n

AIAA 89-0340 Numerical Simulation of the Effect of Spatial Disturbances on Vortex Asymmetry

David Degani and Lewis B. Schiff NASA Ames Research Center

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

b-

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 3 370 L’Enfant Promenade, S.W., Washington, D.C. 20024

Numerical Simulation of the Effect of Spatial Disturbances on Vortex Asymmetry .-

David Degani' and Lewis B. Schifft NASA Ames Research Center, Moffett Field, CA. 94035

Abstract cone-cylinder at large incidence can he seen in the clas- sic photo taken by Thomson and Morrison3. With further increase in incidence, the asymmetric flow becomes non- steady, and, as the angle of attack tends toward goo, the flow pattern approaches that of a circular cylinder in cross- flow.

The steady asymmetric vortex pattern on slender bodies of revolution at large angle of attack was investigated using fine.grid Navier.Stokes computations. The ComDuted results demonstrate, for the first time, the marked asymmetry which has been observed in experi- ments. To obtain asymmetry, i t was found essential to in- troduce a space-ked, time-invariant perturbation into the computation. If the perturbation was removed, the asym- metric flow returned toward symmetry. The perturbations were found to he more effective when located close to the nose. Taken together with the experimental observations, the computational results suggest that vortex asymmetry is forced by amplification of small disturbances, such as those due to surface roughness, occurring within the body viscous boundary layer.

In t roduct ion

The onset of vortex asymmetry on the forebody of a fight vehicle maneuvering at large angles of attack can generate unwanted yawing moments, which can lead to de- parture from controlled flight. This phenomenon is typified by the vortex asymmetry observed in the flow about slen- der bodies of revolution at large incidence. As a result, over the past three decades, a large number of researchers have investigated high-angle-of-attack flow over slender axisym- metric bodies. The recent survey articles by Hunt' and Ericsson and Reding' provide extensive bibliographies of these investigations.

-

The flow observed about a slender of body of r e v 5 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

Flow over a slender axisymmetric body at u = 90" is (except for the region near the nose), the classical flow observed about a two-dimensional cylinder in crossflow. It is well known that at low Reynolds numbers, flow about the cylinder is steady and symmetric. However, when the Reynolds number is increased beyond a critical value (RQ % 50) the flow becomes nonsteady. Vortices are shed periodically from alternate sides of the cylinder and are convected downstream to form the classic Karman vortex street. This transition from a steady to a nonsteady, peri- odic, flow is known as a Hopf bifurcation (cf Ref. 4 for a discussion of bifurcations in aerodynamics). It may be rep- resented schematically by the bifurcation diagram shown in Fig. 2, where stability of the steady flow at Reynolds numbers below the critical value Re, is indicated by the solid line along the horizontal axis, while instability of the flow at Reynolds numbers greater than Re, is indicated by the dashed line.

Timcaccurate computations of two-dimensional flow about the circular cylinder'-' (using algorithms that are unbiased in the circumferential direction) have found that, unless a perturbation was introduced into the flow to trig- ger the asymmetry, the solutions remained symmetric and steady. The perturbations were typically introduced for a short period of time, then removed, and the flows advanced in time until the periodic eolution developed.

In Ref. 8, we utilized the time-accurate thin-layer Navier-Stokes code (F3D) reported by Steger et do to study the three-dimensional subsonic flow surroundinn a

I

inridcnce, the symnietric vortex pair is observed IO become asvmmetrir. but remains steadv in time. At still hieher

slender body of revolution at large incidence. Initial coni- outed rcsults for the flow surroundine an ocive-cvlinder at -

incidence, the steady asymmetric pair evolves to a steady pattern of multiple vortices that leave from alternate sides of the body with increasing distance downstream. Steady alternate shedding of (at least) six vortices on a slender

National Research Council Senior Research Amo date. Associate Profeesnor, on leave from Tecboion IIT, Faculty of Mechdeal Engineering. Member AUA.

tllesesuch Scientist, AppUed Computational Fluids Branch. Assodate Fellow AIAA.

Copyright @ 1888 by the American Institute of Aeronau- tics and Astronautics, be. No copyright 1. asserted in the United States under Title 17, U.S. Code. The U.S. Govern- - ment has a royalty-free license to exercise all rights onder the copyright claimed herein for Governmental purpose.. All other rights w e reserved by the copyright owner.

I - .

