Dynamics and mixing of vortex rings in crossflow

Transcription

1 J. Fluid Mech. (8), vol., pp c 8 Cambridge Universit Press doi:.7/s88 Printed in the United Kingdom 89 namics and mixing of vortex rings in crossflow RAJES SAU AN KRISHNAN MAHESH Aerospace Engineering & Mechanics, Universit of Minnesota, Minneapolis, MN, USA (Received September 7 and in revised form Februar 8) irect numerical simulation is used to stud the effect of crossflow on the dnamics, entrainment and mixing characteristics of vortex rings issuing from a circular nozzle. Three distinct regimes exist, depending on the velocit ratio (ratio of the average nozzle exit velocit to free-stream crossflow velocit) and stroke ratio (ratio of stroke length to nozzle exit diameter). Coherent vortex rings are not obtained at velocit ratios below approximatel. At these low velocit ratios, the vorticit in the crossflow boundar laer inhibits roll-up of the nozzle boundar laer at the leading edge. As a result, a hairpin vortex forms instead of a vortex ring. For large stroke ratios and velocit ratio below, a series of hairpin vortices is shed downstream. The shedding is quite periodic for ver low Renolds numbers. For velocit ratios above, two regimes are obtained depending upon the stroke ratio. Lower stroke ratios ield a coherent asmmetric vortex ring, while higher stroke ratios ield an asmmetric vortex ring accompanied b a trailing column of vorticit. These two regimes are separated b a transition stroke ratio whose value decreases with decreasing velocit ratio. For ver high values of the velocit ratio, the transition stroke ratio approaches the formation number. In the absence of trailing vorticit, the vortex ring tilts towards the upstream direction, while the presence of a trailing column causes it to tilt downstream. This behaviour is explained. In the absence of crossflow, the trailing column is not ver effective at entrainment, and is best avoided for optimal mixing and entrainment. However, in the presence of crossflow, the trailing column is found to contribute significantl to the overall mixing and entrainment. The trailing column interacts with the crossflow to generate a region of high pressure downstream of the nozzle that drives crossflow fluid towards the vortex ring. There is an optimal length of the trailing column for maximum downstream entrainment. A classification map which categorizes the different regimes is developed.. Introduction Jets in crossflow are central to a variet of important applications such as dilution holes in combustors, fuel injectors, pollutant dispersion from smoke stacks, thrust vectoring of turbojets, and V/STOL aircraft. There is therefore considerable incentive to activel control jets in crossflow. This paper is motivated b experimental observations on the use of pulsing to control the mixing characteristics of jets in crossflow (e.g. Wu, Vakili & Yu 988; Chang & Vakili 99; Eroglu & Briedenthal ; Blosse, Naraanan & Bewle ; M Closke et al. ; Karagozian; Cortelezi & Soldati ). It is observed that pulsing the jet results in the formation of vortex rings whose strength and spacing depend on the frequenc and dut ccle of the jet, for a given jet and crossflow combination (Eroglu & Briedenthal ). The resulting flow appears to improve mixing rate and increase the entrainment. Some workers (e.g. M Closke et al. ; Shapiro et al. ) relate the optimal pulse width for maximum penetration of vortical structures, to the formation number proposed

2 9 R. Sau and K. Mahesh Crossflow z x z 8 v inflow 8 x/ Figure. Schematic of the problem. Horizontal slice of the mesh. b Gharib, Rambod & Shariff (998). Johari () presents scaling arguments, based on the motion of individual vortex rings in stationar fluid, for the penetration and mixing of pulsed jets in crossflow. He proposes classification schemes using the formation number and stroke ratio obtained from the frequenc and dut ccle of the pulsed jet. Sau & Mahesh (7) used NS to stud optimal mixing b a single vortex ring in stationar fluid. The considered two Schmidt numbers and a range of stroke ratios. The formation number was found to ield the maximum entrainment for both Schmidt numbers. This behaviour was explained b noting that the entrainment was a combination of that due to the leading vortex ring, and that due to the trailing column of vorticit. The entrainment b the trailing column was negligible compared to that b the vortex ring, as a result of which the relative contribution of the vortex ring to overall entrainment decreases beond the formation number. The above stud ignores the effect of crossflow. This paper therefore studies the effect of crossflow on the dnamics, mixing and entrainment characteristics of vortex rings. The paper is organized as follows. Section discusses the problem and details of the simulations. The effect of crossflow on ring dnamics is discussed in. The case of ver low velocit ratio is presented in.. Entrainment characteristics of vortex rings and hairpin vortices are presented in. The paper concludes with a classification map of the different flow regimes in.. Simulation details.. Problem statement Figure shows a schematic of the problem, in which a slug of fluid is pushed through a clindrical nozzle with : diameter ratio. The nozzle fluid forms a vortex ring as it exits the nozzle. The vortex ring interacts with the crossflow which is directed along the x-direction. The crossflow is modelled as a laminar boundar laer over a flat plate. Note that the origin of the coordinate sstem is located at the centre of the nozzle exit plane, and the nozzle axis points in the -direction. In experiments, vortex rings are often generated b a piston pushing a column of fluid of length L through an orifice of diameter. In simulations, this process is modelled b specifing a top-hat velocit profile (U in ) at the nozzle inflow for a

