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AlAA 99-3809 PREDICTING COMPLEX VORTEX STRUCTURES - RECENT ADVANCEMENTS IN SIM’ULATIONS Egon Krause AE!rodynamisches lnstitut RWTM Aachen Aachen, Germany

30th AIAA Fluid Dynamics,.Conference 28 June - 1 July, 1999 / Norfolk; VA

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AIAA-99-3809

PREDICTING COMPLEX VORTEX STRUCTURES - RECENT ADVANCEMENTS IN SIMULATIONS

Egon Krause* Aerodynamisches Institut, RWTH Aachen, 52062 Aachen, Germany

ABSTRACT

Results of- recent numerical simulations of five incompressible and compressible flows contain- ing vortex structures are presented in comparison with experimental results. The examples chosen are: (1) the leeward vortices on delta wings at large angles of attack, computed with a solution of the parabolized Navier-Stokes equations and compared with experi- mental data for supersonic speeds; (2) the counter- rotating vortex pairs, similar to Goertler vortices, observed in experiments downstream from shock- induced boundary-layer separation in hypersonic flow, computed with a solution of the Navier-Stokes equations, so are the following three examples, for which numerical simulations were compared with recent experimental visualizations; (3) the oblique collision and interaction of two vortex rings, studied experimentally in incompressible flow by T. T. Lim; (4) the formation and interaction of vortex rings in cylinders of automotive engines during the suction and compression stroke; (5) finally, the bubble-type breakdown of swirling decelerated pipe flow will be compared with earlier experiments of J. H. Faler and S. Leibovich, including the transition from bubble- to spiral-type breakdown of a free slender vortex.

1. INTRODUCTION

The marked increase in storage capacity, computational speed, and substantial improvements in solution techniques enabled continuous advancement of numerical simulation of time dependent vertical flows in three dimensions during recent years. To demonstrate the quality of the simulations that can now be obtained, some numerical results are com- pared with experimental data. Problems considered are two three-dimensional steady external flows and three unsteady flows, containing one or more vortex structures in external or internal flows, for which experimental data were either available or obtained recently for the comparison. The major aim of the paper is to seek direct comparison of numerical and experimental results and not to describe the details of the numerical solutions employed.

The problems reported in the following five sections were subjected to numerical and experimen- *

* Professor em., Aerodynamisches Institut, RWTH Aachen, Member DGLR, Germany.

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tal investigations for some time at the Aerodynamis- ches Institut. The experiments were specifically de- signed to provide data for the comparison with nu- merical results.

The first example presented in Sec. 2 is the supersonic flow over a delta wing with rounded lead- ing edge. Flow over delta wings have been studied for a long time, but the International Vortex Flow Ex- periment concluded in 1988 with the statement that the investigations are “neither complete nor conclu- sive“, despite the large experimental efforts provided for this campaign [ 11. Although in previous experi- mental investigations, the influence of the rounding of the leading edge on the flow was studied to some extent, more recent measurements [2] show a varia- tion of the local pressure as a function of the Rey- nolds number, that was unknown until then. The study of the leeward vortices therefore seemed to be an interesting topic for a combined numerical and ex- perimental investigation.

The second example presented in Sec. 3 came from an experimental observation of longitudi- nal flow structures downstream from hypersonic oblique shock boundary interaction with infrared thermography [3]. The thermograms seemed to sug- gest vortex structures, with their axes aligned with the direction of the main flow, but a positive confirmation of this assumption could not be established from the experiments alone. It was therefore decided to seek clarification with the aid of numerical solutions. The major results of this study will be discussed in this section.

A numerical solution of the Navier-Stokes equations also enabled the detection of the formation and interaction of vortex rings in cylinders of auto- motive engines during the suction and compression stroke [4]. The results were experimentally verified with a specially designed test stand [5], in which the flow was visualized in axial and radial sections, for which also velocity measurements were obtained with the Particle-Image Velocimetry. Closely related to the problem of describing the formation of vortex rings is the question of describing the details of the interaction of vortex rings. This problem was studied experi- mentally by T. T. Lim in 1989 for the case of oblique

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collision of two vortex rings ]6]. This experiment was numerically simulated and compared with the avail- able experimental data by J. Hofhaus in 1996, [7]. Some of the major results of this comparison will be shown in Sec. 4. The numerical simulation for vor- tices in engine cylinders will be compared with ex- perimental results in Sec. 5.

