AlAA 92-3666 Numerical Study of Secondary Separation in a Glancing ShockRurbulent Boundary Layer Interaction A.G. Panares and E. Stanewsky Institute for Experimental Fluid Mechanics DLR-Gottingen Germany

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AIM-92-3666 NUMERICAL STUDY OF SECONDARY SEPARATION I N GLANCING

SHOCWTURBULENT BOUNDARY LAYER INTERACTIONS

Argyris G. Panaras 1) and Egon Stanewsky 2)

German Aerospace Research Establishment e.V.

Institute for Experimental Fluid Mechanics

Bunsenstr. 10, D - 3400 Gottingen

Germany

Abstract

Experimentally it has been found that in moderate strength glancing shocWnubulent boundary layer inter- actions, as they occur, e.g., in supersonic and hypersonic intakes, a secondary separation line appears in the surface flow pattern. In the present paper, a flow of this type, studied at the Pennsylvania State University, is simulated numer- ically. It is shown that if the turbulence model of Baldwin & Lomax is applied according to the physics of the flow, the resulting solution agrees very well with the experimental evidence (wall pressure, skin friction, flow angle). Then, post-processing of the solution reveals that in this type of interaction the secondary separation phenomenon is similar to that obsetved in flows around bodies at high incidence. Furthermore, it has been found that the secondary separation adversely affects the conical nature of the flow. The dynamic characteristics of the conical vortex which are known to appear in these types of flow change in such a way that the various flow parameters exhibit a variation along conical rays in the region of the conical vortex, instead of remaining constant, a requirement for a purely conical flow.

-

ing to this model, if the interaction is strong, the separated flow is made up of two vortices. A tight, vigorous conical one in the comer and a weaker, elongated one above it (Figures la , b). Subsequent studies have confirmed the qua- si-conical nature of the interaction. A detailed review has been presented recently by Settles and Dollingz. A remark- able visualization of the conical nature of the flow has been given by Alvi & Settles3. Using conical shadowgraphy (placing a light source at the virtual origin of the approxi-

1. Introduction (a) weak interaction

The three-dimensional shock boundary layer interaction which results when a sharp or blunt obstacle is attached normal to a surface has been extensively studied over the last 20 years. There is now a renewed interest because this type of comer flow appears in parts of a supersonichypersonic vehicle and is of particular importance to the efficiency of air intakes. The oblique shock wave which is generated on or upstream (detached) of an obstacle crosses the boundary layer, which grows along the surface, and due to the subsonic part of the latter the shock pressure increase is smeared out

covers a significant part of the flow upstream and down- stream of the inviscid shock position. If the shock is suffi-

along the wall so that a disturbed flow pattern appears which Ibl

(b) strong interaction

ciently strong the flow separates and the topology of the flow changes significantly. The currently accepted flow model has been proposed by Kubota and Stolleryl: Accord-

Figures la, b Flow Stollery '

by Kubota ' ,) Visiting Scientisl, Member AlAA 2) Head, High Speed Aerodynamics Branch, Member AlAA

Copyright B American Inslilutc of Aeronaulics and Aslronautics, Inc. 1992. All rights rcserved '-,

1

VCO =bJRTUAL CONICAL ORIGIN

t

iMENT

\\ \ INVlSClO SHOCK LOCATION

SEPARATION LINE

'UPSTREAM INFLUENCE LINE

Figure IC: Surface flow pattern (121 modified)

mate conical flow field), they obtained pictures in which the various cross-sections of the conical flow field almost col- lapse in a single two-dimensional shock boundary layer interaction image.