3 namics and mixing of vortex rings in crossflow 9 duration of time τ (referred to as piston time duration). The inflow velocit is zero for time greater than τ. The stroke length and non-dimensional time are determined using the mean nozzle exit velocit, U exit, which is equivalent to the piston velocit. ThestrokelengthL is therefore equal to U exit τ, and the stroke ratio (L/) isequal to U exit τ/. The stroke ratio is varied b changing the piston time duration τ. The velocit ratio r is defined as the ratio of mean nozzle exit velocit (U exit ) to the freestream crossflow velocit (u ). The velocit ratio is varied b changing the crossflow velocit. NS of single vortex rings are performed for stroke ratios varing from. to 8 and crossflow velocit ratios ranging from to. The Renolds number based on U exit and nozzle exit diameter () is in all cases, except where noted below... Numerical details The governing equations are the incompressible Navier Stokes and continuit equations, u i + u iu j t x j and the passive scalar equation C t = p x i + ν u i x j x j, + Cu j x j = ν Sc u i x i =, (.) C x j x j. (.) Here u i, p and ν denote the velocities, pressure and kinematic viscosit respectivel. C is the concentration of the scalar. The densit of the fluid is assumed constant and is absorbed into the pressure. The numerical scheme used to solve the Navier Stokes equations is described in detail b Mahesh, Constantinescu & Moin (). The algorithm stores the Cartesian velocities and pressure at the centroids of the cells (control volumes), and stores the face-normal velocities at the centroids of the faces. The algorithm is a predictor corrector scheme which emphasizes discrete energ conservation on unstructured grids. This propert makes the algorithm robust at high Renolds numbers without numerical dissipation. The predictor velocities at the control-volume centroids are obtained using the viscous and the nonlinear terms in equation (.); the predictor face-normal velocities are then obtained. The predictor face-normal velocit is then projected so that continuit is discretel satisfied. This ields a Poisson equation for pressure which is solved iterativel using a multigrid approach. The pressure field is used to update the Cartesian control volume-velocities. Implicit time-stepping is performed using a Crank Nicholson scheme. The algorithm has been validated for a variet of problems over a range of Renolds numbers (Mahesh et al. ). The passive scalar is computed using a predictor corrector method (Muppidi ). The scalar field is first advanced using a second-order central difference scheme. The predicted scalar field is corrected in regions of scalar overshoot using a firstorder upwind scheme. This corrector step ensures that locall, the passive scalar concentration is bounded (i.e. C [, ]). The scalar is advanced in time explicitl using the second-order Adams Bashforth scheme in an inner loop. The fluid emerging from the nozzle exit has C =. and the ambient fluid has C =... Computational domain and boundar conditions The computational domain spans above the nozzle exit in the x-, - and z-directions respectivel, and includes a length of nozzle. The computational mesh consists of unstructured hexahedral elements. A grid refinement stud was performed on grids containing approximatel.7,. and 7 elements.

