B.M. Cetegen and J.C. Hermanson- Mixing Characteristics of Compressible Vortex Rings Interacting with Normal Shock Waves

8/3/2019 B.M. Cetegen and J.C. Hermanson- Mixing Characteristics of Compressible Vortex Rings Interacting with Normal Sh

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M ixing Ch aracter is t ics o f Com press ib le Vortex RingsInteract ing wi th Norm al Shock WavesB . M . C E T E G E N *Mecha nical Engineering Departm ent, U niversity o f Connecticut, Storrs, C T 06269 -3139

J . C . H E R M A N S O NUnited Technologies Research C enter, East Hartford, C T 06108T h e i n s t a b i l i t y m e c h a n i s m s a n d m i x i n g e n h a n c e m e n t a r i s i n g f r o m t h e i n t e r a c t i o n o f a c o m p r e s s i b l e v o r t e xr i n g w i t h a n o r m a l s h o c k w a v e w e r e s t u d i e d i n a c o l i n e a r , d u a l - s h o c k t u b e . T h i s f l o w g e o m e t r y s i m u l a t e sf e a t u r e s o f t h e i n t e r a c t i o n o f a s h o c k w a v e w i t h a j e t c o n t a i n i n g s t r e a m w i s e v o r t i c i t y , a c o n f i g u r a t i o n o fs i g n if i c a n t i n t e r e s t f o r s u p e r s o n i c c o m b u s t i o n a p p l i c a t i o n s . F l o w v is u a l i z a t io n a n d q u a n t i t a t i v e c o n c e n t r a t i o nm e a s u r e m e n t s w e r e p e r f o r m e d b y p l a n a r l a s e r R a y l e i g h s c a t t e ri n g . F o r a g i v e n p r i m a r y s h o c k s t r e n g t h ,i n t e r r a c i a l i n s t a b i l i t y i s m o r e e v i d e n t i n a w e a k v o r t e x r i n g t h a n i n a s t r o n g e r v o r t e x r i n g . I n a l l c a s e s , t h ei d e n t i t y o f t h e v o r t e x r i n g i s l o s t a f t e r a s u f f i c ie n t ly l o n g t i m e o f i n t e r a c t i o n . T h e p r o b a b i l i t y d e n s it y f u n c t i o no f t h e m i x e d f l u i d c h a n g e s r a p i d l y f r o m a b i m o d a l d i s t r i b u t i o n t o a s i n g l e p e a k u p o n p r o c e s s i n g b y a s h o c kw a v e . T h e m o s t p r o b a b l e c o n c e n t r a t i o n d e c r e a s e s w i t h t i m e , i n d i c a t i n g a r a p i d i n c r e a s e i n m i x i n g a n dd i l u t i o n o f t h e v o r t e x f l u i d . T h e m i x i n g e n h a n c e m e n t i s m o s t r a p i d f o r t h e c a s e o f a s t r o n g v o r t e x r i n gi n t e r a c t i n g w i t h a s t r o n g s h o c k w a v e , s o m e w h a t s l o w e r f o r a w e a k v o r t e x r i n g a n d a s t r o n g s h o c k w a v e , a n ds i g n if i c an t ly s l o w e r f o r t h e c a s e o f a s t r o n g v o r t e x r i n g a n d a w e a k e r s h o c k w a v e . T h e s e o b s e r v a t i o n s a r ec o n s i s t e n t w i t h t h e e a r l i e r n u m e r i c a l p r e d i c t i o n s .

I N T R O D U C T I O NC u r r e n t i n t e r e st i n t h e i n t e r a c ti o n b e t w e e nc o m p r e s s i b l e v o r ti c a l f l ow s a n d s h o c k w a v e s i sl a rg e l y m o t i v a t e d b y t h e n e e d t o p r o m o t e r a p i d ,l o s s- e f fe c t iv e m i xi n g a n d c o m b u s t i o n o f h y d r o -g e n a n d h y d r o c a r b o n f u e l s f o r s u p e r s o n i c c o m -b u s t o r a p p l i c a t i o n s [ 1 ]. W h i l e n u m e r o u s p r e v i-o u s s t u d i e s h a v e e x a m i n e d t h e p e n e t r a t i o n a n dmix ing cha rac te r is t i c s o f t ransve rse gas j e t s i s-su ing in to supe rson ic p r ima ry s t ream s [2, 3 ],re la t ive ly l i t t l e i s known abou t the in te rac t iono f t h e s e j e t s w i t h c o m p r e s s i o n w a v e s . O f s p e -c i f i c impor tance fo r mix ing i s the impac t o fs h o c k w a v e s o n t h e s t r e a m w i s e v o r t i c a l s t r u c -t u r e t h a t a p p e a r s t o p l a y a n e s s e n t i a l r o l e i nt h e m i x i n g p r o c e s s o f t r a n s v e r s e j e t s i n c o m -press ible f low [3 , 4] .