Finally, in Sec. 6, bubble-type breakdown of swirling decelerated pipe flow and the transition from bubble- to spiral-type breakdown of a free slender vortex were simulated numericaIly with solutions of the Navier-Stokes equations for time dependent, in- compressible three-dimensional flow. The breakdown of swirling pipe flow was already studied experimen- tally by several authors. For the comparison with the numerical data the experiment of J. H. Faler and S. Leibovich was chosen [S] .

The simulation of the flow over a delta wing and the comparison with experimental data will be discussed first in the following section.

2. LEESIDE VORTICES ON A DELTA WING

Delta wings generate a vortex system on the leeward side of the wing, which enhances -the lift at high angle of attack. Although this problem has been studied in numerous investigations, the various flow modes observed for example for supersonic flow, cannot completely be understood from experimental considerations alone [l]. It was, however, found out in extended combined numerical and experimental studies, that the vortex structures can be determined uniquely and the accuracy of the numerical results can be verified by comparing the pressure surface distri- butions with experimental data.

The numerical solution employed in previous studies is a space marching technique for solving the parabolized Navier-Stokes equations for supersonic free-stream conditions. This approach is well suited as long as the flow does not separate in the direction of the main flow. Since the associated second deriva- tives of the flow variables can be neglected, the stor- age capacity required is relatively small and the num- ber of grid points can therefore be rather- large so that high accuracy can be attained by small grid sizes [9]. The solution must be iterated because of the non-line- arities in the continuity, momentum, and energy equations. In [9] the iteration was facilitated with a time-like operator, approximated with a 5-step Runge-Kutta method.

In the numerical solution presented in E91 The convective terms are d&ret&d with a finite: volume technique, in which the second-order flux- vector splitting proposed in [lo] is employed, com- bined with the limiter function of Albada. The viscous

terms are as usual discretized with central differences. In the subsonic part of the flow, very close to the wall, where the pressure gradients in the direction normal to the surface are also retained, the flow vari- ables are extrapolated to the neighboring downstream points with second-order accuracy. Details of the solution may be found in [9].

Fig. 1 shows an example of vortex formation in the cross-flow on the leeward side of a delta wing as obtained with the solution of [9] in comparison to experimental results.

Fig. 1: Formation of leeward vortices on a delta wing with a round leading edge; sweep angle cp = 75”; geometry specified in [2]. Ma, = 2.0, Re, = 4x106, angle of attack a = 24’; shown are computed cross- flow streamlines at 61 percent chord. Primary, secon- dary, tertiary, quaternary, and shear-layer vortex [9] (upper picture), and an experimental visualization of incompressible flow (lower picture) [ 111. The incom- pressible vortex structures are very similar to those computed for supersonic flow; differences are noted in the cross-sectional shape of the individual vortices.

The wing has sweep angle cp= 75” anda

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rounded leading edge. The complete geometry of the wing is described in [2]. The free-stream .conditions correspond to a Mach number Ma, = 2.0, a Reynolds number Re, = 4x106, and an angle of attack a = 24”. Shown are the streamlines of the cross-flow at a posi- tion of 61 percent chord. At this relatively high angle of attack, five vortices can be identified, as obtained from the computation with the method of [9]: The primary, secondary, tertiary, quaternary, and the shear-layer vortex.

Since it was not possible to simulate the flow at a free-stream Mach number Ma, = 2.0 in the wind tunnel available, an experimental visualization was attempted with the same model in incompressible flow in [ll]. The picture taken with the Laser-light sheet technique is shown in the lower part of Fig. 1. The primary, secondary, tertiary and the shear-layer vortex are clearly recognized. The quaternary vortex can also be seen in the original photo, but is only vaguely visible in the picture of Fig. 1. It seems that the number of vortices formed is not influenced by the density variation in the supersonic flow, but only the cross-sectional shape of the individual vortices.