4 9 R. Sau and K. Mahesh v Ω z...8. x/ x/ Figure. v-velocit profiles at the nozzle exit in the smmetr plane at t = U exit t/=.:, without crossflow (r = );, r =;, r =;, r =.;, r =.;, r =.. Corresponding spanwise vorticit, Ω z / profiles:, without crossflow (r = );, r =;, r =.;, r =.. The profiles of nozzle-exit velocit and vorticit for the three grids were examined. The maximum v velocit obtained for the. grid differs b. % from the.7 grid. The 7 grid deviates less than.% from the. grid. The vorticit obtained from the. grid deviates less than.% from the 7 grid. Results from the 7 grid are presented in the paper. Figure shows the horizontal cross-section of the mesh (with. elements for clarit). Ver fine mesh elements are used near the nozzle exit and along the direction of the crossflow. The crossflow is simulated as a laminar flow over a flat plate. The velocit field from the self-similar Blasius boundar laer solution is specified at the inflow plane of the crossflow, upstream of the nozzle exit. The velocit field is such that in the absence of nozzle fluid, the crossflow has prescribed δ % at the centre of the nozzle exit. On the spanwise boundaries (z/ = ±8), the velocit field corresponding to laminar crossflow over a flat plate is prescribed. Free-stream velocit boundar conditions are specified on the top boundar at / =. At the nozzle inflow plane (/ = ), a top-hat velocit profile is specified: { Uin if t τ v(x,z,/ =,t)=v inflow = if t>τ. A zero-gradient boundar condition is used at the outflow (x/ = ).. Effect of crossflow on ring dnamics When the ambient fluid is stationar, the shear laer that emerges from the nozzle rolls up into an axismmetric vortex ring. It propagates awa from the nozzle and entrains the ambient fluid radiall inwards as it does so. The formation number (Gharib et al. 998) marks the transition between two possible flow structures: vortex ring and vortex ring with trailing column of vorticit. A single vortex ring is produced for stroke ratios less than the formation number, while a vortex ring with trailing column is produced for larger stroke ratios. Both the formation and propagation of vortex rings are affected b presence of crossflow. The crossflow also breaks the axismmetr of the ring. Figure shows

5 namics and mixing of vortex rings in crossflow 9 the nozzle exit velocit (v) profiles at the smmetr plane for different velocit ratios at the same instant of time (t =.). The exit velocit in the absence of the crossflow is also shown. The crossflow momentum decelerates the nozzle fluid on the upstream side (x/ =.). As a result of mass conservation, the fluid near the downstream side (x/ =.) accelerates as shown in the figure. The peak velocit on the downstream side increases with increasing crossflow velocit. For r =., there is a ver small amount of reverse flow at the upstream side of the nozzle. This is because the adverse pressure gradient outside the nozzle exit overcomes the favourable pressure gradient created b the nozzle throat. Profiles of spanwise vorticit at the nozzle exit in the smmetr plane are shown in figure. Note that vorticit increases significantl on the downstream side due to acceleration of the nozzle fluid. In contrast, the vorticit on the upstream side decreases. Also, the location of peak vorticit on the upstream side shifts to the nozzle right with increasing crossflow velocit. For ver low velocit ratios (r<), the nozzle shear laer at the upstream side does not roll up and a complete vortex ring does not form. The low-velocit-ratio case is discussed in detail in later sections. The following section examines the effect of crossflow on a complete vortex ring. Here, the velocit ratio is greater than... Velocit ratio > For velocit ratio greater than, the simulations show that the vortex ring tilts in the presence of crossflow. Tilting is defined in terms of the angle between the plane of the vortex ring and the exit plane of the nozzle. The ring tilts towards the upstream direction for low stroke ratio and tilts towards the downstream for high stroke ratio. This behaviour is discussed in more detail below.... Low stroke ratio: upstream tilting Figure shows contours of Ω z vorticit in the smmetr plane (x, ) for stroke ratio and velocit ratio at different instants of time. Note that the vortex ring tilts towards the upstream direction as it penetrates into the crossflow. The induced velocit of the vortex ring opposes the crossflow velocit which helps the ring to penetrate deeper into the crossflow. This behaviour is consistent with the experiments of Chang & Vakili (99), who observed the upstream tilting of vortex rings in low-frequenc pulsed jets in crossflow. The fluid in the vortex ring initiall does not have mean horizontal momentum. The relative horizontal velocit between the crossflow and the centroids of the vortex ring combines with ring circulation to produce Kutta Joukowski lift. It is readil seen that the upstream portion of the ring experiences a downward lift force relative to the downstream portion as shown in figure. As a result, the ring tilts upstream. Note that this reasoning is two-dimensional at ever cross-section of the ring. Twodimensional simulations of a vortex pair in crossflow were therefore performed to test this hpothesis. The vortex pair was found to tilt upstream as observed here. A quantitative discussion of Kutta Joukowski lift on vortex cores is provided b Ting & Tung (9) who consider a vortex core of uniform vorticit embedded in a two-dimensional non-uniform stream. The match the near-field and far-field solutions, and use the Kutta Joukowski theorem to obtain the resulting force on the vortex. Figures and(c) show vortex rings of stroke ratio in velocit ratios of and. Note that the tilting increases with increase in the crossflow magnitude. This behaviour is consistent with the Kutta Joukowski lift being responsible for the tilting.

