NASA Technical Memorandum 102796 

Analytical Study of the Origin 

and Behavior of Asymmetric 

Vortices 

Murray Tobak, David Degani, and Gregory G. Zilliac 

April 1990 

(NASA-TM-202795) ANALYTICAL STUDY QF THE N90- 2 1746 

OQIrJIN AND RtHAVIf?A OF ASYHMETRIC VnRTICES 

(NASA) 19 p CSCL 01A 

Uncl as 

03/02 0280238 

National Aeronautics and 

Space Administration 
NASA Technical Memorandum 102796 

Analytical Study of the Origin 

and Behavior of Asymmetric 

Vo r t i ce s ~ 

Murray Tobak, Arnes Research Center, Moffett Field, California 

David Degani, Technion, Israel Institute of Technology, Haifa, Israel 

Gregory G. Zilliac, Ames Research Center, Moffett Field, California 

April 1990 

National Aeronautics and 

Space Administration 

Ames Research Center 

Moff ett Field, California 94035 1 OOO 
SUMMARY 

An hypothesis advanced originally to explain computational observations is supported by 

theoretical considerations: The asymmetric mean flow observed on bodies of revolution at moderate 

to high angles of attack is the result of a convective instability of an originally symmetric flow to a 

time-invariant space-fixed disturbance. Additionally, the time-dependent fluctuations characteristic of 

the flow at higher angles of attack (up to 90°) are the result of an absolute instability of an originally 

steady flow to a small temporal disturbance of finite duration. Within a common domain, the instability mechanisms may coexist. 

- 

The experimentally confirmed existence of bistable states, wherein the side-force variation 

with nose roll angle approaches a square-wave distribution, is attributed to the dominant influence of 

a pair of trailing vortices from the ogival forebody. Their existence is made possible by the appearance of foci of separation in the skin-friction line pattern beyond a critical value of angle of attack. The 

extreme sensitivity of the asymmetric flow orientation to nose geometry, demonstrated experimentally, is attributed to the presence of an indeterminate phase in the family of possible solutions for 

the three-dimensional wave system. 

INTRODUCTION 

It is well known that when a body of revolution is inclined at a sufficiently high angle to an 

oncoming stream, the steady-state component of the flow over the body becomes asymmetric with 

respect to the angle-of-attack plane. The origin of the asymmetry has remained a mystery for many 

years, but recent advances based on computational results have opened the path toward a rational 

explanation of the phenomenon. 

In references 1 and 2, a time-accurate, thin-layer Navier-Stokes code was used to investigate 

the three-dimensional flow fields over ogive-cylinders (LID = 6.5 to L/D = 12.5) at several angles of 

attack. At a = 40°, the mean flow was found to be symmetric. Placement of a small space-fixed and 

asymmetrically disposed perturbation near the tip of the body caused the mean flow to become 

asymmetric. Upon removal of the perturbation, the mean flow returned to a symmetric state (cf. fig.1, 

taken from ref. I). At a = 60" and a = 80°, without a fixed perturbation at the tip, a small temporal 

disturbance of finite duration was sufficient to set off an unsteady flow involving vortex shedding, 

which evolved to a finite-amplitude fluctuation around a symmetric mean flow. At a = 80°, the 

Strouhal frequency was about 0.2, which is consistent with the value expected from vortex shedding 

in the case of flow past a two-dimensional cylinder under the same oncoming flow conditions (cf. 

fig. 2, taken from ref. 2). At a = 60°, when a permanent disturbance was also added near the tip, a 

pair of asymmetrically disposed vortices emerged near the nose, and correspondingly, the fluctuating 

flow developed an asymmetric mean component, just as at a = 40" (cf. fig. 3, taken from ref. 2). 