The computations were carried out under the assumption that the flow is laminar such that closure assumptions did not have to be introduced for the Reynolds stresses and the turbulent heat transfer. Since it is also not known, under what conditions the flow undergoes transition to turbulent flow, it was felt best, to simulate the flow in a first attempt for laminar conditions, and then conclude from the comparison of the numerical data with the experimental data on the influence of the turbulent momentum and energy transport.

The comparison of the data is shown in Fig. 2, as an example for the accuracy of the numerical simulation. The spanwise pressure distributions are plotted for Ma, = 2, Re, = 4x106, and an angle of attack 0: = 10” for seventy, eighty, and ninety percent chord. The data shown here were chosen from several experimental campaigns, which are in good agree- ment with the numerical results within the limits of the error bounds. The details of this comparison may be found in [ 121.

The pressure distributions, given in the form of dimensionless pressure coefficients as a function of the dimensionless spanwise coordinate y/s, are com- pletely smooth for small angles of attack. The onset of vortex formation on the leeward side of the wing can be noted in the step-like change in the pressure distri- bution for 0.4 < y/s I 0.6. With increasing angle of attack, the step-like decrease in the pressure develops a pressure minimum, clearly recognized in the upper two diagrams of Fig. 2. The location of the pressure minimum indicates the position of the core of the

primary vortex. The symbols show the pressure coef- ficients of the experiment and the solid lines the nu- merical results, which completely dublicate the for- mer for all three stations. Obviously the displacement effect of the boundary layer is adequately simulated.

RE+4.OOE+06 MA== 2.0 ALPHA= 10. X+0.70

i I 0.0 02 0.4 0.6 0.0 I.0

‘us

RE=4.OOE+O6 MA- 2.0 ALPHA- 10. X&=0.60 I

REd.OOE+06 MA- 2.0 ALPHA= 10. x/L=o.eo

0.0 02 0.4 0.B 08 1.0 y/s

Fig. 2: Comparison of numerically and experimen- tally determined spanwise surface pressure distribu- tions for a delta wing with a round leading edge and a sweep angle cp = 75”. The geometry of the wing is specified in [2]. Shown are the dimensionless pressure coefficients for Ma, = 2.0, Re, = 4x106, an angle of attack a = lo”, at seventy, eighty, and ninety percent chord. The symbols indicate the experimental data, the solid lines the numerical results. Data are from WI

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3. LONGITUDINAL VOTICES IN HYPERSONIC SHOCK-BOUNDARY LAYER INTERACTION

Oblique shock boundary-layer interaction is usually assumed to be a two-dimensional phenome- non, analyzed in numerous theoretical and experi- mental investigations. More recent experimental studies of thermal loads on flaps in hypersonic flow also included the problem of oblique-shock boundary- layer interaction [3]. In these studies infrared ther- mography was used to obtain top-view pictures of the surface flow pattern at a free-stream Mach number Ma, = 6 and a Reynolds number Re, = 2.8~10~. The thermograms revealed temperature variations on the surface immediately. downstream from the separated region. The temperature variations formed striation- like contours of finite length with their longitudinal axis aligned with the direction of the mainflow. Re- peated inspection and examination of the experimen- tal data resulted in the conclusion, that the striations possibly were formed by longitudinal vortex struc- tures in the boundary layer. The streamline curvature downstream from the separated region was offered as a possible explanation for the formation of the longi- tudinal vortices. It was further concluded that then the vortex structures should be similar to Goertler vortices, and a new three-dimensional model resulted from these considerations. It is depicted in Fig. 3.

Fig. 3: Schematic drawing of flow model for the for- mation of longitudinal vortices in a hypersonic boun- dary layer, downstream from a region of oblique- shock boundary-layer interaction, caused by longitu- dinal streamline curvature. The vortices were there- fore termed Goertler vortices [3].

