boundary layer interactions

Pergamon

0376-0421(95)00005-4

Prog. AerospaceScL Vol. 32, pp. 173-244, 1996 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0376-0421/96 $29.00

REVIEW OF THE PHYSICS OF SWEPT-SHOCK/BOUNDARY LAYER INTERACTIONS Argyris G. Panaras HAF Academy, P.O. Box 64053, Athens 15710, Greece

Abstract--The interaction of a swept-shock wave with a boundary layer appears in those regions of a highspeed vehicle in which two surfaces intersect. Simple configurations have been studied which resemble these regions. The simplest and most studied configurations are: the fin/plate, the wedge/wedge (or axial corner), the wedge/plate {or intake-type) and the swept compression corner. In these configurations, the conditions near the plates or the wedges are similar: a swept-shock impinges on a two-dimensional boundary layer. If the shock is sufficiently strong, the boundary layer separates and the topology of the flow changes significantly. Peak heating and high values of pressure have been measured in the region of intersection of the two surface:~. Early oil-flow visualizations have revealed the existence on the surface, below the separation bubble, of a separation and a reattachment line, which away from the apex of the configuration are straight and intersect upstream of the apex and close to it. The trace of the inviscid shock also passes through this intersection. It has been proposed that the separation bubble is actually a conical flat vortex. By this assumption, the peak heat transfer is explained, because the vortex carries to its reattachment region high energy air from the external flow. More than 20 years of experimental and computational research were required for proving these early hypotheses. Today it is believed that the flow is quasi-conical. The major features of the flow field, like the shock structure and the shape of the conical vortex have been visualized efficiently, both experimentally and computationally. The gradual progress in the understanding of the nature of the flow field is reviewed, and issues that require further research are identified.

CONTENTS 1. OBJECT OF THE REVIEW 2. EXPERIMENTAL STUDIES 2.1. Early eXl:¢rimental results 2.2. The phenomenon of separation in three-dimensional flows 2.3. Appearance of the first flow models 2.4. Period of improvement of the flow models 2.4.1. Investigations of the flow structure 2.4.2. Scaling laws 3. NUMERICAL SIMULATIONS 3.1. Description of the first computational studies 3.2. Period of refinement and validation of the numerical methods 4. RECENT PROGRESS 4.1. Visualization of the flow field, conical similarity 4.1.1. Experimental studies 4.1.2. Computational results 4.2. Secondary separation in a fin/plate configuration 4.3. Multiple :~eparation 5. SWEPT COMPRESSION CORNERS AND CROSSING SHOCKS 5.1. Swept compression corners 5.2. Crossing :shocks 6. CONTROL OF SWEPT-SHOCK/BOUNDARY LAYER INTERACTION 7. CONCLUDING REMARKS REFERENCES APPENDIX The elements of a numerical simulation Equations used for the numerical simulation of the reviewed flows Numerical schemes Dissipation and dispersion errors Computational domains and boundary conditions Turbulence models applied to swept-shock/boundary layer interactions Algebraic models Two-equations models 173

174 177 177 183 186 191 191 196 199 199 2O4 213 213 213 215 22O 222 225 225 227 230 231 232 235 235 235 236 238 239 241 242 243

174

A.G. Panaras 1. O B J E C T O F T H E REVIEW

Swept-shock boundary layer interactions occur in the axial supersonic or hypersonic flow between two intersecting surfaces as well as in the transonic flow over a sweptback wing. Many investigations of the phenomenon, both experimentally and more recently, computationally, have been undertaken in the last 30 years. Most of these research studies have involved configurations that can be classified as having either one or two surfaces producing a compression in the flow field. The simplest example of the former type of configuration consists of a sharp fin (or wedge) attached normally to a flat plate at a certain distance behind its leading edge as shown in Fig. la. The swept compression corner (Fig. lb) is another simple example, while if the wedge on the plate is replaced by any three-dimensional object such as a dihedron or a half circular cone (Degrez) ~13~ that is capable of generating a conical shock, this can be regarded as a generalized geometry of the first type. With all these configurations, the oblique shock wave generated by the compression surface intersects the boundary layer on the plate and, as a result, the increase in pressure through the shock is smeared out on the plate and a disturbed flow pattern is observed for a considerable distance both upstream and downstream of the shock position predicted for inviscid flow. If the shock is strong enough to cause the boundary layer to separate, the topology of the flow changes significantly. The intensity of the interaction depends on the flow conditions (Mach number, flow angle), on the geometry of the fin (swept or not) and on whether the boundary layer is laminar or turbulent. If the distance of the wedge from the leading edge of the flat plate is zero (Fig. lc), the configuration is still considered as belonging to those with only one surface providing compression, although there is a weak shock which appears around the leading edge of the plate due to the displacement effect of its boundary layer, and this can interact with the swept-shock. This configuration is sometimes called an intake-type configuration. In the present review, it will be called a 'wedge/plate' configuration, while the geometry of Fig. la will be referred to as 'fin/plate' configuration. The corner formed by the intersection of two wedges (Fig. ld, Hummel, 1989)~34~is the basic configuration of the flows with two surfaces providing compression. This configuration is usually called an 'axial corner'. Since shock waves appear on both surfaces, the flow is characterized by a shock-shock interaction, in addition to the shock/boundary layer interactions which occur on both surfaces of the corner. In particular, the shock waves that are generated at the leading edges of the wedges interact in the corner region and between them a corner shock is formed (Fig. ld). At each point of the corner shock intersection with the wedge shock waves, an additional shock (called embedded) and a slip surface appear and pass toward the wedge surface and the corner, respectively. On the surface of each wedge, the embedded shock interacts with the boundary layer. The intensity of the interaction depends, in addition to the flow conditions, on the value of the wedge angle, the existence or not of sweep and on the angle of the corner. If the wedges are symmetric, the flow field is also symmetric. Asymmetric flow occurs in cases where the wedges have different sweep or wedge angle. Recently, interest has been shown in an extension of the single fin geometry, consisting of two fins or wedges attached normally to a flat plate (Fig. 2). This is called a 'crossing shock' configuration. All the forementioned configurations are simplified shapes of various elements of high speed vehicles (junctions between wing/body or fin/body, intake ducts of engines, and so on). Following other reviewers of the topic (Korkegi (1971),~47~ Hummel (1989)~34~), we demonstrate in Fig. 3, a schematic high speed vehicle on which the possible regions of occurrence of swept-shock/boundary layer interactions are denoted. If these interactions are strong, undesirable effects such as local high heat transfer rates and static pressures can be present. In both types of interactions, with one or with two compression surfaces, the conditions close to the surface are similar: a swept-shock impinges on a boundary layer developing along a flat surface. Thus, we would expect that the major features of the viscous-inviscid interaction part of the flow field are the same in both configurations. This fact was

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considered as obvious during the early years of study of the swept-shock/boundary layer interaction. For example, Korkegi (1976) t4a) analyzed the experimental results of the flow in a wedge/plate model and assumed that there is no distinction between this model and an axial corner contiguration regarding the structure of the viscous flow. Later, the research in each configuration developed independently. Numerous studies of the swept-shock/boundary layer interactions, most of them experimental, have been performed over the last 30 years. The experimental techniques offer reliable results on a bounding surface or on a section of a flow. A numerical solution has the advantage that it provides the values of the various flow parameters in the flow field (more particularly at the grid points of the computational domain); then these data can be transformed into images of the flow field by applying post-processing techniques. The

A. G. Panaras

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obtained images may efficiently describe the flow field and possibly give an integrated view of the phenomena which occur in complex flow conditions. Regarding the flow conditions, in the fin/plate configuration, emphasis has been given to supersonic speeds and turbulent boundary layers. In the case of the axial corner, though there are early experimental results on flows with turbulent supersonic conditions, the majority of the studies are related to laminar hypersonic flows. Classic references on the subject include the review of corner flows prepared by Korkegi (1971)(47) and the brief, but precise information on the swept-shock/boundary layer interactions included in the comprehensive review of three-dimensional interactions and vortical flows of Peake and Tobak (1980).(75) These early reviews considered both the one and the two compression surfaces configurations. One recent review by Hummel (1989)(34) has as its objective the investigation of the flow about the axial corner configuration, while the structure of the flow in configurations with one compression surface is described in detail in the very recent reviews of Settles (1993) (83) and Knight (1993).(43) The laminar sweptshock/boundary layer interactions are reviewed by Degrez (1993).(~3) In the present review, the physics of the swept-shock/boundary layer interactions will be examined in a unified approach for the configurations that are shown in Fig. 1, so that the common features of their flow fields will be emphasized. A preliminary short version of the present review has appeared in a more general review of the shock/boundary layer interactions prepared by Delery and Panaras (1994).(~6) For completeness, a brief description of the flow features of the crossing shock configuration will also be given. We will review the most critical features of the flows under consideration. In general, the interaction domain in these types of flows is quasi-conical, i.e. it grows almost linearly in the

Swept-shock/boundary layer interactions

177

downstream direction. Various manifestations of the conical nature of the flow have been observed experimentally, e.g. in the footprint of the interaction on the surfaces of a comer or on the plate of a fin/plz,te configuration, even in the early period of research when only limited experimental techniques were available. Nevertheless, recently, strong evidence was provided for the existence of quasi-conical vortices in swept-shock/boundary layer interactions. As regards the organization of the review, experimental results are presented related to the physics of the flow, from the early sixties till the late eighties in Section 2. The third section reviews the numerical simulations produced in the same period. In the fourth section, the most recent experimental and computational developments are reviewed. The swept compression corner and the crossing shock configuration are examined in Section 5. In the Appendix, the elements of a numerical simulation are described: the equations and the various approximations, the most important numerical techniques, the domains of calculation, the boundary conditions and finally, the turbulence models.

2. EXPERIMENTAL STUDIES 2.1. EARLY EXPERIMENTAL RESULTS The works of :~tainback (1960),(91) on a plate/plate configuration (M = 8, laminar) and of Stalker (1960),c9°) on sweptback configurations (M = 3, turbulent), were the first attempts to study the phenomenon of swept-shock/boundary layer interaction. Stainback measured the pressures and the heat transfer rate in the corner region of the intersecting plates and he found a considerable increase of both parameters, compared to flat plate data. Stalker concluded that the peak pressure rise at separation, the upstream influence ahead of separation and the pressure rise at reattachment for moderate sweep angles can be understood by simple examples of two-dimensional theory, but the data cannot be correlated in this way if the sweep angle is large. The first syst,~matic study on the topic was published by Charwat and Redekeopp (1967), (11) who performed extensive measurements in symmetric and asymmetric axial corners in the Mach number range from 2.5 to 4. The flow was laminar. Their experimental results included pitot pressure surveys, surface-flow visualization and surface-pressure measurements. Their measurements enabled them to identify the shock system which is established in t]aese types of flows. Their results are shown schematically in Fig. 4a. Charwat and Redekeopp (1967)(11~were the first to observe that due to the interaction in the corner region of the shock waves, which are generated at the leading edges of the wedges, the so-c~dled 'corner shock' appears. Shock patterns similar to that of Charwat and Redekeopp (1967)(11) were presented later by Cresci et al. (1969)(1°) for the M = 11.2 flow about an axial corner and by Watson and Weinstein (1971) (97) for a M = 20 flow in a similar geometry. At this point, note that although the outer border of zone III is marked by a line (Fig. 4a), it is termed an 'outer' compression. Actually in the description of the wave structure, Charwat and Redekeopp (1967)cl 1) describe this region as a compression fan. Thus, if the outer compression line is not considered as a shock wave, and if Fig. 4a is compared with the models of Hummel (1989)(34) and of Watson and Weinstein (1971)(97) shown schematically in Figs ld and 6, respectively, we see that the shock formation of Charwat and Redekeopp represents the flow structure in the inviscid part of an axial corner flow quite accurately. In addition to the definition of the shock structure, Charwat and Redekeopp (1967)(1j~ have addressed the issue of the conical character of the flow in the central sector of the flow, which is formed by the comer shock and the shear layers (zone 1). For that purpose, they presented the results of their pitot pressure crossflow measurements at several streamwise locations in conical coordinates (Fig. 4b). They have found that the flow appears not to be conical close to the vertex, upstream of the station x = 0.3 in, where the value of the hypersonic viscous interaction parameter (M3/Re I/2) becomes larger than 0.1. Downstream

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surface-flow visualization, and took electron-beam photographs of cross-sections of the flow. Measured surface pressure and heat transfer distributions at various wedge angles are shown in Fig. 5. The data have been normalized by calculated viscous induced pressures for sharp plates and wedges, and by measured wedge heating values. The data are plotted in conical coordinates (z/x) and correspond to various streamwise stations. The collapse of the data is quite good, indicating the nearly conical nature of the flow: Regarding the conical character of the flow, Watson and Weinstein (1971) ¢97) have, in addition, performed calculations which indicated that close to the leading edge, where the surface-flow patterns are highly non-cc,nical, the hypersonic parameter Ma(C/Re 1/2) is greater than 4, while when

180

A . G . Panaras Approximate wedge b( layer thickness Outer disturbance (undetected)

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this parameter becomes smaller than 4 (moving downstream), the patterns are nearly conical. Watson and Weinstein (1971) t971 note that pressures approximately 4.2 and 3.2 times local wedge pressure were measured in the 10° and 5° wedge angle cases, respectively, in the immediate corner region. The corresponding heat values are 10 and approximately 7 times the local undisturbed wedge value. Watson and Weinstein (1971) t97~ observe that near the location of high peak heating, the pitot surveys of Charwat and Redekeopp (1967)tl 1~contain a small region of low pitot pressure. They suggest that this region of low pressure represents the core of a vortex, which is also the cause of the high peak heating found in other corner flows. Watson and Weinstein (1971) t97) present more arguments for the existence of corner vortices. They note that in their electron-beam pictures, in addition to the good visualization of the shock structure, visible are what appear to be single vortices on the surface of the wedges for the 10° and 5° cases. Furthermore, in their oil-flow photographs, a distinct pattern of S-shaped streak lines appear for the 10° case, which "is thought to indicate the presence of vortices". They observed that outside the S-shaped lines, a strong crossflow region exists, terminating in an oil accumulation line, beyond which the flow gradually approaches two-dimensional undisturbed wedge flow. It is clear that the description given by Watson and Weinstein (1971) ~97) corresponds very well to the phenomenon of threedimensional separation, according to which a linearly growing vortex exists between the separation (accumulation) and reattachment lines (where the S-shaped lines terminate). In the late sixties, the knowledge in the field of topology of three-dimensional separations was reaching a state of maturity. In 1970, when Watson and Weinstein were performing their measurements, Green ~25) published a very good schematic presentation of the flow in a separated swept-shock/turbulent boundary layer interaction (see Section 2.2). We will see in Section 2.3 that in the early seventies, more experimental evidence accumulated to support the theoretical models of separation, leading to the development of the first accurate flow models of the swept-shock/boundary layer interaction. Interpretation of their measurements led Watson and Weinstein (1971) (97) to construct the cross-section of the M = 20 flow field shown in Fig. 6. The shock structure found by these researchers is quite similar to the M---3.17 flow field derived by Charwat and Redekeopp (1967).tl~) According to Watson and Weinstein, a feature common to both models is the central zone, which at M = 3.17 is bounded by almost straight slip lines, whereas at M~= 20, the region is squeezed away from the corner by the large vortices. As regards the 2-shocks, which exist in the flow model of Watson and Weinstein, their outer

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legs coincide with a second lower peak heating. The authors mention that two counterrotating vortices could also produce the kind of surface pattern which exists in the vicinity of the second peak in heating, however, they finally propose the existence of a A-shock for explaining the local flow. The existence of separation vortices, was also supported by Korkegi in his 1971 review/47~ As an illustration he used the oil-flow pictures of Watson and Weinstein and the M = 20 pressure and heat transfer distributions of Bertram and Henderson (1969),tr~ Fig. 7, which are similar to those of Watson and Weinstein. Korkegi (1971)t'~7)observes that the pressure distributions exhibit the sharp rise and corner plateau associated with the inner shock of Charwat and Redekeopp's shock structure, following a smaller pressure rise over a broad span as reflected by zone III in Fig. 4a. The heat transfer distributions exhibit a marked dip or trough to values well below those for a two-dimensional wedge, followed by a sharp rise to two distinct peaks, the second of which is 10 times the wedge value for the M = 20 case. Korkegi (1971)c47) notes that the comer heat transfer distribution has distinct features in common with two-dimensional, shock-wave/laminar boundary layer interaction: a drop in heat transfer rates beyond separation (the trough) followed by a rise to high values at reattachment. The pressure distribution also exhibits common features: an initial rise to a plateau followed by a sharp peak. From these features of similarity coupled with the JPAS 32-2/3-H