6 9 R. Sau and K. Mahesh t = 9. t = 7. t =. x/ t =. Figure. Vorticit contours for L/ = and r = at different time instants. The solid and dashed lines denote positive and negative values respectivel. (c) U rel x/ x/ Figure. Schematic of vortex ring tilting. (b, c) Vorticit contours for L/ = at two different crossflow velocities: u =.; (c) u =.. Note that the tilting increases with increase in the crossflow velocit. The vortex ring also deforms in the presence of crossflow. The thickness of the ring becomes non-uniform as it propagates into the crossflow. Figure shows the λ surface of the vortex ring. Note that the downstream side of the ring is thicker than the upstream side. This deformation can be attributed to the strain field experienced b the ring in crossflow. Figure plots the in-plane velocit magnitude along a curve which encompasses the circumference of the ring from upstream to downstream side on the plane of the ring. The velocit field around the ring suggests that a positive velocit gradient is set up along the ring on the upstream side and a negative velocit

7 namics and mixing of vortex rings in crossflow 9 z x z x.. u.8 s x s (c) u s x s > u s x s < Extension Contraction Crossflow Crossflow u s x s > u s x s < Extension Contraction Figure. λ -surface of the vortex ring at a distance of around from the nozzle exit for L/ of at r =. In-plane velocit magnitude on the circumference of the ring from the upstream to downstream side. (c) Schematic to illustrate the velocit gradients along the ring. gradient is set up along the ring on the downstream side. The corresponding velocit gradients experienced b the ring and the resulting deformation is shown in figure (c). The upstream side of the ring experiences a positive velocit gradient and stretches as a result. The vortex tube becomes thinner due to stretching. The downstream side exhibits the opposite behaviour.... High stroke ratio: downstream tilting For large stroke ratios, vortex rings along with a trailing column of vorticit are generated. Figure shows the time evolution of Ω z contours in the smmetr plane for stroke ratio of and velocit ratio of. Note that a vortex ring followed b a trailing column is produced. Also, the leading vortex ring tilts towards the downstream direction, unlike its behaviour at lower stroke ratios. One other significant difference is that the ring diameter expands as it propagates downstream (figure b). This behaviour is due to the enhanced downstream entrainment of crossflow fluid, which is explained later. The ring s induced velocit has a component along the direction of the crossflow which favours downstream movement. The trailing column acts as an obstacle to the crossflow and as a result, the Kutta Joukowski lift is not produced. Instead, the crossflow bends the leading vortex ring and the trailing column along the direction of the crossflow; finall the leading vortex ring pinches off from the trailing column. For a fixed velocit ratio, the stroke ratio determines whether the ring tilts upstream or downstream. This suggests that there is a transition stroke ratio which separates these two regimes. So, the transition stroke ratio for r = is sought and the effect of crossflow velocit ratio on the transition stroke ratio is examined.

8 9 R. Sau and K. Mahesh t =. t = 9. t =. x/ t = 7. Figure. Vorticit contours in the smmetr plane for L/ = and r = at different time instants.... Transition stroke ratio For a fixed crossflow velocit ratio, the transition stroke ratio is obtained from simulations at different stroke ratios. The transition stroke ratio for r = is found to be L/ =.. The vortex ring tilts in the upstream direction for stroke ratios below.. For stroke ratios larger than., the vortex ring tilts downstream and the ring diameter expands. The ring dnamics at the transition stroke ratio for r = are shown in figure 7. Initiall the leading vortex ring tilts in the downstream direction, but eventuall it pinches off and tilts in the upstream direction slightl. The effect of velocit ratio on the transition stroke ratio is examined in figure 8. For r =, the transition stroke ratio is found to be.; the transition stroke ratio decreases as the velocit ratio decreases. We estimate a transition curve which defines the transition stroke ratio as a function of velocit ratio. Figure 8 shows the three transition stroke ratios obtained at r =,. and respectivel. The ambient fluid approximates stationar fluid when the velocit ratio is increased to ver high values. Recall that in stationar fluid, the formation number (L/ =.) defines the transition between single vortex ring and a vortex ring with trailing column. So, the transition curve has an asmptote: L/ =.. An exponential curve is fitted which passes through the three data points and has an asmptote at L/ =.. This ields the transition stroke ratio (L/) tr as a function of velocit ratio r: (L/) tr = F A exp ( A r) where F =. denotes the formation number in stationar fluid, and A and A are constants which are obtained as. and. respectivel. The transition curve is onl