Degani (ref. 2) recognized the relevance of the concepts of convective and absolute instability 

to the observed flow behavior, and advanced an explanation of the observations in terms of the two 

categories. He suggested that the asymmetric mean flow observed in the moderate to high angle-of- 
attack range was the result of a convective instability of an originally symmetric flow to a timeinvariant, space-fixed disturbance. On the other hand, Degani suggested that the periodic fluctuations characteristic of the flow at higher angles of attack (up to 90") were the result of an absolute instability of an originally steady flow to a small temporal disturbance of finite duration. 

In references 3 to 6, various shear flows were studied by means of linear stability analysis 

techniques. Typically, the linear analysis of a flow that is unstable to a class of disturbances allows 

one to characterize the initial growth of the disturbed flow subsequent to a perturbation. The terms 

convective and absolute instability were coined to distinguish between the two classes of instability 

as determined by examining the eigenvalues of the stability analysis. The specifics of the stability 

analysis are beyond the scope of this paper, but in any event, it is believed that if the presence of an 

absolute instability is indicated, the fluctuations that are triggered by a short-duration excitation will 

persist and grow to an equilibrium state even after the excitation has been removed. In contrast, if 

the presence of a convective instability is indicated, the initially growing fluctuations that are triggered by a short-duration excitation will be washed downstream and, in the absence of further excitation, the flow will return to its original unperturbed state. 

I 

The notion of instability, either of the convective or the absolute type, is a mathematical construct. In the computational realm, it is possible to follow the mathematical prescription and, at least 

in principle, prove that an observed flow actually represents the stable state resulting from the 

response of an unstable original flow to a disturbance. A similar proof in the experimental realm 

would be considerably more difficult, in view of the necessity of providing a disturbance-free environment in which the original flow could persist until it was deliberately disturbed. In reference 7 (an 

experimental companion piece to ref. 2), the aim was more limited. One of its implicit objectives was 

to demonstrate that the experimentally obtained observations and measurements on a pointed body 

of revolution at incidence were consistent with the explanation advanced in reference 2 of the computational observations in terms of convective and absolute instabilities. Another objective was to 

address a set of issues that had been raised previously by the results of other experimental investigations. In particular, the question of whether bistable states existed (refs. 8-1 l), and if so, under 

what conditions, was addressed. Additionally, some of the issues raised by previous investigators 

(refs. 10-13) about the role played by the tip geometry in determining the asymmetric orientation of 

the flow field were confronted. 

. 

The study undertaken here is intended as an analytical counterpart to references 2 and 7. The 

discussion is aimed at providing a theoretical basis, both to the explanation of the computational 

observations in reference 2 and the experimental observations in reference 7. 

THEORETICAL CONSIDERATIONS 

This section contains ideas from a forthcoming article by D. C. Hill and M. Tobak on the stability of flow past an inclined cylinder. The features that we believe need to be captured in a theoretical treatment are evident in the set of flow-visualization photographs provided by Ramberg (ref. 14). 

In figure 4, these features are emphasized schematically, and, at the same time, the problem is 

simplified by allowing the pointed cylinder to be infinitely long. 

2 
In figure 4, one sees that there are three distinct regimes of flow. The first is far enough 

downstream from the nose that the influence of the nose on the flow there is negligibly small. 

Accordingly, the flow in regime (1) is essentially the flow past an inclined cylinder, for which the 

independence principle (ref. 15) holds. In the crossflow direction normal to the cylinder axis, we have 

simply the two-dimensional flow past a cylinder. It has now been verified both theoretically (ref. 16) 

and experimentally (ref. 17) that the steady two-dimensional flow past a cylinder becomes unstable 

beyond a crossflow Reynolds number of about 50. As a result of the instability, the disturbed flow 

grows up to a new equilibrium state that replaces the original steady flow. 

The event is describable by means of bifurcation theory. When, as here, the steady flow is 

replaced by one containing a periodic time-dependent component, the event is known as a Hopf bifurcation. Physically, the periodic time-dependent component manifests itself by the appearance in the 

wake of vortex shedding parallel to the cylinder axis. Computations by Yang and Zebib (ref. 6) have 

confirmed that when the disturbance exciting the instability has an antisymmetric component relative 

to the symmetry plane and is situated in the near wake (the region enclosing the separation bubble), 

the instability is absolute. A disturbance of finite duration is sufficient to set off the vortex shedding 

which continues indefinitely. 