M, = 0.2, u = 40", and Reynolds number (based on free- stream conditions and cylinder diameter) RQ = 200,000 indicated that the computed flow remained symmetric at angles of attack where experimental measurements showed the presence of large asymmetry. Guided by the results for the two-dimensional cylinder flow, a transient symmetry- breaking perturbation was applied to induce asymmetry. When the perturbation (a small surface jet blowing nor- mJ to the surface and perpendicular to the angle of attack plane) was introduced, the solution started to evolve to an asymmetric state. However, in contrast to the compu- tations for two-dimensional Karman shedding, when the perturbation WBS removed, the asymmetric solution re- turned to symmetry. When a space-iixed, time-invariant perturbation was used (i.e., the jet, once introduced, was kept constant for all time), the computed flow reached and maintained an asymmetric state.

1

Even with the use of methods based on the Navier- Stokes equations, computation of the asymmetric vortex flow on bodies at large incidence has not been straightfor- ward. Graham and Hankey'O used a Navier-Stokes code based on MacCormack's noncentered algorithm to wm- pute three-dimensional viscous flow around a slender cone- cylinder body at M, = 1.6,a = 30°, and Reynolds num- ber (based on body length) R ~ L = 400,000. Their re- sults showed only a tiny asymmetry in the computed flow field, which they attributed to built-in asymmetries in the numerical algorithm. On the other hand, experimental measurements for this case' show a marked asymmetry. Zilliac", using an implicit incompressible Navier-Stokes method to compute flow over a short ogive-cylinder body at angles of attack up to 45' at R ~ L = 1000, found that the computed flow fields were perfectly symmetric.

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 yields"J*

u

B,Q+B,P+B,B+8(&=Re-'B(S (2)

where only viscous terms in C are retained. These have been collected into the vector 8 and the nondimensional Reynolds number Re is factored from the viscous flux

several upe,.jmental studies of bodies of revolution at large incidence (cf, Refs. 12 ~ 16) have shown that mul-

de of atta& and Reynolds number, depending on the roll orientation of the body, ~n particular, ~ ~ ~ ~ t ~ z , ~ s and Dexter and ~ ~ ~ t 1 4 found evidence of symmetric flows nt angles of attack where most of their measurements showed large asymmetry. H ~ I S , analyzing Lamontts data showed the existence of a range of side force values a* a fixed incidence. These computational and obser- wtions lead us to believe that three-dimensional vortex known a constant number.

term. The coefficients of viscosity and thermal conduc- tivity Which appear in Eq. (2) are specified from auxiliary

obtained using Sutherland's law. For cases of turbulent flow the eddy-viscosity turbulence model reported by De- gani and s&@'-'' is used. This model extends the twc- layer model Baldwin and Immax" to Permit accurate computation of turbulent flows having large re- gions of crossflow separation. The coefficient of thermal conductivity is obtained once the viscosity coefficient is

tiple values of side force can be obtained at a fixed an- relations. For hDlinXI flown, the COdfident Of viscosity is

asymmetry on slender bodies at large incidence is forced by space-fixed, time-fixed disturbances, such as those due to surface roughness.

In the current work we have applied Navier-Stokes computations to further investigate the phenomena gov- erning onset of vortex asymmetry. Time-accurate solutions were obtained for flow over the ogive-cylinder considered in Ref. 8, using the F3D code. To examine whether mul- tiple asymmetric states can be obtained computationally, a small jet acting normal to the angle-of-attack plane was used as a space-fixed, time-inwriant perturbation. The effects on the resulting flow of differing jet locations was investigated. In addition, the effect of wrying free-stream Reynolds numbers on the computed flows was examined.

Theoretical Beekeround

Governing Equat ions

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

8,9 + Be(@ + R) + B,(G + 8.) + B((B + B.) = 0 (1)

where I is the t,ime and the independent spatial vari- ables, e, 7, and C are chosen to map a curvilinear body- conforming grid into a uniform computational space. In Eq. (1) 0 is the vector of dependent flow variables; P = fi(Q), G , = G(Q) , and & = &(Q) are the invis- cid flux vectors, while the terms fi-, e-, and 8- are fluxes containing the viscous derivatives. A nondimensional form

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

L

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 diffcr- CDCC approximations of the spatial metric8 c a n be reduced. In the current application, the Gee stream is used ES the base solution and the right-hand side of Eq. (3) is zero.

