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AIAA 89-0195 Numerical Simulation of Vortex Unsteadiness on Slender Bodies of Revolution at Large Incidence

Lewis B. Schiff and David Degani NASA Ames Research Center Moffett Field, California and Sharad Gavali AMDAHL Corporation Sun nyva le, California

27th Aerospace Sciences Meeting January 9-12, 19891Ren0, Nevada

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Numerical Simulation of Vortex Unsteadiness on Slender Bodies of Revolution at Large Incidence

Lewis B. Schiff and David Deganit NASA Ames Research Center, Moffett Field, CA. 94035

and Sharad G a d $

Amdahl Corporation, Sunnyvale, CA

Abstract merge with the main primary vortices. The frequency of the oscillations in the computed force and moment histories' was found to be that of the appearance of suc- cessive small-scale Vortices.

Tjme-accurate, fine>& Navier.Stokes solutions obtained for flow over a slender omve-cvlinder bodv of " . revolution at angles of attack ranging from 01 = 10" to a = 40". The results indicate the progressive growth of crossflow separation and the development of the leeward side vortex pattern with increasing incidence. The com- puted flows show good agreement with experimental mea- surements. As the angle of attack was increased, the flows become less damped, and at 01 = 40' a nonsteady flow ex- hibiting self-sustained fluctuations was observed. The non- steadiness was linked to the presence of small-scale three- dimensional vortices moving along the primary surfaces of crossflow separation. The behavior of the fluctuations with incidence parallels the trend observed in experiments.

Introduct ion

The introduction of recent supercomputers has per- . mitted a quantum increase in the size of computational

grids. As a result it is now possible to compute high- angle-of-attack flows over bodies with codes based on the Reynolds-averaged Navier-Stokes equations, using suffi- cient grid points to adequately resolve the main features of the three-dimensional separated flows (cf, Refs. 1 - 7). These computational studies have focused on the steady structure of the high-incidence flow. In contrast, recent time-accurate Navier-Stokes computations' of laminar flow about a slender ogive-cylinder body of revolution at large incidence in subsonic flow (M, = 0.2. 01 = 40") have in- dicated the existence of nonsteadiness in the flow. This nonsteadiness was found to be linked to the appearance of multiple small-scale vortices, each having the same sense of rotation as the primary vortices, which emanate from the line of primary crossflow separation. These vortices appeared to travel along the surfaces of separation origi- nating on the primary crossflow separation lines, and to

Research Scientist, Applied Computational Fluids Branch. Assodate Fellow AIAA.

tNational Research Council Senior Rei-& Asso- Asiodate Professor, on leave from Techdon UT, date.

h e d t y of Mechanical Engineering. Member AIAA.

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The flow observed about a slender of body of rev* lution placed at incidence to an oncoming stream exhibits a wide variety of phenomena. As the angle of attack is increased from zero, a steady, symmetric pair of vortices is observed in the leeward side flow. With further increase in incidence, the symmetric vortex pair is observed to become asymmetric, but remains steady in time. At still higher incidence, the steady asymmetric pair evolves to a steady pattern of multiple vortices that leave from alternaie sides of the body with increasing distance downstream. With further increase in incidence, the asymmetric flow becomes nonsteady, and, as the angle of attack tends toward go", the flow pattern approaches that of a circular cylinder in crossflow.

A large number of experimental studies have been conducted over the past three decades to investigate flow over bodies of revolution at large incidence. An exten- sive bibliography may be found in the survey articles of HuntB and Ericsson and Reding". However, these investi- gations have also focused mainly on the overall features of the steady flows, such as the growth of the leeward vortex pattern and onset of vortex asymmetry with increasing in- cidence, and did not report observations of nonsteadiness. Thus, in order to determine whether the nonsteadiness ob- served in the computations' was real or merely an artifact of the computational method, a complementary experi- mental investigation was initiated by Degani and Zilliac," and carried out in the indraft wind tunnel of the Ames Fluid Mechanics Laboratory. Guided by the nonsteady numerical results, this experiment included nonsteady sur- face pressure measurements, limited off-surface hot-wire velocimeter measurements, and high-speed smoke-flow vi- sualizations of the vortices. The experimental results" for a = 30" showed that nonsteadiness, having a band of frequencies centered at about 2400 Hz, was present in the surface pressures. As the angle was increased from a = 30" to a = 60" the magnitude of the pressure fluctu- ations increased. Smoke-flow visualizations indicated that the nonsteadiness is associated with moving, small-scale three-dimensional vortices on the free shear layers, similar to those observed in the computations.

To gain insight into the effects of changes in flow conditions on the computed vortex nonsteadiness, in this paper we have applied Navier-Stokes computations to con- duct a parametric investigation of the effects of changes in incidence and flow Reynolds number. A series of time- accurate solutions were computed using the thin-layer

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Navier Stokes code, F3D, developed by Steger et alla. An- gles of attack considered ranged from a = 10' to a = 40'. In most cases investigated, the computed flows reached a steady (time-invariant) state. However, the flows were ob- served to become less damped with increasu, in incidence, and at a = 40", the flow evolved to an unsteady, fluctu- ating solution even though the body was kept at a 6xed angle of attack.

Theoretical Background

Governing Equations

The conservation equations of mass, momentum, and energy can be represented in a flux-vector form that is convenient for numerical simulation as''

8,0 + a(($+ k-) + 8,(G + G.) + 8((E + &*) = 0 (1)

where T is the time and the independent spatial vari- ables, (, q, and { are chosen to map a curvilinear body- conforming grid into a uniform computational space. In Eq. (1) 4 is the vector of dependent flow variables; E' = $(Q), 6 = G(Q), and E = &(a) are the invis- eid flux vectors, while the terms k,,, G., and He are fluxes containing the viscous derivatives. A nondimensional form of the equations is used throughout this work. The con- servative form of the equations is maintained chiefly to capture shock waves in transonic and supersonic flows as accurately as possible.