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oil-flow pattern, Korkegi (1971) c47) suggests that the corner flow boundary layer separates at the oil accumulation of the experiments of Charwat and Redekeopp. He explains the drop of the heat transfer between the two peaks by the presence of a region of outflow from the surface. Then he postulates the possible existence of a system of three vortices, a weak outer one and two strong ones inboard. Finally, Korkegi notes that it is conceivable that much of the original boundary layer mass flow is scavenged and carried downstream by these vortices. West and Korkegi (1972)tgsJ studied experimentally the effect of Reynolds number on the size of the separation region of an axial comer flow. For this, they measured the pitot pressure in crossflow planes of a M = 3 flow (wedge angle 6 = 9.5 °) over a wide range of Reynolds numbers (0.4-60 x 106). Their measurements also included surface pressures and surface-flow visualization. For the various Reynolds numbers, they constructed the shock structures using the shock surveys. Comparing the results of these compositions, they found that for Reynolds numbers larger than 3 x 104, theflow structure, except very near the wedge surfaces, was conically invariant, from which they concluded that it represents the purely inviscid flow structure (turbulent regime). The flow structure and surface pressures varied significantly with Reynolds number for values much below 3 x 106. This variation is indicative of the dominant influence of laminar viscous-inviscid interaction and boundary layer displacement effects at low Reynolds numbers. The flow structure of a laminar case is compared in Fig. 8 to the invariant turbulent structure. It is noted that in the laminar flow case not only the shock structure moves outward due to viscous effects, but the oil accumulation line also lies much further outboard than in the turbulent regime. This feature ted West and Korkegi (1972)t98~ to argue that the disturbed region outboard of the embedded shock is entirely due to viscous effects and would not be present in an inviscid flow. Hence, it results from flow separation due to the interaction of the embedded shock wave with the boundary layer. As regards the spanwise surface pressure distributions, West and Korkegi (1972) t98~ observed that in the turbulent regime, they are remarkably similar and virtually identical within the accuracy of measurements when plotted in non-dimensional form. Furthermore, they observed that the spanwise surface pressure distributions for laminar and turbulent interactions (Fig. 9) have features identical with those found for

Swept-shock/boundary layer interactions

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two-dimensional separation due to compression ramps or shock wave/boundary layer interaction. Thus, the corner flow may be viewed locally as a two-dimensional shock wave/boundary layer interaction. Drawing an ~,rbitrary line, we may close the early period of research with the paper of West and Korkegi (1972).(gs) During this period, the shock structure established in an axial corner flow was described completely and strong evidence was provided that theflow away from the apex, where viscous effects are small, is nearly conical. In addition, various experimental indications and physical arguments were provided for the existence of one or more separation vortices in the corner region. It remained for future researchers to visualize these vortices and to correlate them with the surface-flow observations. The research in the direction was enhanced in the early seventies by parallel progress in the understanding of the phenomenon of three-dimensional separation. This topic is reviewed in the next section. 2.2. THE P H E N O M E N O N OF SEPARATION IN THREE-DIMENSIONAL FLOWS The existence of pressure gradients in the transverse direction as well as along the external streamline direction is the essential characteristic which distinguishes a threedimensional flow from a two-dimensional or axisymmetric one. The boundary layer responds to the transverse gradient by the development of a crossflow or secondary flow. Though the cro,,~sflow may be small near the outer edge of the boundary layer, it may be substantially close to the surface where the momentum deficit is large. Thus, the velocity vectors close to the surface can progressively rotate resulting in a 'skewing' of the boundary layer. When the distance from the surface tends to zero, the velocity vectors reach a limiting direction which is co-linear to the skin friction vectors. At the same time, the streamlines tend to a limit position which is also a trajectory of the skin friction lines. For this reason they are called 'limiting streamlines'. In general, the limiting streamlines in three-dimensional flows follow paths that are different in direction from the external streamlines. The concept of limiting streamlines was introduced by Sears (1948)(a2) in a paper discussing the laminar boundary layer on a yawed cylinder. According to Sears, limiting streamlines origJLnate at nodal points of attachment, and after circumscribing the body surface disappear into nodal points of separation. This topic was further developed by Maskell (1955) t6:t) who described the mechanism of separation in detail. He suggested that there were two main types of separated flow structure: a bubble type (Fig. 10a) and a free shear layer type (Fig. 10b). Fluid accumulating at a three-dimensional separation line, which in general is set obliquely to the direction of the external flow leaves the separation usually as a free-shear layer (Fig. 10b) and rolls up in the process of passing downstream.

184

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)~" ~'Limiting ¢' streamlines

Fig. 10. Three-dimensional separation (Maskell, 1955).161)(a) bubble; (b) free shear layer.

Maskell assumed that in both types of separation, a limiting streamline would join the separation streamline 'tangentially' or would have a 'cusp' on it. Limiting streamlines from opposite sides that join the separation streamline at a single point were hypothesized to merge and to leave the surface as single streamlines along a surface of separation. Lighthill (1963) ts3) further improved the concept of a separation line by replacing the limiting streamlines by the skin friction lines as the set of lines which asymptotically tend to a separation line (which is actually also a skin friction line)• MeCabe (1966) t62) was the first investigator to apply the concept of a separation line as an envelope of surface streamlines in the study of swept-shock/turbulent boundary layer interactions. His results are reproduced here from the paper of Kubota and Stollery (1982)t49) in Fig. 11. In this figure, McCabe's model incorporates all the expected vital features of a three-dimensional separated flow. Thus, according to this schematic figure, even for attached flow, the pressure gradient in the transverse direction deflects the boundary layer fluid through a larger angle than the more vigorous external flow (Fig. 11(1)). When the deflected surface flow becomes aligned with the inviscid shock, as shown in Fig. 11(2), incipient separation occurs. For fully separated flow McCabe sketched the surface-flow patterns shown in Fig. 11(3). Korkegi (1976) t4a) verified the incipient separation criterion of McCabe over the Mach number range, 2-6. Some objections about the validity of the separation criterion of McCabe were expressed by Oskam and coworkers (1975)36s) Examining their own oil-flow pictures and viscous layer surveys, they concluded that even when the surface-flow deflection exceeded the shock angle, the flow in the interaction region could be attached. They felt that the alignment of the surface flow with the inviscid shock is a necessary condition for the establishment of incipient separation, but is not a sufficient one. A complete schematic presentation of the flow in a swept-shock/turbulent boundary layer interaction, which in addition to the separation line shows how the skin friction lines emanate from the reattachment line was proposed by Green (1970).~2s)The results of a weak shock/boundary layer interaction are shown in Fig. 12a. In this case, although the skin friction lines are deflected substantially more than the inviscid streamlines, they do not

Swept-shock/boundary layer interactions

185

I. Attached flow D

Free-stream

~x,__ Shock

surface flow

~

External flow

2. Incipient separation ~J--Separation -~ ~ f line surface flow

:

flow

3. Separated flow

Free-stream surface flow

Separation Shock

~

E

x

t

e

r

n

a

l flow

Fig. 11. S~arface-flowpattern in swept-shock/boundary layer interaction (McCabe, 1966).t62)

i._.r'-'-,,~ Streamlinesoutside boundary layer •

-

•

.

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.

.

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.

.

.

.

.

.

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= ,,,~,

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.~ ~..-

(b) Separated flow Fig. 12. Schematic pattern of skin friction lines beneath swept-shock (Green).¢25)

186

A.G. Panaras

converge. Hence, there is no three-dimensional separation. On the right-hand side of Fig. 12a, the view along the shock wave is depicted. The pattern is similar to that of a two-dimensional weak normal shock wave/boundary layer interaction. When the shock strength is increased, a stage is reached in the flow development where three-dimensional separation begins. Further increase of the static pressure ratio results in fully separated flow. The pattern of the skin friction lines for the separation case, according to Green, are shown in Fig. 12b. The separation bubble commences at the three-dimensional separation line upstream of the inviscid shock wave position. Skin friction lines emanating from the reattachment line, downstream of the shock pass through the projected line of the inviscid shock to run asymptotically with the separation line. The upstream skin friction lines run with the separation line from ahead of the shock. The view along the shock wave indicates that a lambda foot to the shock and a vortical slip line passing downstream of the triple point appear. According to Green (1970),~2s) the skin-friction line patterns in laminar flow are qualitatively similar to those in turbulent flow. The developments described above in the understanding of the phenomenon of threedimensional separation, by interpretation of the skin friction line patterns, led to the appearance of the first flow models of the swept-shock/boundary layer interaction. This topic is reviewed in the next section.

2.3. APPEARANCE OF THE FIRST FLOW MODELS The first schematic flow model of the swept-shock/boundary layer interaction has been proposed by Cooper and Hankey (1974),t12~ and it is based on interpretation of their own experimental investigation of the laminar flow in a wedge/plate configuration under hypersonic flow conditions (M = 12.5, c5= 15°). Their measurements included oil-flow visualizations, surface pressure distributions and pitot pressure surveys. Their model is shown in Fig. 13, it involves only a single triple point in the shock structure and not two, as Charwat and Redekeopp (1967)tl~) had found for a symmetric corner configuration (Fig. 4a). This happens because the flat plate shock is weak and consequently, at the region of its interaction with the strong wedge shock, the corner shock is not formed. Instead, a triple point appears from which the embedded shock emanates. This shock interacts with the plate boundary layer and induces separation. In the viscous part of the flow, the model of Cooper and Hankey (1974)t~2~includes a pair of separation bubbles on the fiat plate with two attendant separation shocks (which intersect with the embedded shock) and two reattachment lines. Associated with the reattachment lines, surface heating and pressure measurements have shown two peaks of which the inboard value was significantly higher than that found at the outboard location. According to Cooper and Hankey, an important feature of three-dimensional separation is the fact that, unlike what happens in a twodimensional separation, the dividing streamline is not the same streamline that reattaches. As a result, flow entering the bubble is constantly scavenged away and must be replenished by a portion of the separated boundary layer. Hence, it is a higher energy streamline in this layer of scavenged flow that reattaches and not the low-energy separating streamline. It is this feature that accounts for the high value of three-dimensional interference heating. Particularly for the reattachment point R2, where the maximum heat rate is observed, Cooper and Hankey (1974)~ 2~state that "an inviscid finger extends down from the region of the triple point causing extremely high heating in this region". The trajectory of this 'inviscid finger' is denoted by the dashed line which envelops the two separation vortices (Fig. 13). Finally, Cooper and Hankey (1974)c12~in their model include a small separation bubble ($3) in the corner of the wedge, with a reattaching streamline striking the plate near R2. They felt that this small bubble was necessary to round off the corner and was not considered as a dominant feature of the flow. In the flow model of Cooper and Hankey (1974),t~2J for the first time the separation and reattachment lines, which have been observed on the surface of the flat plate, are related with the primary and the secondary vortex. For this reason, although these vortices in their model

187

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.

t ' I:

/,

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~

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/ :

x=

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e

o

Re = .93 x 106

'~/,

/ ', m: i I

.

Quast-comc,,I flow M = 12.5

i2.50

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~ = i.8

Plate shock

~ - - ~ _ _ .

S3 1// 0

R2

S 2

RI

Plate surface

Sm

Z - mn 15°

Fig. 12;. Structure of a wedge/plate flow a ~ o r d i n g to Cooper and Hankey (1974). (t2)

lie next to each other, and not with the secondary one within the primary one, as is now known to be correct, the early modeling effort of Cooper and Hankey (1974)(12) was a significant development. In addition, these authors provide a logical explanation of the peak heating observed in the region of the comer by assuming that it is caused by the impingement in this region of a high energy inviscid finger, or supersonic jet, which exists between the triple point and the separation vortices. In their conclusions they mention: "an inviscid supersonic jet, similar to the Edney type IV, was detected and found to impinge upon the plate surface near the second reattachment point. This jet resulted in high impact (pitot) pressure and heating at this point". It was the firslLtime that the Edney IV shock formation (Fig. 14a) was related to the peak heating observed in the comer region. This issue has been addressed again more recently. For this reason we will briefly describe this formation. Edney (1968) c2°) has found experimentally that, i~Lthe region of intersection of a strong shock, AB, (Fig. 14a) with an oblique one, AS, a third shock, AR, appears, the flow behind which under certain flow conditions is supersonic and terminates at a free shear layer, SRMO. If the flow beyond shear layers ATN and SRMO is subsonic, a supersonic jet is formed between them (behind shock AR). If this jet impinges on a surface, a strong shock appears at the region of impingement (MN in Fig. 14b) resulting in peak pressures and heating. The flow conditions for the existence of supersonic flow behind shock AR have been calculated by Henderson (1965)(zT) using shock polars, and are given in the diagram of Fig. 14c. We see in this diagram that for the appearance of z supersonic jet, the free stream Mach number must be at least equal to approximately 2.4. The element~; of the Edney IV shock formation may be related to the sweptshock/boundary layer interaction as follows: AB is the swept shock, AS the separation one and AR the eml:¢dded shock. The shear layer, SRMO, corresponds to the upper part of the separation vortex, and the flow has to be considered in the normal to the wedge shock direction (M1 = M,). In this particular case, the minimum Math number for the appearance of the supersonic jet is not given from the diagram of Henderson. Instead, the value observed in two-dimensional shock/boundary layer interactions should be considered. It is known that in these types of flows, where a 2-shock appears when extensive separation occurs, a 'supersonic tongue' exists behind the rear quasi-normal shock and above the separation bubble (see right-hand side of Fig. 12b), if the freestream Mach number (M, in the present case) is larger than 1.47. The length of the supersonic jet increases if the Mach number becomes higher. The supersonic tongue appears because the flow is non-uniform and away from the triple point, the shock waves are curved. A detailed analysis of this subject is included in the review of Delery and Marvin (1986).(*5) In the case of the fin/plate configuration, the footprint of the flow (turbulent or laminar) on the surface ~ff the plate also consists of a separation line, lying ahead of the inviscid position of the shock, and of a reattachment line close to the corner. The first flow model of

188

A. G. Panaras K/

0

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/13

,

/

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/

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(b)

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peak

sec. peak

B

81

50° 40°

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s

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~

30° 20° 10o

2 (c)

3

4

5

Moo

Fig. 14. The basic elements of the Edney IV shock formation. (a) Type IV shock formation; (b) jet impingement region, (c) Henderson's diagram.

Swept-shock/boundarylayerinteractions

189

the fin/plate co~Lfiguration was proposed in 1974 by Token. (92) For explaining the high heat transfer peak ~aeasured on the flat plate, near the root of the fin, he suggested that a separation conical vortex appears between the separation and the reattachraent lines (Fig. 15). In the same period, Kipke and Hummel (1975) (4°) published the results of their experimental investigation on symmetrical corners formed by unswept wedges at Mach numbers 12 and 16, respectively, Reynolds numbers, ReL = 5 x 106 and ReL = 1.7 x 106. The wedge angle (6) and the corner angle (0) have been varied systematically. The measurements included pitot pressure surveys, wall-flow visualization and the distribution of static pressure and heat transfer on the walls. Kipke and Hummel (1975) t4°) derived a schematic presentation of the axial corner flow, shown here in Fig. 16. The shock pattern proposed by these authors is quite similar to that of Charwat and Redekeopp (1967).(x1) Close to the surface, the model shows that the embedded shock intersects the wedge surface causing separation of the crossflow there. The superposition of the separated cross flow with the longitudinal flow component leads to vortices on both sides of the corner, which start at the leading edge of the corner and increase linearly downstream. In the region of reattachment of the separated cross flow, high static pressure and very large values of local heat transfer were measured. These effects increase with decreasing corner angle.

~ S e p a r a t i o n

line

Shockgenerator ~ ~ ~ - S i d e

wall

Fig 15. Fin/plateflowmodelof Token.(92) (a)

ylx

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t

~~ I Grcnzschichtrand I,/I/I

/ I / ( 1 1 1 ~i'l i ' l l -

(b)

z

t

J

i

,

Fig. 16. SeZhematicpresentationof the axial cornerflow,accordingto Kipkeand Hummel.~4°~

190

A. G. Panaras

In 1976, Korkegi(38) compared existing experimental data on swept-shock interactions with laminar and turbulent boundary layers at supersonic and hypersonic speeds, for the purpose of describing the structure of three-dimensional separated flow regions, from incipient to extensive separation. He considered a wedge/plate configuration, but he pointed out that, “the model could also be a double compression corner which has a more complex shock structure. The common feature of these models is that in both cases a skewed shock interacts with the boundary layer on a planar surface”. Korkegi (1976)(38) compared the surface-flow patterns, pressure distributions and heat transfer. From the comparisons, he concluded that the characteristics of swept-shock induced separated regions do not exhibit any basic difference between laminar and turbulent flow for comparable extents of separation, but of course, the shock intensities needed for extensive separation are much larger for turbulent than laminar flow. Korkegi (1976)(48) presented qualitatively a sequence of flow characteristics (repeated here in Fig. 17) from unseparated flow to extensive separation on a planar surface caused by a wedge-induced shock wave of progressively increasing strength. The lines of interaction on the surface are essentially conical. According to the model of Korkegi, when the wedge angle or the shock strength is small and the boundary layer is attached, the surface shear lines turn through a mild inflexion before assuming their new direction parallel to the wedge (Fig. 17a). The symbol Sh in Fig. 17a indicates the inviscid location of the swept-shock, and B the beginning of interaction, or the line along which the boundary layer begins to respond to the pressure jump (author’s note: this line is now called upstream influence line). In Fig. 17b, the wedge angle has been increased to the point where the boundary layer is on the verge of separating as indicated by the tangency of the surface shear lines at their point of inflexion, to the approximate shock wave direction (indicated by I for incipient separation). Obviously Korkegi applied the separation criterion of McCabe (1966).@” In Fig. 17c for a still larger wedge angle, there is a small region of separation containing a counterclockwise vortex as viewed from upstream (S, R are the points of separation and reattachment, respectively). In strong interactions, the model of Korkegi (1976)‘48’ predicts the

RS

(b) Incipient

(a) Unseparated

y* 1

n

15)

(c) Separated

Legend 1)