Regimes (2) and (3) both lie between the nose and regime (l), and hence the nose influences 

their corresponding flows. The principal change is that the instability mechanism becomes threedimensional, allowing flow properties to become quasi-periodic functions of the axial coordinate. In 

regime (2), the periodic time-dependent shedding of vortices continues as before, but, owing to the 

presence of the nose, the vortices are inclined obliquely to the cylinder axis. Close to the cylinder, 

the set of obliquely shedding vortices composes a wave system that is propagating ygstream relative to the body surface. One member of this system has a wave speed of zero relative to the surface, and this defines regime (3), the asymmetric steady-state component. Thus, the source of the 

asymmetry in the steady flow that is observed in experiments with inclined bodies of revolution may 

be attributed to a three-dimensional outgrowth of the instability mechanism that causes the appearance of vortex shedding in the two-dimensional flow past a cylinder. In contrast to the twodimensional case, however, the three-dimensional wave system requires continuous excitation to 

persist; that is, the instability is convective, as will be demonstrated later. 

In Ramberg's experiments, which involved bodies of high fineness ratio, (LID -loo), all three 

regimes are easily identifiable in the flow-visualization pictures, although the extent of regime 

(1) diminishes rapidly as a approaches zero. In experimental studies involving bodies of lower fineness ratio, such as the one studied in reference 7 (LID = 16), at low to moderate values of a regime 

(1) is virtually nonexistent, because the influence of the nose extends over the length of the body. In 

the virtual absence of regime (l), there is no source of periodic fluctuations to excite regime (2), so 

that it also is virtually nonexistent. Hence, at moderate values of a, the flow is dominated by regime 

(3), the asymmetric, steady-state component. The presence of regime (l), and with it, regime (2), 

becomes identifiable only after a has become sufficiently large (in ref. 7, a - 65'). Notice that within 

this latter range, while regime (3) would be absent in the absence of a fixed perturbation at the nose, 

the three-dimensional component represented by regime (2) still would be present, forced by regime 

(l), as would be evident by the presence of a fluctuating flow (having two dominant frequencies) 

around a symmetric mean flow. Regime (1) then becomes predominant as a approaches 90°, 

. 

squeezing regimes (2) and (3) into an ever-diminishing zone just behind the nose. 

3 
Let US now study the development of the three-dimensional wave system more closely. To 

show why the instability mechanism that sets it off must be convective, consider the following 

thought experiment. Suppose we have, simply, the flow past an infinitely long inclined cylinder, and 

that the cylinder is equipped with a small retractable pin located at a point on the leeward side, 

asymmetrically disposed with respect to the angle-of-attack plane (i.e., the symmetry plane in the 

crossflow direction). When the pin is inserted into the flow, there first will be a transient response 

that is eventually replaced by an equilibrium state. The equilibrium state will consist again of the 

same three regimes that were observed to characterize flow past the inclined pointed cylinder. 

Regime (1) will prevail in a zone far enough downstream of the pin to escape its influence. Regimes 

(2) and (3) will exist together in the intervening zone between the pin and regime (1). The steady 

flow component corresponding to regime (3) may differ in form from the flow characterizing its counterpart on the inclined pointed cylinder, but its origin is the same, nevertheless. If the pin is now 

retracted, after a transient period, we must have the stable equilibrium state for flow past an inclined 

cylinder, a state which can sustain only parallel vortex shedding. Both the steady and time-dependent components of the three-dimensional wave system must vanish. This demonstrates that the 

three-dimensional wave system requires continuous excitation to persist; that is, its instability 

mechanism is convective. 