Numerical Algorithm

The implicit scheme employed in this study is the F3D algorithm reported by Stc er et al in Ref. 9. The a l p

encing for the convection terms in one coordinate hrection (nominally streamwise). As discussed in Ref. 9, schemes using upwind differencing can have several advantages over methods which utilize central spatial differences in each direction. In particular, such schemes ean have natural numerical dissipation and better stability properties. By using upwind differencing for the convective terms in the streamwise direction while retaining central differencing in the other directions, a twwfactor implicit approimately- factored algorithm is obtained, which is unconditionally stable" for a representative model wave equation. The

rithm uses flux-vector splitting % and upwind spatial &ffcr-

L./

scheme may be written for the thin-layer Navier-Stokes equations in the form

I

[I+ h$(&)"+ h@" - hRe-'b(J-'&"J - Dil(]

x [ I + h6{(A-)" + h6,B" - D,lv]A@' =

- At{6:[(P+)n - @L] + 6{[($-)" - fi;] + 6,(Gn - 8,) + S((B" - B,)

Re-'$ -n - ((5' - &)I -De(@' - Qm) (4)

where h = At or At12 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-joint, 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 @ has been split into P+ and P-, according to its eigenvalues2', and the matrices, Ai,B,8, and & 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 Di, are used in the central space differencing directions. Full details of the development of the algorithm may be found in Refs. 9 and 24.

- Numerical Smoothing

The finite-difference scheme uses flux-splitting in the e direction and central differencing in the 7 and ( direc- tions. As a consequence, numerical dissipation terms de- noted by D, and D. in Eq. (4) are employed in the 7 and { directions, and are given as combinations of second and fourth differences. The smoothing terms are of the form

where P = & and where [hl is the absolute value of

the matrix B or an approximation. Here p is the nondi- mensional fluid pressure and €2 is 0(1 + M-") while is O(0.01). 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 g/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 Conflgurations and Computational Grids

Computations were performed for subsonic flow over the two ogive-cylinder bodies shown in Fig. 3. Configura- tion A consisted of a 3.5 diameter tangent ogive forebody with a 7.0 diameter cylindrical afterbody extending aft of the nose-body junction to x/D = 10.5. Configuration B was identical to body A, but had a 0.5 body diameter sting which extended an additional 5.5 diameters downstream, to x/D = 16.0. These bodies were selected for study be- cause an ogive-cylinder body having a 3.5 diameter tan- gent ogive nose and a 4.0 diameter cylindrical afterbody had been extensively tested by LamontXs in the Ames 12- Ft Pressure Wind Tunnel. In that experiment, detailed surface pressure distributions were obtained at Reynolds numbers ranging from R a = 200,000 to Ra = 4.0 x lo', and at angles of attack ranging from a = 20" to a = 90".

The grid used for numerical simulation of the flow about configuration A is shown in Fig. 4. The grid con- sists of 120 equispaced circumferential planes (A4 = 3") extending completely around the body. In each circum- ferential plane the grid contained 50 radial points between the body surface and the computational outer boundary, and 59 axial points between the nose and the rear of the body. For configuration B, an additional 10 axial points were used to model the sting.

Boundary Conditions and Init ial Conditions

An adiabatic nc-slip boundary condition was applied at the body surface, while undisturbed free-stream condi- tions were maintained at the computational outer hound- ary. An implicit periodic continuation condition WBF im- posed at the circumferential edges of the grid, while at the downstream boundary a simple zero-axial-gradient extrap- olation condition was applied. This simple extrapolation boundary condition is not strictly d i d in subsonic flow, since the body wake can affect the flow on the body. How- ever, by letting both computed body lengths extend be- yond the physical length of the experimental model", and neglecting the portion of the flow near the downstream boundary, we can minimize the effect of the boundary. On the upstream axis of symmetry an extrapolation boundary condition was used to obtain the flow conditions on the axis from the cone of points one axial plane downstream.

In these computations, a time-accurate solution is required. Thus, the second-order time-accurate algorithm is used with a globally-constant time step. The flowfield was initially set to free-stream conditions throughout the grid, or to a previously obtained solution, and the flowfidd was advanced in time until a solution WBS obtained.

A weak jet wan imposed on the body surface M symmetry-breaking perturbation. Two jet locations were used, an upstream jet located at x/D 0.12, and a down- stream jet located at x/D = 1.20. Both jets were located 90" arcumferentially from tbe windward meridian, that is, acting perpendicular to the angIe of attack plane. A small normal momentum WBS imposed, and maintained constant an the solutions evolved.