(d) Secondary

incipient

I) Surface shearlines 2) Surface shear component normal

y

(e) Secondary separation Fig. 17. Plow model of Korkegi.‘48)

to lines of interaction

Swept-shock/boundarylayerinteractions

191

development of a secondary separation region within the primary one, as shown in Fig. 17e. S1 and R1 are, :respectively, the primary separation and reattachment lines, and $2, R2, the secondary ones. The component of surface shear normal to the lines of interaction, Zn, undergoes three reversals in sign. At this point, we note that in the actual surface oil-flow photographs presented by Korkegi, the secondary reattachment line exists only in the examined laminar flows. In the case of turbulent flows, the photographs show that between the primary separation and reattachment lines, only the secondary line exists. The reattachment line does not appear clearly. In conclusion, during 1974-6, flow models of the swept-shock/boundary layer interaction have been proposed by various investigators for all the basic configurations, except for the swept corner, wlhich in that period of time had not been studied. As regards the separation bubble, all the models considered it as a conical flat vortex formed by the rolling up of the boundary layer along the separation line. One of these models (Korkegi, 1976)(4s) has also anticipated the existence of a secondary vortex below the conical one, for both laminar and turbulent flows. The peak heating observed close to the corner has been attributed to the existence of the :~eparation vortex. Cooper and Hankey (1974)(12) assumed that it is caused by the impingement in this region of a supersonic jet, which exists between the triple point and the separatJion vortex, similar to the Edney type IV. 2.4. PERIOD OF IMPROVEMENT OF THE FLOW MODELS In the late seventies and during the eighties, remarkable progress occurred. More details of the structure of the swept-shock/boundary layer interactions were detected, the models were refined and[ scaling laws of various flow quantities were developed. All these significant improvements will be reviewed. During the eighties, the simple configuration of a sharp fin attached normally to a flat plate became the primary tool of investigation of the swept-shock/boundary layer interactions. It was chosen on the grounds of the simplicity of the initial flow field: a planar swept-shock impinges on a two-dimensional boundary layer, laminar or turbulent. Most of the studies are related to turbulent boundary layers with supersonic free streams. The majority of the work in this field, experimental and numerical, has been performed in the United States at the Universities of Princeton, Penn State and Rutgers (Professors Bogdonoff, Settles, Knight and their associates) and at NASA/Ames (Professor Horstman). The same group ha:~ also studied the swept corner and more recently the crossing shock configuration. This research group combined experimental and computational results in a complementary way. As regards the axial comer, it became the object of a systematic experimental research programme at the Technical University of Braunschweig by Professor Hummel and his associates. Recently, the Braunschweig group started also investigating the intake-type configuration (Petzel and Hummel, 1992),(76) under laminar hypersonic conditions. In ~Lddition, continuous research has been undertaken at the yon Karman Institute (Professor Degrez and associates), in the field of laminar swept-shock/boundary layer interactiom (various configurations). This acknowledgement of specialized and planned research does not mean that the progress in the field has only originated in the mentioned institutions. Many significant investigations have also been performed in other institutions.

2.4.1. Investigations of the Flow Structure The first major improvement in the modeling of the swept-shock/boundary layer interactions occurred in 1982 when Kubota and Stollery(49) improved the vortex-dominated flow model of the fin/plate configuration, which had been proposed by Token (1974).(92) These investigators studied experimentally the interaction between a swept-shock and a turbulent boundary layer using a variable incidence wedge (fin) mounted on the side wall of a supersonic wind tunnel (M = 2.3). The study included oil-flow pictures, vapour and smoke-screen photographs, wall pressure distributions and heat transfer measurements. According to Kubota and Stollery (1982),(~9) their experimental measurements confirmed the model of Token (1974)(92) as regards the convergence and divergence oil-flow lines, for

192

A . G . Panaras

large fin angles, and the high heat transfer measured near the reattachment line. However, the model cannot explain some of their observations. For example, long before the appearance of a side wall separation line, there was evidence of an attachment line near the root of the fin. The oil-flow pattern on the fin showed a corresponding separation line, while the smoke pictures indicated the existence of a small corner vortex regardless of whether the rest of the side wall flow was separated or not. In addition, high heat transfer was measured in the corner region, even when the fin angle was small. This heat transfer, according to Kubota and Stollery (1982)¢49) is associated with flow reattaching above the corner vortex. Kubota and Stollery (1982)¢49) proposed the flow model shown in Fig. 18, which is fully consistent with their experimental observations. The separated flow in their model is characterized by two counter-rotating vortices, a tight, vigorous, roughly circular one in the corner with a weak, very elongated one above it (Fig. 18b). The small fin vortex is always present according to Kubota and Stollery, even when a weak interaction is established, in which no primary separation vortex is formed (Fig. 18a). The investigators noted that in their experiments, the size and shape of vortices were such that they still did not occupy a region much thicker than the upstream boundary layer thickness, even at the trailing edge of the fin. Thus, the disturbance that the viscous flow caused to the external inviscid flow

Incomplete / convergence line A • ~ °

-

.

°

°.

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.°

"

°.

"

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°

°

.

~

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--Vortex "~ Convergence line B

Shock generator

~Side

wall

(a)

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" ~ I~

i /

Complete convergence line A

----Vortical free layer i j

~

. ~

Reattachment line V o r t e x

- - - Convergence line B

Shock generator Side wall (b) Fig. 18. Flow model of Kubota and Stollery. ¢49) (a) Separated flow field; (b) attached flow field•

Swept-shock/boundarylayerinteractions

193

was of a minor nature. However, the vortices would be expected to grow and thus have a larger effect on the outer flow further downstream. Kubota and :gtollery (1982)~49) in addition to discussing the separation vortices, investigated the structure of the shock system. Of paramount importance for the development of the flow model is the appearance or not of the A-foot bifurcation of the fin shock at the region of its interaction with the boundary layer, exactly as it happens in a strong two-dimensional shock wave/boundary layer interaction. For resolving this question, Kubota and Stollery applied the vapour screen technique to their M = 3 tests. According to their vapour screen pictures, when 1Lheangle of the fin is smaller than the one required for the appearance of separation, there exists no A-shock, but for larger angles, when the corresponding oil-flow picture suggests a separated flow, there is evidence of the shock splitting into a A-shape near the edge of the boundary layer (Fig. 19). However Fig. 19 shows that the resolution of the pictures of the vapour screen technique is not adequate for a firm conclusion. As for the corner flow configuration, M611enst/idt~63) performed at the Technical University of Braunschweig, experiments similar to those of Kipke and Hummel (1975),t4°) but with corners; having swept leading edges. His measurements revealed the structure of the flow field for various sweep and corner angles. In almost all cases primary and secondary vortices have been found. He discovered that with increasing sweep angle, the strength of the primary and secondary vortices were reduced, and for large sweep angles, the secondary separation disappeared. The Braunschweig group has also investigated the issue of the conicity of the axial corner flow. Hummel (1989)t34) indicates that the inviscid comer flow should be conical, since the boundary conditions are conical. However, due to the displacement effect of the non-linear boundary layer, the wedge shock waves as well as the corner shock are located more outboard. In addition, Hummel notes that in the vicinity of the apex, the viscous interaction parameter reaches large values, thus the flow there is nonconical. Moving downstream, the viscous interaction parameter reduces more and more and the flow b:comes fairly conical with respect to a center which is located slightly upstream of the corner apex. Interpretation of the results of Mtllenst/idt (1984)t63~provided the synthesis of the cross flow shown in t]ae right part of Fig. 20a (Hummel, 1989),t34~ for a M = 12.3 laminar flow. The inviscid part of the flow has been calculated by the shock relations and a few key experimental data. We note that the shock system has been calculated in a plane normal to

i

~- Side-wall boundary , layer edge i

(Complete oil-flow conv. A)

__~ Side-wall boundary layer

02

e,

b-"

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,

v/ (Oil-flow div.)

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Oil-flow conv. B)

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(b) Separated interaction region Fig. 19. Detectionof the 2-shockfromvapor screenpictures(Kubotaand Stollery).~49~

194

A . G . Panaras

the conical ray through the triple point and it has been transformed in conical coordinates (I7 = Y / X , ~ = Z / X ) with respect to the conical center, which is located upstream of the corner along the line of intersection of the wedges. The similarity of the results of these simple calculations with the flow surveys (left part of figure) is remarkable. The flow structure close to the wall was drawn from evaluation of the shear stress pattern (Fig. 20b). The qualitative features of the viscous part of the flow, including the variation of flow parameters close to the wall (skin friction lines, shear stress), are similar to those shown in the Fig. 17 model of Korkegi (1976).(3s) The distributions of wall pressure and heat transfer rates are shown in Fig. 21 together with the flow model in the same length scale: this figure shows that the measured quantities are well correlated with the corresponding flow structure. As for the flow in the vicinity of the peak heating, Mrllenst/idt (1984) (63) and Hummel (1989) (a4) give the following description: "at the lower end of the flow in region 2 (see Fig. 20a), the pitot pressures decrease considerably, and this can be interpreted to be due to an expansion fan originating from the point of intersection J between the embedded shock and the boundary layer. The expansion waves are reflected from the slip surface and the bisector plane as compression waves and this leads again to a steep increase of the pitot pressure towards the corner center. Finally, the flow is deflected parallel to the bisector plane and parallel to the wedge surface in the immediate neighborhood of the corner by an l l}~ .23] ~ 0.~

0"16 0.92

~ II~ 2.75 ~ I ~ 2.29

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Swept-shock/boundary layer interactions (a)

195

~-

jll, 00j

0.~ R 3

(b) 6

0.3 7

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Pw

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-.-o-- .b__c ~ I I

(c) Io /

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inner shock or by a strong steady compression as indicated on the right-hand side". This description is very accurate because it is based on actual measurements. We note in Fig. 20a that in the cross flow plane, the flow behind the embedded shock (region 2) is supersonic, while that behind the corner shock (region 3) is subsonic. Thus, the channeled flow between the surface of the separation bubble and the shear layer is actually a supersonic jet, surrounded by subsonic flow. We observe that though the shock structure in an axial corner flow is different compared with an Edney IV shock formation, the resulting supersonic jets are quite similar (compare Figs 14 and 20a). Finally, we note that in Fig. 20a, the maximum value of the pitc,t pressure occurs within the supersonic jet, just before its impingement on the surface of the wedge. This fact explains the high pressure and heat transfer measured in the region of impingement. The flow in a ,~wept compression-corner configuration has been examined experimentally for the first time by Settles and co-workers in 1980.(87) A more detailed study was performed a few years later by Settles and Teng (1984).(ss) They found that depending on the flow conditions, two different flow regimes are established: a conical one, similar to that observed in a fin/plate configuration, and a cylindrical one. In Fig. 22, an example of the surface-flow pattern is shown for these two flow regimes (Settles and Teng, 1984).(ss) Interpretation of the surface pattern leads to the conclusion that in the case of the conical regime, a quasiconical vortex is developing in the corner region, while the shape of the separation vortex must be cylindrical in the other regime. Settles and Teng (1984)(sS) hypothesized that the cylindrical regime is associated with attached shocks and the conical regime with detached ones. Examining Fig. 22, we note that according to the experimental evidence in the surfaceflow pattern of both flow regimes, the separation and the reattachment lines are curved and not straight close to the apex. The initial region of deviation from conical or cylindrical behavior has been called the 'inception zone'. Another significant line, which has been denoted in Fig. 22 by AM, is the 'upstream influence line', i.e. the line which is formed by the points where the skin friction lines first start to curve from the initial free stream direction.

196

A. G. Panaras

M

-~z~Virtual

S

C

R

conicalorigin

M B

Fig. 22. Conical and cylindricalflowregimeon a sweptcorner configuration (Settlesand Teng, 1984).(85~

We have mentioned in Section 2.3 that the upstream influence line corresponds to the first increase of the wall static pressure. It is shown in Fig. 22 that in the case of a conical flow regime, the extrapolation of the upstream influence line intersects the swept comer line upstream of the configuration, at a point named 'virtual conical origin'. From this origin the separation and the reattachment lines also pass. A similar behavior has also been observed in the case of the fin/plate configuration. Experimental studies at Princeton and Penn State University have verified the quasi-conical nature of the interaction. It has been found that in addition to the separation and reattachment lines, the extrapolation of the undisturbed oblique shock also converges approximately to the virtual origin of the conical flow. This origin lies upstream of the leading edge of the fin, like in the swept corner configuration (see Fig. 23, taken from Settles, 1993).(83) At this point, it is useful to remind the reader that according to the Braunschweig research group, also in the axial corner configuration the origin of the conical flow lies upstream of the apex of the corner.

2.4.2. Scaling Laws The United States research group has made considerable efforts to scale the spanwise extent of the upstream influence line, as well as the inception length with appropriate physical and geometric parameters of the flow (see for example: Settles et al. (1981),~sS~Dolling and Bogdonoff (1983),(19) Lu et al.(S4)). Some results of this effort are shown in Fig. 24a taken from the paper of Settles and Lu (1985)i(s4) In this figure, the upstream influence lines

Swept-shock/boundary layer interactions

197

Fig. 23. Sketch of footprint of fin/plate flow (Settles). (83)

of 41 swept or unswept fin interactions (all at M = 3, turbulent flows) have been plotted in non-dimensional form: Ls is the length along the inviscid shock wave trace from the fin leading edge and Lus is the upstream influence length normal to the inviscid shock position trace (see Fig. 24b). Scaling Ls and Lus by (Re6L)I/3MN and (Re6L)1/3, respectively, accounts for the effect of shock strength and Reynolds numbers (Ms is the Mach number component normal to the shock, Re6L is the Reynolds number based on local boundary layer thickness). According to the data of Fig. 24a, the non-conical inception length, Li, may be calculated from the approximate relation: (LdfL). (Re6L)t/3 = 1600. Beyond the inception zone, the co~rrelation of the lengths Ls and Lus is linear. Settles and Lu (1985) (s4) have found that in this conical zone, these lengths are connected by the relation: Lus/(Ls + ALs) = 0.09Ms, where ALs is the displacement of the virtual conical origin from the leading edge of the fin. The scaling of the upstream influence in case of laminar boundary layers was examined by Degrez and Ginoux (1984)314) To that purpose, they performed an experimental investigation of the flow in a fin/plate configuration at various Reynolds numbers and at M = 2.25. They collected experimental data in the form of surface pressures and flow visualization. Scaling the upstream influence with parameters similar to those used in the scaling of the turbulent flows by the United States research group (the power of the Reynolds number was equal to 3/2 and not 1/3), Degrez and Ginoux ~14) found that the various values collapse on a single curve, which close to the apex is curved, but then it becomes straight i.e. the flow exhibits a conical behavior. In the same period, Inger (1987) ~3s~ presented a fundamental paper on the analytical calculation of the spanwise propagation of upstream influence in conical sweptshock/boundary layer interactions. Writing the boundary layer equations in conical arclength coordinates, Inger proved that while at an arbitrary distance from the conical origin the viscous flow cannot be conical, the far-field behavior at large radial distance, beyond a certain inception length, approaches a quasi-two-dimensional state. This state is comparable with the corresponding far-field conical behavior of the overlying inviscid flow, including viscous-inviscid interaction effects. Based on an order of magnitude analysis of the JPAS 32-|/J-!

198

A . G . Panaras

Inception . q . . _ ~

Conical

zone

0

zone

10

20

30

L s _ 1/3

(a)

~ KeSI X 10-2 Shock

r,,/,v, -%

;'

(b) Fig. 24. Scalingof the spanwiseextent of the upstream influenceline (Settlesand Lu, 1985);(s4)(a) correlationof experimental data; (b) nomenclature.

governing flow equations for large radial distance, Inger (1987) tas) deduced an estimate of the inception length, showing it to be proportional to the upstream influence length and the tangent of the shock angle. Implicitly, by means of a constant of proportionality, the inception length depends on the characteristics of the interactive boundary layer (shape factor), on the Reynolds and Mach numbers and on the wall temperature. The results of the analysis of Inger (1987)(35) apply to both laminar and turbulent boundary layers. Recently Lu and Settles (1991)tssl studied the effect of Mach number on the length of the inception zone by correlating data from a broad range of Mach numbers (2.47-3.95). Plotting of the normalized inception length--~ = (Ls/6)" Re61/3 (where 6 is the undisturbed boundary layer thickness) versus the shock angle, fie--resulted in a rather good correlation of the data, with a slight decrease of ~ as M increases at a fixed fie (Fig. 25), Lu and Settles (1991) tss) observed that according to their correlation, the inception length to conical

Swept-shoekfl~otmdary layer interactions

199

60.

4O

2.47 o 2.95 •

~1

3.95 •

2oi 10. O. 20

25

30

35

to

45

$o' deg. Fig. 25. Plotting of the normalized inception length ~ = (Ls,/6)" R~' (where 6 is the undisturbed boundary layer thickness) versus the shock angle flo (Lu and Settles)J ss)

symmetry for fin-generated shock-wave/boundary layer interactions is weakly dependent on Math number, but it depends strongly on shock sweep. More specifically, the inception length is constant an.d small for shock angles larger than approximately 35 °, but it increases with decreasing shock angle. As for the q~testion of the existence of the inception zone in speeds higher than those considered by Lu and Settles (1991),~55~we note that Dolling examined some M = 6 data in one of his correlation studies (Dolling, 1985).tls~ He came to the conclusion that whereas in supersonic flow there is always a curved inception zone, in the hypersonic test case, the virtual origin and leading edge were found to be coincident. On the other hand, we have already mentioned that in the M = 12 laminar comer flow studied by the Braunschweig research group, the conical center is located upstream of the comer along the line of intersection of the wedges. In conclusion, we feel that the issue of the dependence of the size of the inception zone on the flow parameters is not yet completely resolved. More data are required covering a broad range of Mach and Reynolds numbers.