INTERPRETATION OF RESULTS 

Convective and Absolute Instabilities 

Theoretical analysis fully supports the hypothesis advanced in reference 2. Each of the computational observations leading to the hypothesis can be given a firm theoretical basis. The asymmemc mean flow (fig. 1) observed in the moderate angle-of-attack range is the result of a convective 

instability of an originally symmetric flow to a time-invariant, space-fixed disturbance. The virtual 

absence of time-dependent fluctuations in this range is consistent with the absence of regime (l), in 

view of the finite length of the body. At higher values of a, when regime (1) is present, in the 

absence of a fixed perturbation at the nose, the observed fluctuating flow around a symmetric mean 

flow (fig. 2) is the result of an absolute instability of an originally steady Symmetric flow to a small 

temporal disturbance of finite duration. In effect, the disturbance triggers regime (l), which in turn 

acts as a forcing to sustain regime (2). This is also the situation for the solid curve in figure 3(a) and 

in figure 3(b). Adding a fixed perturbation at the nose (dashed curve, fig. 3(a) and fig. 3(c)) then triggers regime (3). This demonstrates that the instability mechanisms triggering regimes (1) and (3) 

may exist either separately or together. In the absence of an external periodic source, however, it 

appears that regime (2) cannot persist without the presence of regime (1) to continually excite it. 

Results of the experimentally obtained observations and measurements of reference 7 also 

are consistent with the hypothesis and provide an indirect confirmation of it in the following sense. It 

was demonstrated experimentally (cf. fig. 5, taken from ref. 7) that when the nose geometry was 

made symmetric with respect to one or more planes, orthogonal orientations of these planes with 

respect to the angle-of-attack plane were coincident with the occurrence of zero-crossings in the 

side-force variation with nose roll angle. That is, an imposed symmetry at the nose with respect to 

the angle-of-attack plane suppressed the appearance of the asymmetric flow field. This indirectly 

I 4 
confirms the theoretical assertion that the development of an asymmetric flow field is the result of an 

instability, requiring an asymmemcally disposed, space-fixed disturbance to trigger it. 

Bistable States 

The experimental results of reference 7 confirmed the existence of bistable states both for 

bodies having " natural" tips and tips with imposed symmetry planes. As an aid in understanding 

the conditions that bring about bistable states, it may be helpful to consult the results of reference 18, which show in considerable detail how the stationary component of the three-dimensional 

wave system develops downstream from the nose. Drawn from the original photographic evidence, a 

set of sketches of the topology of the flow in crossflow planes at a succession of closely spaced 

streamwise stations is presented. The crossflow planes are oriented normal to the free-stream 

direction. 

One sees that the stronger of the two primary vortex structures grows continually in successive stations downstream, assimilating the newly formed vortical structures that are born beneath it. 

The growth of this structure eventually reaches a saturation point. The other vortical structure then 

becomes dominant and undergoes a similar history of growth to saturation. The two vortical structures gradually lose their link'to the structures beneath them as the angle of attack increases and 

eventually become the trailing vortex system of the ogival forebody. 

In a set of experimental results obtained under similar conditions, it is this pair which can be 

identified in the hot-wire survey plots of reference 19 (cf. fig. 6) and in the smoke/laser sheet pictures 

of reference 7 (fig. 7). These results, obtained for a = So, must also be associated, however, with 

another factor which has already entered at a slightly lower value of a. Surface oil-flow visualization 

studies (not reported in refs. 7 or 19) reveal the presence of foci of separation in the skm-friction line 

patterns at angles of attack beyond about a = 50". An example for a = 55" is shown in figure 8 

(Degani, D. and Zilliac, G. G., unpublished data, 1988). A focus is visible (with a second barely discernible) on the line of separation just aft of the ogive/cylinder junction. A similar focus appears on 

the line of separation on the other side, forward of the ogivekylinder junction. 