P r e v i o u s r e s e a r c h o n t h e i n t e r a c t io n o f c o m -p r e s s i o n w a v e s w i t h v o r t i c a l f l o w s h a s e x a m -ined the dens i ty f i e ld , shock p ropaga t ion , andt h e p r e s s u r e a m p l i f ic a t io n o f a c o u s t i c w a v e s ,w i th m u c h o f th e r e s e ar c h b e i n g m o t i v a te d b y

* C o r r e s p o n d in g a u t h o r .P r e s e n t e d a t t h e T w e n t y - F i f t h S y m p o s i u m ( I n t e r n a t i o n a l )o n C o m b u s t i o n , I r v i n e , C a l i f o r n i a , 3 1 J u l y - 5 A u g u s t 1 99 4.0 0 1 0 - 2 1 8 0 / 9 5 / $ 9 . 5 0SSDI 0010-2180(94 )00058-Z

t h e n e e d f o r n o i s e r e d u c t i o n f r o m h i g h - s p e e dje t s [5 , 6 ]. Ea r l i e r expe r imen ta l s tud ie s o f thei n t e r a c ti o n b e t w e e n a v o r t ic a l r in g a n d a s h o c kw a v e h a v e a l so i n d i c a t e d a s t r o n g t u r b u l i z a t io ndue to the in te rac t ion , sugges t ing a po ten t i a limpa c t on mix ing [7 ] . M ore recen t ly , de ta i l edn u m e r i c a l s i m u l a t i o n s h a v e e x a m i n e d s h o c ks t ruc tu re and vor t i c i ty ampl i f i ca t ion re su l t ingf r o m t h e i n t e r a c ti o n o f s h o c k w a v e s a n d v o r t e xpa i r s [8, 9 ]. Th e re su l t s sugg es t tha t the m os tp r o n o u n c e d v o r t i c i t y a m p l i f i c a t i o n o c c u r s f o rthe case o f a s t rong vor tex in te rac t ing wi th as t rong shock . Somewha t l e s s vor t i c i ty ampl i f i -c a t i o n i s p r e d i c t e d f o r t h e c a s e o f a s t r o n gs h o c k w a v e c o m b i n e d w i t h a w e a k v o r t e x ; c o n -s ide rab ly l e s s ampl i f i ca t ion i s s een fo r the caseo f a w e a k s h o c k a n d s t r o n g v o r te x . E x p e r i m e n -t a l v e ri f ic a t io n o f t h e s e t r e n d s i s o n e f o c u s o ft h e c u r r e n t w o r k .A k e y f e a t u r e o f t h e v o r t e x / s h o c k i n te r a c-t ion i s the vor t i c i ty gene ra ted by the misa l ign-m e n t o f t h e a d v e r s e p r e s s u r e g r a d i e n t a s s o ci -a t e d w i t h t h e s h o c k a n d t h e d e n s i t y g r a d i e n tb e t w e e n t h e v o r t e x r i n g a n d t h e s u r r o u n d i n gf low [4 , 10, 11] . Th e ac t ion o f th i s ba roc l in ict o r q u e i n c o m p r e s s i b l e fl o w h a s b e e n s t u d i e dn u m e r i c a l l y [ 1 1 - 1 3 ] a n d e x p e r i m e n t a l l y

C O M B U S T IO N A N D F L A M E 1 0 0 : 2 3 2 - 2 4 0 ( 1 99 5 )C o p y r i g h t 1 99 5 b y T h e C o m b u s t i o n I n s t i t u t eP u b l i s h e d b y E l s e v i e r S c i e n c e I n c .


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MIXING CHARACTERISTICS OF VORTEX RINGS 233

[11, 14-16]. Yet to be established is how thevorticity generated by shock passage interactswith the existing vorticity in a vortex ring, andwhat the implications of the instabilities andvortex interactions for fuel/air mixing mightbe. The instability characteristics of an acceler-ated volume of light gas and one of heavy gascan differ by more than simply the sign of thevorticity generated by baroclinic torque [14].For example, acceleration of regions of heaviergas can lead to inertial (Rayleigh-Taylor) in-stabilities and .strong, non-uniform accelera-tion can lead to shear (Kelvin-Helmholtz) in-stabilities [14, 15]. Thus the growth of actualdisturbances may be a combined result of sev-eral instability mechanisms. Achieving an in-creased understanding of these phenomena isof obvious and direct importance to scramjetapplications, given the inevitability of shockwaves and regions of non-uniform density in ascramjet combustor.The current research examined the instabili-ties and mixing enhancement induced by thepassage of a normal shock wave through acompressible vortex ring consisting of propanegas injected into air. The visualization of thevortex structure and quantitative measure-ments of jet gas concentration during and aftershock passage allowed study of the time evolu-tion of the resulting instabilities and their im-pact on fuel/air mixing. The current workfollowed a study in the same facility of themixing enhancement resulting from the inter-action between shock waves and axial turbu-lent jets [17].

EXPERIMENTAL APPARATUSF l o w F a c i l i t yThe test facility consisted of two colinear, op-posed shock tubes as shown in Fig. 1. A circu-lar shock tube of 10 mm inside diameter with adriven section 325 mm in length was used togenerate the compressible vortex ring. Theopen end of the vortex shock tube was situatedwithin the test section of a larger, primaryshock tube 50 50 mm in internal cross sec-tion, in which a normal shock wave was gener-ated. The test section of the primary shocktube was fitted with quartz windows for optical

access and discharged vertically upwards to theatmosphere. The driven and test sections ofthe primary shock tube had a combined lengthof 1.9 m. The diaphragm of the primary shocktube, consisting of Mylar film of either 25 or 76#m in thickness, was burst by impulsivelycharging the driver section with compressedair. The diaphragm of the vortex shock tubeconsisted of either a single or double sheet of5-1xm-thick Mylar film placed in contact with apair of 50-txm-diameter copper wires. Synchro-nization of the vortex ring generation with thearrival of the primary shock wave was achievedby rupturing the diaphragm of the vortex shocktube, after an appropriate time delay, by dis-charging an electric current through the upperwires.