For body-conforming coordinates and high-Reynolds number flow, if C is the coordinate leading away from the surface, the thin-layer approximation can be applied, which yield^"^'^

arQ + a$+ a,&+ a,E = Re-'a($ ( 2 )

where only viscous terms in { are retained. These have been collected into the vector and the nondimensional Reynolds number Re is factored from the viscous flux term. The coefficients of viscosity and thermal conduc- tivity which appear in Eq. (2 ) are specified from auxiliary relations. For laminar flow, the coefficient of viscosity is obtained using Sutherland's law. For cases of turbulent flow the eddy-viscosity turbulence model reported by De- pari and Schiff'6-'* is used. This model extends the two- layer model developed by Baldwin and Immax" to permit accurate computations of turbulent flows having large re- gions of crossflow separation. The coefficient of thermal conductivity is obtained once the viscosity coefficient is known by assuming a constant Prandtl number.

In differencing these equations it is often advanta- geous to difference about a known base solution denoted by subscript o as follows

where 6 indicates a general difference operator, and 8 is the differential operator. If the base state is properly chosen, the differenced quantities can have smaller and smoother variation and therefore less differencing error. In particu- lar, errors introduced into the solution by the finite differ- ence approximations of the spatial metrics can be reduced. In the current application, the free stream is used as the base solution and the right-hand side of Eq. (3) is zero.

\d

Numerical Algorithm

The implicit scheme employed in this study is the F3D algorithm reported by Steger et al in Ref. 12. The algorithm uses flux-vector splitting'O and upwind spatial differencing for the convection terms in one coordinate di- rection (nominally streamwise). As discussed in Ref. 12, schemes using upwind differencing can have several ad- vantages over methods which utilize central spatial dif- ferences in each direction. In particular, such schemes can have natural numerical dissipation and better stabil- ity properties. By using upwind differencing for the con- vective terms in the streamwise direction while retaining central differencing in the other directions, a twc-factor im- plicit approximately-factored algorithm is obtained, which is unconditionally stable" for a representative model wave equation. The scheme may be written for the thin-layer Navier-Stokes equations in the form

[ I + h6,b(ii+)" + h6(@ - hRe-'&J-'MnJ - nil(] x [ I + h6[(i-)" + h6,L?' - D;l,]A@' =

- Ai{6:[(fii+)" - $21 + 6[[(k-)" - k;] + 6,(Gn - G,) +a,(&" - 8,) - Re-'&(S" - 3,)} -De(&" - Qm)

(4) where h = Ai or Ail2 for first or second order time ac-

curacy, and the free stream base solution is used. Second- order time accuracy is used throughout this work, since a nonsteady solution is required. In Eq. (4) 6 is typically a three-point, second-order-accurate, central difference oper- ator, 6 is a midpoint operator used with the viscous terms, and the operators 6: and 6: are backward and forward

three-point difference operators. The flux k has been split into fi+ and k-, according to its d g c n v a l ~ e s ~ ~ , and the

matrices, A*,&,6, and M result from local linearization of the fluxes about the previous time level. J denotes the Jacobian of the coordinate transformation. Dissipa- tion operators, D. and D;, are used in the central space differencing directions. Full details of the development of the algorithm may be found in Refs. 12 and 20.

Numerical Smoothing

The finite-difference scheme uses flux-splitting in the E direction and central differencing in the q and { diree- tions. As a consequence, numerical dissipation terms de- noted by D; and D, in Eq. ( 4 ) are employed in the q and

v'

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C directions, and are given as combinations of second and fourth differences. The smoothing terms are of the form

where P = & and where 1k1 is the absolute value of

the matrix B or an approximation. Here p is the nondi- mensional fluid pressure and 61 is O(1+ Mm7) while €4 is O(O.01). In this form the second-order smoothing terms act to control numerical oscillations across shock waves, while the fourth-order smoothing is effective elsewhere. In order to improve the accuracy of the solutions, the fourth-order numerical smoothing terms are further scaled by q/q=. This has the effect of reducing the numerical smoothing in the viscous layer adjoining the body sur- face where viscous dissipation controls the dispersion, and where additional numerical smoothing terms can adversely affect the accuracy of the solution by modifying the phys- ical viscous terms.

Body Configurations and Computat ional Grids

Computations were performed for subsonic (M, = 0.2) flow over the three ogive-cylinder configurations shown

. in Fig. 1. Configuration A consisted of a 3.5 diame- ter tangent ogive forebody with a 7.0 diameter cylindri- cal afterbody extending aft of the nose-body junctiou to

'-- x/D = 10.5. Configuration B had the same tangent ogive nose as body A, and a 12.5 diameter cylindrical afterbody extending to x/D = 16.0. Configuration C was identical to body A, but had a 0.5 diameter sting which extended an additional 5.5 diameters dowustream, to x/D = 16.0. These bodies were selected for study because an ogive- cylinder body having a 3.5 diameter tangent ogive nose and a 4.0 diameter cylindrical afterbody had been extensively tested by Lamont2] in the Ames 12-Ft Pressure Wind Tun- nel.