In addition to the experimental studies, numerical sini- ulations have been performed which have revealed critical features of the flow field. Concerning the existence of the conical vortex, Knight et have shown that the trajectories of streamlines that originate upsweam of the separation line close to the wall rise, cross the separation line and rotate in the direction in which the separation vortex should rotate. Very recently, Panarass using appropriate post-processing tools (Vollmers6) has confirmed the model of Kubota & Stolleryl, the conception of which was based mainly on sur- face data. In his illustration, the flow is dominated by a large vortical strncture (separation vonex) which lies on the plate surface and whose core has a remarkably conical shape with a flattened elliptical cross-section. Along the fin and close to the corner, a slowly growing smaller vortex develops. On top of the conical vortex and along it a h-shock is formed. In addition, Panaras has found that in spite of the apparent conical shape of the separation vortex the flow is only sec- tionally conical and not in the total. This means that tlie various flow variables do not remain constant along rays that Stan at the origin of the conical flow field, but they vary slowly. Using a conical projection technique similar to the one used experimentally by Alvi & Settles), Panarass has shown that this deviation from conical behaviour is mainly due to the effect of the different rate of development of the boundary layer along the plate, compared to the development of the conical vortex.

For validating the numerical procedure that he used, PanarasS has compared his results with the experimental data of Shapey & BogdonofP. The type of the flow described by these data is simple in the sense that tlie surface visualization indicates that only primary separation exists. Close to the

sulface, the flow particles move between the separation and the reattachment line (Fig. IC). In this case the separation vortex system is also simple. The primary vortex develops conically downstream, similar to the structure of the flow investigated by Kubota & Stolleryl. L/

Very recently G . S. Settles and his associates (Settles8, Kim et 211.9) have presented experimental data which include surface evidence of a secondary separation. According to Alvi & Settles), in supersonic turbulent swept interactions the line which is called the "secondary-separation line" has been an enigma. Usually it is too weak to qualify unequivo- cally as a secondary separation of the reversed flow within tlie primary separation zone. Besides, it appears only for moderate strength interactions and not for weak or very strong ones.

In this paper one of the flows which the Pennsylvania State University researchers have found to include a sec- ondary separation is studied numerically so that some fea- tures of this extraordinary phenomenon can be clarified. Fortunately, the experimental data include skin friction dis- tributions and the angle of the skin friction lines with the free streams. Thus, before trying to study the physics of the flow, the numerical technique and especially the turbulence model can be validated. Kim et a1.9 have already tested the various existing turbulence models. They have found that the alge- braic models are superior to the k-E model as far as the pre- diction of the skin friction is concerned. In the present cal- culations the two-layer algebraic model of Baldwin & Loinaxlo is used. This model predicts too much viscosity in cases of flows with large vortical structures. Therefore, some details of the flow, like the secondary separation, are sup- pressed, unless the contribution of the vortices to viscosity is disregarded when the turbulent viscosity of the outer layer is estimated. Degani & Schiff" have introduced this concept for the calculation of low-speed vortical flows. In addition, Panaras & StegerlZ have developed a simple numerical pro- cedure for the implementation of this concept within an algorithm. In this paper it is shown that the same principles can be applied to high speed vortical flows.

L./

/ TEST CASE : M,=3.0

6,=2.9 mm / Sketch of geometric characteristics

2

2. Numerical Method

2.1 Algorithm, Grid, Boundary Conditions

The numerical algorithm which was used in the present calculations has been developed for calculating laminar or turbulent comer flows. The Navier-Stokes equations in dimensionless conservation law form and in general coordi- nates are

-

W

where Q = J-'(p, pu, pv, pw, e)' is the vector of the con- servative variables, and J is the Jacobian of the aansforma- tion from Cartesian to general coordinates. E, F, G are the inviscid and E,, F,, G, are the viscous fluxes in the stream- wise 5-, lateral q-, and normal (-direction, respectively. The expressions for the fluxes are given in certain textbooks. In the case of the fidplate configuration, walls exist in two directions (q and 6). Thus it is necessary to apply the concept of the thin layer approximation to both of them. Furthermore, in the comer region, the viscous terms associated with cross derivatives may be of the same order of magnitude as the normal derivatives which are the only ones retained in the thin-layer approximation. Considering these facts the viscous terms F,, G, are retained.

Stokes' hypothesis is used for relating the two viscosity coefficients which appear in the viscous fluxes. Furthermore, the Sutherland law is employed for the estimation of the molecular viscosity. The above system of equations is valid for laminar as well as turbulent flows. In the case of the latter, the molecular transport coefficients are only replaced by their turbulent counterparts. The turbulence model is described in Paragraph 2.2.