Piezoelectric pressure transducers, situatedin the wall of the primary shock tube 35 cmupstream and downstream of the test sectionwindow, allowed determination of the primaryshock wave velocity. A third transducer situ-ated 127 cm upstream of the test section win-dow served as the timing reference for thevortex shock tube, laser pulse, and the imagingsystem. A fourth pressure transducer was situ-ated in the wall of the vortex shock tube at thetube exit, allowing determination of the time-of-flight of individual vortex rings from the exitof the vortex tube into the test section. Thedriven section of the vortex shock tube waspurged with propane, which was then expelledinto the surrounding test section air, giving riseto the vortex. A slow air co-flow was employedto keep the test section clear of propane priorto the initiation of the vortex ring.D i a g n o s t i c sPlanar Rayleigh scattering was employed toprovide visualization of the flow structure anddetailed quantitative measurements of the de-gree of mixing between the propane vortex andthe surrounding air. The significantly higherRayleigh scattering cross section of propanerelative to that of air allowed measurements ofvortex fluid concentration. The utilization ofRayleigh imaging as a quantitative, spatiallyand temporally resolved means to determinemixing has been demonstrated by earlier inves-tigators [16-18].


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234 B.M. CETEGEN AND J. C. HERMANSONh

~ o r t e x Shock ube

lionJ FDriver

Fig. 1. Experimentalsetup.

g

Illumination for Rayleigh scattering was pro-vided by a pulsed Nd:YAG laser (ContinuumYG-681-10) at its third harmonic wavelengthof h = 355 nm with a pulse energy of 120 mJ.The nominal 10 ns duration of each laser pulsewas sufficiently fast to effectively freeze theflow. Quartz cylindrical optics were used toform the laser beam into a 0.1-mm-thick lightsheet of roughly 30 mm breadth at the testsection centerline. A single Rayleigh scatteringimage for each test run was acquired by animage intensified CID camera (Xybion ISG-204-U-2) equipped with a Poynting Productsframe grabber. Corrections for nonuniformlaser illumination, optical system response andbackground were made for all images. For thequantitative concentration measurements, datawere also corrected for number density changesdue to shock passage. The number densitycorrections were invoked using the relationNSTP//N = ( e o / P ) ' y 1 /~ 'm i x, where the specifichea t ratio of the mixture, Ymix, was estimatedfrom concentrations and unmixed fluid specific

heats. The typical signal-to-noise-ratio of theimages was approximately 10.

F l o w C o n d i t io n sThe Mylar primary shock tube diaphragms wereburst at pressures of 3.0 and 6.4 atm, produc-ing average measured primary shock Machnumbers in the test section of M, = 1.21 and1.44, respectively. These will be referred to asthe "weak" (Ms = 1.21) and "strong" (M s =1.44) shocks. The primary shock Mach num-bers were generally repeatable to within 5%.For all runs the initial temperature of all gaseswas 300 K. The diaphragms of the vortex shocktube were burst at 2.4 and 4.4 atm, producingaverage Mach numbers of the shock waves inthe tube of 1.03 and 1.1, respectively. Theresulting vortex rings are referred to here as"weak" and "strong." The average convectivevelocity of a vortex ring, U0, was calculatedfrom the position of the forward stagnationpoint of the vortex ring and the time differencebetween the illuminating laser pulse and theinitiation of the vortex ring as determined fromthe pressure transducer at the vortex tube exit.For the region imaged (approximately 2 to 5cm downstream of the tube exit), the convec-tive speed of the vortex ring ranged from U0 --44 to 170 m/s for the weak vortex and U0 = 71to 286 m/s for the strong vortex. The corre-sponding Reynolds numbers, based on the dis-tance between the cores of the vortex pair andthe gas properties of propane, were 5 104 to19 104 for the weak vortex ring and 8 104to 32 104 for the strong one respectively.

In the following, weak and strong vortex ringinteractions with a strong shock wave and thestrong vortex ring interaction with a weak shockwave are described. Due to the relatively largevariations in the weak vortex and the weakshock speeds, the reliable imaging of weakvortex/weak shock case was not possible.RESULTSVortex Ring Stabi l i ty and StructurePlanar Rayleigh scattering images of a weakvortex ring interacting with the strong normalshock wave are shown in Fig. 2. Each of the


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( a )

( b )F i g. 2 . P l a n a r R a y l e i g h i m a g e s o f w e a k v o r t e x r i n g / s t r o n gs h o c k i n t e ra c t io n . B l a ck a n d w h i t e c o r r e s p o n d t o m i n i m u ma n d m a x i m u m i n t e n s i t i e s , r e s p e c t i v e l y . T h e s h o c k p o s i t i o ni s s h o w n b y t h e a r r o w s . ( a ) b e f o r e i n t e r a c t i o n ; ( b ) d u r i n gi n t e r a c t i o n ( A t = 3 7 p .s , ~ = 0 . 2 9) . T y p i c a l r e g i o n s f o r t h ep d f c o m p u t a t i o n s a r e s h o w n a s d a s h e d r e c ta n g l e s.

three images shown was taken during a differ-ent test run. In all cases, the direction ofvortex ring propagation is from top to bottomand the shock propagation is from bottom totop. Each imaged region corresponds to 2.7 2.7 cm in physical space, with the exit of thevortex shock tube 2.2 cm above the top edge ofeach image shown. The vortex ring prior tointeraction with the normal shock is shown inFig. 2a, in which the vortex structure is clearlyvisible. While significant mixing had taken placein the vortex cores, there is a region of rela-

tively high propane concentration in the for-ward stagnation region.