Initial computations, performed using a grid which completely encircled the body circumferentially, showed that in the absence of a symmetry-breaking perturbation, the nonsteady computed flow always remained symmet- ric about the angle of attack plane. Since the objective of this study was to investigate vortex nonsteadiness (see Ref. 22 for a companion numerical study investigating vortex asymmetry on axisymmetric bodies at large incidence), a half-body grid and a plane of symmetry boundary condi- tion were used. The grid used for numerical predictions of the flow about configuration A is shown in Fig. 2. The grid consisted of 61 circumferential planes (A4 = 3') equispaced between the windward and leeward symmetry planes. In each circumferential plane the grid contained 50 radial points extending between the body surface and the computational outer boundary, and 59 axial points be- tween the nose and the rear of the body. For configurations B and C, an additional 10 axial points were used to model the added body length or ding.

Boundary Conditions a n d Init ial Conditions

For these computations an adiabatic ne-slip bound- ary condition was applied at the body surface, while undis- turbed free-stream conditions were maintained at the com- putational outer boundary. An implicit symmetry plane boundary condition WBB imposed at the circumferential edges of the grid, while at the downstream boundary a simple Zero-axid-gradient extrapolation condition was ap- plied. This simple extrapolation boundary condition is not strictly valid in subsonic flow, since the body wake can affect the flow on the body. However, by letting both computed body lengths extend beyond the physical length of tbe experimental model", and neglecting the portion of the flow near the downstream boundary, we can mini- mize the effect of the boundary. On the upstream axis of symmetry an extrapolation boundary condition was used to obtain the flow conditions on tbe axis from the cone of points one axial plane downstream.

In these computations a time-accurate solution was required. Thus, the second-order time-accurate algorithm was used, with a globally-constant At. The flowfield was initially set to free-stream conditions throughout the grid, or to a previously obtained solution, and the flowfield was advanced in time until a solution was obtained. Note that the need to resolve the thin viscous layers for high- Reynolds-number flow requires that the computational grids have a fine radial spacing at the body surface. As a result, the allowable computational nondimensional time steps were found to range from 0.005 to 0.010.

A series of laminar and turbulent time-accurate com- putations were carried out for subsonic flow (M, = 0.2) over the ogive-cylinder bodies shown in Fig. 1. The lami- nar cases were computed at Reynolds numbers (based on free-stream conditions and cylinder diameter) of Rs, = 75,000 and Rs, = 200,000, for angles of attack every 5' between u = 10' and (11 = 40'. These computations were all for body configuration B, except for the cases at u = 40" which were for configuration C. The turbulent computations were carried out at Rs, = 5 x lo', at an- gles of attadr of 20", 25", and 30", and for configuration A. The computations were started from undisturbed free- stream conditions, or from a solution previously obtained at a nearby Reynolds number or angle of attack, and the governing flow equations were advanced in time to obtain time-histories of the evolution of the flows. In most cues investigated, the computed flows reached a steady (time- invariant) state. However, in several instances the flows evolved to a nonsteady, fluctuating solution even though the body was kept at a fixed angle of attack.

Steady Flow

Laminar Flow, u = 20'

Typical results for laminar flow over the long ogive- cylinder (configuration B) at u = Z O O , R m = 200,000 are presented in Figs. 3 - 7. Computed helicity density con- tours over the forward portion of the body (0 < x/D 5 6.3) are shown in Fig. 3, while the computed surface flow pat- tern is shown in Fig. 4. All computations presented in this paper were for a half body, and the resulting solu- tions were reflected about the plane of symmetry in the

figures. As discussed in Ref. 22, identical symmetric s* lutions would be obtained using a full-body grid, even at a = 40°, if no symmetry-breaking perturbation is included in the cornnutation. The corresnondinn off-s-surface instan-

by Lamont" in the Ames 12-Ft Pressure Wind Tunnel. In Lamont's experiment detailed surface pressure measure- ments were obtained for an ogive-cylinder model consisting of a 3.5 diameter tangent oaive nose with a 4.0 diameter

taneous &amlines (for (0 5 x iD 5 1K5) an presented in Fig. 5. These streamlines are lion, which are everywhere parallel to the instantaneous local velocity vector. Recall that inStantaoCOus streamlines, streaklines, and partide paths all have unique definitions for nonsteady flow. Bow- ever, for steady flows, they are coincident.

The surface flow pattern (Fig. 4) shows a well- established primary crossflow separation line emanating from the nose tip and running the lengtb of the body. Boundary layer fluid is s u n to converge arcumferentially along the body surface toward this line from both the windward and leeward directions. Note that toward the rear of the cylindrical portion of the body, the primary crossflow separation line is located approximately 90' from the windward symmetry plane. The boundary layer fluid leaves the body surface along a surface of separation em- anating from the crossflow separation line, and rolls up to form the primary vortex structure on the leeward side of the body. The primary vortices grow in strength with in- creasing distance along the body and, when they are SUI. ficiently strong, induce a secondary crossflow separation. This secondary crossflow separation is observed in Fig. 4 to originate at x / D == 2.9, upstream of the ogive-cylinder junction.

Belicity density contours (Fig. 3) confirm the pres- ence of both primary and secondary vortices above the leeward side of the body. nelicity density is defined as the scalar product of the local velocity and vorticity vectors. Since it indicates both the strength and sense of rotation of the vortices, helicity density has been found to be an excellent method of visualizing the position and strength of the vortex pattern." By marking the positive and neg- ative d u e s of helicity with different colors it is easy to differentiate between the primary and secondary vortices. Although both positive and negative belicity contours are presented in Fig. 3, the monochromatic figures cannot ex- plicitly indicate the sense of rotation of the vortices. How- ever, the location of the primary crossflow separation on the side of the body, and the growth in size and height of the vortices with increasing distance downstream is eas- ily seen. This growth can also be seen in the off-surface streamline pattern (Fig. 5).

In order to assess the effects of Reynolds number on the computed flows, a computation was carried out for laminar flow at a = 20", Rq, = 75,000. The surface flow pattern and belicity density contours (not shown, see Figs 10 and 11 of Ref. 18) are almost identical to those for Rq, = 200,000 and confirm that over this range of Reynolds numbers there are only small differences between the two laminar flows.