3. NUMERICAL SIMULATIONS Numerical simulations of experimentally-studied swept-shock/boundary layer interactions have appeared in the literature since the early seventies. The character of the solutions has followed th,~ progress in the development of algorithms for the solution of the flow equations, as well as the increase of the processing speed and memory size of the computers. In the Appendiix, the following basic elements of a numerical simulation are briefly analyzed: equations and algorithms, computational domains and boundary conditions, turbulence models. 3.1. DESCRIPTION OF THE FIRST COMPUTATIONAL STUDIES The first numerical solution of the flows examined in this review was presented by Kutler (1974) ~5°~and its object was the calculation of the inviscid flow in an axial symmetric comer which had previously been investigated experimentally by West and Korkegl (1972)39s} The results of Kutler came at a time where, as Korkegi (1971) c'7~ had noted and Kutler repeated in his paper, "there existed no adequate method of predicting even the inviscid flow structure". Kutler (1974) {5°} transformed the steady-state flow equations into non-orthogonal conical coordinates (~ = x, r/= y/x, ~ = z/x). The cross flow mesh (t/, 0 had a size of 30 x 30 points. The second-order accurate predictor-corrector scheme of MacCormack (1969) ~56} was used to solve the Euler equations iteratively along the l-direction, until the solution became independent of the axial direction (implying the establishment of a conical flow field). In the numerical scheme of MacCormack, shock waves and contact discontinuities (slip surfaces) that exist in a flow are formed automatically in the solution

200

A . G . Panaras

(shock-capturing technique). Calculations similar to those of Kutler (1974)tSm were also performed a few years later by Marconi (1980)t6°) who, however, used a shock-fitting procedure to solve the conical Euler equations. Marconi (1980) ~6°)compared his results not only with the experimental data of West and Korkegi (1972),(gs) but also with the equivalent numerical results of Kutler (1974).(50) The wave structures are compared in Fig. 26 for two M = 3 cases; one for low and one for high Reynolds number. In Fig. 26a, both calculations are in good agreement with the experiments when the flow is turbulent, but when the flow is laminar (Fig. 26b), both inviscid flow predictions are not able to demonstrate the experimentally-observed considerable outward displacement of the wave system, which is due to the viscous-inviscid interaction. In judging the results, we should keep in mind that the purpose of those computational studies was the assessment of innovative computational techniques and not the investigation of the physics of the axial corner flows. As regards the wall pressure distribution at the region of impingement of the embedded shocks (Fig. 26c), the shock-fitting procedure gave the abrupt jump which is expected for inviscid flows, while in the shock-capturing method, the pressure exhibited an oscillation, which was caused by the numerical dispersion of the solution. This type of oscillation can be suppressed by the addition of proper numerical smoothing (see Appendix). In general, the shock-capturing methods are more efficient, especially when complicated flows are simulated because the shock waves appear naturally in the solutions. Shock fitting seems to be appropriate for blunt-body calculations using space marching numerical techniques. The first Navier-Stokes solution of a swept-shock/boundary layer interaction was performed by Shang and Hankey (1977)389) They simulated numerically the supersonic laminar intake-type flow (M--- 12.5, ~ = 15°), which Cooper and Hankey (1974)~12) had studied experimentally (see Section 2.3). They solved the unsteady compressible threedimensional Navier-Stokes equations, expressed in a generalized conical coordinate system similar to that used previously by Kutler (1974).(50) The time splitting predictor-corrector scheme of MacCormack (1971) (57) was used to solve the equations, subjected to the Courant-Friedrichs and Lewy (CFL) number regarding the permitted size of the time-step. In order to suppress the numerical oscillations of the shock waves, which are due to the truncation error, Shang and Hankey (1977)t89) incorporated in their numerical procedure a fourth-order smoothing term, originally introduced by MacCormack, which included the second-order derivative of the pressure or of the density. The smoothing term was only significant in regions of pressure oscillations. The grid of Shang and Hankey (1977) tag) is shown in Fig. 65 (see Appendix). It is the frustrum of a rectangular pyramid, appropriate for the calculation of conical flows. Various grid sizes were tested; the larger one had 8 x 32 × 36 points. The very few grid points in the direction of the,freestream had smaller spacing at the leading edge of the configuration in order to adequately resolve the rapidly developing flow field. For resolving the boundary layer, exponential stretching had been applied in the crossflow directions, ensuring that about ten grid points were contained in the boundary layer. The numerical results of Shang and Hankey (1977)(a9) are compared in Fig. 27 with the experimental surface pressure measurements on the plate (in the crossflow direction). The agreement between the measurements and calculation is very good, especially in the case of the finer grid. In Fig. 28, the experimental and the computed crossflow pitot pressure maps are shown. A comparison indicates that the numerical results duplicate all the essential features observed in the experiment. However, as noted by Shang and Hankey (1977),tag) the computations underpredict the thickness of the viscous dominant region, and, as a consequence, the shock wave system is displaced inward to the surface of the wedge. Also, the calculations exhibit a certain amount of smearing of the shock wave. A finer mesh would result in a sharper shock. Furthermore, the authors compare the experimental surface oil-film flow with the calculated shear stress vectors in their paper. The calculation predicts the separation and the reattachment line, which are indicated by the existence of two convergent rays originating from the comer, but it fails to capture the secondary vortex. For studying the conical aspect of the flow, Shang and Hankey (1977)tag~ projected the conical erns.~flow v e l o c i t y e o m n a n e n t

a n t n t h e n n r m n l t o t h e f r e e ~ t r e n m ninny, 1]171a 9(1~ 117'i~11ra 90

Swept-shock/boundary layer interactions

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-0.2

(c)

o:

o.z

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Fig. 26. Wave patte:m in symmetric corner. Comparison of calculations (Kutler (1974),(s°) Marconi (1980)(~°))with experiments (West and Korkegi, 1972).(gs) (a) Turbulent flow; (b) laminar flow; (c) wall pressure distribution.

202

A. G. P a n a r a s Z = (Z + x t ~ Ow~'X

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0.80

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M==12.5 ReL = 121x 10° 0w =15° TW = 660OR 1355 PTS Y(inch) /

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0

(a)

Wedge shock wave

!/

wave

"

/

20

40

60

80

! oo

z (inch)

M = 12.5 ReL =121x 106 0w=15 ° TW = 660OR 1355 PTS

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20

40 60 z (inch)

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Fig. 28. Crossflow wave structure (Shang and Hankey, 1977). (89) (a) Experimental; (b) computational.

shows in that the unperturbed flow region (upper and right portion of the graph), the conical crossflow velocity converges to the origin of the conical flow, which is to the left of the corner. The vortex is clearly shown by the recirculatory flow above the flat plate. Overall, the agreement between the calculation and the experimental data is remarkable considering the limited number of grid points. Critical features of the flow field, like the

Swept-shock/boundary layer interactions

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existence of the conical vortex and the shock wave system were identified in the paper of Shang and Hankey (1977).(sg) The small processing power and memory of the computers in the seventies is reflected in their results. A more complicated configuration has been calculated by Hung and MacCormack (1977), (32) primarily for testing in three-dimensional flows, the hybrid explicit/implicit numerical scheme of MacCormack (1976).(Ss) A M = 3 laminar flow in a compression corner with side wall was calculated for which no experimental data are available. In the hybrid scheme of MacCormack, the external flow is calculated using his original two-step explicit code, while close to the surface, an implicit scheme is used. The grid had 30 x 30 x 30 points and stretching was applied close to the surfaces. According to the authors, the hybrid code required an order of magnitude less time than the purely explicit scheme. For the particular configuration considered in these calculations, upstream of the wedge a shock-shock interaction appears (like in an axial corner), while in the region of the wedge a swept-shock~,oundary layer interaction is established on the surface of the vertical plate (side wall). The leading shocks were very weak (because they were formed only by the displacement eflect of the boundary layer) and Hung and MacCormack (1977)(32) were not able to resolve clearly the detailed structure of their interaction. As for the sweptshock/boundary layer interaction, the calculated isobars on the vertical plate reveal the quasi-conical character of the flow field. Encouraged by the successful application of the hybrid scheme to the forementioned three-dimensional low Reynolds number flow, Hung and MacCormack (1978)(31) applied their numerical scheme to turbulent fin/plate interactions. This time, the experimental data of Law (1975)(5:') were available for comparison (M = 5.9, fin angle: ~ - 6° and 12°). The algebraic turbu'~Ience model of Escudier (1965)(21) was primarily used; however, for the = 6° interaction, the model of Baldwin and Lomax (1978)~4)was also considered. The two models gave similar results. A significant feature of the work of Hung and MacCormack (1978)(31) was the introduction in the algebraic turbulence models of the length-scale, which they termed modified distance, to account for the turbulent mixing length near the comer of the configuration (see Appendix). Surface parameters were used to validate the code. Though the grid was rather coarse for the calculation of turbulent interactions (21 x 36 x 32 points), good agreement was found between the computed results and the experimental data, for both the surface pressures and the heat transfer in the case of the weak interaction (~ = 6°). For the case of ~ = 12°, the computed results do not demonstrate the experimentally observed existence of peak pressure near the comer; however, the range of interaction and the value and location of peak heat transfer were well predicted (Fig. 30). For visualizing the separation vortex, Hung and MacCormack (1978)(ax) used a crossflow velocity plot, which clearly indicated its existence even for a = 6° (Fig, 31).

204

A. G. Panaras

6-oI o oo 5.0

4.0

-'!

\

Invi~id

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Experimental data Present result (LX= 1.763)

a= M =S.9012"

[//solution

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3.2. PERIOD OF REFINEMENT AND VALIDATION OF THE NUMERICAL METHODS The work of Hung and MacCormack (1978)tall was the prototype for many subsequent numerical investigations of swept-shock/boundary layer interactions. Shang et al. ~1°2) simulated a laminar-transitional-turbulent corner-flow. The full Reynolds-averaged, Navier-Stokes equations were solved using the time-split scheme of MacCormack (1971)t57) in a grid of 17 x 33 x 33 points. An algebraic two-layer turbulence model of the Cebeci-Smith type was employed, in which the modified distance was used as length scale for the inner layer. The simulated corner was symmetric (wedge angle: 6 = 9.5°), and it had been tested previously by West and Korkegi (1972)tga) at M = 3 (see Section 2.1). As regards the transition, they extended the empirical model of Dhawam and Narasimha (1958)tl 7) to three dimensions. This model requires specification of the location of onset of transition. It is worth noting that in addition to wall pressure, field pitot pressure measurements were available for validating the code.

Swept-shock/boundarylayerinteractions tl ~

. . . . . .

= 6 deg, at X/L = 1.763 r

I

m .....

205

•

(V-Z)

]

I Z

......

~,~ Inviscid shock position

.......

t

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Y Fig. 31. Crossflowvelocityfieldin transverseplane (Hung and MacCormack,1978).(31)

The shock wave structure of the entire calculated field, visualized by the density contours, is shown in perspective in Fig. 32a. The structure is similar to that found already in many previous studies. Shang et al. (1°2) observed that as the flow progresses downstream, undergoing transition from laminar to turbulent, the shock wave system re-adjusts itself to accommodate the change in length scale. In Fig. 32b and c, the shock structure for laminar and turbulent ttlows correspondingly, are compared to the experimental data. Good agreement is observed regarding the location of shock waves, triple points and slip surfaces. The agreement is also good for the surface pressure distribution and the pitot pressure surveys (Fig. 33). In cortclusion, though the grid was not sufficient to study the flow separation in detail, the calculations of Shang et al. t1°2) demonstrated the emerging capability of simulating efficiently three-dimensional compressible turbulent flows. Test cases including more field data were selected by Horstman and Hung,(29) who studied several turbulent swept-shock/boundary layer interactions, from unseparated to highly separated ones, generated in a fin/plate configuration. The experimental cases studied by Peake (Ta) and by Oskam (1976)(64) were computed. From the study of Horstman and Hung, t29) some results will be presented related to a weak interaction (Oskam, M = 3, t~ = 10°) and to a strong one (Peake, M = 4, 6 = 16°). Horstman and Hung used the numerical procedures of Hung and MacCormack (1978)t31) for solving the time-averaged full Reynolds-averaged, Navier-Stokes equations. The authors emphasized the serious problem of computer memory that existed by mentioning that in their grid (21 × 36 x 28 points), the mesh spacing was approximately 16o in the x direction and 0.560 in the y and z directions (away from the wall). Nevertheless, the agreement with the experimental evidence was very good. Figure 34 shows the streamwise and crossflow velocity profiles computed by Horstman and Hung (1979)(zg) and measured by Oskam (1976)364) Some velocity profiles for the M = 4 experiments of Peake (73) are compared in Fig. 35. Other flow quantities which were compared in the paper of Horstman and Hung (1979)(29) are wall pressure, skin friction, heat transfer and angle of skin friction lines. Horstman and Hung (1979)(29) were the first to use computed streamlines for visualizing the flow near the separation line. To that purpose, they calculated streamline trajectories that originate near the surface at z = 0.05~o and at various y positions. The resulting trajectories are shown in Fig. 36 for three of the test cases of Peake, (7a) who from surface oil streaks identified them as examples of, respectively, non-separated, incipient separation and separated, flow. For each case, a top view of the streamlines is shown on the left-side of the

A. G. Panaras

206 Moo = 3.0 ReL = 1.10 x 10 6 Wedge angle 9.48 °

I

i0 6

I.O

West and

0.8

1.Oi|

Korkegi(98)

0.8II

lation

6 lest and Korkegi (g

~

,n

0.6

0.6 Y/X

Y/X 0.4

0.4

0.2

0.2

0.0

(b)

. . . . . .

0.2

0.4 0.6 zrx

0.8

t.o

0.0

(c)

0.2

0.4 0.6 zrx

0.8

LO

Fig. 32. Visualization of the flow, structure by density contours (Shang et al. 1979)/1°2) (a) Perspective view; (b) crossflow, laminar; (c) cross flow, turbulent.

figure and a three-dimensional view of the same streamlines on the right-side of the figure. The z scale has been expanded relative to x and y scales. In the case of the non-separated flow (Fig. 36a), the streamlines do not converge significantly and show very little lift-off from the surface. On the contrary, in the case of the separated flow (Fig. 36c), the streamlines rapidly converge and lift off the surface. The convergence of the streamlines occurs ahead of the inviscid shock position, and there is a massive eruption of streamlines away from the surface along the separation line. The incipient separation case (Fig. 36b) shows results similar to the separated flow case, but with a much smaller lift-off. A few years later, Knight (1984) t4t) modified the hybrid explicit/implicit algorithm of MacCormack (1971) {57)by replacing the implicit part of it. Instead of solving the complete Navier-Stokes equations, he used a limiting form of them which is applicable in the viscous sublayer and transition wall region of a turbulent boundary layer. Within the sublayer, a limiting form of the Navier-Stokes equations was employed, in which the u, v velocity components vanish (zero tangential velocity on the wall) and the pressure is constant across the layer (in the z direction). Also the temporal derivative terms are omitted. The sublayer equations were solved by the implicit box scheme of Keller (1974).taa) The turbulence model of Baldwin and Lomax (1978) t4) was used with the length scale in the inner layer determined by the modified distance formula introduced by Hung and MacCormack (1978).°t) The method was used for simulating swept-shock/turbulent boundary layer interactions (Knight, 1984, 1985).t41,42~ Good agreement was found between the calculations and the experiments (wall pressure and heat transfer, yaw angle, pitot pressure).

Swept-shock/boundary layer interactions P-Pw - Pw 1.0

i

207

Rex o 0.39- 106 2 3.2 - 39.9 - 106 West and Korkegi (95) Present result laminar Present result turbulent

,,.

0.8 0.6 0.4 0.2 0.0 -0.1

(a)

0

0.2

0.4

0.6

0.8

1.0

Z/X

0.5 PZT 0.3 [ P

Y/X ........

0.60 0.3 West and Korkegi(98)

0:.,7

Present result

0.6 [ ' 04t 0

(b)

0.2

0.4

0.6

Z/X

Fig. 33. Comparison of surface pressure and pitot surveys (Shang et al., 1979). (1°2) (a) Surface pressure; (b) pitot distribution.

We have mentioned in Section 2.4 that during the eighties, a research group was formed in the United States which investigated experimentally and computationally many sweptshock/boundary layer interactions (fin/plate, swept-corner, crossing shocks). The research group combined experimental and computational results in a complementary way. The first combined paper was prepared by Knight et al. (1987).(46) The object of the cooperation was the study of the structure of the turbulent flow past a sharp fin. The M -- 3, ~ - 20 ° experiments of Shapey and Bogdonoff (1°3) were simulated by Knight and Horstman. The first author used his previously described algorithm, while the second one applied the calculation scheme of Hung and MacCormack (1978)(31) employing the k - t turbulence model of Jones and Launder (1972) (37) coupled with the wall-function model of Viegas et al. (1985). (93) Knight used a grid consisting of 32 x 48 x 32 points; Horstman employed in his calculations more streamwise stations (64 x 32 x 44 points). Knight et al. (1987) (*6) compare profiles of yaw angle, pitot pressure, x-component velocity and wall pressure distributions. The agreement with the experiments is good. In Fig. 37 comparisons are shown between the computed and the measured yaw angle and the pitot pressure at Station 5, which is located between the separation line and the shock. The same quantities are compared in Fig. 38 at Station 7, which is located slightly behind the shock. The discrepancy in the computed pitot pressure outside the boundary layer according to the authors is associated with the shock-capturing nature of the numerical algorithms, the difference in the streamwise grid spacing for the two computations and the proximity of Station 7 to the shock. It is remarkable to note that the flow field predictions of Knight and of Horstman are similar, except in the computed yaw angle in a narrow region of the boundary layer adjacent to the wall, though the turbulence models employed by them provide eddy viscosities which differ by as much as a factor of 14 (Fig. 39). According to Knight (1993) (*3) this is an

208

A. G. Panaras o • - XShock ~-o

-2

Experiment, Oskam(64) Computations

-I

0

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2

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q

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~l • 0

k" 0

o

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V/U e

Fig. 34, Comparison ofcomputed by H o r s t m a n and H u n g (1979) (29) and measured by O s k a m (64) velocity profiles. (a) Streamwise profiles; (b) crossflow profiles.