Two parameters are utilized here to charac-terize the "age" of the interaction. At is theelapsed time from the first contact of the nor-mal shock with the vortex ring. The nondimen-sional parameter ---At/(R/Uo) is obtainedby normalizing At by a characteristic time ofthe large-scale circulation of the vortex ring,( R /Uo ) , where R is half the distance betweenthe visible vortex cores. ~ serves as a roughmeasure of the number of rotations undergoneby the vortex cores during the period of theshock/vortex interaction. An image taken af-ter the interaction (At = 37 ~s, ~ = 0.29) isshown in Fig. 2b. Although the vortex structureis still recognizable, there appears to be in-creasing instability and turbulization at theflanks of the vortex. The decrease in overallsignal intensity (especially after correction forpressure and temperature, not performed inthe images shown) is suggestive of increasedmixing. The degree of mixing enhancementappears greatest at the flanks and least in thestem region immediately beyond the vortexring. The turbulization of the vortex ring in-creases with increasing time, with the vortexring eventually losing its identity. The lateralextent of the vortex does not appear to begreatly influenced by the interaction, in agree-ment with previous experimental and numeri-cal results [7, 8].A series of interaction images for the case ofa strong vortex ring/strong shock wave isshown in Fig. 3. In this case, the normal shockwave is of the same strength as the previouscase (Fig. 2) but the vortex ring is stronger.The qualitative features of the strong vortex(Fig. 3a) are similar to its weaker counterpart,although somewhat more mixing appears tohave taken place in the vicinity of the vortexcores. In Fig. 3b, the shock wave has propa-gated past the vortex cores, as indicated (At =31 tzs, ~ = 0.31). This appears to have resultedin flattening of the vortex ring. In addition,there is a crescent-shaped region characterizedby relatively high signal strength, suggestive ofhigh propane concentration. The relativelylower signal strength in the region behind theleading edge of the vortex ring indicates thatthe region of high intensity is not entirely an


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2 3 6 B . M . C E T E G E N A N D J . C . H E R M A N S O N

(a )

(b )

a r t i fa c t o f t h e d e n s i t y r is e d u e t o t h e p a s s a g eo f t h e n o r m a l s h o c k w a v e . A n i m a g e c o r r e -s p o n d i n g t o A t -- 4 5 / z s , ~ = 0 . 55 i s s h o w n i nF i g . 3 c . I n t h e c a s e a " Y - s h a p e d " s e c o n d a r ys t r u c t u r e c o n s i s ti n g o f r e l a t iv e l y p o o r l y m i x e dp r o p a n e i s a p p a r e n t i n t h e s t e m r e g i o n o f th ev o r t e x r in g .

L a s t l y , a s i n g l e i n t e r a c t i o n i m a g e f o r t h ec a s e o f a s t r o n g v o r t e x r in g a n d a w e a k s h o c kw a v e i s s h o w n i n F i g . 4 . I n t h i s c a s e , t h e v o r t e xr i n g is o f t h e s a m e s t r e n g t h a s t h e p r e v i o u sc a s e ( F ig . 3) b u t t h e n o r m a l s h o c k w a v e isw e a k e r ( M s = 1 .2 1 ). T h e s t r u c t u r e f o r t h isp o s t - s h o c k c a s e ( A t = 8 1 / ~ s, ~ = 1 . 0 6) d o e sn o t e x h i b it th e s e c o n d a r y " Y - s h a p e d " s t ru c -t u r e s i m i l a r t o t h a t f o r t h e c a s e o f a s t r o n g e rs h o c k s t r e n g t h . A t e a r l i e r i n t e r a c t i o n t i m e s( n o t s h o w n ) , a f l a t t e n i n g o f t h e v o r t e x r in g ,a c c o m p a n i e d b y a c r e s c e n t - s h a p e d r e g i o n o fh i g h p r o p a n e c o n c e n t r a t i o n , i s o b s e r v e d a s i nt h e p r e v i o u s c a s e o f t h e s t r o n g v o r t e xr i n g / s t r o n g s h o c k i n t e ra c t i o n .

T h e b a r o c l i n ic t o r q u e r e s u lt in g f r o m t h ed e n s i t y g r a d i e n t i n t h e v o r t e x r in g a n d t h es h o c k p r e s s u r e r i s e a c t s i n a d i r e c t i o n t o a u g -m e n t t h e v o r t i c it y i n t h e v o r t e x c o r e s . A l -t h o u g h t h e h i g h e r d e n s it y o f t h e p r o p a n e v o r -t e x r i n g g a s r e l a t i v e t o t h e s u r r o u n d i n g a i rs u g g es ts t h a t t h e v o r t e x r in g i s R a y l e i g h - T a y l o ru n s t a b l e , t h a t t h e r e l a ti v e l y s m o o t h c a p s o f t h ev o r t e x ri n gs p e r s is t w e ll i n to t h e s h o c k / v o r t e xr i n g i n t e r a c t i o n s u g g e s t t h a t t h i s i n s t a b i l i t yg r o w s r e l a t i v e l y s lo w l y . T h u s t h e s t r u c t u r a l f e a -

(c )Fig. 3. Planar Rayleigh images of stron g vortex ring/strongshock interaction. Shock positions are show n by arrow s. (a)before interaction; (b) du ring interaction (a t = 31 /xs,~: = 0.31); (c) shoc k go ne by (At = 45 /xs, ~ = 0.55).