Comparison with Experiment

The computed surface pressure coefficient distribu- tion on the windward plane of symmetry (a = Z O O , R- = 200,000) is shown in Fig. 6. Corresponding circumfer- eutial pressure coefficient distributions are shown for two axial stations, xJD = 0.5 and xJD = 6.0 in Figs. 7a and 7b, respectively. Also shown in Figs. 6 and 7 are uperi- mental surface pressure Coefficient measurements obtained

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cylindrical afterbody.-Flow total pressure wried from ap- '4 proximately 0.25 to 5.0 atmospheres, resulting in Reynolds numbers ranging from R a = 200,000 to & = 4.0 x lo6, while the angle of attack ranged from a = 20" to a = 90".

comparing Lamont's dataz1 with related experimental dataa6 and theoretical resultsa', has sug- gested that there is a systematic offset of approximately 1.5 psf in Lamont's data. Hartwich and Hallz5 have used a correction of 1.3 psf in utilizing the data. We, bow- ever, have used a correction of 1.6 psf in order to adjust the windward C, d u e s on the downstream portion of the cylinder to match those of related experiments. Note that this offset will have less effect on the higher Reynolds num- ber (higher dynamic pressure) data.

It can be seen in Fig. 6 that the agreement between the computed and measured windward pressures is very good over the entire length of the body. Similarly, the circumferential pressures are in good agreement at the for- ward station (Fig. 7a), and are in good agreement on the windward side of the body at the aft station (Fig. 7b). A discrepancy is noted between the computed and exper- imental pressures on the leeward side of the body, in the region of large crossflow separation. This is attributed to the fact that at a = 20" the experimental flow is tran- sitional toward the rear of the body, while the computed flow is considered to be laminar. Hartwich and Hall" have conducted an extensive computational study of this case, and have shown that improved agreement can be obtained by switching on an eddy-viscosity turbulence model part way along the body.

A series of turbulent flows were computed for the ogive-cylinder (configuration A) at R a = 5.0 x 10' and (I = Z O O , 25', and 30'. Results of these computations are presented in Ref. 18. The eddy-viscosity model reported by Degani and Schiff'J' was used to model the effects of turbulence. Computed longitudinal surface pressures along the windward plane of symmetry for the a = 20' CBM are shown in Fig. 8, together with high-Reynolds- number experimental measurements". The corresponding circumferential surface pressure distributions for x/D = 6.0 are shown in Fig. 9. In this case the experimental flow is fully turbulent, and the agreement between the computed and measured pressures is excellent, even on the leeward side where extensive crossflow separation exists.

The agreement between the computed and measured turbulent flows lends further credence to the explanation that the discrepancies between the laminar computation and the measured flow at Rq, = 200,000 is due to tran- sition effects present in the experiment. Clearly, a set of fully-laminar experimental data would be of use to condu- sively demonstrate the accuracy of the computed laminar flows. Despite this, the laminar computational results can be used to demonstrate the effects of changes in incidence on the flow.

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Fffects of Incidence

It is well known that the crossflow separation pattern on bodies of revolution becomes more pronounced as the angle of attack is increased, and originates further forward

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on the body. This progression is graphically demonstrated by the computed flows. Computed helicity density, sur- face flow patterns, and off-surface streamlines for laminar flow over the ogive-cylinder (configuration B) at a = lo',

,- Rq = 200,000 are presented in Figs. 10 ~ 12, respec- tively, while similar results obtained for a = 25' are shown in Figs. 13 - 15. As expected, the flow obtained at the lower incidence shows a smaller crossflow separation than that observed at a = 20". The surface flow pattern (Fig. 11) indicates that the primary crossflow separation line is not well developed until aft of the ogive-cylinder junction (x/D x 3.8) and no secondary crossflow separation line is observed. This is confirmed by the helicity density con- tours (Fig. 10) and off-surface streamlines (Fig. 12)

At the higher incidence, a = 25', the crossflow sepa- ration is more pronounced. The increase in angle of attack causes the primary crossflow separation to occur closer to the windward meridian. Also, the primary vortices de- velop more quiddy with distance along the body, and are stronger. As a result, secondary crossflow separation orig- inates farther upstream. These effects can readily be ob- served upon comparing the surface flow pattern obtained at a = 25" (Fig. 14) with that for a = 20' (Fig. 4). One sees that the secondary crossflow separation line originates much farther forward (x/D x 0.5) on the body. The com- puted helicity density contours (Fig. 13) and off-surface streamlines (Fig, 14) confirm that the leeward vortices are larger and are located higher above the body surface.

The surface flow pattern for a = 25" (Fig. 14) shows evidence of several short lines of crossflow separation 1- cated between the primary and secondary crossflow sep- aration lines. As will be discussed below, these lines are believed to be associated with small-scale flow nonsteadi-

-< ness.

The steady-state normal-force coefficients obtained from the laminar computations at RQ = 75,000 and R a = 200,000 are summarized in Fig. 16. The normal force was obtained by integrating the surface pressure and surface shear-stress distributions over the entire length of the body. All values shown in Fig. 16 were obtained us- ing a half-body (configuration B), and were doubled to convert them to the normal force that would be measured on a full body. As seen in Fig. 16, tbe normal force in- creases smoothly as the angle of attack increases. Further, the results confirm that over the range of Reynolds num- ber investigated, the computed steady-state laminar flows are insensitive to Reynolds number. The computed tur- bulent results are not shown in Fig. 16, since they were obtained for the shorter body (configuration A) and cannot be compared directly. However, as expected, normal-force coefficients show a similar trend with increasing incidence.