The F3D code requires approximately 8 x lo-' sect iterationfgrid point on the NAS CRAY-2 computer. This

S

.-. _.

translates t o approximately 27 sec/iteration for the smaller grid, and approximately 32 sec/iteration for the larger grid. Further, the need to resolve the thin viscous layers for high- Reynolds-number flow requires that the grids have a fine radial spacing at the body surface. As a result, the allow- able computational nondimensional time steps were found to range from 0.005 to 0.010.

Results The F3D code has been utilized to compute both

laminar and turbulent high-incidence subsonic flows about bodies at large incidence, and has been demonstrated to give good results compared to experiment". The four computations presented below were all obtained for M, = 0.2 and a = 40'. Except for case 4, the computations were all carried out for a Reynolds number based on body di- ameter, RQ, of 200,000. Case 1 is the symmetric flow which was obtained in grid A when the jet strength was set to zero. Case 2 is the asymmetric flow obtained in grid B when the symmetry-breaking jet was located ap- proximately 1.20 body diameters downstream from the nose. Case 3 is similar, and was obtained using grid A with the jet located further upstream, approximately 0.12 body diameters from the nose. For case 4, grid B was used with the jet located 0.12 diameters from the nose, and with RQ = 26,000. This Reynolds number was cho- sen to match the conditions used by Degani and Zilliac" in a complementary experimental investigation of vortex asymmetry and vortex nnsteadiness.

Case 1 - Symmetr ic Flow.

Results for case 1 are presented in Figs. 5 - 8. In this computation the jet strength was zero, and the computed flow remained symmetric about the angle of attack plane. Although the computed flow was found to be nonsteady, the nonsteadiness produces only small changes relative to mean flow, and thus the main flow features can be seen in the "snapshots" shown below. A companion compu- tational study of vortex unsteadiness is contained in Ref. 26.

Computed helicity density contours in several cross sections on the forward part of the body (0 5 x/D 5 9.9) are shown in Fig. 5. Details of the cross section located at x/D = 5.6 are repeated in the upper right corner of the figure. Helicity density is defined as the scalar prod- uct 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 excel- lent means of visualizing the position and strength of the vortex patternz'. By marking positive and negative values of helicity with different colors it is easy to differentiate between the primary and secondary vortices. Although both positive and negative helicity contours are presented in Fig. 5 , the monochromatic figures cannot explicitly in- dicate the sense of rotation of the vortices. However, the location of the primary crossfiow separation on the side of the body, and the growth in size and height of the vortices with increasing distance downstream is easily seen.

The location and rotation of the primary leeward- side vortices are also evident in the off-surface streamline pattern presented in Fig. 6. The pair of vortices which originate at the nose run almost parallel t o the body up- per surface as they grow with distance downstream. In

contrast to the results obtained (see also Fig. l), the computed vortices do not appear to lift away from the body surface. It should be noted that the lift-off of the vortices seems to be linked with the asymmetry in

in the smoke tunnel using a splitter plate to enforce sym- metry of the leeward side flow indicated that only one pair of primary vortices was present, and that they tended to remain dose to the body surface.

A top view of the computed surface flow pattern is shown in Fig. 7, and confirms that the computed flow remains symmetric. However, the flow is not steady. As reported previously8, the shear layers which emanate from the primary separation lines and roll up to form the pri- mary vortices &bit a small-scale nonsteadiness. For the current computations, a change in the code reduced the ef- fect of the numerical smoothing terms near the wall. As a result, the nonsteadiness exhibited more frequencies than was observed previously. This can be seen from the seg- ment of the normal-force coefficient time history shown in Fig. 8. In contrast to the almost periodic behavior of the normal force obtained previously, the normal force is oscillating in a more arbitrary manner. The scale of the fluctuations relative to the mean value of the normal-force coefficient can also be seen in Fig. 8.

Case 2 - Asymmetric Flow, Downstream J e t .

the experimental flow. Indeed, experiments" conducted u

At 40" angle of attack the experimental flow was asymmetric, while the computed flow remained perfectly symmetric. To break the symmetry of the solution, a time- invariant, space-ked disturbance was added to the flow- field discussed above. A small jet, having a strength of about 0.1% of the total normal force was introduced at x/D ~ i : 1.2, blowing normal to the body axis and oriented perpendicular to the angle of attack plane. Although the resulting flow is no longer symmetric, the asymmetry is rel- atively small. This can be seen from the computed helicity density contours shown in Fig. 9, the off-surface stream- line pattern shown in Fig. 10, and the surface flow pattern shown in Fig. 11. In this case the vortices still lie close to the leeward body surface.