1.4

-2.8 1.5

x - XShock = - I I.I

-5.6

1177

1.0

I<

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z/8 o

Io/~

C 0.5

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°

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/o

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Swept-shock/boundary layer interactions 20[- [

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210

A.G. Panaras

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indication that in the shock/boundary layer interaction, there is a triple-deck structure. The viscous effects are restricted to a thin layer adjacent to solid boundaries, while the remainder of the boundary layer is effectively rotational and inviscid. For studying the structure of the flow, Knight et al. (1987) ~46)computed mean streamlines originating upstream of the interaction and close to the wall (y = 0.526o). Their results are shown in Fig. 40, where the trajectories of streamlines which originate upstream of the separation line rise close to the wall, cross the separation line and rotate in the direction in which the separation vortex should rotate. These data strongly support the model of the fin/plate flow which has been proposed by Token (1974) ~92~and which is shown in Fig. 15. However, critical issues, such as the shape of the conical vortex and whether the longitudinal vortex detected by Kubota and Stollery (1982)~49Jexists, remain unanswered. More recently, Bogdonoff (1990) ~7) tried to clarify some critical features of the fin/plate interaction, including the question of the 2-shock. To that purpose he used experimental (Shapey and Bogdonoff)~1°3~and calculated (Knight et al., 1987)346) Math-number iso-lines. Review of the pictures of Bogdonoff (Fig. 41) shows that it is hard to say that a 2-shock formation is depicted in them, though in the examined interaction (M = 3, ~ = 20 °) the surface oil-flow pictures indicate the existence of separation and reattachment lines. Thus, the question of the existence of a 2-shock formation in a fin/plate configuration, as Kubota and Stollery (1982) ~49~ suggest, had not been resolved, evidently because the available computer resources have not permitted the use of fine grids. Regarding the numerical simulation of the comer flows, the results of Scriba (1988),~81J who used the conical flow solver developed by Qin and Richards (1987) ~78~to calculate the M = 12.8 flow in a axial comer configuration, which had previously been studied by Mrllenst~idt (1984),~63~are significant. Some results are shown in Fig. 42. The agreement with the experiments of Mrllenst~idt (1984)~63~is very good (Fig. 42a). Also the streamlines

Swept-shock/boundary layer interactions Experiment-Shapey Theory-Horstman Theory-Knight

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Fig. 40. Computed streamlines in a fin/plate flow (Knight et al.). (46)

A. G. Panaras

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Swept-shock/boundarylayer interactions

213

in the calculated cross-section (Fig. 42b) support the model of Mrllenst~idt (1984) (63) and Hummel (1989),(a4) which is shown in Fig. 20a. It is noted that the solution of Scriba predicts a corner vortex which had not been reported earlier.

4. RECENT PROGRESS

As we have seen in the previous section, significant progress occurred during the eighties. More details of the flow field were observed, the models were refined and numerical simulations appeared which efficiently duplicated the experimental data and provided strong indications of a helical motion within the separation region below the swept-shock. However, critical questions still remained unanswered. As for the conical vortex, even its cross-section has not been visualized accurately, while it has largely been named conical because the outer inviscid flow is conical and because it was implied by the surface/flow visualization. No perspective visualization has however been presented. On the other hand, while the shock formation has been clearly visualized experimentally and computationally in the case of the corner flows, in the simpler fin/plate configuration, the question of the existence of a 2-shock formation remained unresolved. Evidently, the number of the points which have been used for measuring the pitot pressure or for computing the flow parameters have not been enough for the task of providing clear pictures of the cross-section of the flow. Fortunately, in the early 1990s, non-intrusive experimental techniques became available to depict clear cuts of any flow (unlimited number of points), while the efficiency of computers permitted the use of high resolution grids and more advanced codes in the simulation of flows. The parallel progress in the development of flow visualization techniques in the area of post-processing of the numerical solutions must also be acknowledged. The new developments will be described in the next section. It will be shown that indeed the new experimental techniques produce very clear pictures of the cross-section of the flow field in the normal to the shock direction, and provide quantitative data, such as the skin friction distributions, which are suitable for direct comparison with computational predictions. In addition, the new numerical simulations verify all the critical elements of the flow structure of a swept-shock/boundary layer interaction, some of which up to now were merely objects c,f speculation.

4.1. VISUALIZATION OF THE FLOW FIELD, CONICAL SIMILARITY 4.1.1. Experimental Studies An important consequence of the quasi-conical nature of the fin/plate flows is the presumption that their features can be projected upon the surface of a sphere whose center is the conical origin. Alvi and Settles (1990) (2) have demonstrated this feature of the conical flow by using conical shadowgraphy. Focusing a light beam at the origin of the approximately conical flow field and aiming it such that the resulting conical light beam coincided with the rays of the swept interaction, they obtained clear pictures of the flow field of fin/plate interactions. An example is shown in Fig. 43, for a M = 2.91, 0t = 20 ° interaction. The cross-section of the flow is normal to the shock. On top of the separation bubble a well-bifurcated shock is present. From the shock triple point, a shear layer emanates and moves towards the corner. The rather good collapse of the flow field features in the conical optical frame is considered by Alvi and Settles (1990) (2) as a proof that the interaction is almost conical in nature. Recently, Alvi and Settles (1991)(3) obtained clear cuts of the flow field normal to the shock in a fin/plate configuration using the non-intrusive planar laser scattering (PLS) technique, also known as vapor screen technique. This technique originated in the sixties, I

JPA$ 32-21]-J

214

A.G. Panaras

Fig. 43. Conical shadowgram of M = 3, ~t = 20°, fin/plate flow (Alvi and Settles).t2~

Fig. 44. Planar Laser Scattering image of M = 3, • = 20°, fin/plate flow (Alvi and Settles, 1991)33~

however, the recent development of powerful lasers enabled high quality images to be obtained (Settles, 1993). ts 3~ Usually the flow is seeded with water vapor; the light is scattered by the aerosol particles as the flow passes through the light screen. A P L S image is shown in Fig. 44, where this technique also visualizes very clearly the ),-shock and the shear layer. Alvi and Settles (1991) t3~ c o m b i n e d the P L S results with previous wall pressure and skin

Swept-shock/boundary layer interactions M 12

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215

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friction measurements to construct a physical flow field model. An example is shown in Fig. 45 for a M =-- 3, • = 16° test case. Below the flow field map, the corresponding surface distributions of static pressure and skin friction are plotted in the same conical angle scale. Alvi and Settles (1991) °) observe that the inviscid air processed between the triple point and the separation vortex is curved downwards and impinges upon the flat plate. This is accomplished through reflected Prandtl-Meyer expansion and compression fans. The authors call it 'impinging-jet' and attribute to it the peak heating, pressure and skin friction, observed in the comer region. In the strongest interactions (M, > 2), they include in the model a normal shock in the region of impingement of the jet. Settles (1993)(s3} notes the similarity of the conditions with the Edney type IV shock formation. The similarity of the flow model of Alvi and Settles (1991)(a) with that of Hummel (1989), (34) Fig. 20, is evident. There are only two differences: first, the lack of the separation shock in the comer flow case, where according to Hummel, in an axial comer, compression waves are generated at the region of separation of the boundary layer instead of a shock wave (because the thickening and separation of laminar boundary layers are smoother for turbulent boundary layers); and second, in the fin/plate configuration, no secondary reattachment line exists. 4.1.2. Computational Results Clear images of the swept-shock/boundary layer interaction were provided by the present author (Panaras, 1992),(67) who has calculated the flow which has been studied experimentally by Shapey and Bogdonofl~1°3) and computationally by Knight et al. (46) The Reynolds averaged Navier-Stokes equations, with the (axial) l-derivative of the viscous terms ignored, were solved at the interior grid points of the mesh shown in Fig. 66 of the Appendix (45 x 95 x 79 points). The Baldwin and Lomax (1978) (4) algebraic turbulence model was used in which the modified distance introduced by Hung and MacCormack

216

A. G. Panaras

(a)

y

I 0

5

I0

15

20

25 30 35 40 Yaw angle (deg.)

06

08 1" Pitot pressure

45

50

55

60

(b)

y

02

~

04

I0

12

Fig. 46. Comparison of pitot pressure and yaw angle at Station 5, located between the separation line and the shock (Panaras). t67) (a) Pitot pressure; (b) yaw angle.

(1978)~31) was incorporated as the length scale for the inner layer. A second-order central differencing was applied to the implicitly-treated viscous fluxes. The inviscid fluxes were determined by the upwind total variational diminishing (TVD) scheme of Yee and Harten (1987),~99) which uses Roe's approximate Riemann solver (Roe, 1981)~79) and Harten's second-order modified flux approach. Alternating Gauss-Seidel relaxation in the streamwise direction was employed. In Figs 46 and 47, the comparison is shown between the computed by Panaras (1992) ~67) and measured by Shapey and Bogdonofft1°3) yaw angle and the pitot pressure at Stations 5 and 7: the agreement is very good. Compared to the calculations of Knight et al. (1987),~46~ the results of Panaras (1992)~67)are closer to the experimental data reflecting the effect of the finer crossflow grid and probably the less dissipative character of the TVD scheme. After this validation of the results, Panaras (1992) ~67) used various post-processing techniques which have been developed at DLR in order to visualize the elements of the flow structure. As regards the visualization of the vortices, an effective technique has been proposed by Vollmers et al. (1983)395) They have demonstrated that vortices exist in those parts of a flow in which the discriminant of the tensor of the gradient of the velocity indicates complex eigenvalues. The discriminant is evaluated numerically at all the points of the grid. Then contour surfaces of constant values are created and displayed. These contours indicate where in the field there are vortices. The discriminant technique has been incorporated by Vollmers (1989)~95) in a graphic system called Comadi. Figure 48 (Panaras, 1992),~67)includes all the critical elements of the swept-shock/boundary layer interaction. The vortices which are expected to appear in this type of flow are visualized in the three-dimensional space by the contours of the eigenvalues of the velocity gradient field. In addition, three cross-sections have been drawn on which the density contours are displayed (visualization of shock waves). It is observed that, as expected, the

217

Swept-shock/boundary layer interactions 41-

:- (a)~,~

Yaw angle (deg.)

(b)

Y

l

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2

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O

O --

I

0.4

0.6

0.8

1.0 1.2 Pitot pressure

I 1.4

I 1.6

1 1.8

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{iii)

/ Cross-section of the vortices Fig. 48. Fin/plate flow. Perspective view of the conical vortices and of the shock waves (Panaras)/6~

218

A.G. Panaras

Fig. 49. Visualizationof the vorticitysheet of the boundary layer (Panaras). 167)

Fig. 50. Streamlinespassing through the core of the vortices(Panaras).¢67)

flow is dominated by a large vortical structure, which lies above the flat plate and whose core has a remarkable conical shape with aflattened elliptical cross-section. Also on the flat plate, on the side of the main vortex, a thin vortex has developed in the direction of the flow. This is not an independent vortex, but the core of the vorticity sheet which lifts-off the surface, along the separation line, and rolls up to form the conical vortex. Along the vertical fin and close to the comer, the longitudinal vortex, mentioned by Kubota and Stollery (1982), t49) is observed. It also develops quasi-conically, but with a smaller rate of increase, compared to the prime conical structure. In the lower part of the figure, a cross-section of the vortices is shown. There, it is indicated that indeed the flat ground vortex constitutes the initial part of the primary conical vortex. In Fig. 49 the vorticity sheet of the boundary layer is shown, as evidence that the boundary layer lifts-off the surface of the plate along the separation line to form the conical separation vortex. Also in Fig. 49, the formation of the corner vortex along the fin is clearly observed. The induction characteristics of the vortices of the flow field are indicated in Fig. 50 where some of the streamlines which pass through their cores are shown. The conical vortex

Swept-shock/boundary layer interactions

219

Fig. 51. Visualization of the flow structure using iso-contours of the absolute value of vorticity (Panaras). t67~

completes more than one turn in the calculated field, and it is hard to determine if the fin vortex turns within the extent of the field. The density contours on three cross-sections of the flow, which are also displayed in Fig. 48, visualize the shock system which is formed along and on top of the conical vortex. The system is composed of the swept-shock wave, the separation shock, produced by the coalescence of the compression waves, and the rear quasi-normal shock, which extends from the bifurcation point to the surface of the conical vortex. The shear layer which is expected to originate at the triple point is not visible in Fig. 48. This is because the density change across the shear layer is smaller than the threshold of the density contours used in this figure. Howeve,r if the iso-contours of the absolute value of the vorticity is used as a visualization parameter, the shear layer appears. This type of visualization is presented in Fig. 51, where a remarkable similarity exists between the calculated cross-section of the flow and the model of Alvi and Settles, ta~Fig. 45. Knight et al. ~4~ presented refined calculations of the test case of Alvi and Settles (1991) t3~ in which the A-shock is evident in various colored contour plots, particularly the density plots. The slip line emanating from the triple point is visible in the total pressure contours of Knight et al. According to the authors, the computed flow field exhibits most of the features of the model of Alvi and Settles. The computations do not exhibit the normal shock, transonic shocklets and secondary separation observed in the experiment. In the conical projections of Alvi and Settles (1991) tzJ and of Charwat and Redekeopp, (1967)~11~ the flow appears to be quasi-conical and not conical. Then the question arises, what causes this deviation? In this context, Panaras (1992) t67~ observed that the different rate of thickening of the conical vortex and of the boundary layer of the plate is expected to affect the conical similarity adversely. For studying this effect, Panaras conically projected the sections (i) and (ii) shown in Fig. 48 on the outflow section (iii). The iso-Mach lines have been used as vi,;ualization parameters. The results of that correlation are shown in Fig. 52. It is observed in Fig. 52b, where section (i) is conically projected on the outflow section, that while good coincidence is observed between the separation bubbles, the swept-shocks and the shock triple point, the feet of the A-shocks are correlated fairly, especially the separation shocks, but there is no equivalence at all between the boundary layers in the two crosssections. In particular, the boundary layer of the first cross-section is approximately 40% thicker than the boundary layer of the third section. This large difference in the scaling is due to the lower rate of development of the boundary layer compared to that of the vortex. The deviation from the conical behavior is smaller if the two cross-sections which are conically correlated are closer. This is demonstrated in Fig. 52c, where section (ii) is conically projected on section (iii). At this point note that in the shadowgraph pictures of Alvi and Settles (Fig. 43), multiple shock formations exist, one close to the other. However, in this case, the effect of the boundary layer is not visible because in the optical technique used by Alvi and Settles, it is not possible to distinguish the various cross-sections of the flow. Panaras ~6~'~,in addition to the forementioned illustrations, has presented quantitative data which verify the observation from Fig. 52b that the deviation from conical similarity is greater at the part of the flow between the separation shock and the plate.

220

A.G. Panaras (a)

~

~

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.

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(b)

y 6 8 4

0

5

10

15

20

25

30

35

40

25

30

35

40

SI8

(c)

y 8

6 4

0

5

10

15

20 S/8

Fig. 52. Conical projection of sections of the flow on the outflow plane. (a) Technique of conical projection; (b) projection of Section (i) of Fig. 48; (c) projection of Section (ii) of Fig. 48.