Since.a proof ofthis conjecture could not be extracted from the experimental data, numerical methods were employed. In order to ensure reliability of the, results, ‘two- different solutions were used for the description of the three-dimensional flow. The solutions are described in [14] and [15]. Both meth- ods resulted in a good overall agreement with respect to the simulation of the front shock of the slightly blunted plate, the impinging and the reflected shocks.

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This can be seen in Fig. 4, where the Schlieren image of the shocks (upper picture) [16] is compared with results of the two methods. The- picture in the middle of Fig. 4 shows the computed shock shapes, obtained with themethod described in [15] for a wall tem- perature .of T, = 300 “K, while the lower picture shows the computed shock shapes, obtained with the method of [14] for an adiabatic wall. The Mach num- ber of the experiment was Ma, = 6 and the Reynolds number had the value given before, Re, = 2.8~10~.

450 500 x M i

Fig. 4: Comparison of computed shock patterns with Schlieren image (upper picture), obtained in .[ 161. The numerical data were simulated with the method de- scribed in [15] (picture in the middle) for a wall tem- perature of T, = 3.00 OK, and-for an adiabatic wall with the method described in [14] (lower picture). The Mach number of the experiment was MC = 6 and the Reynolds number Re, = 2.8~10~.

Three-dimensional flow was enforced by pe- riodic lateral boundary conditions. Fig. 5 shows the projections of the velocity vectors in’ a cross-flow plane at a dimensionless streamwise position x = 0.269, measured from the leading edge.

(9 . . . . . . . , . . . . . . . . . . . . . . .

I ____............................,.....: . . . . . . . . . . . . . . . . . . . . . . . . . . 1 : ? I ,..,__.................,_..............,................______... y! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 1 . . . . , . . . . . . . . . . . . . . . 1

Fig. 5: Projections of the velocity vectors. in a cross- flow plane at a dimensionless streamwise position x = 0.269, measured from the leading edge. ‘Four vortex structures can be recognized.

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The vortices depicted in Fig. 5 are counter- rotating vortex pairs, embedded in the boundary layer, with relatively large velocity components in the di- rection normal to the wall. As a consequence, the local temperature exhibits substantial changes, as can be seen in Fig. 6, taken from [ 14).

Fig. 6: Dimensionless wall temperature distribution in a cross-flow plane at a dimensionless streamwise position x = 0.269, measured from the leading edge. Data are taken from [ 141.

I

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

. . . . * * - ~.. . . . . . . . . . ._. ._ _ - . . . . . . . .____ .._____ . . . . . . . . .

Fig. 7: Comparison of a thermogram with the com- putation. The striations observed in the experiment are depicted in the upper part of the Fig.

In Fig. 7 a direct comparison of a thermo- gram with computed data is shown. In the upper part, the striations as observed in the experiment are de- picted, and in the lower picture, Fig. 5 is shown again plotted to scale of the experimental data. It is seen, that the counter-rotating vortex structures are clearly confimed by the numerical simulations.

In concluding this paragraph it is pointed out again, that the lateral boundary conditions, which cannot be obtained from the experiment, were as- sumed to be periodic. This assumption required how- ever the knowledge of an even multiple value of the diameter of one of the Goertler vortices, which could easily be estimated from the experimental observa- tions.

In the next example, time dependent incom- pressible flow of two colliding vortex rings will be considered.

4. COLLIDING VORTEX RINGS

Vortex rings are particularly apt for studying and understanding vortex motion. In [6] T. T. Lim described his experimental studies in which he let two vortex rings of equal strength collide with each other, starting under a certain prescribed angle. In order to find out, what would happen to the two individual rings, he dyed them with different colors. The ex- periments showed, that the two rings first join to a single ring, one half of the new ring having the color of one of the starting rings, and the other half that of the second starting ring. After some time, the newly generated vortex ring is stretched in the direction normal to direction of the original motion, and two new vortex rings are being formed. They separate from the intermediate one-ring structure, again in the direction normal to the direction of the initial motion of the two starting vortices.