Nonsteady Flow

Each of the previous computations evolved to a steady state. However, as the angle of attack increased the solutions took a longer time to achieve the steady state value. At a = 40' the nature of the computed results changed radically. The flow never reached a steady-state, but instead fluctuated about a mean (symmetric) flow. An oscillatory behavior of the computed flow at a = 40" and RQ = 200,000 had been observed earlier by Degani and Schiff, and reported in Ref. 8. In the current study the nu- merical smoothing was smaller than that used previously,

I

and Rome differences were observed between the present and the previous nonstcady results, as indicated below.

Force Histories

The time-history of the normal-force coefficient ob- tained using the older numerical smoothin is shown in Fig. 17. This force history was computed using a full body of configuration A. Although the computed flow was unsteady, it remained perfectly symmetric for all time. As seen in Fig. 17, after the initial transient dies away, the force history exhibits a smooth, periodic, oscillation about a mean d u e of 4.3. Note that this mean d u e is less than the steady-state d u e s shown in Fig. 16, because the body used in the earlier investigation is substantially shorter.

Time-histories of the normal-force coefficients for the present laminar half-body computations at Ren = 75,000 and ReD = 200,000 are summarized in Fig. 18. With the exception of the a = 40' case, these show the response of the flow in time following a small change in the free-stream. The 40" case is the most recent segment of a longer time history (not shown) where the forces evolved to meander about a mean d u e of 2. 16, all d u e s shown in Fig. 18 were doubled to convert them to the normal force that would be measured on a full body.

8

As was done in Fig.

Two main trends can be seen from Fig. 18.

1. At fixed Reynolds number, the time required for the aerodynamic response to reach steady state in- creases as the angle of attack is increased. Similarly, the aerodynamic responses appear to be less damped with increase in incidence. At a = 10" the aerody- namic responseis well damped, and smoothly evolves to the steady state value. Increasing numbers of os- cillations required to damp to the steady state are observed for a = 20" and 25', respectively. The response shown for a = 25" has not yet reached a steady state, but it appears to be decaying toward a steady-state solution. However, at a = 40" no steady state is obtained.

2. At fixed incidence (a = 20") the aerodynamic re- sponses become more damped as the Reynolds num- ber is decreased. This is the expected trend, since a decrease in Reynolds number is eqnident to an increase in the effect of viscosity. Similarly, a turbu- lent flow history computed at RQ = 5.0 x 10' and a = 30' (not shown) was more heavily damped than any of the laminar cases. Here the eddy viscosity pre- dicted by the turbulence model more than offsets the decrease in laminar viscosity caused by the increase in Reynolds number.

The time history obtained at a = 40' differs from the one obtained previously.' In particular, the aerody- namic response is seen (compare Figs. 17 and 18) to con- tain several frequencies, in contrast to one frequency ob- served in the earlier work. The presence of multiple fre- quencies can also be seen from the power spectrum of the normal force history shown in Fig. 19. In this figure the nondimensional frequencies obtained in the computation are corrected, that is made dimensional, to correspond to the flow conditions used by Lamont (M, = 0.2, a = 40°, Rq = 200,000, D = 0.5ft, and T, ~ j , 520"R). The main frequency is seen to occur at 500 Hz, with additional, smaller amplitude peaks at higher and lower frequencies.

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Nonsteady Flow Structure

Ideally, the nonsteady structure of the flow at (I = 40' should be visualized with movies. In particular, non- steady partide traces of fluid elements released along the lines of separation at the body surface would show the nn- steady behavior of the vortex structure. Although straight- forward in concept, such movies are currently difficult to construct. One can either store the three-dimensional flow at selected times during the computation for later vi- sualization post-processing or, alternatively, generate the movie frames simultaneously with the flow solution. Both approaches have drawbacks; storing the computed solu- tions at more than a few time instants will overwhelm current storage devices, while combining the visualization with the flow solver assumes the researcher knows precisely what flow features will be of interest in advance of carrying out the computation.

Consequently, we are currently limited to looking at instantaneous "snapshots" of the nonsteady flow. S n a p shots of the helicity density, surface flow pattern, and in- stantaneous streamlines of the flow computed at a = 40" are shown in Figs. 20 - 22, respectively. Since the normal force excursions are relatively small compared to the mean value, we can justifiably infer the large-scale structure of the computed flow from such snapshots.

At a = 40" the crossflow separation is more pr* nounced than that observed at lower incidence. The sur- face streamline pattern (Fig. 21) indicates that the pri- mary crossflow separation line on the ogive is displaced toward the windward meridian in comparison with that observed at a = 25" (Fig. 14). The primary crossflow vor- tices are much stronger, but are located approximately the same height above the leeward body surface. As a result, the secondary crossflow separation (Fig. 21) is induced much closer to the leeward meridian.

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The surface flow pattern shows evidence of many short lines of crossflow separation located between the pri- mary and secondary crossflow separation lines. The cor- responding off-surface streamlines (Fig. 22) indicates a series of small-scale vortices, all emanating from the pri- mary separation line and having the same seuse of rota- tion as the primary vortices, which appear to wrap around the primary vortex originating near the nose. The short crossflow separation lines in the surface flow pattern ap- pear to be secondary separations induced by these small- scale vortices. A similar vortex nonsteadiness was observed previously' and was found to be periodic with frequency corresponding to that of the normal-force history shown in Fig. 17. In the present case, however, more small-scale vortices are present, and correlation between the vortex patterns and the fluctuating force history (Fig. 18) is not straightforward.