4.2. SECONDARY SEPARATION IN A FIN/PLATE CONFIGURATION In Section 2.3, we noticed that according to the published data, while in a laminar strong swept-shock/boundary layer interaction, a secondary reattachment line appears between the primary separation and reattachment lines, no secondary reattachment line is visible in a turbulent interaction. In addition to this difference between laminar and turbulent flows, there exists another one perhaps which is more fundamental. Thus, while in the laminar corner flows studied by Professor Hummel and his associates, the secondary separation phenomenon appears progressively with increasing shock strength, in the turbulent fin/plate flows studied by Professor Settles and his associates at the Penn State University, the existence of a secondary separation line in the surface-flow pattern has been detected only in interactions of moderate strength and it disappears in stronger ones. The first data regarding this strange phenomenon were published recently by Kim et al. (1990), ¢39~and since no other evidence that the separation line has been detected related to the phenomenon of the secondary vortex, Alvi and Settles (1990) t2Jstate that "this

Swept-shock/boundary layer interactions

221

Fig. 53. Skin friction lines and cross-section of the vortices (Panaras and Stanewsky, 1992)36s)

separation is too weak to qualify unequivocally as a secondary separation of the reversed flow within the primary separation zone". Panaras and Stanewsky (1992)t68~have simulated numerically one of the flows of Alvi and Settles (1991) t3J in which the secondary separation line appears. Panaras and Stanewsky have shown tha.t if the algebraic turbulence model of Baldwin and Lomax is interpreted according to the physics of the flow (Panaras and Steger, 1988), <69J the resulting numerical solution agrees well with the experimental evidence (wall pressure, skin friction). Then post-processing of their solution revealed that actually in this type of interaction, the secondary separation phenomenon is similar to that observed in flows about bodies at high incidence. The Panaras-Steger interpretation of the Baldwin-Lomax turbulence model is based on a physical understanding of the flow and is described in the Appendix. In Fig. 53, a perspective view of the skin friction lines along with some cross-sections of the vortices is shown (visualized by the discriminant technique). In this figure it is demonstrated that the secondary vortex is gradually formed along the secondary separation line. No secondary reattachment line is observed. It appears that a finer grid is necessary in the region of the secondary separation, for resolving the local details of the flow. In Fig. 54, a cross-section of the vortices and the corresponding variation of the wall pressure and of ~Lheskin friction coefficient are shown. The secondary vortex coincides with a peak in the skin friction distribution in the direction normal to the shock, ci,, exactly as it has been postulated by Korkegi (Fig. 17). Furthermore, it lies close to a secondary peak in the total skin friction coefficient, cI. It is noted that the pressure and the skin friction take their maximum values outside the core of the vortex, closer to the corner, as it is shown in the Fig. 45 mod,~l of Alvi and Settles (1991).ta> This condition supports the expressed view that the high values of the heat transfer close to the comer are due to the impingement of a supersonic jet, which is formed between the separation vortex and the shock triple point. In Fig. 55, the velocity vectors have been drawn in a cross-section normal to the axis of the secondary wgrtex. A weak circulatory motion exists close to the surface of the plate at z/6 = 12 - 15.5. Above the secondary vortex a nearly 'dead-air' region of triangular shape exists. Panaras ~,nd Stanewsky conclude that in the crossflow, the lower part of the conical vortex is channelled between the fin and the secondary vortex, which appears as a 'bump' of triangular shape. In his recent review, Settles (1993)ta3) includes some significant experimental data from the Institute for Theoretical and Applied Mechanics in Novosobirsk (Russia). These data have originally been published in Russian. A part of these data is related to secondary separation. We transfer here some of the conclusions of the Russian scientists, as they have

A. G. Panaras

222

3.5

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been quoted by Settles. According to Zheltovodov et al. (1987),(1°1) the secondary separation line first appears once the interaction has achieved a certain strength, showing up in the conical region, but not in the inception zone (exactly as it is shown in Fig. 53 of Panaras and Stanewsky)/6s} Its spanwise extent grows with increasing shock strength, but then diminishes again, eventually appearing only in the inception zone and then disappearing altogether. Secondary separation then reappears in the strongest interaction yet observed (M = 4, g = 30.6°), but in a different position, closer to the fin than previously. The experimental results of Zheltovodov et al. (1°1) demonstrate that the initial behavior of secondary separation is related to laminar, transitional and then turbulent reverse-flow in the swept separation bubble. They also ascribe the reappearance of secondary separation to the development of supersonic reverse flow in the separated region with an imbedded normal shock wave. Regarding the secondary reattachment line, Zheltovodov and his associates have obtained evidence of its existence only in the forementioned extremely strong interaction (very close to the secondary separation line). 4.3. MULTIPLE SEPARATION In Section 2.1 we stated that in the M = 20 experiments of Watson and Weinstein (1971), (97) the heat transfer distribution in the normal to the flow direction is characterized

Swept-shock/boundary layer interactions 12

223

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6 5 4 3

,

11

,

12

-

,

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-

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,

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14

,

15

.

,

16

17

-

-

,

-

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18

,

19

.

,

20

,

21

+

,

.

22

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by two peaks (Fig. 5). Korkegi (1971)<47)explained the drop of the heat transfer between the two peaks by the presence of a region of outflow from the surface. Then he postulated the possible existence of a system of three vortices, a weak outer one and two strong ones inboard. No olLher details were given at that time. However, in 1976 when Korkegi(4m presented his flow model (described here in Section 2.2), more experimental data were available and he observed that there exist two peaks in the case of an extensive laminar separation, which are associated with two reattachment lines. Very recently, Petzel and Hummel (1992)(76) and Papuccuoglou (1993)<70)provided evidence of existence of multiple vortices in a swept-shock/laminar boundary layer interaction. Petzel and Hummel (1992)<76)performed experiments on an wedge/plate configuration at M = 12.6. The model consisted of a horizontal wedge of 8° and a vertical flat-plate, intersecting at a corner angle of 90°. The measurements included the pressure, the heat transfer and the local direction of shear stress at the wall, as well as the pitot pressure distribution in a plane normal to the freestream. Interpretation of the experimental data led to the construction of the flow model shown in Fig. 56a. The shock system consists of the wedge and the plate shocks, which interact through a corner shock. From the triple points, the secondary shocks (which impinge on the boundary layers of the surfaces) and the slip surfaces emanate. Petzel and Hummel observe that on the wedge side, the secondary shock is weak and the related flow separations take the form of a primary and a secondary vortex, as in a symmetric corner flow. On the plane side, the secondary shock is rather strong and therefore extensive separations with a primary, secondary and tertiary vortex have been

A. G. Panaras

224 S)

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detected. In the heat transfer data (Fig. 56b), a second peak exists in the region of reattachment of the secondary vortex; however, no peak is observed in the reattachment of the tertiary vortex. Papuccuoglou (1993) (70) has investigated experimentally the heat transfer rate and flow pattern in an axial corner and in an wedge/plate, at M = 6 over a range of Reynolds numbers that result in laminar and turbulent flows. Papuccuoglou discovered that while in

225

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Flow

Fig. 57. Multiple separation in an axial corner flow (Papuccuoglou)JTM

the turbulent-flow cases only a single peak existed in the heat transfer distribution, in some laminar-flow cases, three and even four peaks were observed. Papuccuoglou, relating the heat transfer distributions with oil-flow visualizations, constructed a qualitative flow model that includes up to four vortices and separated zones, depending on the number of the peaks in the heat transfer distribution. Following Korkegi (1976),~4s~he assumes that each peak coincides with the reattachment of a vortex. An example of the flow model of Papuccuoglou (1993)F °} for three-vortices, is shown in Fig. 57. At this point we note that the surface lines have been drawn qualitatively by hand, based on the actual oil-flow picture. In that picture, according to Papuccuoglou (1993),~°) only the primary and secondary separation lines are seen clearly, while between them two oil accumulation lines are present that are not very dear. It appears that the model proposed by Papuccuoglou is basically correct, especially as regards the correlation of the heat transfer peaks with the multiple reattachment points. A numerical simulation of the particular flow is necessary for studying in detail the pattern of the surface flow.

5. SWEPT COMPRESSION CORNERS AND CROSSING SHOCKS 5.1. SWEPT COMPRESSION CORNERS In Section 2.4 we stated that according to the experimental data of Settles and Teng (1964), t8s) the surface pattern in a swept compression corner is very similar to that observed in a fin/plate covLfiguration. Using a specialized kerosene-lampblack streak technique, they were able to detect on the surface of the configuration a straight separation line (formed

226

A . G . Panaras

o ° °

S oo

0

Tunnel

o

floor

z = 10.16 cm Data Cebeci-Smith model

....

Jones-Launder

I

I

-I0

10

model

I

20

(cm) Fig. 58. Comparison of measured and computed surface pressure (Settles et al.). tar)

upstream of the inviscid position of the swept-shock) and a reattachment line (visible on the corner). A schematic presentation of the surface-flow results of Settles and Teng (1984)t85~is given in Fig. 22. More recently, some collaborative experimental/computational papers of the United States researchers have appeared. The first paper was prepared by Settles and co-workers (1986).ta6~The sweepback angle of the comer was 40 °, the compression comer angle 24 ° and the freestream Math number equal to 2.95. The implicit method of MacCormack (1982) t59~ cast in a finite-volume formulation prepared by Hung and Kordulla (1983)ta°~ was used, employing the algebraic Cebeci-Smith (1974)t9~ model and the two-equation Jones-Launder t37~ model for turbulence closure. The computational mesh had a size equal to 40 x 35 x 27 points. Variable mesh-point spacing was applied in all directions. An example of a comparison between the experimental and calculated wall pressure distributions is shown in Fig. 58. The two turbulence models gave similar results. Both computations significantly overpredict the pressure level in the vicinity of the corner line. Settles et al. (1986),~86~also compare the experimental and computational surface streamlines. The authors note that both turbulence models succeeded in predicting the qualitative features of the interaction footprint. Discrepancies occur in the immediate vicinity of the upstream influence and separation lines, where both calculations underpredict the forward influence of the interaction. Also, the streakangles within the interaction calculated by the algebraic model are in error by up to 20 °. Figure 59 shows some computed streamlines originating near the model apex in the incoming boundary layer. The streamlines nearest the wall are drawn into the separated flow, where they follow a helical vortex motion. The streamlines at higher levels in the boundary layer do not follow the separation zone, but they are yawed across the model and are swept downstream. More details regarding the features of the flow appear in a combined experimental/computational paper by Knight and co-workers (1992).t45~ initially, the authors compare the calculated results with appropriate experimental data. The agreement is found to be rather good. Then they draw the skin friction lines and some streamlines (Fig. 60a,b), which led them to the development of the mean streamline model shown in Fig. 61. The dominant feature is a large vortex approximately aligned with the corner. A three-dimensional surface of separation originates from the line of separation and spirals into the core of the vortex. The streamlines in this surface are strongly skewed in the spanwise direction. Another three-dimensional surface, originating within the upstream boundary

Swept-shock/boundary layer interactions

227

Fig. 59. Computed streamlines in a swept corner (Settles etal.). ¢s6)

z

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(a)

Line of coalescence

(b)

Fig. 60. Computed skin friction lines and streamlines in a swept corner flow (Knight etal., 1992).¢45)(a) Skin friction lines; (b) streamlines.

Y

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nco zr

Line~°gence ~ (separation)

\'Surface No. 1 (attachment) Not to scale

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layer, intersects tlhe compression surface at the line of attachment. This surface marks the extent of the flow entrained into the vortex. Within the upstream boundary, fluid beneath this surface is entrained into the vortex, while fluid above this surface passes over the vortex and up the compression ramp. 5.2. CROSSING SHOCKS In this section, we will present some basic information regarding the physics of the crossing shock/boundary layer interaction (Fig. 2). A precise review of computed and experimental results is included in the paper of KuightJ 43) In addition, the paper of

228

A.G. Panaras

Fig. 62. Perspectiveviewof the crossingshockflow,accordingto Garrison and Settles(1992)3TM F, fin;p, plate; r, reflectionplane; 1, incidentseparationshock;3, incidentinviseidshock;5, separationvortex;6, reflectedseparation shock (6a, centerline segment;6b, bridge segment;6c, main segment);7, reflectedrear shock;8, reflectedinviscid shock.

Garrison and Settles (1992) ~13) is important; it presents a detailed visualization of the flow field using the Planar Laser Scattering technique. The flow in a crossing shock geometry represents the interaction (or confluence) at the plane of symmetry of two counter-rotating quasi-conical vortices generated by the individual single fins. According to the measurements of Garrison and Settles (1992), ~z3) the initially fiat conical vortices gradually become vertically oriented (Fig. 62) in a mushroomshaped separation region, which occupies a major portion of the exit area of the dual-fin crossing shock geometry. Garrison and Settles (1992) ~23) observed that though the two conical vortices are reflected and distorted, the overlying 2-shock structure still persists throughout. Knight (1993) ~43) presents a model of the streamlines and shock-wave structure in which the two counter-rotating vortices, generated by the individual single fin interactions, interact at the plane of symmetry and rise above the surface (Fig. 63a). The shock structure of the model of Knight (1993) ~43) is shown in Fig. 63b at various cross-sections of the flow. Near the leading edge of the fins, the flow field contains two single fin interactions, which are characterized by a 2-shock (Fig. 63b(1)). Downstream of the leading edge, the separation shocks '2' intersect, forming a reflected shock with two segments '4a' and '4b' (Fig. 63b(2)). The 'bridging' segment '4a' rises with increasing downstream distance. The remaining segment '4b' moves towards the fins and interacts with the rear segment '3' of the original 2-shock, forming a localized high pressure region '6'. A separate curved shock '5' forms on the centerline near the surface, and is associated with the turning of the flow near the surface along the plane of symmetry (Fig. 63b(3)). Downstream of the intersection of the

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230

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inviscid shocks '1', an expansion region '8' forms, while the separate shock '5' is still present (Fig. 63b(4)). The reflected shocks '7' move towards the fins.

6. CONTROL OF SWEPT-SHOCK/BOUNDARY LAYER INTERACTION The basic principle of control of the separation of a boundary layer is to increase the resistance to separation by decreasing the momentum deficit close to the wall. This may be achieved either by increasing the energy of the lower part of the boundary layer by injecting air tangentially to the wall, or by removing the low momentum layer by suction. Both techniques have been used successfully for the control both of incompressible and compressible separations. The particular case of control of two-dimensional shock/boundary layer interactions is reviewed in detail by Delery3 *°4} To our knowledge, a very limited number of studies exist which have as their objective the control of the separation in swept-shock/boundary layer interactions, though it is desirable to reduce, if not to eliminate, in some practical configurations the conical separation vortex which appears in these types of flows. Peake and Rainbird (1975)tTa~applied tangential air injection through a longitudinal rearward facing step to a M = 2, ~t = 8° fin/plate turbulent flow. The exit Mach number of the two-dimensional nozzle was equal to 3. The jet excess momentum was slightly more than the momentum deficit of the undisturbed boundary layer. The nozzle was located upstream of the fin and it was possible to change its angle relative to the freestream, because the nozzle was a part of a turntable installed on the wind-tunnel floor, which served as the flat plate of the configuration. In the absence of control, surface-flow visualization has indicated that a substantial skewing of the boundary layer was occurring, but separation was not observed. Peake and Rainbird (1975)~74} considered the examined flow as being close to incipient separation. They observed that when the jet was activated, the skewing of the boundary layer was reduced. The optimum direction of injection was found to be along a line somewhere between the deflected surface of the fin and the line of the swept-shock (inviscid position). The use of suction for the control of separation has been considered experimentally by Barnhart et al. (1988){s~ and computationally by Gaitonde and Knight (1991).~22~ The former authors have presented limited results of control of a M = 3, 0t = 8° fin/plate turbulent flow, which was close to separation (weak interaction), while the latter authors in addition to that case have also studied a strong interaction (M = 3, ct = 20°). In what follows, some data are presented from the work of Gaitonde and Knight (1991)3TM For the 8° fin, the area of suction considered was roughly rectangular and covered a strip on both sides of the footprint of the inviscid shock. Gaitonde and Knight (1991)t22~applied suction in two regions in the 20° case: between the upstream influence line (of the regular flow) and the swept-shock (region I), and between the shock and the surface of the fin (region II). For the numerical simulation, the hybrid explicit/implicit algorithm of Knight (1984)t41~ was used, employing the turbulence model of Baldwin-Lomax (1978)t4~ with the mixing length specified by the formula of Buleev (1963).~8~The effect of bleed was incorporated through the bleed corection factor of Cebeci and Smith (1974).~9}The mesh had a size equal to 32 x 38 x 54 points. The suction was accounted in the boundary conditions by specifying the normal mass bleed at the appropriate areas. Gaitonde and Knight (1991)~22}in both interactions observed a reduction of the surface turning angles in the presence of bleed. The local skin friction was increased in the suction area. In the case of the strong interaction, the separation line persisted for all suction magnitudes and areas considered. The upstream influence line moved towards the fin when suction in region I (case D) was applied (Fig. 64b), however, the vortical structure persisted, indicating that the studied high Reynolds interaction was robust and could not be easily manipulated by suction. Actually, the only effect of suction was to ingest particles whose no-bleed trajectory passed close to the bleed area. The overall flow field was not affected significantly. Application of suction in region II (case E) had no effect in the dimensions of the separation vortex (Fig. 64c).

231

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by suction, surface streamlines (Gaitonde and Knight, 1991).(22)(a) case C, no control; (b) ease D, control upstream of the shock; (c) case E, control aft shock.

7. CONCLUDING REMARKS The physics of the flow field which is generated when a swept-shock wave impinges on a two-dimensional boundary layer has been examined by reviewing experimental and computational studies. Emphasis has been given to the most widely examined configurations on which this phenomenon appears, i.e. the fin/plate, which resembles external parts of a high speed vehicle, and the axial corner (two wedges attached normally), which simulates high speed intakes. The supersonic/hypersonic flows about these geometries are characterized by the appearance of straight separation and reattachment lines on the surfaces (if the intensity of the interaction is strong), and peak heating and pressure near the corner. The peak heating reaches values which are 10 times greater than in the equivalent twodimensional wedlGleflows.

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A.G. Panaras

The progress in the understanding of the physics of these interactions has accelerated in the last few years, primarily due to the availability of non-intrusive experimental techniques and of powerful computers, for the efficient simulation of these flows. However, it is remarkable that the essential elements of the structure of the flow field in a swepts h o c k / b o u n d a r y layer interaction were identified successfully in the early seventies, when these means did not exist, mostly by interpretation of the patterns of the skin friction lines and of the pitot pressure c o n t o u r s on cross-sections of the flow. T o d a y it is k n o w n that the separated flow in a s w e p t - s h o c k / b o u n d a r y layer interaction is d o m i n a t e d by a conical fiat vortex, which appears beneath the swept-shock. This vortex carries high energy fluid (a supersonic jet) in the region of its reattachment close to the corner causing the measured peak heating and high pressure. U n d e r certain conditions, which are not yet clearly known, secondary and even multiple vortices appear beneath the primary vortex. In case of configurations with one surface providing compression, on top of the separation bubble a A-shock structure appears. This description applies to all the configurations examined. The basic features of the flow are the same in laminar or turbulent regimes. Very recently, clear pictures of cross-sections of fin/plate turbulent flows were published. These pictures were obtained experimentally using planar Laser scattering (PLS), or computationally using second-order N a v i e r - S t o k e s solvers and fine grids. These capabilities have not yet been applied to axial corner flows, whose simulation must be easier, because the examined flows are laminar. In addition, some details of the flow field have not been clarified completely, e.g. the question of the secondary vortex in a fin/plate and of the multiple vortices in a axial corner flow.