The normal-force Coefficient time history for this case is shown in Fig. 12a, while the side-force coefficient history is shown in Fig. 12b. These force histories were previously presented in Ref. 2, but are included here for the sake of completeness. In this wmputation, an earlier type of numerical smoothing was used, which resulted in a greater smoothing near the wall. As a result, after the initial tran- sient dies away, both force histories show a smooth periodic oscillation. The oscillatory excursions are small compared to the mean values of the coefficients.

As shown in Fig. 12b, the mean value of the side- force coefficient is approximately -0.7. This side force is about 100 times larger than the net force due to jet itself. Two numerical experiments were conducted to investigate the stability of the asymmetric solution. Both experiments started from the solution obtained at time = 27.5. In the first, the jet was turned off and the computation was con- tinued in time. The solution returned toward the symmet- ric state. In the second experiment the jet strength was reduced to half its original value and the computation was continued. The resulting flow evolved toward .a smaller asymmetry.

u'

4

Case S - Asymmetric Flow, Upstream Jet.

Several experimental studies of high inadence flow have suggested that small surface imperfections near the nose can have a large effect on the resulting flow. Recently, Moskovitz et alzo showed that Wrious d u e s of side force could be obtained by using various sizes of beads as ge- ometric perturbations. They found that as the perturba- tions were located farther from the nose, larger heads were necessary to induce the same degree of flow asymmetry.

-

To assess the sensitivity of the computed flow to the spatial location of the symmetry-breaking disturbance, in this computation the jet was located closer to the nose (x/D % 0.12) and had the same strength as in case 2. The effect on the computed flow was much more dramatic. The flow field became highly asymmetric, as shown in the helicity density contours presented in Fig. 13. The corre- spondingoff-surface streamline pattern (Fig. 14) shows the presence of two pairs of primary vortices. The upstream pair is highly asymmetric and curves away from the body surface near the ogive-cylinder junction in a manner sim- ilar to the experimentally-observed vortex patterns. Note that the vortex emanating from the near side of the body in Fig. 14 leaves the surface further downstream. The trailing vortices are not parallel to the free stream, but are inclined approximately 5' toward the body from the free stream direction, consistent with the behavior observed ex- perimentally. The computed surface flow pattern (Fig. 15) confirms the presence of a large asymmetry where the vor- tices wash across the leeward meridian to the opposite side of the body..

Time-histories of the normal-force and side-force c* efficients following a small change in flow conditions are shown in Figs. 16a and 16b respectively. As was seen in case 1, the flow field is nonsteady and several dominant fre- quencies can be noted. The oscillations in the normal force are small compared to the mean value of the coefficient. However, the side-force coefficient (Fig. 16b) undergoes a much larger excursion in response to tbe kick occurring at time zero, and appears to evolving toward an oscillation about a mean d u e of -2.5. This mean side force in this case is approximately half as large as the normal force, and is about 350 times larger than tbe net force due to the jet itself.

-

Case 4 - Asymmetric Flow, R ~ D = 26,000

In this case the Reynolds number was changed to R- = 26,000 to match the experimental conditionsof De- gani and Z i l l i a ~ ~ ~ * ~ ' , and the jet was located at x/D x 0.12. In general, the flow field looks similar to that obtained in case 3 (see Figs. 17 - 19). However, small differences can be observed in the the off-surface streamline pattern shown in Fig. 18. By comparing Figs. 18 with Fig. 14, one can see that the upstream pair of vortices curves away from the body surface further downstreamin the lower Reynolds number case. This is consistent with the behavior observed experimentally by Lamontls, who found that as Reynolds number was reduced from 1.25 x 10' to 0.20 x 10' the first maximum in the measured side-force distribution moved downstream.

A front quarter view of the off-surface streamlines is shown in Fig. 19. The adOgOUS surface flow pattern is

-

shown in Fig. 20. The asymmetric disposition of the lee- ward vortices can be clearly seen. The downstream vortex from the far (right) side of the body is seen to lie close to the body surface and to curve across the leeward merid- ian in an "8" shaped curve. The effect this vortex has on the surface flow dan he seen in the top view shown in Fig. 20b. The computed patterns are strikingl similar to the experimental surface oil-flow visualization$ shown in Fig. 21. Although the computed results are not in perfect agreement with the experiments, the main features of the comouted flow structure aDDear to be correct.