Pressure Histories

the corresponding normal-force hi8tory shown in Fig. 18. The fluctuations in the pressure histories arc of higher fre- quency than those observed in the force history. In addi- tion, the pressure fluctuations vary in magnitude and na- ture along the length of the body. This can be seen more readily in the power spectra of the C, histories presented in Fig. 24. The power spectrum for x/D = 3.5 (Fig. 248) indicates strong peaks at 1000 and 1100 He. The corre- sponding spectrum for x/D = 5.3 (Fig. 24b) indicates only a single peak at 1000he. The magnitude of this peak grows with increasing distance downstream, reaches a maximum in the spectrum obtained at x/D = 6.3 (Fig. 24c), and then decreases at larger downstream distances along the body.

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Comparison with Experimental f iends

A complementary experimental study was carried out by Degani and Zilliac''*" in the indraft smoke tunnel of the Fluid Mechanics Lab at the Ames Research center. Nonsteady surface pressure measurements, and limited off- surface hot-wire measurements were obtained for an ogive- cylinder having a 3.5 diameter nose and a 12.5 diameter cylindrical afterbody, at R a = 26,000. Although the measurements were conducted at a lower Reynolds number than the computations, the computed results are in general agreement with the trends of tbe experimental data.

Power spectra of the experimental surface pres- sures:' measured at x/D c 5.4, 60' from the leeward meridian, and angles of attack ranging from a = 0" to a = go", are shown in Fig. 25. As the angle of attack is increased from zero, a peak is observed to develop in the power spectra, centered at approximately 2400 Ha. This magnitude of the peak is seen to grow with increasing in- cidence up to a c 50", and then to decrease with further increases in incidence. This trend is paralleled by the com- putational results, where the force histories (Fig. 18) show less damping with increased angle of attack, and become oscillatory for a = 40".

Discussion and Summary

W

The present laminar computational results demou- strate the development of the steady (mean) flow struc- tures on a slender ogive-cylinder with increasing incidence between a = 10" and a = 40" and, further, show a grad- ual development toward the nonsteadiness observed in the results for a = 40". At a = 10" the primary crossflow separation starts downstream of the ogive-cylinder junc- tion, the primary crossflow vortices are small and lie dose to the leeward body surface, and no secondary separation is observed. With an increase in incidence, the primary crossflow separation develops close to the nose, the primary vortices become larger (and stronger) and lie farther above the body. As the strength of the primary vortices increases, the secondary crossflow separation and the associated de- velopment of secondary vortices also occurs closer to the nose. This devdoDment of the vortex structure with in- creasing incidence'parallels that observed experimentally. firther, computed surface pressure distributions for the laminar (and turbulent) flow are in good agreement with uperimental measurements, b h n g experimental bound- ary layer transition effects, and demonstrate the accuracy of the computed results.

Time histories of the surface pressure coefficient were obtained at selected points along the body during the com- putation at a = 40", R a = 200,000. These results were obtained at Points located at X / D = 3.5, 5.3, 6.3, 7.59 8.4 and 9.3, all located 60' drcumferentially from the lee- ward meridian. Typical C, histories for x/D = 3.5 and

These histories span the same time interval M that of x/D = 9.3 are shown in Figs. 238 and 23b, respectively. The flows were computed using a half-body grid, v

and were constrained to remain symmetric. Thus the

results cannot duplicate the change from a symmetric vortex pair to the steady asymmetric vortex pattern ob- served experimentally with increase in incidence. However, e ~ p e r i r n e n t s ~ ~ conducted in the smoke tunnel using a split- ter plate in the leeward side flow indicated that the ensuinn

' Kordnlla, W., VoIlmers, H., and Dallmann, U., "Sim- dation of Three-Dimensional Transonic Flow with Separation Past a Hemisphere-Cylinder Confignra- tion," AGARD CPP-412, Applications of Compu- tstional Fluid Dynamics in Aeronautics. Paper 31. .- & m e t r i c vortex pattern was similar to t h o b observed in

the computational results.

As the angle of attack was increased at fixed Reynolds number, the computed flows exhibited decreased damping and increased nonsteadiness. With increased incidence the force-history response to a small change in boundary con- ditions had larger oscillations and took longer to reach a steady state. At a = 40" no steady-state solutions were ob- tained, and the flow exhibited a self-sustained fluctuation having multiple frequencies. The structure of this non- steady flow showed evidence of multiple small-scale vor- tices moving along the surfaces of separation. The move- ment of the vortices is mirrored in the fluctuations ob- served in the computed surface pressure histories.

The trend of the computed solutions toward in- creased nonsteadiness with increased incidence parallels the trend observed experimentally", where as the angle of attack was increased from a = 0" to a = 60" the magni- tude of pressure fluctuations increased. The experimental measurements"'2B were performed at F h = 26,000 while the computations were for Rs, = 200,000. Thus, a eom- parison between the computed and measured frequencies is not justified. However, we think it is encouraging that the trends of the computed and experimental results are the same. Just as the computed results give us added in- sight into the details of the steady (mean) flow structures, the computed nonsteady results show promise of enhancing our understanding of the physical phenomena underlying - the vortex nonsteadiuess.

Acknowlednements

The authors wish to thank Murray Tobak and Gre- gory Zilliac for their helpful discussions. We would also like to thank Yuval Levy and Ken Gee for their assistance in preparing the graphical representations of the computed flows. Finally, we would like to gratefully acknowledge the interest and support of the Amdahl Corporation in allow- ing us to use the AMDAHL 1200 Vector Processor in the conduct of this investigation.

References

Fujii, K. and Obayashi, S., "Computations of Three- Dimensional Viscous Transonic Flows Using the LU-AD1 Factored Scheme," Technical Report of the Japanese National Aerospace Laboratory, TR-889T, 1985.