The equations in (1) are solved at the interior grid points of a boundary-fitted structured mesh. A second order central differencing is applied to the implicitly treated viscous flux- es, while the terms with the cross derivatives are considered explicitly. The inviscid fluxes are determined by the upwind total variation diminishing (TVD) scheme of Harten & Yeel3 which uses Roe's approximate Riemann solver and Harten's second order modified flux approach. Employing the first order Euler implicit formula, the inviscid fluxes are approximately linearized. Since the resulting linear system is block-diagonally dominant with respect to the spectral radii, it may be solved by a relaxation method. In the present case, where time marching was applied for calculating the steady-flow solution, alternating Gauss-Seidel relaxation in the streamwise direction was employed. More numerical details are given by Miiller'4 who describes the basic thin layer Navier-Stokes algorithm on which the present corner- flow code is based.

The present mesh was generated algebraically. For an adequate resolution of the viscous effects a clustering was applied close to the plate and to the fin. In Figure 2 the mesh which was used is shown. In each crossflow plane (z, y directions) 95 x 79 points were used, while in the streamwise x-direction there were 45 grid planes uniformly spaced with Ax = 0.858-. The input plane was located at a distance equal

L,

to 26- upstream of the leading edge of the fin, and the downstream boundary at x 35&. The height of the com- putational domain was 6.68-. The width was uniform before the fin, viz., lo&, but from there on it increased to z = 298- at the outflow plane. This widening was necessary to insure that laterally the computational domain extended more than the interaction domain so that the boundary conditions could be applied.

A boundary layer profile was prescribed at the inflow plane which approximately corresponds to the experimental one, given by Settles8. This profile was also used as the ini- tial condition of the flow field. At the outflow boundary as well as in the far field (upper and lateral boundaries), the

-direction 65 mesh paints

Figure 2 Computational grid

gradients of the flow were set equal to zero. The wall was assumed impermeable and no-slip boundary conditions were applied. Also, the wall was assumed adiabatic and the pres- sure gradient normal to the wall was set to zero. At the symmehy boundary (upstream of the fin, at z=O), the normal component of the velocity was set to zero; the normal derivatives of the remaining flow quantities were also assumed to be zero

2.2 Turbulence Model

It has k e n mentioned in the introduction that the two- layer model developed by Baldwin and Lonp 'o was used. The calculation of the eddy viscosity is carried out in the crossflow planes (y-z planes), for successive streamwise positions (x-direction). In each y-z plane, the computational domain is an orthogonal rectangle formed by the plate and by the fin. In the inner region the Prandtl-van Driest formu- lation is used

(k),,,, = P(KDq)2W (2)

where K is the von K m a n constant, D is the van Driest damping factor, w is the absolute value of the vorticity and

is the distance normal to the wall. In the particular case examined here, where there are two walls, a modified dis- tance developed by Hung and MacCormackl5 is used:

This modified distance accounts for the turbulent mixing length near the comer under the influence of both walls. In the outer region, the following equation is used

3

The quantity F,, is the maximum value of the function F(q)=qwD and qm. is the value of q at which it occurs. The Klebanoff intermittency factor p is given by

The quantity udlr is the difference between maximum and minimum velocity in the velocity profile (in the present case it is equal to uJ. The constants appearing in the previous relations are

Ccp = 1.6 C,, = 0.25 C ~ l ~ b = 0.3

3. Results

The flow to be simulated here has been well documented by Settless. The characteristic parameters are: M_ = 3.03, Re = 6,19xlO'/meter, f in angle a = 16' and thickness of the boundary layer upstream of the fin 6- = 3.02mm.

In the following section, the important effect of the tur- bulence model is fust studied and it is shown how the Baldwin & Lomaxlo turbulence model has to be interpreted in these types of flow for a reliable simulation. Then, visual images of the flow are presented which reveal the vortex system. The secondary vortex is well illustrated.