This problem was studied with a numerical solution of the Navier-Stokes equations by J. Hofhaus at the Aerodynamisches Institut. His results and a comparison with the experimental data given by T. T. Lim in [6] were reported in his dissertation [7] and in u71.

One of the problems associated with the nu- merical integration of the Navier-Stokes equations for studying the motion of vortex rings consists in fInding suitable initial conditions. It is certainly possible to follow the experiment and release a certain volume of a incompressible fluid through an orifice. Such an approach would, however, require a large number of grid points for the description of the flow during the initial phase. For later phases of the time-dependent flow these points can no longer be used. Another disadvantage of this approach is. that for circular

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cross-sections with corresponding coordinates, grid singularities may give rise to large errors.

Another possibility consists in initiating the computation with the use of Hill’s vortex [IS]. Therein an analytic solution of the governing equa- tions was derived for incompressible inviscid flow, The solution consists out of two parts, an outer part and an inner. The outer part represents the inviscid irrotational flow around a sphere, and the inner the inviscid rotational flow of a vortex ring. Although the two parts of the solution exhibit discontinuous veloc- ity derivatives at the common boundary, numerical experiments showed, that Hill’s vortex can be used as initial condition for the simulation of vortex rings with finite core size..

Since vortex structures, as for -example vor- tex rings, tend to move in the flow field, it must be avoided, that they touch the boundary of the domain of integration during the numerical simulation, as the solution can no longer converge then. In [7] this diffi- culty was circumnavigated by introducing a moving system of coordinates. Since the instantaneous drift velocity of the vortex structures can always be deter- mined, it is possible to keep the vortex structure in the center of the domain of integration, or at least at a safe distance away from the boundaries.

Before comparing numerically predicted collision of vortex structures with experimental visu- alizations, the collision of two vortex rings is sche- matically explained in Fig. 8, taken from [17]. The components of the vorticity vectors parallel to the plane of contact have opposite signs and cause the opening of the initially closed vortex rings. The com- ponents normal to the plane of contact have the same sign and connect the two vortex rings to a single structure. If the strength of the vortices is large

Fig. 8: Schematic of collision of two vortex rings. Opposite signs of vorticity vectors cause the opening.

vortices is large enough the distant parts of the newly generated structure, which have opposite signs, begin to interact, and two new vortex rings may be gener- ated, separating in the direction normal to the original direction of motion.

The shape of the vortex structure which is generated in the collision depends to a large extent on the angle under which the two vortex rings are set in motion. This can be seen in Fig. 9, where the results of numerical simulations are shown for a Reynolds number Re = 500 and the initial angle between the ring axes, cp are shown. Plotted are the vortex lines for initial angles cp = 40°, 50”, 60°, 70”, SO’, and 90”. The results indicate that the formation of two new vortex rings along the vertical is observed only for initial angles cp > 70’. For cp = 80” almost all vortex lines are separated and two new vortex rings are formed. Only for cp = 90” is the separation of vortex rings complete within the numerical accuracy [17]. The formation of two new vortex rings was aheady observed experi- mentally by Fohl and Turner[ 191, although for differ- ent initial angles and Reynolds numbers.

60’

Fig. 9: Formation of vortex structures after collision of two initial rings at a Reynolds number Re = 500 and various initial angles of the collision.

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In the process of redistribution of vorticity, only a portion’ of the strength of the original vortex rings is found in the newly formed vortex pair. As already indicated in Fig. 9 for an initial angle cp = 80’ through the connecting vortex lines, a third vortex structure with closed vortex lines is formed on the leeward side of the vortex pair. The comparison with the experimental data of [6] in Fig. 10 shows, that this redistribution of vorticity is also observed in the ac- tual flow. The third structure results from the interac- tion of the vortex pair generated in the collision, as the two vortex rings (dark structures in Fig. 10) do not lie in one and the same plane. They can therefore continue spatial redistribution of vorticity through interaction.

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Fig. 10: Comparison of computed with the experi- mental data, of [6]. A third vortex structure with closed vortex lines is formed on the leeward side of the vortex pair, (dark spots) as a consequence of their interaction.