' Newsome, R. W. and Adams, M. S., "Numerical Sim- ulation of Vortical-Flow over an Elliptical-Body Mis- sile at High Angles of Attack," AIL$ Paper 86-0559, AIAA 24th Aerospace Sciences Meeting, Jan. 1986.

a Pan, D. and Pulliam, T. H., "The Computation of Steady Three-Dimensional Separated Flows over Aerodynamic Bodies at Incidence and Yaw," AIAA Paper 86-0109, AIAA 24th Aerospace Sciences Meet- ing, Jan. 1986.

Y

. - April, 1986. "

Ying, S. X., Schiff, L. B., and Steger, J. L., "A Nu- merical Study of Three-Dimensional Separated Flow Past a Hemisphere-Cylinder," AIAA Paper 87-1207, AIAA 19th Fluid Dynamics, Plasma Dynamics, and Lasers Conference, June, 1987.

e Ying, S. X., Steger, J. L., Schiff, L. B., and Baganoff, D., "Numerical Simulation of Unsteady Viscous High-Angle-of-Attack Flows Using a Partially Flux- Split Algorithm," AIAA Paper 86-2179, AIAA 13th Atmospheric Flight Mechanics Conference, August 1986.

' Vatsa, V. N., Thomas, J. L., and Wedan, B. W., "Navier- Stokes Computations of Prolate Spheroids at Angle of Attack," AIAA Paper 87-2627, AIAA Atmospheric Flight Mechanics Conference, August, 1987.

Degani, D. and Schiff, L. B., UNumerical Simulation of Asymmetric Vortex Flows Occurring on Bodies of Revolution at Large Incidence," AIAA Paper 87- 2628, AIAA Atmospheric Flight Mechanics Meeting, August, 1987.

Hunt, B. L., "Asymmetric Vortex Wakes on Slen- der Bodies," AIAA Paper 82-1336, AIAA 9th Atmo- spheric Flight Mechanics Meeting, August 1982.

lo Ericsson, L. E. and Reding, J. P., UAerodynamic Ef- fects of Asymmetric Vortex Shedding from Slender Bodies," AIAA Paper 85-1797, AIAA 12th Atmo- spheric Flight Mechanics Meeting, August 1985.

Degani, D. and Zilliac, G. G., "Experimental Study of Unsteadiness of the Flow Around an Ogive-Cyl- inder at Incidence," AIAA Paper 88-4330, AIAA Atmospheric Flight Mechanics Conference, August, 1988.

l2 Steger, J. L., Ying, S. X., and Schiff, L. B., "A Par- tially Flux-Split Algorithm for Numerical Simulation of Unsteady Viscous Flows," Proceedings of a Work- shop on Computational Fluid Dynamics, University of California, Davis, 1986.

I' Viviand, H., "Conservative Forms of Gas Dynamics Equations," La Recherche Aerospatiale, No. 1, Jan- Feb (1974), 65-68.

I' Baldwin, B. S. and Lomax, H., "Thin Layer Approfi- mation and Algebraic Model for Separated Turbulent Flows," AIAA Paper 78-257, AIAA 16th Aerospace Sciences Meeting, Jan. 1978.

Steger, J. L., "Implicit Finite-Difference Simulation of Flow About Arbitrary Two-Dimensional Geome- tries," AIAA Journal 16 (1978), 679-686.

le Degani, D. and Schiff, L. B., "Computation of Su- personic Viscous Flows Around Pointed Bodies at Large Incidence," AIAA Paper 834034, AIAA 2lst Aerospace Sciences Meeting, Jan. 1983.

. .

Degani, D. and Schiff, L. B., “Computation of Turbu- lent Supersonic Flows Around Pointed Bodies Having Crossflow Separation,” J . Comp. Phys. 60 (1986), 173-196.

Schiff, L. B., Degani, D., and Cummings, R. M., “Nu- merical Simulation of Separated and Vortical Flows on Bodies at Large Angles of Attack,” Fourth Sym- posium on Numerical and Physical Aspects of Aero- dynamic Flows, Jan. 1989.

Steger, J. L. and Warming, R. F., “Flux Vector Split- ting of the Inviscid Gasdynamic Equations with Ap- plications to Finite-Difference Methods,” J. Comp. Phys. 40 (1981), 263-293.

’’ Ying, S. X., “Three-Dimensional Implicit Approd- mately Factored Schemes for Equations in Gasdy- namics,” Ph.D. Thesis, Stanford University, 1986, (also SUDAAR 557, June, 1986).

’I Lamont, P. J., “The Complex Asymmetric Flow Over a 3.5D Ogive Nose and Cylindrical Afterbody at High Angles of Attack,” AIAA Paper 82-0053, AIAA 20th Aerospace Sciences Meeting, Jan. 1982.

22 Degani, D. and Schiff, L. B., “Numerical Simula- tion of The Effect of Spatial Disturbances on Vor- tex Asymmetry,” AIAA Paper 89-0340, AIAA 27th Aerospace Sciences Meeting, Jan. 1989.

23 Levy, Y., Seginer, A., and Degani, D., “Graphical L’ Representation of Three-Dimensional Vortical Flows by Means of Helicity Density and Normalined Helic- ity,” AIAA Paper 88-2598, AIAA 6th Applied Aero- dynamics Conference, June 1988.

24 Hall, R. M., (private communication), 1988.

25 Rartwich, P. M. and Hall, R. M., “Navier-Stokes Solutions for Vortical Flows Over a Tangent-Ogive Cylinder,” AIAA Paper 89-0337, AIAA 27th Aero- space Sciences Meeting, Jan. 1989.