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4. Baldwin, B. S. and Lomax, H. (1978) Thin layer approximation and algebraic model for separated turbulent flows. AIAA Paper No 78-257. 5. Barnhart, P., Greber, I. and Hingst, W. (1988) Glancing shock wave-turbulent boundary layer interaction with boundary layer suction. AIAA Paper No 88-0308. 6. Bertram, M. H. and Henderson, A. (1969) Some recent research with viscous interacting flow in hypersonic streams. ARL 75-0212. 7. Bogdonoff, S. M. (1990) The modeling of a three-dimensional shock wave turbulent boundary layer interaction. The Dryden Lectureship. AIAA Paper 90-0766. 8. Buleev, N. (1963) Theoretical model of the mechanism of turbulent exchange in fluid flows. Atomic Energy Research Establishment, TR 957, Harwell, England. 9. Cebeci, T. and Smith, A. (1974) Analysis of Turbulent Boundary Layers. Academic Press, New York. 10. Cresci, R. J., Rubin, S. G., Nardo, C. T. and Lin, T. C. (1969) Hypersonic interaction along a rectangular corner. AIAA J. 7(12), 2241-2246. 11. Charwat, A. F. and Redekeopp, L. G. (1967) Supersonic interference flow along the corner of intersecting wedges. A1AA Jr. 5(3), 480-488. 12. Cooper, J. R. and Hankey, W. L. (1974) Flowfield measurements in an asymmetric axial corner at M = 12.5. AIAA J. 12(10), 1353-1357. 13. Degrez, G. (1993) Swept shock wave/laminar boundary layer interactions, experimental and numerical results. AGARD-R-792, Paper No. 2. 14. Degrez, G. and Ginoux, J. J. (1984) Surface phenomena in a three-dimensional skewed shock wave/laminar boundary-layer interaction. AIAA J. 22(12), 1764-1769. 15. Delery, J. M. and Marvin, J. G. (1986) Shock-wave/boundary layer interactions. AGARD-AG-280. 16. Delery, J. M. and Panaras, A. G. (1994) Shock wave/boundary layer interactions in high Math number flows. A GARD-AR- 319.

17. Dhawan, S. and Narasimha, R. (1958) Some properties of boundary layer flow during the transition from laminar to turbulent motion. J. Fluid Mech. 3, 418-436. 18. Dolling, D. S. (1985) Upstream influence in conically symmetric flow. A/AAJ. 23(6), 967-969. 19. Dolling, D. S. and Bogdonoff, S. M. (1983) Upstream influence in sharp fin-induced shock wave/turbulent boundary-layer interaction. AIAA J. 21(1), 143-145. 20. Edney,B. (1968) Anomalous heat transfer and pressure distributions on blunt bodies at hypersonic speeds in the presence of an impinging jet. FFA, Report 73-157.

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(1987) Numerical experiments with hypersonic flows beneath a cone-delta-wing combination. AGARD-CP-428, Paper 20. 79. Roe, P. L. (1981) Approximate Riemann solvers, parameters vectors and different schemes. J. Comp. Physics 43, 357-372. 80. Rubesin, M. W. and Viegas, J. R. (1985) A critical examination of the use of wall functions as boundary conditions in aerodynamic calculations. Third Symposium on Numerical and Physical Aspects of Aerodynamic Flows, 20-24 Jan. 1985, California State University, Long Beach. 81. Seriba, K. W. (1988) Exploration of the importance of viscous effects on waverider configurations in simulated re-entry flow. Techn. Report, University of Glasgow, Dept. of Aeronautics and Fluid Mechanics. 82. Sears, W. R. (1948) The boundary layer of yawed cylinders. J. Aero. Sci. 15(1), 49-52. 83. Settles, G. S. (1993) Swept shock/boundary-layer interactions--sealing laws, flowfield structure, and experimental methods. AGARD-R-792, Paper No. 1. 84. Settles, G. S. and Lu, F. K. (1985) Conical similarity of shock/boundary-layer interactions generated by swept and unswept fins. AIAA J. 21(7), 1021-1027. 85. Settles, G. S. and Teng, H. (1984) Cylindrical and conical flow regimes of three-dimensional shock/boundarylayer interactions. AIAA J. 22(2), 194-200. 86. Settles, G. S., Horstman, C. C. and McKenzie, T. M. (1986) Experimental and computational study of a swept compression corner interaction flowfield. AIAA J. 24(5), 744-753. 87. Settles, G. S., Perkins, J. J. and Bogdonoff, S. M. (1980) Investigation of three-dimensional shock/boundarylayer interactions at swept compression corners. AIAA J. 18, 779-785. 88. Settles, G. S., Perkins, J. J. and Bogdonoff, S. M. (1981) Upstream influence scaling of 2D & 3D shock/turbulent boundary layer interactions at compression corners. A/AA Paper No. 81-0334. 89. Shang, J. S. and Hankey, W. L. (1977) Numerical solution of the Navier-Stokes equations for a threedimensional corner. AIAA J. 15(11), 1575-1583. 90. Stalker, R. J. (1960) Sweepback effects in turbulent boundary-layer shock-wave interaction. J. Aerospace Sci. 27(5), 348-356. 91. Stainback, P. C. (1960) An experimental investigation at a Mach number 4.96 of flow in the vicinity of a 90° interior corner alligned with the free stream velocity. NASA TND-184. 92. Token, K. H. (1974) Heat transfer due to shock wave/turbulent boundary layer interactions on high speed weapon systems. AFFDL TR-74-77. 93. Viegas, J., Rubesin, M. and Horstman, C. (1985) On the use of wall functions as boundary conditions for two-dimensional separated compressible flows. AIAA Paper No 85-0180. 94. Vollmers, H., Kreplin, H. P. and Meier, H. U. (1983) Aerodynamics of vortical type flows in three dimensions. AGARD CP-342, Paper 14. 95. Vollmers, H. (1989) A concise introduction to Comadi. DLR IB 221-89 A 22. 96. Warming, R. F. and Hyett, B. J. (1974) The modified equation approach to the stability and accuracy analysis of finite-difference methods. J. Comp. Phys. 14, 159-179.

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97. Watson, R. D. and Weinstein, L M. (1971) A study of hypersoniccorner flow interactions. AIAA J. 9(7), 1280-1286. 98. West, J. E. and Korkegi, R. H. (1972)Supersonicinteractions in the comer of intersectingwedges at high Reynolds nurabers. AIAA J. 12(5), 577-578. 99. Yee, H. C. and Harten, A. (1987) Implicit TVD schemes for hyperbolic conservation laws in curvilinear coordinates. AIAA J. 25, 266-274. 100. Yee,H. C. (1989)A class of high-resolutionexplicitand implicit shock-capturingmethods. NASA Technical Memorandum 101088.

101. Zheltovodov,A. A., Maksimov, A. I. and Shilein,E. K. (1987)Developmentof turbulent separated flows in the vicinityof sweptshock waves.In The interactions of Complex 3-DFlows, pp. 67-91 (ed.A. M. Kharitonov) USSR Academy of Sciences,Institute for Theoreticaland Applied Mechanics, Novosobirsk. 102. Shang,J. S., Hankey, W. L. and Perry, J. S. (1977)Three-dimensionalsupersonic interacting turbulent flow along a corner. AIAA J. 17(7),706-713. 103. Shapey,B. and Bogdonoff,S. M. (1987)Three-dimensionalshock wave/turbulentboundary layer interaction for a 20° sharp fin at Macli 3. A/AAPaper 87-0554. 104. Delery,J. M. 111985)Shockwave/turbulentboundary layer interactionand its control.Prog. AerospaceSci. 32, 209-280. 105. Beam, R. and Warming, R. F. (1976) An implicit finite-differencealgorithm for hyperbolic systems in conservation law form. J. Comp. Phys. 22, 87-110. 106. Degani,D. arLdSchiff,L. B. (1986)Computation of turbulent supersonicflowsaround pointed bodieshaving crossflowseparation. J. Comp. Phys. 66, 173-176.

APPENDIX

THE ELEMENTS OF A NUMERICAL SIMULATION Equations Used for the Numerical Simulation o f the Reviewed Flows

The equations of mass, momentum and energy conservation are used for the numerical simulation of a flow. If the flow is turbulent, the equations are considered in their Reynolds-averaged form. Usually, the forementioned equations are written in a flux-vector conservative form and are collectively called Navier-Stokes equations. The conservative form is particulkarly necessary when shock waves are present in the simulated flow, because it has been found that this form guarantees better capturing of the shocks. In a Cartesian system of coordinates the Navier-Stokes equations are: aQ

aE

aF

cOG

dEv

cOF~ aG~

(A1)

where Q = (p, pu, pv, pw, e) T is the vector of the flow variables, E, F, G are the inviscid and Ev, Fv, G~ are the viscous fluxes in the streamwise x-, lateral y- and normal z-direction, respectively. TILe expressions for the fluxes are given in certain textbooks: Anderson et al., m Hoffmann and Chiang. (2s) The accuracy of a numerical simulation is enhanced and the implementation of the boundary conditions is simplified if a transformation from the physical space to a computational space is performed. This transformation allows clustering of grid points in regions where flow variables undergo high gradients, like in a boundary layer. The computational space is a cube divided into equally-spaced grid points. One or two sides of the cube correspond to the surface(s) of the body, about which the flow is simulated. It is noted that while the sides of the cube are planar, the surface of the examined body may be curved. For solving the equations of motion in the computational space, they must be transformed into generalized coordinates (z, ~, r/, 0. The resulting equations in a flux vector form are: aQ

aE

aF

aG

aT = where (~ =

,

/~ =

(~xE + ~yF + ~zG),

a/~

OP~

at~

+ a-T

ff = j(r/xE + rbF + t/zG), etc.,

(A2)

236

A. G. Panaras

J is the Jacobian of transformation

defined by: J

=

w>II, 0 ah, Y, 4

Equations (A2), which are called the complete Navier-Stokes equations, may be solved to either provide a time-accurate solution (unsteady flow) or a steady-state solution (r = t). Thus, even for steady flows a time-dependent solution procedure is normally used. This means that the Navier-Stokes equations are integrated in time until a steady-state solution is achieved. In this case, however, the time-steps required are less than those needed for obtaining the solution of an unsteady flow, because the intermediate solutions have no physical meaning, so a rather large integration time-step is used, subjected only to the limits of the stability of the numerical scheme which is used. For solving the complete Navier-Stokes equations, a very large amount of computer time and storage is required. In the majority of the published papers which simulate swept-shock/boundary-layer interactions, the complete Navier-Stokes equations have been used: however, some reasonable approximations have also appeared. A reasonable approximation is to ignore the r-derivatives of the viscous terms because they are small compared to the derivatives in the normal to the two surfaces directions. In addition, due to the quasi-conical nature of the flows, conical solutions have been presented in which the flow parameters (velocity, pressure, temperature) are invariant with radial distance from the virtual origin of the quasi-conical flow. The conical solutions are useful for obtaining clear images of the flow in cuts, because the calculations are done in two-dimensions and consequently a fine grid may be used. The weak point of these solutions is the a priori assumption that the simulated flows are conical. For high Reynolds number flows, viscous effects are confined to the vicinity of a surface, i.e. within the boundary layer. Outside the boundary layer, the velocity gradients are negligible resulting in zero shear stresses. In this inviscid region, the right-hand side of eqs. (A.2) may be set equal to zero. The resulting reduced equations are the well-known Euler equations. It may be argued that the Euler equations have nothing to do with the strong viscous-inviscid interactions that characterize the presently examined flows; however, they have still been used for the investigation of particular aspects of the inviscid flow structure in the case of the axial corner flows. Numerical Schemes

For solving the partial difSerentia1 equations (PDE) of fluid dynamics numerically, the partial derivatives appearing in the equations are converted to approximate finite difference expressions. These expressions are used for rewriting the PDE as algebraic equations and are referred to asjinite diference equations (FDE). Subsequently the FDE are solved at the discrete points of the mesh. The finite difference expressions of the partial derivatives are usually derived by expanding them in Taylor series. Examining only the first derivatives of a function ZJin the direction r (denoted in the grid as i), we note that the most commonly used difference expressions are the following ones: central difference approximation

+ O(A&

au

(-IX i

Ui -

Ui-

At

1

+ O(A&

643)

forward difference approximation

backward difference approximation

(A%

Swept-shock/boundary layer interactions

237

The meaning of these expressions is that the derivative of the parameter u at the grid point i is equal 1Lothe difference of the value of the parameter at the two indicated points in each scheme. We note that the order of approximation of the central difference scheme is higher than the,' order of the forward and backward schemes (which usually are called upwind schemes). For the solution of steady-state problems, the time-derivative of the vector Q (eq. A2) is approximated by a forward difference scheme (Qt = Q" + 1 _ Qn, n: time-step). If the convective and viscous terms of eq. (A2) are considered at the time-step n, then the value of the vector Q at the time step n + 1 at a point (i,j, k) of the grid is defined explicitly by an algebraic equatJLon, as a function of the value of the flow parameters at the time step, n, at some neighboring points (which are determined by the difference schemes which are used for the discreti~:ation of the spatial derivatives). Methods based in this formulation are defined as explicit. On the contrary, if the other terms of eq. (A2) are defined at the time-step, n + 1, (or equivalently if a backward difference scheme is used for the discretization of the time derivative), then more than one unknown appear in the difference equation (for example examining only the /-direction, at the n + 1 time-step, the values of the flow parameters may appear at the points i - 2, i - 1, i, i + 1, i + 2. Hence, a set of simultaneous algebraic equations needs to be solved, in order to define the values of the flow parameters at these points at the time-step 'n + 1'. Methods based in this formulation are defined as implicit. Explicit methods, compared to the implicit ones, are characterized by the simplicity of solution and by the small computer-time per step. However, the size of the permitted time-step, At, in an explicit method is restricted by stability considerations. On the contrary, the implicit methods are unconditionally stable. Theoretically there is no restriction in the size of the time-step, but in practice, accuracy considerations limit the permitted size. Still for steady-state calculations, a larger time-step may be involved in an implicit method, leading to conv,~rgence to the steady-state solution with smaller number of steps. A very popular explicit method among aerodynamicists is that of MacCormack (1969). ~56~It is a two-stage, second-order accurate method (the equations are solved twice during a time-step) appropriate for solving non-linear equations. It is known as a 'predictorcorrector' method because at the first stage of each step, a forward differencing is applied to predict a tempo~ral value of the flow parameters, while a backward differencing is applied at the second stage of the step to correct the temporal solution. MacCormack (1971) (57) modified his original method by incorporating time splitting, i.e. the scheme is split into a sequence of one-dimensional operators. As a result, the stability condition is based on a one-dimensional scheme which is less restrictive than the original three-dimensional scheme. Hence, it is possible to advance the solution in each direction with the maximum possible time-stc,'p. Time-stepping, i.e. solving the equations at many stages during a timestep, has been used by Jameson et al. (1981) t26~ to develop a four-stage Runge-Kutta explicit method, which has been considered as very successful in solving the flow field about complete aircraft. The MacCorrnack (1971) t57) explicit method is suitable for solving both steady and unsteady flows .at moderate to low Reynolds numbers. However, it should be used with caution for solving high Reynolds numbers flow where the viscous regions become very thin and require a highly refined grid to resolve the viscous region. This leads to small time-steps and subsequently long computer times. To overcome this difficulty, MacCormack (1976) tSa~ developed a hybrid explicit/implicit scheme, which is implicit close to the wall and explicit away from it. Some of the early applications of the hybrid scheme were related to swept-shock/boundary layer interactions (Hung and MacCormack (1977) and (1978) t3 t, 32~ Section 3.1). Knight (1984) t41~ for simulating various swept-shock/boundary layer interactions, modified the hybrid explicit/implicit algorithm of MacCormack (1971) tSaJ by replacing the implicit part of it. Instead of solving the complete Navier-Stokes equations, he used a limiting form of them which is applicable in the viscous sublayer of a turbulent boundary layer, and which is solved implicitly (details are given in Section 3.2). As a matter of fact most of the United States numerical studies of swept-shock/boundary layer interactions are based on MacCormack's schemes or computational strategies.