5. VORTICES IN FLOWS IN PISTON ENGINES

A better understanding of the unsteady, three-dimensional, compressible flow generated in piston engines during the intake and compression stroke is, crucial for future development and im- provement of piston engines with higher performance and lower emission rates. The main reason for this requirement is that the flow at the end of the compres- sion phase determines the flame propagation speed in homogeneous charge spark-ignition engines, and the

fuel-air mixing and burning rates in Diesel engines. The flow has a major impact on the emission values and on the breathing capacity of the engine and there- fore also on the maximum available power.

During the suction phase, the flow in the cylinder is strongly influenced by the formation of vortex rings and their interaction. The rings are gen- erated by flow separation at the valve seat and head, and the strength of the vortices generated depends on the seat angle, the rounding of the seat comers, the seat width and the swirl of the flow in the intake ports. What is even more important is, that the char- acteristic flow times are of the order of a few milli- seconds and that there are rapid transient changes in the flow.

These rather complex flow phenomena are presently studied in many research institutions. Also at the Aerodynamisches Institut extensive numerical and experimental investigations were carried out in the past, These studies resulted in a numerical solu- tion of the Navier-Stokes equations suitable for the simulation of the time-dependent, three-dimensional, compressible flow arising during the intake and com- pression stroke. The results were compared with ex- perimental data, obtained in specially designed test stands [5]. The major aim of the comparison with the experimental data was to confirm the accuracy of the numerical predictions. Another aim of the investiga- tion was to find a method, with which the vortex structures can be detected and followed in the course of time.

An accurate description of the flow requires the resolution of all important details of the intake and cylinder head geometry. Although other possibilities exist, a boundary-fitted block-structured moving grid system was implemented in the algorithm. The grid is refined during the opening and closing of the valve and during the up- and downward motion of the pis- ton, such that the resolution of the flow can be con- trolled to be approximately constant. In addition to the solution of the conservation equations for mass, momentum and energy, a conservation equation for the cell volume is also solved during the integration, The solution rests on an explicit finite-volume discre- tization method of second-order accuracy in time and space, used for all differential equations. Details may be found in [4].

Presently a grid with two million grid points is used, still not fine enough for a Large-Eddy- Simulation of this problem. It is, however, possible to capture the large vortex structures in qualitative agreement with experimental observations [5] without additional assumptions. Capturing and tracing of the vortex structures is achieved with an interactive method, which consists of the following algorithmic

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. - . - _ - - - - _ l .

(c ) l 9 9 9 A m e ri c a n In s ti tu te o f A e ro n a u ti c s & A s tro n a u ti c s

e l e m e n ts : F i rs t, th e v e l o c i ty v e c to rs c o m p u te d a re p ro j e c te d o n a n a rb i tra ry p l a n e , e i th e r a m e r i d i o n a l o r a c ro s s -s e c ti o n a l p l a n e . T h e n e l e m e n ts o f th e s tre a m - l i n e p ro j e c ti o n s i n th e p l a n e c h o s e n a re c o n s tru c te d fro m th e p ro j e c ti o n s o f th e v e l o c i ty v e c to rs (F i g :‘H )): V o rte x c o re s c a n b e i d e n ti fi e d b y s e a rc h i n g .fo r c l o s e d o r s p i ra l i n g s tre a m l i n e s e g m e n ts . A fte r i d e n ti fy i n g a c o re , v o rti c i ty v e c to rs a re c o m p u te d i n th e n e i g h b o r- h o o d o f th e c o re . E l e m e n ts o f v o rte x l i n e s a re th e n o b ta i n e d b y n u m e ri c a l i n te g ra ti o n , i n w h i c h th e v o r-

F i g . 1 1 : V o rte x c o re s i n d i c a te d b y c l o s e d o r s p i ra l i n g s tre a m l i n e s e g m e n ts i n m e r i d i o n a l p l a n e (u p p e r p i c - tu re ). T w o v o rte x r i n g s (p i c tu re i n th e m i d d l e ), c o m - p u te d w i th v o rti c i ty v e c to rs , i n th e c y l i n d e r o f a p i s to n e n g i n e ( l o w e r p i c tu re ) [4 ].

ti c i ty v e c to rs a re u s e d . T h e s te p -s i z e o f th e i n te g ra ti o n i s d e te rm i n e d b y th e d i s ta n c e to a n e i g h b o r i n g p l a n e , fo r w h i c h v o rti c i ty v e c to rs a re a v a i l a b l e fro m th e i n te - g ra ti o n o f th e c o n s e rv a ti o n e q u a ti o n s . T h e m e th o d i s d e s c r i b e d i n d e ta i l i n [4 ].