Fig. 4. Planar R ayleigh image of a strong vortex ring/weakshock interaction. Post-sho ck case (At = 86 p.s, ~ = 1.06).


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M I X I N G C H A R A C T E R I S T I C S O F V O R T E X R I N G S 237

t u r e s a n d m i x i n g c h a r a c t e r i st i c s , o b s e r v e d d u r -i ng t h e v o r te x r i n g / s h o c k w a v e i n t e ra c t io n a n dd e s c r i b e d q u a l i t a t i v e l y a b o v e , a r e l i k e l y t o r e -s u l t f r o m a c o m b i n a t i o n o f s h e a r i n s t a b i li t ya n d b a r o c li n ic t o r q u e .S p a t i a l P r o b a b i l it y D e n s i t y F u n c t i o n so f C o n c e n t r a t io nR e p r e s e n t a t i v e p r o b a b i l i t y d e n s i t y f u n c t i o n s( p d f 's ) o f t h e m i x t u r e f r a c t i o n o f p r o p a n e a r es h o w n i n F i g s . 5 - 7 . A v a l u e o f C = 0 c o r r e -s p o n d s t o p u r e a i r ; C = 1 0 0 , t o p u r e p r o p a n e .T h e c h a r a c t e r o f a s p a ti a l p d f n a t u r a ll y d e -p e n d s o n t h e s e l e c t e d r e g i o n . A l l p d f s p r e -s e n t e d h e r e w e r e o b t a i n e d i n r e g i o n s c o m -p l e t e l y c o n t a i n i n g t h e v o r t e x r i n g b u t e x c l u d i n gt h e f a r s t e m r e g i o n , a s s h o w n i n F i g . 2 .

P d f s f o r t h e c a s e o f a w e a k v o r t e x r i n gi n t e r a c t i n g w i t h a s t r o n g s h o c k a r e s h o w n i nF i g. 5. T h e p d f o f t h e w e a k v o r t e x r i n g p r i o r t ot h e s h o c k i n t e r a c t i o n is c h a r a c t e r i z e d b y tw op e a k s , o n e a t l o w a n d a n o t h e r a t h i g h c o n c e n -t r a ti o n . T h e l o w - c o n c e n t r a t io n m i x e d fl u id p e a ki s b e l i e v e d t o c o r r e s p o n d t o t h e v o r t e x c o r er e g i o n s , w h i l e t h e h i g h e r p e a k i s d u e t o t h er e l a ti v e l y t e ss m i x e d f o r w a r d s t a g n a t i o n r e g i o n .T h e t w o p e a k s a r e o f c o m p a r a b l e h e i g h t w h i c hi s i n d i c a t i v e o f t h e s i m i l a r e x t e n t s o f t h e s er e g io n s . U p o n p r o c e s s in g b y t h e s h o c k w a v e,t h e n a t u r e o f th e p d f c h a n g e s d r a m a t i c a ll y ,w i t h th e b i m o d a l p d f o f t h e u n d i s t u r b e d v o r t exr i n g r a p i d l y c h a n g i n g t o a p d f w i t h a s in g l e

0.050 .045

0 .04

0 . 0 3 5

0 .03

0 0.0250.02

0 .015

0.01

0 . 0 0 5

p e a k . T h i s s h i f t is i n d i c a t iv e o f a r a p i d i n c r e a s ei n m i x i n g a n d d i l u t i o n o f t h e v o r t e x f l u i d . F o rt h e c a s e o f th e w e a k v o r te x , t h e p e a k m i x e df l u i d c o n c e n t r a t i o n i s f o u n d t o h a v e a v a l u e o fr o u g h l y 3 0 % f o r ti m e s i n th e r a n g e o f A t =3 1 - 3 7 / ~ s ( ~ = 0 . 2 1 -0 . 29 ) .

T h e p d f s f o r t h e c a s e o f a s t r o n g v o r t e x p a i ri n t e r a c t i n g w i t h a s t r o n g s h o c k a r e s h o w n i nF i g . 6 . T h e p d f o f t h e m i x e d f l u i d in t h e v o r t e xr i n g b e f o r e t h e s h o c k i n t e r a c t i o n i s q u a n t i a -t iv e l y v e r y s i m i la r t o t h a t o f t h e w e a k v o r t e x .T h e p d f o f t h e s t r o n g v o r t e x a ls o s h o w s a r a p i dc h a n g e t o a s in g l e p e a k . T h e l o c a t i o n o f t h ep e a k i s s e e n t o c l e a r l y m a r c h t o w a r d s i n -c r e a s e d m i x e d n e s s ( i . e . , l o w e r v a l u e s o f C )w i t h t i m e . S p e c i fi c a ll y , t h e p e a k m i x e d f l u idc o n c e n t r a t i o n d e c r e a s e s f r o m a p p r o x i m a t e l y26% a t a t i me o f A t = 31 #s (~ : = 0 . 31 ) to20% a t a t i me o f A t = 45 / z s (~ = 0 .55 ). Th i sa m o u n t o f d e c r e as e a p p e a r s t o be s o m e w h a tg r e a t e r t h a n t h a t e x h i b i t e d b y t h e p d f s f r o mt h e w e a k v o r te x r i n g / s t r o n g s h o c k c as e s h o w ni n F i g . 5 , sugges t i ng t ha t , f o r a g i ven shocks t r e n g t h , m o r e r a p i d m i x i n g o c c u r s f o r t h e c a s eo f a s t r o n g v o r t e x t h a n f o r a w e a k e r o n e .