We make the following conjectures regarding their function. First, by considering the analogous situation for a finite-length wing or body, we argue that the foci play the same role in the two 

cases: They act as the anchor points that allow the stream surfaces containing the cast-off vortical 

structures to roll up in the wake and form the trailing vortex system. An attempt to portray schematically how a pair of foci might carry out this role on an ogive-cylinder is shown in figure 9. Alongside 

is a sketch which we have used as a model, depicting how foci cany out the same role on a finitelength wing or body (cf. ref. 20). Notice that the ogive-cylinder may require an additional pair of foci 

to originate the lines of separation on the cylinder. 

. 

Second, we argue that by allowing the stream surfaces containing the pair of cast-off vortical 

structures to roll up in the wake, the foci enable the formation of a trailing vortex system which is virtually independent of the details of the nose geometry. The change in surface flow topology thereby 

signals a significant change in the properties of the flow field. Before the foci appear, flow-field properties vary continuously with variations of the roll-angle position of the nose. After the foci appear, 

5 
the discontinuous behavior characteristic of bistable states becomes possible. The following argument suggests that it is the freeing of the pair of cast-off vortical structures that accounts for the 

onset of bistable states. 

In the early stage of development (incidence angles below the critical value for the appearance of foci, or near the nose), all of the new vortical structures are linked and they influence each 

other. When the nose is rolled to a new position, all of the structures move accordingly, so their net 

contribution to the side force varies continuously with variation of the roll-angle position of the nose. 

In the later stage of development (incidence angles beyond the critical value for the appearance of foci, and downstream of the foci), where the stream surfaces containing the pair of cast-off 

vortical structures have rolled up to form the trailing vortex system, the situation is different in two 

respects. First, because the influence on side force of the remaining linked vortices decays with distance downstream, the pair of trailing vortices emanating from the tip becomes the dominant factor in 

determining the magnitude and sign of the side force. Second, this pair of vortices no longer responds 

to the particular orientation of the nose, but only to the ability of the nose to set the right-hand or 

left-hand pattern of the flow field behind it. A simple application of the Biot-Savart law in a crossflow 

plane will suggest that the trailing vortices are able to find stable positions at which the balance of 

forces on them is zero, allowing them to remain stationary relative to the cylinder and to each other. 

Thus, as long as the flow-field pattern does not change hand as the nose is rolled to new positions, 

the side-force contribution attributable to the pair of trailing vortices remains virtually constant. 

Then, this side-force contribution abruptly shifts to the opposite sign when the pair of trailing vortices responds to a change in hand of the flow-field pattern by shifting to its mirror image orientation 

relative to the angle-of-attack plane (cf. fig. 7). These two factors appear to be sufficient to explain 

why the side-force variation characteristic of the later stage of development approaches a bistable 

state, having virtually a square-wave variation with roll-angle position of the nose. 

We note that for a finite-length body, the flow at low to moderate values of a is dominated by 

the early stage of development, whereas the later stage becomes dominant as a approaches 90'. 

The side-force variation with roll-angle position of the nose changes accordingly from continuous at 

low to moderate a to a square-wave pattern for larger a, after the first appearance of foci in the skinfriction line pattern. Finally, we note again that as a approaches 90°, regime (3), the regime consisting of an asymmemc steady flow, becomes confined to an ever-diminishing zone just behind the 

nose. The magnitude of the square-wave pattern of side force characteristic of this regime therefore 

approaches zero as a approaches 90'. 

The existence of extremes of side-force magnitude that are independent of tip geometry and of 

a maximum of all extremes both are consistent with the virtually independent behavior of the pair of 

trailing vortices that we have postulated. The magnitude of their contribution to the side force was 

said to be independent of the tip geometry, to become dominant as a increases, but to approach zero 

as a approaches 90". Hence, it is natural that a maximum of all magnitudes should exist at some 

particular value of a. 

