the flow pulsates;

b) when the shock intersection point approaches the shoulder of the body, the flow, instead of becoming oscillatory, is stabilized; c) varying the initial position of the shock intersection point by testing flat ended spikes or spikes with a semi cone angle of at

=

7.5°, was veryrevealing. The results (plus the spiked flat ended cylinder) are shown in Fig. 17. It is observed that for a radius of curvature

%

= 0.25 and a flat spike, no instability occurs.

32 -4.4 Concave conic bodies

A series of concave bodies having a form as that shown in plates 2e-f-g, were tested as follows :

a) influence of the existence of the forebody in the unsteady character of the flow. Possibly the limitation of the dead air region, because of the existence of the forebody (Fig. 18), would eliminate the pulsation mode, as the shock intersection point would reach the limit of the sheulder before the trapped.air in the coni-cal region becomes excessive. The tests have shown that there is no influence; the phenomenon develops as in a spiked flat ended cylinder. Critical is only the position of the shock intersection point (initial) relative to the edge of the shoulder, and the slope of the afterbody.

b) stabilization of flow by optimization of the shape of the afterbody. By varying the angle of the afterbody and the

curva-p 1

ture o·f its shoulder, it was found that for Ot

=

70° and ëf

=

TG

the separated flow was steady and independent of the shape of the forebody andjor of the initial position of the shock discontinuity

33

-5. CONCLUSIONS

1. Two distinct modes of instability have been observed in the ~'gh speed flow about a concave body. They have been termed "pul-sation" and "oscillation". By measuring the pressure fluctuation of the unsteady flow and performing sublimation tests we have shown that the governing mechanism of these two instabilities

is quite different.

2. The flow conditions prevailing about a concave body, at the starting phase of each cycle of pulsation, resemble an impulsive flow, in which the shock envelope retains a position corresponding to an inviscid flow field. The impulsive nature of the flow field is due to the violent radial expansion of the separation bubble which occurs during the final phase of each cycle of pulsation.

3. An examination of the impulsive flow about a concave body

indicates that the forting mechanism for the appearance of pulsa-tion is an annular supersonic jet which, under certain condipulsa-tions, is formed at the intersection point between the conical foreshock, which envelopes the nose of the body, and the strong aftershock, which appears in front of the blunt afterbody. This supersonic jet feeds large high pressure mass flows into the conical separa-tion bubble, causing it to explode' and th en collapse in a repe-titive manner.

4. The sharpness of the shoulder of the afterbody in conjunction with high values of the velocity gradient of the shear layer

seem to be the reasons for the occurrence of the oscillation mode of instability. Both parameters affect the stability of the flow adversely by increasing the reattachment pressure to a level greater than the total pressure along the dividing streamline.

5. According to the proposed mechanism, for the appearance of

the pulsation mode of instability, the following conditions should be simultaneously satisfied :

a. the mean inclination angle of the afterbody must be greater than a critical value. This value seems to