3.1 Effect of Turbulence Model

In both layers of the turbulence model, p, depends on the absolute value of the local vorticity vector, w, and on the distance from the wall, q (see Eqns. 2. 4). Moreover, in the outer layer, w, depends on the maximum value of the momentum of vorticity, F(q). In the case of attached boundary layers, the profile of F(q) has a single maximum. However in the presence of crossflow 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 sur- face to determine the peak of F(q), it will select the second maximum. This will cause an underestimation of the extent of the crossflow whose magnitude depends on the strength of the vortical structure. For eliminating the forementioned problem, Degani & Schiff" modified their algorithm so that the fust peak of F(q) is selected in each profile. But gener- ally, even a boundary layer may have two peaks of the vor- ticity momentum, one in the suhlayer and one far from the wall. Panaras & Steger'z applied a simpler procedure for the estimation of the flow about a prolate spheroid which, how- ever, requires more computation time. In a first step they calculated the flow without considering the distinction between the multiple maxima of F(q). Then, after reviewing the profiles of that function, they defined a cut-off distance that divides the boundary layer and the vortical structure. The same procedure was applied in the present calculations.

First, the flow was calculated without considering a cut-off distance in the turbulence subroutine. Then the pro- files of F(q) were reviewed in order to determine the cut-off distance. In Figure 3a profiles of F(q) are shown at some typical positions within the crossflow plane x = 34.56.. In Figure 3b the contours of the density distribution at the same cross section have been drawn. Thus it is easy to judge to which part of the shock boundary layer interaction region each profile corresponds. It is observed in Fig. 3a that the

LI

VORTlCllY FUNCTION Fl l l

a. Profiles of the vorticity function F(q)

b. Density contous at d6.. = 34.5

Figure 3: Profiles of the vorticity function F(q) and density contours in the crossflow plane xI6" = 34.5

maximum value of the vorticity momentum of the separation vortex is more than five times greater than that of the boundary layer. It is also indicated that the cut-off distance that divides the boundary layer from the vortical structure is approximately equal to K = 32. This value was considered for the final calculations.

For the flow that we have simulated, Settles8 gives the measured skin friction coefficients and the surface flow angles (p) along a single circular arc at radius R = 89mm and the surface pressure at radius R = 101.6mm from the fin leading edge. In Figure 4 the initial approximate solution is compared with the experimental data. It is observed that in the case of the skin friction the agreement is reasonable, but the agreement is poor in the case of the surface flow angle. Regarding the pressure (Fig. 4c), the agreement of the cal- culation with the experimental data is marginal. The low- pressure region between the two peaks, i.e. the one at the separation shock and the one in the reattachment region, is not present in the calculation. Also, the skin friction com- ponent normal to the shock (Fig. 4d) does not approach zero between the separation and reattachment points as is indi- cated by the experimental data. b

4

@Skin Friction @ Surface Flow Angle Figure 4 Comparison of 1 A 1 ,*ih. calculated and measured

58 data; approximate solution

,885

,881

W 1 fi @ Static Pressure

: ,883 " .e82

@Skin Friction Normal to the Shock

The same type of comparison is shown in Figure 5 for the final calculations where the turbulence model has been applied correctly, i.e., according to the physics of the flow. In this case the agreement in all parameters is very good. Especially we note how accurately the details of the flow are simulated, e.g., the low pressure region (Fig. 5c) and the two separation regions (Fig. 5d), and the good agreement in the extent of the interaction (Fig. 5b). The dishibution of the skin friction component normal to the shock shows, in addition, some interesting feature of the flow structure: The normal component goes to zero at the primary separation point (line), becomes negative and touches zero again where secondary separation seems to occur. In the region of the main vortex, a very strong negativ skin friction component is indicated which again goes to zero at reattachment. The behaviour of the skin friction coefficient C," is similar to the development in a two-dimensional shock boundary layer interaction and seems to justify the definition of a three-di- mensional separation by the criterion that the normal com- ponent of the skin friction becomes zero.

b

Figure 5: Comparison of calculated and measured data: final solution

- Calculation --.-*i--Measurement

3.2 Analysis of the Flowfield

The interpretation of the pattern of the skin friction lines provides essential information concerning flow separation. Thus we start the analysis of the flow field by considering the calculated skin friction lines on the flat plate, Figure 6. It is seen that the surface pattern is different from the one shown in Fig. IC which corresponds to flows without sec- ondary separation. Between the separation and reattachment Lines (S, ,R,) a secondary separation line (&) has been formed. It originates downstream of the apex of the fin and appears as an asymptote to the skin friction lines which leave the reattachment line. This pattern is very similar to the one established in flows about bodies at high incidence (see for example Ref. 12).