F i n a l l y , s a m p l e p d f s f o r t h e c a s e o f a s t r o n gv o r t e x p a i r i n t e r a c t i n g w i t h a w e a k s h o c k a r ep r e s e n t e d i n F i g . 7 . T h i s i n t e r a c t i o n a p p e a r s t ob e c h a r a c t e r i z e d b y a c o n s i d e r a b l y s l o w e r r a t eo f m i xi n g t h a n t h e c a s e o f e i t h e r t h e s t r o n g o rt h e w e a k v o r t e x r i n g s i n t e r a c t i n g w i t h a s t r o n gs h o c k . F o r e x a m p l e , a t a r e la t i v e ly lo n g t i m e o fA t = 86 / xs (~ = 1 .06 ), t he pd f o f t he w ea k

......... i . . . .- - No shock

. . . . . . . - - - - At = 31 p.s. , ~ = 0.214... . At = 37 gs. , ~ = 0.289

i !c / J " / I1 " ,

/ /,, ~ " ~ . . . . .. .

10 20 30 40 50 60 70 80 90 10(F i g . 5 . S p a t i a l p r o b a b i l i t y d e n s i t y d i s -t r i b u t i o n s o f c o n c e n t r a t i o n f o r t h ew e a k v o r t e x r i n g - s t r o n g s h o c k i n t e r -a c t i o n .


7/9

2 3 8 B . M . C E T E G E N A N D J . C . H E R M A N S O N

0.0250.02

0.0150.01

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5 / i i /0.045 t ........... ... ...... . ~ ... ... . ............. ............ ~...............~...............~............... ............... ... ... .../ i i i - - N o s h 0 e k i

0 0 4 1 - . . . .. . . .. . .. . . . . . .. . . . .. . . .. . i . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . .. . . . .. . . . i . . . .. . . .. . . .. . . i - - - A t = 3 1 ~ s . , ~ , = 0 3 1 0' / :: :: . . . . . . A t = 4 5 p a . , ~ = 0 5 4 6 /o o ~ l ~ . . . .. . . . . . . . . .. . . . . . . . . i . . . . . . . . . . . . i . . .. . . .. . . .. . .. i . . . .. . . .. . . .. . . '~ . . . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . t

L. . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . i . . . . . . . . . . . . . . i . . . .. . . .. . .. . . . i . . .. . . .. . . .. . .. ! . . . .. . . .. . . .. . . ~ . . . .. . . .. . .. . . . ; . . . . . . . . . . . . . . . . . . . . . . . . . . .

" t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

10 20 30 40 50 60 70 80 90 100C

F i g , 6. S p a t i a l p r o b a b i l i t y d e n s i t y d i s -t r i b u t io n s o f c o n c e n t r a t i o n f o r t h es t r o n g v o r te x r i n g - s t r o n g s h o c k i n t e r-a c t i o n .

s h o c k c a s e s t il l e x h i b it s r e m n a n t s o f t h e t w op e a k s c h a r a c t e r i s t ic o f t h e u n d i s t u r b e d v o r t e xr i n g , i n c l u d i n g a s i g n i f ic a n t f r a c t i o n o f m i x e df lu i d c o n c e n t r a t i o n w i t h v a l u e s e x c e e d i n g 4 0 % .T h e s e t r e n d s a r e in q u a l i t a t i v e a g r e e m e n t w i thn u m e r i c a l p r e d i c t io n s o f t h e s a m e p h e n o m e n a[8 ]. I t s h o u l d b e n o t e d t h a t f o r t h e e x p e r i m e n -t a l r e s u l ts p r e s e n t e d h e r e , t h e s t r e n g t h o f t h ev o r t e x r i n g d i f f e r e d b y r o u g h l y 6 0 % b e t w e e nt h e s t r o n g a n d w e a k v o r t e x r in g s , w h i l e t h ed i f f e r e n c e in s h o c k M a c h n u m b e r s a m o u n t e dt o o n l y a b o u t 1 5 % . T h i s u n d e r s c o r e s t h e o b -s e r v a t i o n t h a t t h e m i x i n g i s i m p a c t e d t o am u c h g r e a t e r e x t e n t b y s h o c k s t r e n g th t h a n b yv o r t e x s t r e n g t h .

S U M M A R YT h e i n s ta b i l i t y m e c h a n i s m s a n d m i x i n g e n -h a n c e m e n t a r is in g f r o m t h e i n t e r a c t i o n o f an o r m a l s h o c k w a v e w i th a c o m p r e s s i b l e v o r t e xw e r e s t u d i e d i n a c o l i n e a r , d u a l - s h o c k t u b e .P l a n a r l a se r R a y l e i g h s c a t t e r i n g w a s e m p l o y e dt o p r o v i d e f l o w v i s u a l iz a t i o n a n d q u a n t i t a t i v em e a s u r e m e n t s o f t h e v o r t e x f lu id c o n c e n t r a -t i o n .