238

A.G. Panaras

In the implicit methods, large systems of linear algebraic equations have to be solved. It is known that there are two approaches for such a task, direct methods and iterative ones. The direct methods give the solution of the simultaneously solved sets of equations during a time-step in a single application of a finite and pre-determined number of operations using an algorithm. Iterative methods consist of a repeated application of a simple algorithm, during each time-step. Most widely used are the direct algorithm of Thomas and the iterative algorithms of Gauss-Seidel and of Jacobi. The required number of operations in the Thomas algorithm is limited if tridiagonal matrices are involved in the calculations. This led to the development of the called Alternating-direction implicit (ADI) or to the fractional steps methods for solving the fluid mechanics equations, because in these methods, a threedimensional set of equations is split into a product of three one-space dimensional factors, which have tridiagonal coefficient matrices, and which are solved sequentially. Thus, the calculation procedure is simplified and the execution time is considerably reduced. A wellknown ADI method is that of Beam and Warming, tl°s) which has been employed in many codes. Widely used is the ARC3D code of Pulliam and StegerF 7) The iterative methods are efficient if the coefficient matrices are diagonally dominant. Thus, they apply well to unfactored implicit upwind schemes, which indeed have diagonally dominant coefficient matrices, but not necessarily tridiagonal (for the application of direct methods). For example, the present author (Panaras, 1992)t67~ applied Gauss-Seidel relaxation in the upwind scheme which he has used for the calculation of the flow past a fin/plate configuration.

Dissipation and Dispersion Errors The conversion of the partial differential equations (PDE) into finite difference equations (FDE) by substitution of the spatial derivatives by finite difference approximations introduces errors in the numerical solutions due to the fact that the FDE are not equal to PDE. The difference between two corresponding PDP and F D E i s called truncation error and Warming and Hyett (1974) ~96) have developed a method for deriving the called modified equation, whose LHS is the PDE and the RHS is the truncation error. By this type of analysis, using scalar model equations, it has been found that errors associated with first-order accurate methods tend to smear sharp gradients within the solution domain. These errors are called dissipation errors. Errors associated with second-order methods are known as dispersion errors and they cause oscillations of discontinuities, like the shock waves. At this point, we remind the reader that central difference schemes are second-order accurate, while upwind schemes are first-order accurate. Hence an upwind scheme simulates the shock wave more efficiently, provided that the mesh is sufficiently fine to reduce smearing of the sharp gradients. Central difference schemes are less dissipative, but they have the problem of dispersion. Fortunately, various techniques have been developed that deal with these problems. A simple technique to reduce dispersion and consequently to efficiently capture shock waves by a second-order, central-difference scheme, is to add explicitly fourth-order or second-order damping (or smoothing) terms. In applications where viscous effects are confined to particular regions (like the boundary layer), viscosity may be sufficient in itself to prevent oscillation in these regions. Therefore, the damping terms are turned on only in the inviscid regions where formation of shock waves are expected. Sensors have been developed which detect (calculate) the pressure gradients and activate the damping terms when their values are large. This technique captures the shock waves quite efficiently (within three to four grid points). Much better results are obtained by using Time Variation Diminishing schemes (TVD). These schemes capture a shock wave within a grid point at the expense of complexity and larger computer time and memory. The TVD schemes ensure monotonic behavior of the calculated flow parameters, by employing special functions in the difference schemes which approximate the convection terms of the equations. A variety of TVD schemes has been developed. They may employ upwind or central differences, they

Swept-shock/boundary layer interactions

239

may be of first- or of second-order, appropriate for explicit or implicit methods. Some second-order TVD schemes employ a limiter parameter, called flux limiter, which ensures that the scheme is second-order accurate in the smooth region of the field and switches to first-order accurate in regions where large gradient exist: the aim here is to prevent oscillations in the results. The works of the present author which are reviewed in Sections 4.1.2 and 4.2, for the fin/plate configuration, have been obtained with a code based on a TVD second-order scheme developed by Yee and Harten (1987).t99! A variety of TVD schemes, explicit and implicit, are described by Yee (1989)J 1°°)

Computational Domains and Boundary Conditions A common teature of the configurations shown in Fig. l a - d is the existence of a swept shock-wave/boundary layer interaction near the intersection of one planar surface that may or may not provide compression (wedge or fiat plate) with another surface that provides compression (a fin or wedge or a swept ramp). In an experimentally studied fin/plate or swept-corner configuration, the compression surface is attached to the plate far from its leading edge to avoid interaction of the leading-edge shock with the swept-shock wave. Also, care is taken so that the vertical extent of the fin or of the wedges, or the lateral extent of the swept compression corner is much greater than the size of the interaction domain. The construction of the computational domain of a numerical simulation is usually based on the following rules. This domain is confined in the immediate interaction region, and it does not cover the whole extent of the wedges or of the fins. Care is taken to include the shock formation, as well as the lateral extent of the interaction (the upstream influence line) into the comp~atational domain. In the case of a fin/plate, the calculations start a few boundary layer thicknesses upstream of the fin, while in an axial corner or in an intake-type configuration, the inflow plane is located a few grid points upstream of the leading edges. For an adequalle resolution of the viscous effects, clustering is applied close to the surfaces. In the axial direction, in the fin/plate configuration equidistant grids are usually used, while clustering is necessary in the vicinity of the leading edges of an axial corner or of a wedge/plate configuration, for better resolution of the leading edge shock waves. Some typical computational domains are given in Fig. A1 for the axial corner, the fin/plate and the swept corner configurations. It is evident that by restricting the mesh as regards the extent of the irtteraction, the number of the required grid points can be reduced and the calculations considerably simplified. If, alternatively, the compression surfaces were assumed to have finite dimensions, the calculations would be complicated. For example in the case of the fin/plate configuration, the whole fin should be included within the computational domain (like a wing attached on a plate). Apart from the considerable increase of the number of grid points, the square edges on the fin would probably create numerical problems. An all-around conical-flow calculation has been applied by Degrez (1993) tl 3~for the computation of the flow about an asymmetric dihedron (Fig. A2).t66~ For starting the calculations in a fin/plate or a swept corner, it is necessary to know the boundary layer profile upstream of the interaction region. This profile is used as a boundary condition on the inflow plane, as well as an initial condition of the flow field. Furthermore, the thickness of the undisturbed boundary layer is the length scale of the interaction. If the boundary layer profile is not available, knowledge of its thickness is sufficient because one may be able to define a proper two-dimensional fiat-plate profile. If nothing is known about the boundary layer upstream of the interaction, its development along the flat plate must be calculated from its leading edge to the vicinity of the fin. Fortunately, a two-dimensional boundary layer solver will be sufficient for such a task (independent calculation, not part of the computation of the interaction domain). In all the examined configurations, the gradients of the flow parameters are set equal to zero on the farfield (upper and lateral boundaries) and on the outflow boundary. These boundary condiJ~ions implicitly mean that the fins or the compression corners have unlimited dimensions. The walls are assumed impermeable and no-slip boundary conditions are

240

A.G. Panaras Grid-point system 0.? 02 0.

ol 0.2

0.1 0

0.1

0.2

0.3

0.4

0.5 u.o Z (ft)

u. i

(a)

Corner f l o w Fin

Plate Fin/plate

(b)

K ""i

J

.-a

,,' ',

x

L~(" / I,'" / D ,- " " | Z -"" / Compression G .-'" [ /H.-"" / corner

( ........ ~ : ~ ~"c'/c~ net,ine F,~low ! ~ ~ h t (c)

l,ading edge Compression corner

Fig. A1. Typical computation domains; (a) intake-type flow (Shang and Hankey, 1977),(89) (b) fin/plate configuration (Panaras, 1992)(67);(c) compression corner (Knight et aL, 1992).(4~

Swept-shock/boundary layer interactions

241

1.0

Y 0.0

-2.0

-1.0

0.0

x

Fig. A2. Grid used by Degrez (1993)t13) for the calculation of the conical-flow about an asymmetric dihedron.

applied. The p~ressure gradient normal to a wall is taken equal to zero, as well as the temperature, if the wall is assumed adiabatic. If it is required to calculate the heat transfer rates on the walls, the wall temperature must be known.

Turbulence Models Applied to Swept-Shock~Boundary Layer Interactions It is well-known that for the prediction of turbulent flows by numerical solutions to the Reynolds averaged Navier-Stokes equations, it is necessary to model the apparent turbulent stress and heat flux terms. A considerable number of models have appeared through the years for the siraple reason that each of them has acceptable accuracy over only a limited range of flow conditions. The turbulen~I viscosity models constitute one broad category of turbulence models that are used extensively for external or internal flows of practical interest. The basis of this category is the use of the Boussinesq assumption, according to which the apparent turbulent shearing stress might be related to the rate of mean strain through an apparent scalar turbulent or 'eddy' viscosity coefficient #t. The same reasoning is applied in the case of the apparent tu:rbulent conductivity term. This term is assumed to be proportional to the gradient of temperature through the turbulent conductivity coefficient K , Then, Kt is related to the eddy viscosity by means of the Reynolds analogy, which is based on the similarity between the transport of heat and momentum. Experiments confirm that the ratio of the diffusivities for the turbulent transport of heat and momentum, called the turbulent Prandtl number, at = Pt cp/Kt, is a well-behaved function across a flow. For estimating/at, it is assumed that this quantity depends on a characteristic velocity Ut, and a length scale It. Both of these parameters may be defined by algebraic equations, or, one by an algebraic equation and the other by a partial differential equation (PDE), or both by PDEs. Most models currently employed in applications are of the type of the turbulent viscosity/diffusivitymentioned above. This is the rule for the algorithms which have been

242

A.G. Panaras

applied for calculating swept-shock/turbulent boundary layer interactions. However, this hypothesis is not valid for certain flow situations. The assumed isotropy (i.e. the same values of the coefficients are valid independent of direction) is the major weakness of these models. For instance, for the flow in a tube of rectangular cross-section, turbulent viscosity and diffusivity depend on the stress or flux component considered. This is also true in the case of strong influence of body forces which act in a preferred direction. In order to account for the particular development of each individual stress, models are being developed which do not assume that the turbulent shearing stress is proportional to the rate of mean strain. Instead, transport equations for the Reynolds stresses are written and solved. These are the Reynolds stress models, which are characterized by higher complexity and computer requirements. Historically, the evolution of turbulence models started with the development of the algebraic models. Modeling by algebraic equations has the advantage of computational simplicity and the disadvantage of dependence of the model upon only local flow parameters. In reality, it 'is felt that a turbulence model must provide a mechanism by which effects upstream can influence the turbulence structure downstream. This has been the trend for the last 20 years, during which numerous models have appeared, which consider the upstream 'history' by using one or two PDE's for the definition of #t. The computations, of course, become more complicated. It is shown in Section 3 that in the case of the swept-shock/turbulent boundary layer interactions, the algebraic models till now have provided solutions which are closer to the experimental evidence, than the k-e model, which is the typical two-equations model in use today. ALGEBRAIC MODELS The most common among these models are the two-layer model of Cebeci and Smith (1974) (9) and that of Baldwind and Lomax (1978) ¢4). These models provide similar results; however, the latter is more widely used in aerodynamic calculations, because, unlike the former model, it does not require knowledge of the boundary layer thickness. For this reason, we will describe here the Baldwin and Lomax (1978)(4) model and how it has been applied for the numerical simulation of swept-shock/turbulent boundary layer interactions. In the inner-layer, Baldwin and Lomax (1978)(4) use the Prandtl-van Driest formulation: (~,)i . . . . = p (x D ?/)2 co

(A6)

where x is the von Karman constant, D is the van Driest damping factor, co is the absolute value of the vorticity and ?/is the distance normal to the wall. In the outer region, the following equation is used: (/h)out©r = C~p(0.0168 pFwake fl),

Fwake ----the smaller of f~?/max Fmax 2 Cwk ?/maxUdif/Fmax

(A7)

The quantity F~x is the maximum value of the function F(?/) = ?/coD, and ?/~x is the value of ?/at which it occurs. The Klebanoff intermittency factor fl is given by f l : I 1 + 55(Ckleb----~6~-l" ~ ?/max /] _]

(A8)

The quantity Udifis the difference between maximum and minimum velocity in the profile. The constants appearing in the previous relations are: Ccp= 1.6, C,,k = 0.25 and Ckleb = 0.3. In the particular geometries of Fig. 1, where there are two intersecting surfaces, a 'modified distance' (originally developed by Gessner and Pu (1976)(24) for application to square ducts) was proposed by Hung and MacCormack (1978) c31) to account for the turbulent

Swept-shock/boundary layer interactions

243

mixing length near the intersection of the surfaces: 2yz

t/= y + z +

(A9) + z 2)

In all the published numerical simulations of axial comer or fin/plate turbulent flows, in which algebraic turbulence models are used, the modified distance is employed for the calculation o f / # Another point which needs attention when the Baldwin-Lomax model is applied to flows with substantial crossflow separation, like in the cases examined in this review, is the proper calculation of the distance from the wall where the momentum of vorticity, F(t/), takes the maximum value. Actually, in both layers of the turbulence model, /z, depends on the absolute value of the local vorticity vector, to, and on the distance from the wall, t/. Especially in the outer layer,/z,, depends on the maximum value of the momentum of vorticity, F(t/). In the case of attached boundary layers, the profile of F(r/) has a single maximum. However, in the presence of cross flow separation, a second maximum of greater value appears. This second maximum is due to the overlying vortical structure. Thus, if in a code, the computer searches outward along each normal to a surface to determine the peak of F(r/), it will select the second maximum. This will cause an underestimation of the extent of the cross flow (because/z, will take greater values, i.e. the calculated flow will be more turbulent t]aan the real one). Degani and Schiff °°6) modified their alogrithm, so that the first peak of F(t/) is selected in each profile. But usually, even a boundary layer may have two peaks of the vorticity momentum: one in the sublayer and one far from the wall. Panaras and Steger (1988) (69) applied a simple procedure for the estimation of the proper values of F(t/), that requires the drawing of the profiles of this function at various streamwise stations and the selection by the user, and not by the computer, of a single cut-off distance that divides the boundary layer and the vortical structure. This technique proved efficient for the accurate prediction of the secondary separation occurring in the flow about a prolate spheroid, which would otherwise be suppressed by the high values of #,. The PanarasSteger interpretation of the Baldwin-Lomax turbulence model has been applied by Panaras and Stanewsky (1992) t6s) for the simulation of a fin/plate flow in which according to the experiments secondary separation occurred. Calculations of other researchers (Knight et al., 1992)t44~who applied the standard Baldwin-Lomax turbulence model in similar flows, have not exhibited the secondary separation phenomenon (details will be given in Section 4). TWO-EQUATIONS MODELS The so called k-e model is one of the most widely used two-equation models. This model was first proposed by Harlow and Nakayama in 1968.(26~ Subsequently, many variants of this model have made their appearance. The model of Jones and Launder (37~ is very popular. In the k--~ model, the characteristic velocity Ut is proportional to the square root of the kinetic energy of turbulence k, and as length scale, the dissipation length l~is used. The dissipation length is related to the turbulence dissipation rate, ~, through an algebraic equation. The turbulent kinetic energy and the turbulence dissipation rate are obtained by two transport equations, which contain some empirical constants. The turbulence, transport equations for the k-e model have been devised originally for high Reynolds number, fully turbulent flows. Near walls, where viscous effects become dominant, these equations must be modified so that a wall damping effect may be incorporated into the empirical constants of the transport equations. The proposed modifications are based largely upon numerical calculations and comparisons with experiments. Thus, following tlhese modifications, there is no clear set of equations which can be used successfully in a broad spectrum of flows. A systematic evaluation of some representative near wall turbulence models has been done by Patel et al. (1985).(72) They have concluded that some of the methods yield comparable results and perform better than others. However, even these need refinement if they are to be used with confidence to calculate /J/

244

A.G.

Panaras

arbitrary near wall flows~ Actually, the topic of the near wall turbulence modeling is still open to research. Alternatively, the wall-function approach may be used. The vanishing of the convective transport of a fluid near a wall and the resulting dependence of the equations essentially on the normal distance (although corrections will be applied for 'non-equilibrium'), is the theoretical basis of the wall-function approach. Though the dropping of the streamwise convection from the momentum equations is justified over only a thin wall-layer region, nevertheless, this is usually a sufficiently extensive zone to cover the entire low-Reynoldsnumber region (Launder)3 TM Thus, the dependent variables (u, k, ~) may be expressed as functions of the normal distance and of various flow parameters. In practical applications, the flow equations are solved by using a turbulence model in the outer flow region on a relatively coarse grid, and the near wall region is 'patched in' by using the wall functions for developing the boundary conditions at the first computational point away from the wall. In this approach, the zone in which convective transport is relatively unimportant, i.e. the law of the wall, is assumed to be valid in the range 30 < y+ < 200 and the first computational point away from the wall is located in this interval. (y +: normalized distance from the wall = yU~/v), U,; friction velocity = (zw/p)l/2). Various comparisons have been made concerning the accuracy of the results of the two alternative approaches, i.e. that of the wall functions and that of integration directly to the wall by using near wall turbulence modeling. There is no general agreement as to which approach is the best. For example, Patel and Chen ~7~ when comparing the two approaches in the case of the turbulent wake of a fiat plate, report that the direct integration technique gives better results, without a significant increase of the computation time. On the contrary, Rubesin and Viegas (1985)~8°~report that the wall-function approach gives results closer to the experimental values in the case of shock-boundary layer interactions. At this point it is noted that Rubesin and co-workers (1985)~8°~ succeeded in developing an adequate wallfunction model for the types of flows examined, in the present review. Some applications of this model are reviewed in Section 4. The cost-effectiveness is an area where the wall-function approach is definitely superior. Indeed, in the case of the approach integration to the wall, very fine mesh-spacings are ~~i~quired to achieve the necessary resolution, due to the rapid variation of the dependent variable with distance from the wall. Then, the allowable time-step, even in the case of use of an implicit algorithm, is decreased and the number of iterations required for the convergence of the numerical solution increases. Obviously, by using the wall-function approach, the many close-spaced points near the wall are not required. Consequently, the time-step may be increased and the number of iterations required for convergence is reduced.