Concerning the visualization of the vortices, an effective technique of automatic detection of vortices in a three-di- mensional flow has been proposed by Vollmers et al.lS. They have shown that vortices exist in those parts of a flow in

5

,vco

VCO =VIRTU

Inviscid F

which the discriminant of the tensor of the gradient of the velocity (au$a.tj) indicates complex eigenvalues. The dis- criminant is evaluated numerically at all points of the flow field. Then contour surfaces of constant values are created and displayed. These contours indicate where in the field vortices exist. The discriminant technique has been incorpo- rated by Vollmers6 in a graphic system called Comadi.

In Figure 7, the contours of the eigenvalues of the velocity gradient field are shown along with two cross sec- tions for which density contours have been drawn. The den- sity contours visualize very clearly the shock system which is formed along and on top of the vortex which dominates the flow. This vortex lies on the flat plate and its core has a conical shape with a flattened elliptical cross section. Also indicated on the side of the main vortex is a very thin vortex core which has developed in the direction of the flow. This is not another independent vortex but the core of the vorticity sheet which lifts off the surface along the separation line and rolls up to form the main conical vortex. Along the vertical fin and close to the comer, the longitudinal vortex, men- tioned by Kubota & Stollery', is seen. It also develops qua- si-conically but at a smaller rate of increase. Below the vor- ticity sheet and parallel to the prime conical smcture another thin vortex has developed which is, however, better detected in Figure 8a where a perspective view of the skin friction lines together with the vortical structure is shown. It is clearly demonstrated in this figure that the secondary vonex is gradually formed along the secondary separation line. For comparison, the same data are shown in Fig. 8b for the initial approximate solution. In this case the secondary separation line is incomplete and there is no evidence of the secondary vortex. However, the surface flow along the fin is similar in both solutions.

In Figure 9 the cross section of the vortices is shown in a crossflow section close to the outflow plane. In the same figure, the corresponding variation of the surface pressure and of the skin friction coefficient are also indicated. Con- sidering the fact that the height of the calculation domain is 6 .68 , it is easily estimated that the thickness of the second- ary vortex is about 0.56-. Furthermore, the location of the

Figure 7: Perspective view of the vortices and of the shock waves - - A CORNER VORTEX

' PRIMARY VORTEX

a. Final solution

h. Initial approximate solution Figure 8: Skin friction lines and cross section of the vortices

secondary vortex coincides with a peak in the skin friction distribution in the direction normal to the shock, ch. The existence of this peak gives the impression that the sepa- ration bubble is split into two regions. However, we believe that here it is also appropriate to interpret it as region where the flow direction coincides with the direction of the glanc- ing shock (see also Fig. 6). A I other flow conditions the direction of the secondary separation line may not coincide with the direction of the shock and cy. may become positive in the region of the secondary separation. It is furthermore ' -

3 . 5

3 . 8

3a 2 . 5

d

2 .E

.. \. x., , '.

primary vortex '. 1 .B 1

2 4 6 8 io 12 I? 1 6 18 29 21 21 26

I Z/DELTA

,005

E ,004 -. k ,003

,002

.EO1

,088

-.001

- ,0412 0 2 4 5 8 l a I 2 I" I 6 12 26 22 2 4 26

Z/CELTA - Figure 9: Cross section of the vortices and variation of wall pressure and skin friction (~16.. = 32.8) seen in Fig. 9 that the secondary vortex lies close to a sec- ondary peak in the total skin friction coefficient, cy. This fact has already been observed by Kim et al.9. Concerning the wall pressure distribution, there is no direct evidence indi- cating the existence of the secondary separation. However, it will be shown in the next section that the level of the low-pressure region below the core of the conical vortex is strongly affected by the existence of the secondary vortex. Finally it is seen in Fig. 9 that the maximum pressure appears between the conical vortex and the fin in a region where the air processed through the shock system reattaches on the plate.