The Rs, = 26,000 computation was started from a solution obtained for Rs, = 200,000. Time-histories of the normal-force and side-force coefficients following the change in Reynolds number are shown in Figs. 22a and 22b respectively. In contrast to case 3 (Fig. 16), the increased laminar viscosity causes the normal and side forces tend to decay smoothly and relatively quickly toward constant d u e s , with only small-scalefluctuations. The mean value of the side-force coefficient tends toward -1.5, which is about 200 times larger than the net force due to the jet itself.

In computations of twc-dimensional cylinder flow it was found that, once nonsteady asymmetric vortex sbed- ding is initiated, the perturbation can he removed. The vortex shedding will continue without the need for any further perturbations. In order to assess whether the same trend exists for a well-established steady three-dimensional flow (that is, whether the asymmetry would persist in the absence of the perturbation), the jet was turned off and the computation was continued from the solution shown at nondimensional time = 120. Unlike the two-dimensional cylinder case, the asymmetric flow immediately returned toward symmetry.

The fluctuations of the overall forces are moderate in comparison with those observed at the higher Reynolds number, and those fluctuations are not periodic. However, upon examining time histories of surface pressure obtained at discrete points along the body (see Fig. 23 for a typ- ical example), one can see relatively large fluctuation in the pressures. These fluctuations differ, depending on the axial and circumferential location at which they are ob- tained. Undoubtedly, the fluctuations observed at a point are linked to the mean structure of the asymmetric flow existing above that point. Analysis of the computed flow to explain the differences in the pressure fluctuations will he the subject of a future paper.

Discussion and Conclusions

The computational results presented above demon- strate, for the first time, the marked asymmetry that bas been observed experimentally on slender bodies of revolu- tion at large incidence. The following observations can be drawn from the Navier-Stokes results:

1. In the absence of a space-fixed, time-invariant sym- metry-breaking perturbation, the computed solu- tions for flow around bodies of revolution at large incidence do not become asymmetric, in contrast to the asymmetry observed experimentally.

2. The computed symmetric flows are stable to time- varying Perturbations. Even after an asymmetric so- lution develops in response to a perturbation, remov-

ing the perturbation causes the asymmetric flow to relax to the original symmetric state.

3. The degree of asymmetry in the computed solutions is a function of the strength of the applied distur- bance. At a lixed jet position, decrease of the jet strength causes a corresponding decrease in the ob- served asymmetry.

4. The dfcctiveness of a given strength perturbation is a strong function of its point of application. A per- turbation applied dose to the nose induces a much greater asymmetry that the same perturbation ap-, plied further downstream.

The observation that different degrees of asymmetry can be obtained computationally is also consistent with the experimental data base. Experimental studies of bodies of revolution at large incidence have shown that multiple va- ues of side force can be obtained at a k e d angle of attack depending on the roll orientation of the body. In partic- ular, Lam~nt ' ' ~ '~ and Dexter and Hunt" found evidence of near-symmetric flows at angles of attack where most of their measurements showed large asymmetry. Hall'5, analyzing Lamout's data showed the existence of a range of side force values at fixed incidence. These findings are consistent with the idea that space-ked surface roughness elements are inducing the asymmetry. As the body roll angle was varied in the experiments, the surface rough- ness pattern changed relative to the oncoming flow. Rapid growth, with increasing distance downstream, of the per- turbations induced by the roughness elements within the boundarylayer, amplify the effect of the roughness to cause the resulting large asymmetry in the vortex pattern. Fur- ther, Moskovitz et alzD have shown that different degrees of asymmetry can be obtained experimentally hy varying the size of a surface disturbance at a fixed location on a slender body. They also have shown that surface distur- bances were more effective in inducing asymmetry when placed dose to the nose.

.

The present computational results parallel the exper- imental findings and explain why asymmetry is almost al- ways observed in the experiments. Despite all possible care taken in constructing a model, small irregularities (large compared to the adjoining boundary layer thickness) will exist near the tip. Taken together, the computational and experimental observations lead us to believe that thrce- dimensional vortex asymmetry on slender bodies at large incidenceis forced by space-ked, time-ked disturbances, such as those due to surface roughness.

AcknowledKements

The authors wish to thank Murray Tobak and Gre- gory Zilliac for their helpful discussions. We would also like to thank Y u d Levy and On Degani for their assistance in preparing the graphical representations of the computed flows. Finally, we would like to acknowledge the interest and support of the U.S. Army Ballistic Research Labora- tory in the conduct of this investigation.