F i g . 1 1 s h o w s tw o v a l v e s o f a fo u r-v a l v e p i s to n e n g i n e , th e p ro j e c ti o n s o f s tre a m l i n e s e g m e n ts i n a s e l e c te d p l a n e , a n d a fe w v o rti c i ty v e c to rs v i s i b l e n e a r th e c o re o f th e v o rte x o n th e l e ft (u p p e r p i c tu re ). T h e tw o r i n g - l i k e v o rti c e s g e n e ra te d b y th e i n ta k e fl o w th ro u g h th e v a l v e g a p s a re s h o w n b e l o w , a n d i n th e l o w e r p a rt o f th e p i c tu re th e v o rti c e s a re d e p i c te d i n th e c y l i n d e r o f a p i s to n e n g i n e w i th a p e n t ro o f [4 ].

A p l o t o f n u m e ri c a l l y d e te rm i n e d v e l o c i ty v e c to rs [4 ] i s c o m p a re d w i th a p i c tu re o f p a rti c l e s i n j e c te d i n th e fl o w o f a n e x p e r i m e n t [5 ] i n F i g . 1 2 .

F i g . 1 2 : C o m p a r i s o n o f c o m p u te d v e l o c i ty v e c to rs o f [4 ] w i th a p i c tu re o f p a rti c l e s i n j e c te d i n th e fl o w d u r i n g th e i n ta k e s tro k e i n a e x p e r i m e n t [5 ] fo r a c ra n k a n g l e E = 1 2 5 ’ a n d 2 0 0 0 r-p m .

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The lower picture in Fig. 12 just gives the position of the individual particles at a certain instant of time. The vortex cores contain the smallest number of particles and can therefore be identified by the white areas. If a sequence of pictures is taken, the local velocity vectors can be determined and also segments of streamlines. This can be seen in Fig. 13. In the upper picture the two vortices beneath the inlet valves shown in Fig 11 can be recognized. The pic- ture was taken in a water experiment for a crank angle E = 60’ with 0.4 rpm [5]. The lower picture in Fig. 13 shows streamline segments obtained with the Laser light-sheet technique in an air experiment for a crank angle E = 180” with 350 rpm. For this crank angle the valves are closed [20].

Fig. 13: Experimentally determined segments of streamlines, obtained during the intake stroke. The upper picture was taken in a water experiment for a crank angle E = 60” with 0.4 rpm [5], and the lower in an air experiment for a crank angle E = 180’ with 350 rpm. For this crank angle the valves are closed [20]. The streamline segments are obtained with the Laser light-sheet technique

These few examples may suffice to show,

that vortex structures may play a dominant role in internal flows.

6. VORTEX BREAKDOWN

Swirling pipe flows can be subject to vortex breakdown, if the pipe has a diverging cross-section such that a positive axial pressure gradient is gener- ated, finally leading to a stagnation point on the axis. Several modes of breakdown have been observed, see, for example, [21] and (221. Because of its time- dependent, three-dimensional nature, this flow is difficult to predict, but, at the same time offers a challenging problem for numerical flow analysis.

For the reasons mentioned vortex breakdown was simulated with a solution of the Navier-Stokes equations for incompressible laminar time-dependent flows. The numerical method was based upon the concept of artificial compressibility combined with a dual-time-stepping technique.- Block-structured grids were employed to avoid the singularity on the axis of the pipe. An implicit relaxation scheme was used for the integration in each physical time step [23].