In Figure 10a the velocity vectors have been drawn in a cross-section normal to the shock wave at x = 32.8L. The existence of the conical vortex is well shown in the velocity plot. However, no circulatory motion is detected in the regions of the comer vortex and of the secondary vortex. This is so because the axes of these vortices are not parallel to the glancing shock, as is the axis of the primary conical vortex. Trying various angles we have found that the sec- ondary vortex appears in the velocity plots when a cross section normal to a line which forms an angle of 42' with the freestream direction is considered (the shock angle is 33.4' ). An enlargement of the region where the secondary vortex appears is shown in Fig. lob. It is seen that a weak circulatory motion exists very close to the surface of the plate at d6., = 12.2 to 15.5. At ~16.. > 15.5 the flow near the

'-

7

surface is again directed towards the primary separation line which corresponds to the skin friction distribution cln shown in Fig. 9. Note that in Fig. 10% a connection of arrows does not represent streamlines since we are considering a helical motion with an out-of-plane component.

If the vorticity is used as a visualization pardmeter other specific details of the flow field appear. In Figure l l a the iso-contours of the absolute value of the vorticity have been drawn in two cross sections of the flow field. In addition to the shock system, which has already been visualized by the density contours in Fig. 7, the shear layer which originates at the triple-point and impinges in the corner is clearly identified. The vorticity distribution within the conical vortex is another critical feature of the flow which is shown in Fig. l l a . It is seen that the vortex is continuously fed with the vortical fluid of the boundary layer. The secondary vortex is clearly indicated on the surface of the plate by the protruding region of low-vorticity below the triple-point. It is seen that this region increases in size downstream, causing a "chan- neling" of the conical vortex. Alvi & Settles", using data obtained by planar laser scattering, have constructed flow-

ISHOCK

1

a. Velocity vectors in a cross section normal to the oblique shock

~ ~~~

1 5 J-\\\-\\\\\\\\\ \

I, 12 13 I 4 15 16 17 I 8 19 i B 21 22 Z I O E L l A

b. Velocity vectors in a cross section normal to the axis of the secondary vonex

Figure 10 Interaction velocity field at XI& = 32.8

field maps for six cases of swept shock boundary layer interactions, including the one studied in the present paper. Their flowfield map is shown in Fig. l l b for comparison with our calculations. Note that the calculated vonicity con- tours are shown in a cross section normal to the freestream direction, while the experimental results are presented in a plane normal to the shock. Thus, the proportion between the horizontal and the vertical length scale is different in each figure. Yet the similarity between the calculations and the experimental data is remarkable.

3.3 Skin Friction and Conical Similarity

In this section the variation of the skin friction coeffi- cient, of the wall pressure and of the surface flow angle will be studied with respect to conical similarity. It has already been mentioned in the introduction that PanarasS has found the flow to be sectionally conical in the absence of secondary separation, Le., the various flow parameters are not constant along conical rays of the flow, but the variation is small. This variation is visible if the distance between two examined points along a conical ray is larger than 20 to 30 boundary layer thicknesses. In the present paper, the variation of the aforementioned flow parameters along lines which are nor- mal to the flow will be studied. Three lines with a spicing of 10 6- will be considered. Each point of the first two lines is conically projected onto the third line which lies very close to the outflow plane.

For the application of the conical projection, the position of the virtual origin, defined is the point where all the critical lines of the flow meet, must be known. The virtual origin can be determined graphically, using Fig. 6, by finding the point of intersection of the separation line, the reattachment line and the shock line. For eliminating a possible error caused by the graphical determination, a numerical search was carried out assuming that the origin lies on the extrapo- lation of the inviscid shock position upstream of the fin. During this search, the virtual origin was located at succes- sive positions along the shock line and at each position the pressure variation along conical rays was checked. The best conical behaviour defined the virtual origin which was used in the following comparison.