References

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

' Ericsson, L. E. and Reding, J. P., "Aerodynamic Ef- fects of Asymmetric Vortex Shedding from Slender Bodies," AIAA Paper 851797, AIAA 12th Atmo- spheric Flight Mechanics Meeting, August 1985.

' Thomson, K. D. and Momson, D. F., "The Spacing, Position, and Strength of Vortices in the Wake of Slender Cylindrical Bodies at Large Incidence," J. Fluid. Me& SO, part 4 (1971), 751-783.

' Chapman, G. T and Tobak, M., "Bifurcations in Unsteady Aerodynamics - Implications for Testing," NASA TM 100083, March 1988.

Patel, V. A., "Karman Vortex Street behind a Cir- cular Cylinder by the Series k c a t i o n Method," J. Comp. Pbys. 28 (1978), 14-42.

e Lecointe, Y. and Piquet, J., "On the Use of Several Compact Methods for the Study of Unsteady Incom- pressible Viscous Flow Round a Circular Cylinder," Computers & Fluids 12 No. 4 (1984), 255-280.

' Rosenfeld, M., Kwak, D., and Vinokur, M., "A So- lution Method for the Unsteady and Incompress- ible Navier-Stokes Equations in Generalized Coor- dinate Systems," AIAA Paper 88-0718, AIAA 26th Aerospace Sciences Meeting, Jan. 1988.

* Degani, D. and Schiff, L. B., "Numerical Simda- tion of Asymmetric Vortex Flows Occurring on Bod- ies of Revolution at Large Incidence," AIAA Paper 87-2628, AIAA 14th Atmospheric Flight Mechanics Meeting, August 1987.

L

Steger, J. L., Xng, 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.

Graham, J. E. and Hankey, W. L., Vomputation of the Asymmetric Vortex Pattern for Bodies of Revo- lution," AIAA Paper 82-0023, AIAA 20th Aerospace Sciences Meeting, Jan. 1982.

Zilliac, G. G., "A Computational/Experimental Study of the Vortical Flow Field on a Body of Rev- olution at Angle of Attack," AIAA Paper 87-2277, AIAA 5th Applied Aerodynamics Conference, Au- gust 1987.

" Lamont, P. J., "Pressure Around an Indined Ogive Cylinder with Laminar, 'hnsit ional, or Turbulent Separation," AIAA Journal 20 (1982), 1482-1499.

Is 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.

I' Dexter, P. C. and Hunt, B. L., "The Effects of Roll Angle on the Flow over a Slender Body of Revolution at High Angles of Attack," AIAA Paper 81-0358, AIAA 19th Aerospace Sciences Meeting, Jan. 1981.

Hall, R. M., "Forebody and Missile Side Forces and the Time Analogy," AIAA Paper 87-0327, AIAA 25th Aerospace Sciences Meeting, Jan. 1987.

L

L/

Keener, E. R., Chapman, G. T., Cohen, L., and Taleghani, J., “Side Forces on Forebodies at High Angles of Attack and Mach Numbers From 0.1 t o 0.7: Two Tangent Ogives, Paraboloid and Cone,” NASA TM X-3438, Feb. 1977.

Viviand. H.. “Conservative Forms of Gas Dvnamies

v

Equatiohs,”’La Recherche Aerospatiale, No.-l, Jan- Feb (1974), 65-68.

Baldwin, B. S. and L o w , H., “ThinLayer Approxi- mation and Algebraic Model for Separated Turbulent Flows,” AIAA Paper 78-257, AIAA 16th Aerospace Sciences Meeting, Jan. 1978.

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

zn Degani, D. and Schiff, L. B., “Computation of Su- personic Viscous Flows Around Pointed Bodies at Large Incidence,” AIAA Paper 834034, AIAA 21st Aerospace Sciences Meeting, Jan. 1983.

2i Degani, D. and Schiff, L. B., “CompntationofTurbn- lent Supersonic Flows Around Pointed Bodies Having Crossflow Separation,” J . Comp. Phys. 88 (1986), 173-196.