6 
Nose Geometry 

It is true that nose geometry is the dominant factor in determining the leeward-side vortex 

orientation. The orientation’s sensitivity to even the minute changes in nose geometry caused by the 

accumulation of dust particles was clear testimony to that. An explanation of this extreme behavior 

on the basis of theory may become possible as a result of further study of the three-dimensional 

wave system that composes regime (3) of our description of the flow field. We can already see that, 

in general, there will be a family of possible solutions whose dependence on the axial coordinate will 

be wave-like, containing an indeterminate (open) phase. From the infinity of possible solutions, the 

particular tip geometry will pick out the one having the appropriate phase. Hence, every alteration to 

the tip will alter the phase accordingly. The sensitivity of the flow field to size, location, and form of 

perturbation may find an explanation on this basis. 

The application of theory may also clarify the role of other perturbations in triggering the 

asymmetric flow field when tip perturbations are absent. Theory suggests that a minute perturbation 

at the tip will activate growth of the asymmetry in the flow field immediately behind it, which in turn 

will activate that of the succeeding flow field, and so forth downstream. In the absence of a tip perturbation, any one of these activated flow fields could be considered an appropriate trigger for succeeding flow fields. To achieve the same effect as the tip perturbation, however, it is clear that a 

downstream trigger would have to be very large. This discussion suggests that to each position of 

the body there corresponds a critical dimension, characterizing the size of a disturbance at that site 

just sufficient to set off the instability in the flow field behind it. 

CONCLUSIONS 

The analysis carried out here was intended to provide a theoretical basis to the explanation in 

terms of convective and absolute instabilities of the computational observations contained in reference 2 and to the experimental observations contained in reference 7. Both pertain to the phenomena 

of the flow about an ogive-cylinder of finite length placed at incidence to an oncoming stream. Principal conclusions of the analysis follow. 

1. Theoretical considerations support the hypothesis advanced previously on the basis of 

computational observations: At moderate angles of attack, where the nose influences flow over the 

entire body, the observed asymmetric mean flow is the result of a convective instability of an originally symmetric flow to a permanent disturbance applied at the tip, asymmetrically disposed with 

respect to the angle-of-attack plane. At higher angles of attack, where the flow over the aft part of 

the body escapes the influence of the nose, the original steady symmemc flow is additionally absolutely unstable to a temporal disturbance of finite duration, applied asymmemcally with respect to 

the angle-of-attack plane, Triggering the absolute instability results in the appearance of both pardle1 and oblique vortex shedding. The fluctuations may occur about a symmemc mean flow in the 

absence of a fixed asymmetrically disposed disturbance, or about the asymmetric mean flow resulting from the convective instability triggered by the presence of such a fixed disturbance at the tip. 

- 

7 
2. The experimentally confmed existence of bistable states, wherein the side-force variation 

with nose roll angle approaches a square-wave distribution, is attributable to the dominant influence 

of a pair of trailing vomces from the ogival forebody. Their formation and virtual freedom from the 

influence of the particular nose geometry is made possible by the appearance beyond a critical value 

of the angle of attack of foci of separation in the skin-friction line pattern. 

3. At low to moderate incidence angles, the extreme sensitivity of the asymmetric flow orientation to nose geometry (demonstrated experimentally) is attributable to the presence of an indeterminate phase in the family of possible solutions for the three-dimensional wave system. 

8 
REFERENCES 

1. Degani, D.; and Schiff, L. B.: Numerical Simulation of the Effect of Spatial Disturbances on 

Vortex Asymmetry. AIAA Paper 89-0340, Jan. 1989. 

2. Degani, D.: Numerical Investigation of the Origin of Vortex Asymmetry. AIAA 

Paper 90-0593, Jan. 1989. 

3. Triantafyllou, G. S.; Kupfer, K.; and Bers, A.: Absolute Instabilities and Self-sustained 

Oscillation in the Wakes of Circular Cylinders. Phys. Rev. Letters, vol. 59, 1987, 

pp. 1014-1917. . 

4. Chomaz, J. M.; Huerre, P.; and Redekopp, L. G,.: Bifurcations to Local and Global Modes in 

Spatially Developing Flows. Phys. Rev. Letters, vol. 60, 1988, pp. 25-28. 