F l o w v i s u a l i z a t i o n s u g g e s t s t h a t , f o r a g i v e np r i m a r y s h o c k s t r e n g t h , i n t e r r a c i a l i n s t a b i l i t y i sm o r e e v i d e n t i n a w e a k v o r t e x r i n g t h a n i n as t r o n g e r v o r t e x ri ng . T h e s t r o n g e r v o r t e x r in gh o w e v e r a p p e a r s s o m e w h a t m o r e s u b j e c t to

= / / 4 ~ I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . i . . . . . . . . . . . i . . . . . . . . i . . . . . . . 1

i i i !0 , 0 4 ~ - . . . . . . . . . .. , . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . .. . i . . . . . . . . . . . . . . i . . . . . . . . . . . .. . . . i . . . . . . . . . . . . . . . . . 1 ~ ; s h o c k . . . . . . . . . . . . . . . . ./ ! _ _ _ A t = 8 , ~ ~ s . , ~ , = 1 . 0 ~ /

o .o 35 ~ ......... .. i ............ i ............. ......... ........ ......... ......... ........ ....... .............. ................ i . . . .o o ~ . i ~ - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J . . . . . . . . . . . . . . J

0 . 0 2 5 . . . . . . . . / i t . . . . . . . . . . . . . . . i . . . . .. . . . . . . . . i . . . . . . . . . . . . . . i . . . . .. . . . . . . . . . . . . . .. . . . . . . . . ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.02 ..........J t . ~ i .. .. . ................ .... .... ..~ ... i ............... ............ i ............... . . . . .

, - - i i i / ~ t i ~ / - . . \ . . . .. . . .. . . . ~ ; , ~ ; ~ ' - - - ~ : . . . ;

0 005o ~ 0 10 20 30 40 50 60 70 80 90 100

F i g . 7. S p a t i a l p r o b a b i l i t y d e n s i t y d i s -t r i b u t i o n s o f c o n c e n t r a t i o n f o r t h es t r o n g v o r t e x r i n g - w e a k s h o c k i n t e r-a c t i o n


8/9


deformation during the interaction than theweaker vortex ring. In all cases the identity ofthe vortex ring is lost after a sufficiently longtime of interaction.

The probability density function of the mixedfluid of vortex rings changes rapidly from abimodal distribution to a single peak uponprocessing by a shock wave. The value of thepeak concentration further decreases with time,indicating a rapid increase in mixing and dilu-tion of the vortex fluid. The mixing enhance-ment is most rapid for the case of a strongvortex ring interacting with a strong shockwave, somewhat slower for a weak vortex ringinteracting with the same shock wave, and sig-nificantly slower for the case of a strong vortexring and weaker shock wave. These observa-tions are consistent with earlier numerical pre-dictions [8].

The authors acknowledge helpful discussionswith Pro f. E. IC Dabora an d the technical assis-tance o f Mssrs. D. Y Alessandri , P. Boardman,an d J. Fikiet . This w ork wa s supported in par t bythe University o f Connecticut Research Fou nda -tion.R E F E R E N C E S

1. Kay, I . W., Pesch ke, W. T., and G uile , R. N. , A/A A J.Prop. Power Vol . 8 :507-512 (1992) .

2 . Herm anson , J . C . and Winte r , M. ALAA J. 31:129-132(1993).

3 . Ho l lo , S. D. , Har t f ie ld , R . J . and M cDan ie l , J . C .,A I A A P a pe r 90 -1632 , 1990 .

4 . W ai tz , I . A. , Marble , F . E. , and Zu koski , E . E. , A IA APaper 91-2265, 1991.

5 . Dosan jh , D. S ., and W eeks , T. M. , A/A A J . 3 :216-223(1965).

6 . Na um a nn , A . , a nd He r m a nns , E . , i n No i se M e c ha -nisms, AGARD CP-131, 23-1 , 1973.

7. Ibragim , M. A. , Klimov, A. I ., and Shugaev, F. V.,Flu id Dyn . , 13:785-787 (1978).

8. Ellzey, J. L. , Pico ne, M. , and Ora n, E . S., N R LM e m or a n dum R e p or t 6919 , J a nua r y , 1992 .9. Ellzey, J. L. , and He nn eke , M. R. , 19th Inte rnati on al

S ym pos ium on S hoc k W a ve s , M a r se il l e, F r a nc e , Ju ly1993.

10. M arble, F. E. , He ndrick s, G. J. , and Zu kosk i, E. E. ,A IA A Paper 87-1880, 1987.11. W ai tz , I . A. , Marble , F . E., and Zuk oski , E . E. , A IA APaper 92-3550, 1991.

12. Ce teg en, B . M. , and S ir ignano, W. A. , Combus t . Sc i .Techno l . 72:157-181 (1990).

13. Yang, J., Ph.D . thesis, Califo rnia Insti tute o f Te ch-nology, 1991.

14. Haas, J . F. , and Sturtevant, B. , J . F lu id Mech .181:41-76 (1987) .

15. Jacobs, J . W . , J . Flu id Mech . 234:629-649 (1990).16. Budzinski, J . M. , Ph.D . thesis, Califo rnia Insti tute of

Technology, 1992.17. Ales sand ri, D. S., and C eteg en, B. M. , 19th Inte rna-t iona l Sym posium on Shock Waves , Marse i l le , France ,

July 1993.18. Pitts, W. M., and Kashiwagi, T. , J . F lu id Mech .

141:391-429 (1977).19. Shaughnessy, E. J. , and Mo rton, J. B. , J . Fluid Me ch.

80:129 148 (1977).