The distribution of the characteristic flow parameters at the three streamwise sections is shown in Figure 12. Exam- ining first the surface pressure (Fin. 12a), we note that it

,MAIN INVISCID SHOCK

A, =Primary reattachment 5, =Primary separation S,=Secondary separation

M,=2.95

M,=1.64 U

b. Experimental flowfield map of the calculated flow (Alvi & Settlesl7)

Figure 11: Comparison of experimentally and numerically determined features of the interaction

shows a rather small variation along conical rays which cross the three sections in the region occupied hy the initial flat part of the separation vortex and by the secondary vortex. This is indicated by the almost perfect collapse of the three curves for values of zl6.. > 15. In contrast, at the region of the conical vortex itself, i.e. for 5 < z / L < 15, the three curves remain distinct. In particular, the maximum value of the pressure in the reattachment region of the vortex is pro- gressively increasing downsueam while the low-pressure region below the core of the vortex is progressively decreasing. However, the three curves have their maximum or their minimum almost at the same value of I/s_. These '-J

observations lead to the conclusion that, although the sepa- ration vortex remains geometrically conical in the down- stream direction, its strength is increasing which is most probably due to the channeling of the vortex between the fin and the secondary vortex.

The variation of the skin friction is shown in Figures 12b and c. It should first be noted that at the station closest to the apex of the fin (x/& = 12.8). the secondary separation has not yet been formed. This is evident in the lack of a secondary peak in the skin friction distribution at this sec- tion. The other stations lie within the part of the flow where the secondary separation has appeared. One observes that in this case the skin friction is rather well correlated. Differ- ences exist only in the reattachment region of the conical vortex where the skin friction is lowest close to the outflow plane.

Finally, the variation of the surface flow angle is shown in Fig. 12d. Away from the conical vortex the curves almost coincide, but in the vicinity of the main vortex again differ- ences exist. The present data as well as the other flow parameters considered indicate that the appearance of the secondary vortex affects the dynamic characteristics of the main conical vortex so that, although this vortex remains conical in shape, the various flow parameters do not remain constant along conical rays which pass through this vortex,

V Y

a. Iso-contours of the absolute value of vorticity in two cross sections of the flow field

8

3.5 -

3 . 0

E 2 . S - 2 u1 Y)

a 2 . 0

1.5

I . 0 - . . , , . . . . 0 5 I 0 15 20 25 30

2,OELTI

@Skin Friction

0 .00+

tl ,003

- x16.= 32 8 -- x16.=228

/6,=128 z 5 ,002

- ,001

0 5 I 0 15 20 25 30 Z l D E L i h

0 ."I. !.ami' - , ~

5 ,002 -

w- .. ,001

0 5 I 0 15 20 25 30 Z l D E L i h

,0015 @Skin Friction Normal

,0010 t o Shock

0

$ -.0005

9 - .0010

b - .0015

-.0020 V 0 5 I 0 15 20 25 30

2,DELlA

I 0 0 1 0

Figure 1 2 Conical correlation of surface flow parameters

4. Concluding remarks

Experimentally it has been found that in moderate strength glancing shockhrbulent boundary layer inter- actions, as they occur, e.g.. in supersonic and hypersonic aircraft air intakes, a secondary separation line appears in the surface flow pattern. According to Prof. G. Settles, this sec- ondary separation in supersonic turbulent swept interactions still remains enigmatic. In the present paper a flow of this type, studied experimentally at the Pennsylvania State Uni- versity, is simulated numerically. It is shown that if the tur-

bulence model of Baldwin & Lomax is applied according to the physics of the flow, the resulting solution agrees very well with the experimental evidence (wall pressure, skin friction, surface flow angle). Then, post-processing of the solution reveals that in this type of interaction the secondary separation phenomenon is similar to that observed in flows about bodies at high incidence. However, the secondary vortex is very weak. A special numerical visualization tech- nique developed at DLR for the detection of vortices was required to identify this vortex in addition to the other vor- tices that are expected to appear in such a flow.