Special attention had to be paid to the ap- proximation of the in- and out-flow boundary condi- tions. The velocity distribution in the inflow cross- section was taken from the experimental data of [8]. In the outflow cross-section the pressure distribution was computed by solving Poisson’s equation, and the velocity components were extrapolated from the inte- rior of the domain of integration. Some of the results obtained are compared with the experimental data of r81 in Fit? 14 below.

Fig. ‘14: Bubble-type breakdown of swirling pipe flow. The upper picture shows the experimental data of [8], the lower picture the vortex lines and pressure distribution (grey shaded areas) simulated with the numerical solution described in [23]. The Reynolds- number is Re =3220. based on the pipe diameter.

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(c)l999 American Institute of Aeronautics & Astronautics

Shown is the stable bubble-type breakdown in the swirling pipe flow of the experiment described in [8] and [24], in comparison with the numerical results of [23]. The upper picture is an experimental visualization of [8], the lower one showing the vortex lines and pressure distribution (grey shaded a&s) simulated with the numerical solution described in [23] for the conditions of the experiment. The .Rey- nolds-mmrber is Re =3220, based on the pipe diame- ter and mean velocity of the inflow cross-section.

After a stagnation point is generated on the axis, the breakdown-region moves upstream and grows in time. The bubble shown in Fig. 14 is fully developed and stable. Further downstream a second spiral-type breakdown occurs, after the core has been healed. Shapes and positions of the simulated break- down agree well with the experiment.

Fig. 15: Transition from bubble-type to spiral-type breakdown. Comparison of two pictures of a sequence obtained in experiments at the Aerodynamisches Institut (upper and third picture from above); in be- tween the numerically simulated vortex structures of WI.

The transition from bubble-type to spiral- type breakdown (Fig. 15) was first observed in ex-

periments, and also in numerical simulations of a slender vortex in an unbounded domain [22]. Al- though the characteristic values of the Reynolds num- ber and swirl parameter determining the transition are notyetkriown, the spiral-type breakdown evolved in the”&neiical simulations of [22] in qualitative agreement with experimental observations. Fig. 15 shows a comparison of two pictures of the transitional flow, taken in experiments at the Aerodynamisches Institut (upper and third picture from above), and below the experimental visualizations are the com- puted vortex structures. Shown are only two pictures of a sequence, first showing the formation of a bub- ble, which then undergoes transition to spiral-type breakdown. The visualizations clearly confirm the change in the core structure during the transition. While the bubble is almost axially symmetric with a double-ring vortex structure inside, the core- is being distorted during the transition in which the two vortex rings travel downstream, resulting in a spiral-shaped core as can be seen in Fig. 15.The comparison of experimental and numerical results presented here shows, that the generation of complex vertical struc- tures, their interaction and their time-dependent changes can be predicted in agreement with experi- mental observations.

7. CONCLUSIONS

Five incompressible and compressible flows containing vortex structures were investigated with numerical solutions and compared with experimental results. The examples chosen comprise leeward vor- tices on delta wings, the counter-rotating vortex pairs, similar to Goertler vortices, observed downstream from shock-induced boundary-layer separation in hypersonic flow; the oblique collision and interaction of two vortex rings; the formation of vortex rings in cylinders of automotive engines during the suction and compression- stroke; the bubble-type breakdown of swirling decelerated pipe flow, and the transition from bubble- to spiral-type breakdown of a free slen- der vortex.

In all cases, the quality of the numerical simulations of the steady and the time-dependent three-dimensional vertical flows could be demon- strated by good agreement with experimental data. The examples chosen comprise external and internal flows.

The m.un&cal results were obtained with a solution of the parabolized Navier-Stokes equations for example 1 and of the complete conservation equa- tions for examples 2 to 5. Description of the algo- rithmic elements of the solutions employed was left out in order to be able to emphasize the comparison of numerical and experimental data. The interested reader may find the details in the literature cited.

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(c)l999 American Institute of Aeronautics & Astronautics

ACKNOWLEDGEMENT

The author gratefully acknowledges the continuous support of the Deutsche Forschungsge- meinschaft (DFG).

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