5. Sreenivasan, K. R.; Roghu, S.; and Kyle, D.: Absolute Instability in Variable Density Round 

Jets. Exper. Fluids, vol. 17, 1989, pp. 309-317. 

6. Yang, X.; and Zebib, A;: Absolute and Convective Instability of a Cylinder Wake. Phys. 

Fluids, vol. 4, 1989, pp. 689-696. 

7. Zilliac, G. G.; Degani, D.; and Tobak, M.: Asymmetric Vortices on a Slender Body of 

Revolution. AIAA Paper 90-0388, Jan. 1990. 

8. Lamont, P. J.: Pressures Around an Inclined Ogive Cylinder with Laminar, Transitional or 

Turbulent Separation. AIAA J., vol. 20, no. 1 1, Nov. 1982, pp. 1492- 1499. 

9. Lamont, P. J.: The Complex Asymmetric Flow Over a 3.5D Ogive Nose and Cylindrical 

Afterbody at High Angle of Attack. AIAA Paper 82-0053, Jan. 1982. 

10. Dexter, P. C.; and Hunt, B. L.: The Effects of Roll Angle on the Flow Over a Slender Body of 

Revolution at High Angle of Attack. AIAA Paper 81-0358, Jan. 1981. 

11. Hunt, B. L.; and Dexter, P. C.: Pressures on a Slender Body at High Angle of Attack in a 

Very Low Turbulence Level Airstream. AGARD CP 247, Paper 17, 1979. 

12. Keener, E. R.: Flow Separation on Symmetric Forebodies. NASA TM-86016, 1986. 

13. Moskowitz, C. A.; Hall, R. M.; and Dejamette, F. R.: Effects of Nose Bluntness, Roughness 

and Surface Perturbations on the Asymmetric Flow Past Slender Bodies at Large Angles of 

Attack. AIAA Paper 89-2236, Aug. 1989. 

14. Ramberg, S. E.: The Effects of Yaw and Finite Length on the Vortex Wakes of Stationary and 

Vibrating Cylinders. J. Fluid Mech., vol. 128, 1983, pp. 81-107. 

9 
15. Jones, R. T.: Effects of Sweepback on Boundary Layer and Separation. NACA Rep. 884, 1947. 

16. Zebib, A.: Stability of Viscous Flow Past a Cylinder. J. Eng. Math., vol. 21, 1987, pp. 155-156. 

17. Provansal, M.; Mathis, C.; and Boyer, L.: Benard-von Karman Instability: Transient and 

Forced Regimes. J. Fluid Mech., vol. 182, 1987, pp. 1-22. 

18. Ward, K. C.; and Katz, J.: Topology of the Flow Structures Behind an Inclined Projectile: Part 

B. J. Aircraft, vol. 26, no. 11, Nov. 1989, pp. 1023-1031. 

19. Degani, D.; and Zilliac, G. G.: Experimental Study of Nonsteady Asymmetric Flow Around an 

Ogive-Cylinder at Incidence. AIAA J., vol. 28, no. 4, April 1990, pp. 642-649. 

20. Hummel, D.: On the Vortex Formation Over a Slender Wing at Large Angles of Incidence. 

AGARD CP 247, Paper 15, 1979. 

10 
. 

WITH JET 

WiO JET 

0 120 240 360 480 

NON-DIMENSIONAL TIME 

Figure 1.- Side-force coefficient history, M, = 0.2, a = 40°, ReD = 26,000 (ref. 1). 

2, 

0 80 160 240 320 

NON-DIMENSIONAL TIME 

Figure 2.- Side-force coefficient history, a = 80°, M, = 0.2, ReD = 200,000 (ref. 2). 

11 
6 -. W/O DISTURBANCE 

WITH DlSTU R BANCE --- 4 

v '.' 

-4 ' 

0 80 160 240 

(ai NON DIMENSIONAL TIME 

. 

Figure 3.- Side-force coefficient history, a = 60°, M, = 0.2, ReD = 200,000 (ref. 2). a) Side-force 

coefficient, b) off-surface streamlines without disturbance, c) off-surface streamlines with 

disturbance. 