Rec eived 1 Dec em ber 1993; revised 20 Ap ri l 1994

Commen t sE . Hasselbrink, Stanfor d University, US A. I thinkthat it has been too hastily suggested thatmixing has occurred where the propane con-centration is low. Even if the mixture fractionof propane is unity at a given spatial location,the propane scattering may be low due to lowgas density. This explanation is consistent withthe images: the signal is low in the cores wherepressure is low and the signal is high at thefront stagnation point, where the pressure isexpected to be high. My own estimates for thedensity variation in such a vortex ring are~ 20% (references: D. W. Moore, Proc. Roy.Soc. Lond. A., 1989; Mandella, Ph.D. Thesis,Stanford U., 1987; two papers in Fluid Dyn.

Res., v. 12, 1993). Therefore, caution must be'exercised in drawing conclusions about mixingfrom the concentration data and PDFs. Forexample, your interpretation of high mixing inthe vortex ring core contrasts previous studiesof turbulent vortex rings (Glezar, 1981, Ph.DThesis Caltech; Maxworthy 1974 & 1977, J.Fluid. Mech.) which find very thin, persistent,stable cores that mix little with outside fluid.Authors ' Reply . It is true that the pressurevariations in a compressible vortex will lead tovariations in the strength of the Rayleigh scat-tering signal. For the shock/vortex ring inter-actions studied here, however, we believe that


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240 B . M . C E T E G E N A N D J. C . H E R M A N S O N

th is e f fec t i s cons ide rab ly sma l le r than thev a r i a t i o n s i n s i g n a l d u e t o t h e c o n c e n t r a t i o nf ie ld . A n e x a m i n a t io n o f im a g e s p r e s e n t e d i nFig . 2b he lp suppor t th i s v iew. In th i s case , thep r e s s u r e r a t io a c r o s s t h e s h o c k w a v e is a b o u t2 .3 , l ead ing to a change in Ray le igh s ca t t e r ings ig n a l m u c h i n e x c e ss o f t h a t d u e t o t h e e x -p e c t e d 1 0 - 2 0 % p r e s s u r e v a r i a t i o n i n t h e v o r -t ex ring . In add i t ion , ex am ina t ion o f F igs . 2and 3 sugges t s tha t the reg ions o f low signa ls t r en g t h i n th e v o r t e x c o r e s r e p r e s e n t o n l y asma l l f rac t ion o f the im aged f low f i eld . Las t ly ,i t s h o u l d b e n o t e d t h a t t h e t w o s t u d i e s c i t e d( G l e z e r a n d M a x w o r th y ) w e r e c a r r i e d o u t i nl i q u i d s , w h e r e m a s s d i f f u s i o n r a t e s a r e m u c hs m a l l e r t h a n t h o s e o f g a s e s. I n s u m m a r y , w eb e l i e v e th a t t h e q u a l it a t iv e f e a t u r e s o f t h e p d f sa n d t h e q u a n t i t a t i v e c h a n g e s i n m i x e d f l u i dc o n c e n t r a t i o n u p o n s h o c k i n t e r a c ti o n a r e n o tg r e a t ly i n f l u e n c e d b y p r e s s u r e v a r i a t io n i n t h evor tex r ings p r io r to the in te rac t ion .E. J. Gutmark, Na val Ai r Warfare Center, USA .T h e P D F s , c o m p a r i n g f u e l c o n c e n t r a t i o n i nt h e v o r t e x b e f o r e a n d a f t e r t h e i n t e r a c t i o n w i t h

t h e s h o c k , a r e n o t c o n s i s t e n t w i t h t h e c o n s e r -v a t i o n o f fu e l . T h e d a t a s h o w m o r e f u e l i n t h em i x i n g d o m a i n b e f o r e t h e i n t e r a c t i o n t h a n a f -t e r i t. W ha t i s the exp lan a t ion f o r th is appa r -en t incons i s t ency?

Authors' Reply. As s ta ted in the a r t i c l e , eachs p a t ia l p d f w a s o b t a i n e d i n a s i m i la r r e g i o ne n c o m p a s s i n g t h e v o r t e x rin g. T h e q u a n t i t a ti v ec o m p a r i s o n o f th e t o t al a m o u n t o f in j ec t an tb e t w e e n i n d i v i d u a l p d f s i s c o m p l i c a t e d b y t h ee x i s te n c e o f l a rg e p e a k s c o r r e s p o n d i n g t o e i -t h e r p u r e i n j e c t a n t g a s o r p u r e ( o r n e a r l y p u r e )a m b i e n t a i r . T h e s e p e a k v a l u e s w o u l d , i n m a n yc a s e s , b e o f f - s c a l e i f i n c l u d e d i n t h e P D F ss h o w n i n F i g s . 5 - 7 , a n d t h u s s o m e i n j e c t a n tf l u i d m a y n o t b e f u l l y a c c o u n t e d f o r i n t h eP D F s s h o w n . T h e r e m a y a l s o h a v e b e e n s o m es h o t - t o - s h o t v a r i a ti o n s i n t h e q u a n t i t y o f t h ein jec ted gas . In any case , these unce r ta in t i e sd o n o t a f f e c t t h e c o n c l u s i o n s r e g a r d i n g t h ec h a n g e d n a t u r e o f t h e P D F s o r t h e r a p i d sh i fti n m i x e d f l u id t o l o w e r v a l u e s o f c o n c e n t r a t i o nr e s u l t i n g f r o m t h e s h o c k / v o r t e x i n t e r a c t i o n .

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B.M. Cetegen and J.C. Hermanson- Mixing Characteristics of Compressible Vortex Rings Interacting with Normal Shock Waves