2z Schiff, L. B., Degani, D., and Cummings, R. M., “NU- merical Simulation of Separated and Vortical Flows on Bodies at Large Angles of Attadr,” Fourth Sym- posium ou Numerical and Physical Aspects of Aero- dynamic Flows, Jan. 1989. -

a5 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 Appro6 mately Factored Schemes for Equations in Gasdy- namics,” Ph.D. Thesis, Stanford University, 1986, (also SUDAAR 557, June 1986).

as Degani, D. and Zilliac, G. G., “Experimental Study of Unsteadiness of the Flow Around an Ogive-Cylin- der at Incidence,” AIAA Paper 88-4330, AIAA 15th Atmospheric Flight Mechanics Conference, August 1988.

a’ Schiff, L. B., Degani, D., and Gavali, S., LLNnmerical Simulation of Vortex Unsteadiness on Slender Bodies of Revolution at Large Incidence,” AIAA Paper 89- 0195, AIAA 27th Aerospace Sciences Meeting, Jan. 1989.

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

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

” Moskovitz, C. A., Hall, R. M., and DeJarnette, F. R., “Effects of Surface Perturbations on the Asymmetric Vortex Flow Over a Slender Body,” AIAA Paper 8% 0483, AIAA 26th Aerospace Sciences Meeting, Jan. 1988!

Fig. 1 Schlieren Aow visualization; M, = 0.8, a = 40’ (Ref. 3).

7

Fig. 2 Bifurcation diagram.

Fig. 5 Helicity density contours; M , = 0.2, c( = 40', ReD = 200,000 (no jet).

Fig. 3 Ogive-cylinder configurations.

Fig. 6 Off-surface streamlines; M , = 0.2, u = 40", Reo = 200,000 (no jet).

Fig. 4 Tangent ogive-cylinder grid,

Fig. 7 Surface flow pattern; M , = 0.2, u = 40", ReD = 200,000 (no jet).

5 -

-

'N 3 ........................... i .........................................

* ........................... i .......................... .......................... > .........................

1 7

-.5 -

15 20 25 30 -1.0 t ' 0

NONDIMENSIONAL TIME

Fig. 11 Surface flow pattern; Mw = 0.2, u = 40', ReD = 200,000 (jet at x/D % 1.2).

b) Side-force coefficient.

Fig. 12 Force-coefficient histories; M , = 0.2, a = 40°, ReD = 200,000 (jet at x/D % 1.2). - Fig. 10 Off-surface streamlines; M , = 0.2, a = 40°,

ReD = 200,000 (jet at x/D 1.2).

9

Fig. IS Helicity density contours; M , = 0.2, (I = 40°, ReD = 200,000 (jet at x/D ?z 0.12).

Fig. 14 Off-surface streamlines; M , = 0.2, (I 40°, R ~ D = 200,000 (jet at x/D 0.12).

~

.

2 .......................... i ....... ..................; .......................... i 1 J

0 25 50 75 NON.DIMENSIONAL TIME

a) Normal-force coefficient.

0

b) Side-force coefficient,

Fig. 15 Surface flow pattern; M , = 0.2, (I = 40', Reo = 200,000 (jet at x/D o 0.12).

Fig. 16 Force-coefficient histories; M , = 0.2, a = 40", ReD = 200,000 (jet at x/D ?z 0.12). L

10

Y

c) Left side view.

b) Top view.

Fig. 17 Helicity density contours; M , = 0.2, a = 40", ReD = 26,000 (jet at x/D % 0.12).

a) Right side view.

Fig. 20 Surface flow pattern; M , = 0.2, a = 40", ReD = 26,000 (jet at x/D ~i 0.12).

c) Left side view. . ,

Fig. 18 Off-surface streamlines; M , = 0.2, a 40", R ~ D = 26,000 (jet at x/D % 0.12).

b) Top view.

a) Right side view.

Fig. 21 Experimental oil-flow pattern; a = 40", ReD = 26,000 (Ref. 28).

Fig. 19 Off-surface streamlines, front quarter view; M , = 0.2, a = 40", R ~ D = 26,000 (jet at x/D x 0.12).

-

11

0 25 50 75 100 NON-DIMENSIONAL TIME

a) Normal-force coefficient.

CY

0 25 50 75 100 1 NON-DIMENSIONAL TIME

b) Side-force coefficient.

W

5

Fig. 22 Force-coefficient histories; M, = 0.2, a = 40", €&D = 26,000 (jet at x/D 0.12).

1 . 6 1 7 1.2 ................................... i i

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

'% J ......................................................................................................

.4

0 50 100 150 NON-DIMENSIONAL TIME

Fig. 25 Surface pressure coefficient history at x/D = 6.5, 60" from the leeward meridian on side opposite the jet; M, = 0.2, a = 40", R ~ D = 26,000 (jet at x / D M 0.12).

12