“ Champigny, P., “Reynolds Number Effect on the Aerodynamic Characteristics of an Ogive-Cylinder at High Angles of Attack,” AIAA Paper 84-2176, AIAA 2nd Applied Aerodynamics Conference, August 1984.

” Mendenhall, M. R. and Lesieutre, D. J., “Predic- tion of Vortex Shedding from Circular and Noncircu- lar Bodies in Subsonic Flow,” NASA CR 4037, Jan. 1987.

Degani, D. and Zilliac, G . G., “Experimental Study of the Nonsteady Asymmetric Flow Around an Ogive-Cylinder at Incidence,” paper submitted for publication in AIAA Journal, 1989.

Degani, D. and Zilliac, G. G., (private communica- tion) 1988.

W

Fig. 1 Ogive-cylinder configurations. Fig. 2 Tangent ogive-cylinder grid.

8

v

AXIAL LOCATION. xlD

Fig. 6 Longitudinal surface pressure distribution; M , = 0.2, (2 = Z O O , R e D = 200,000.

Fig. 3 Helicity density contours; M , = 0.2, a = 20", ReD = 200,000 (laminar).

a) x/D = 0.5

pig. 4 Surface flow pattern; M , = 0.2, a = 20", ReD = 200,000 (laminar).

4 O 30 60 90 120 150 160

CIRCUMFERENTIAL ANGLE, d q

b) x/D = 6.0

Fig. 5 Off-surface streamlines; M , = 0.2, a = 20", Fig. 7 Circumferential surface pressure distributions; ReD = 200,000 (laminar). M , = 0.2, (2 = 200, ReD = 200,000.

9

. . . .

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Axial location, z J D

Fig. 8 Longitudinal surface pressure distribution; M , = 0.2, a = 20" (turbulent flow), Fig. 11 Surface flow pattern; M , = 0.2, a = lo",

ReD = 200,000 (laminar).

0.4

o-. 0.3 0 Exp., Ref. 22, ReD = 4.0 x 10'

F3D, R e D = 5.0 x IO4 " - c 8 0.2

% 0.1

s

2 -0.1

T3

2 -O.O

a

rn -0.3

t , , , . , , . . , . . , . , , , , , 0 30 6 0 90 120 150 180 R ~ D = 200,000 (laminar).

~ i g . 12 Off-surface streamlines; M , = 0.2, a = IO", -0.4

Circumferential angle, 4 (deg)

Fig. 9 Circumferential surface pressure distribution at x/D = 6.0; M , = 0.2, a = 20" (turbulent flow).

Fig. 10 Helicity density contours; M , = 0.2, a = lo", Reo = 200,000 (laminar).

Fig. 13 Helicity density contours; M , = 0.2, a = 25", Reo = 200,000 (laminar).

10

W

0 5 10 15 NONDIMENSIONAL TIME, r=taJD

Fig. 17 Normal-force coefficient history; M- = 0.2, a = 40", R e D = 200,000 (Ref. 8).

Fig. 14 Surface flow pattern; M , = 0.2, a = 25", R e D = 200,000 (laminar).

2

'N

a

6

CN

4

2

0

Fig ' .5 Off-surface streamlines; M , = 0.2, a = 25", ReD = 200,000 (laminar).

4

'N 3

2

... ...................... ............................... .. .....,,,.,.,....... .........

.:- ---\ ReD-75000 - ?

Re

0 75,000 0 200,000 9'

/ /

(c1 a = 25" ~

2

5 (dl 0 = 40' :

Fig. 16 Variation of normal-force Coefficient with incidence; M , = 0.2, R e D = 75,000 and R e D = 200,000.

-

_ . 0 50 loo 150 200

NON.DIMENSIONAL TIME

Fig. 18 Normal-force coefficient histories; M , = 0.2, R e D = 75,000 and R e D = 200,000,

11

I 1 i

............................................................... lL , ~

1 ................. : ............................. .:

0

...............................

................................

............................... 1 .............................

2000 4000 CORRECTED FREOUENCV. 11-

Fig. 18 Power spectrum of normal force history; M , = 0.2, a = 40", Reo = 200,000.

pig. 22 Off-surface streamlines; M, = 0.2, a = 4O", ReD = 200,000 (laminar).

0 20 40 60 80 u NON-DIMENSIONAL TIME

a) x/D = 3.5

1.2 1.61-

....................... j... .................... i . . .................. .; . . . . . . . . . . . . . . . .

Fig. 20 Helicity density contours; M , = 0.2, CL = 40", Reo = 200,000 (laminar).

......................................................... .8 1' '' "' ~

Fig. 21 Surface flow pattern; M , = 0.2, a = 40", ReD = 200,000 (laminar).

~~

0 20 40 60 NON-DIMENSIONAL TIME

b) x/D = 9.3

Fig. 23 Surface pressure coefficient histories 60" from the leeward meridian; M , = 0.2, a = 40", ReD = 200,000.

12

I

SJ

f ) x/D = 9.3

" a) x/D = 3.

.......................

A r 0

C

b) x/D = 5.3

....................

.................

. . . . . . . . . . . . . . . . i. 0 4 RECTED FI

c) x/D = -

.......

......

. . . . . . 1 .........

-

ENC

e) x/D = 8.4

- .............

....................

0 8000 !C

Fig. 24 Power spectra of B U ~ ~ ~ C C ~ ~ C B S W C histories 60" from the leeward meridian; M, = 0.2, a = 40', Reo = 200,000. -~

1s

....

...

...

M, = 0.2. ReD = 26,000

25000

30"

25000

25000

25000 0

D

0 12500 25000 FRERUENCY. HZ

v

Fig. 26 Surface preasure power spectra measured at x/D = 4.5, 60" from the leeward meridian; M, = 0.2, ReD = 26,000 (Ref. 28).

14