12 
ON AN INFINITELY LONG INCLINED POINTED CYLINDER 

@ PARALLEL SHEDDING 

@ OBLIQUE SHEDDING 

@ STATIONARY TIP VORTICES 

Figure 4.- Leeward-side flow regimes. 

5 

CY 0 

-5 

NATURAL TIP 

I 

90 

9, 

MACHINE[ 

FLATS 

-- 

D 

eg 

0 

Figure 5.- Side force versus nose roll angle. Natural tip and tip with microscopic machined flats, 

ReD = 30,000, a = 53" (ref. 7). 

13 
15 15 

Figure 6.- Hot-wire survey of a plane. XIL = 0.55, a = 55O, ReD = 26,000 (ref. 19). a) Mean velocity, 

b) RMS velocity. 

Figure 7.- Smoke-laser sheet visualization of flowfield, a = So, ReD = 30,000 (ref. 7). a) Left hand, 

Cy = 3.79, b) right had, Cy = -3.57. 

14 
Figure 8.- Surface oil-flow visualization, ReD = 26,000, a = 55" (Degani, D. and Zilliac, G. G., 

unpublished data, 1988). 

15 
Figure 9.- Conjectured mechanism for formation of trailing-vortex system on a) ogive-cylinder, and 

b) finite-length wing or body. 

16 
Report Documentation Page 

1 Report No. 2. Government Accession No. 

NASA TM- 102796 

4. Title and Subtitle 

Analytical Study of the origin and Behavior of Asymmetric 

3. Reclpmnt's Catalog No. 

5. Report Date 

April 1990 

Vortices 

7. Author(s) 

Murray Tobak, David Degani,* and Gregory G. Zilliac 

6. Performing Organization Code 

8. Performing Organization Report No. 

A-90089 

10. Work Unit No. 

505-6 1-7 1 

I 2. Sponsoring Agency Name and Address 

9. Performing Organization Name and Address 

11. Contract or Grant No. 

Ames Research Center 

Technical Memorandum 

Moffett Field, CA 94035- lo00 

Asymmetric vortices, Convective and absolute 

instability, Bistable states, Bodies of revolution, 

figh angles of attack 

13. Type of Report and Period Covered 

Unclassified-Unlimited 

19. Security Chssif. (of this report) 20. Security Chssif. (of this page) 21. No. of Pages 

Unclassified Unclassified 19 

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

22. Price 

A02 

e 14. Sponsoring Agency Code 

I 

5. Supplementary Notes 

Point of Contact: Murray Tobak, Ames Research Center, MS 227-4, Moffett Field, CA 94035-1000 

*Associate Professor on leave from Technion, Israel Institute of Technology, Haifa, Israel 

(415) 604-5855 or FTS 464-5855 

6. Abstract 

An hypothesis advanced originally to explain computational observations is supported by 

theoretical considerations: The asymmemc mean flow observed on bodies of revolution at moderate to 

high angles of attack is the result of a convective instability of an originally symmetric flow to a timeinvariant space-fixed disturbance. Additionally, the time-dependent fluctuations characteristic of the 

flow at higher angles of attack (up to 90') are the result of an absolute instability of an originally steady 

flow to a small temporal disturbance of finite duration. Within a common domain, the instability 

mechanisms may coexist. 

The experimentally confirmed existence of bistable states, wherein the side-force variation with 

nose roll angle approaches a square-wave distribution, is attributed to the dominant influence of a pair 

of trailing vortices from the ogival forebody. Their existence is made possible by the appearance of foci 

of separation in the skin-friction line pattern beyond a critical value of angle of attack. The extreme 

sensitivity of the asymmetric flow orientation to nose geometry, demonstrated experimentally, is 

attributed to the presence of an indeterminate phase in the family of possible solutions for the threedimensional wave system. 

17. Key Words (Suggested by AuthorW) I 18. Distribution Statement 

Subject category - 02 

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