The dynamic characteristics of the conical vortex gen- erated by the interaction of the oblique shock wave with the boundary layer on the plate are affected by the presence of the secondary separation: The maximum wall pressure in the reattachement region increases downstream, while the mini- mum pressure below the core of the conical vortex decreases. This behaviour is probably due to a channeling effect caused by the secondary vortex which appears as a "bump" of vi- angular shape between the fin and the primary separation line.

Examination of the variation of the static pressure, the skin friction and other surface flow parameters along conical rays has shown that in spite of the conical shape of the sep- aration vortex the various flow parameters do not remain constant along these rays. The variation is such that the flow can only be considered as quasi-conical in the vicinity of the vortex.

It should be noted that in the case of the flow without secondary separation, which has been studied by Panaras5, the flow in the region of the conical vortex was found to be essentially conical. The deviation from conicity was prima- rily resbicted to the flow field region between the separation shock and the plate. These deviations have been attributed to the difference in the development of the boundary layer approaching the shock. The boundary layer effect appears also in the case examined here. However, this effect is not significant in the conical correlation of the wall flow parameters considered above. The conical similarity away from the wall has not been adressed here because this subject has been studied in detail in Ref. 5.

References

1. Kubota, H. & Stollery, J.L., An experimental study of the interaction between a glancing shock wave and a turbulent boundary layer. J. Fluid Mech. 116. pp. 431-458, 1982

2. Settles, G.S. & Dolling, D.S., Swept Shock Boundary Layer Interactions - Tutorial and Update. AIAA Paper 90- 0375, January 1990

3. Alvi, F.S. & Settles, G.S., Structure of swept shock wavelboundary-layer interactions using conical shadowgra- phy. AIAA Paper 90-1644, 1990

4. Knight, D.D., Horstman, C.C., Shapey, B. & Bogdonoff S . , Structure of supersonic turbulent flow past a sharp fin. AIAA J. 25, pp. 1331-1337, 1987

5. Panaras, A.G., Numerical investigation of the high-speed conical flow past a sharp fin. J. Fluid Mech., Vol. 236, pp. 607-633, 1992

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6. Vollmers, H., A concise introduction to Comadi. DLR IB 221-89 A 22, 1989

7. Shapey, B. & Bogdonoff S.M., Three-dimensional shock wavelturbulent boundary layer interaction for a 20" sharp tin at Mach 3. AIAA Paper 87-0554, 1987

8. Settles, G.S., Sharp fin shockhoundary-layer interaction data. Penn State Gas Dynamics Lab, Private Communication, 1990

9. Kim, K.S, Lee, Y., Alvi, F.S, Settles, G.S., Horstamm C.C., Laser skin friction measurements and CFD comparison of weak-to-strong swept shockjboundary-layer interactions. AIAA Paper 90-0378, 1990

10. Baldwin, B.S. & Lomax, H.. Thin layer approxiination and algebraic model for separated turbulent flows. AIAA Paper 78-257, 1978

11. Degani, D. & Schiff, L., Computation of turbulent flows around pointed bodies having crossflow separation. J. Comp. Phys. 66, pp. 173-196, 1986

12. Panaras, A.G. & Steger, J.L, A thin-layer solution of the flow about a prolate spheroid. Z . Flugwiss. Weltraumforsch. 12, pp. 173-180, 1988

13. Yee, H.C. & Harten, A,, Implicit TVD schemes for hyperbolic conservation laws in curvilinear coordinates. AIAA J. 25, pp.266-274, 1987

14. Muller, B., Development of an upwind relaxation method to solve the 3D Euler and Navier-Stokes equations for hypersonic flow. GAMM Wissenschaftliche Jahrestagung, Hannover, April 1990.

15. Hung, C.M. & MacCormack, R.W., Numerical solution of supersonic laminar flow over a three-dimensional com- pression comer. AIAA Paper 77-694, 1977

16. Vollmers, H., Kreplin, H.P. & Meier, H.U., Aerodyna- mics of vortical type flows in three dimensions. AGARD CP-342, Paper 14, 1983

17. Alvi, F.S. & Settles, G.S., A physical model of the swept shockjboundary-layer interaction flowfield. AIAA Paper 91-1768, 1991

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