9 namics and mixing of vortex rings in crossflow 97 t = 9. t =. t =. x/ t = 7. Figure 7. Time evolution of vorticit contours at transition stroke ratio (L/ =.) for r =. 8 r 7 8 L/ Figure 8. The transition curve in the (r, L/)-plane separates two regimes of flow in the presence of crossflow. The dotted line represents the transition line in stationar flow. plotted for r> as shown in figure 8. It will be shown later that complete vortex rings do not form for r<. The transition curve has important practical consequences. For a particular velocit ratio, stroke ratios on the left of the curve shown in figure 8 will generate vortex rings

10 98 R. Sau and K. Mahesh (c) x/ x/ x/ Figure 9. Effect of Renolds number for L/ = at r =: Re =, Re = 9, (c) Re =. which tilt upstream and penetrate into the crossflow. The stroke ratios to the right of this curve will generate rings which tilt downstream and have a trailing column. It will be shown later that the trailing column significantl improves entrainment in crossflow. In terms of penetration of the rings into the crossflow, stroke ratios around the transition curve will provide maximum opposition to the crossflow. For stroke ratios higher than the transition stroke ratio, the ring-tilting will produce a component of the induced momentum along the crossflow and favour downstream movement of the ring. Shapiro et al. () performed experiments on a pulsed jet in crossflow and reported maximum penetration at two stroke ratios, one in the range of formation number (..7) and the other ling in the range.8.. Their jet was acousticall pulsed and r was equal to.8. From our results, it can be inferred that the maximum penetration of the pulsed jet is due to the formation of vortex rings which tilt upstream and penetrate deeper into the crossflow. Note that for a velocit ratio r =.8, the stroke ratio from the transition curve is approximatel. which is in agreement with the experimental observation of Shapiro et al. (). The above behaviour is not significantl affected b the Renolds number, at least in the Renolds number regime where the rings are stable. Figure 9 shows the vorticit field in the smmetr plane for the vortex ring with L/ = and r = at three different Renolds numbers. Note that the Renolds number has almost no effect on the tilting of the vortex rings. As the Renolds number increases, the vorticit gradients increase and the instabilities in the trailing jet become more pronounced... Velocit ratio < The increase in crossflow (i.e. decrease in velocit ratio) causes the vorticit in the crossflow boundar laer to increase. Thus, for ver low velocit ratios, the vorticit in the crossflow boundar laer starts interacting with the vorticit of the nozzle boundar laer. The interaction is most significant on the upstream side of the nozzle exit. Figure shows results for three low velocit ratios:,. and. Crossflow boundar laer profiles and vorticit contours in the smmetr plane are shown at t =. for each case. Note that roll-up of the nozzle boundar laer is inhibited on the upstream side. The vorticit in the emerging boundar laer is cancelled b the opposing vorticit from the crossflow boundar laer. The vorticit cancellation becomes more pronounced as the velocit ratio decreases. For r =., the vorticit in the emerging nozzle boundar laer is almost annihilated b the crossflow boundar laer as shown in figure (c). So, a complete vortex ring structure does not form. Instead, a hairpin

11 namics and mixing of vortex rings in crossflow u u (c) u x/. 7 8 x/ Figure. Low velocit ratio cases: r =., r =. and(c) r =.. Crossflow boundar laer profiles (left) and the vorticit contours in the smmetr plane (right) are shown at t =. for each case. Note that the vorticit from the nozzle boundar laers is cancelled b opposing vorticit in the crossflow boundar laer at the upstream side of the nozzle exit. x z x z x z Figure. Hairpin structures are shed for L/ = at r =. The figures correspond to t =.. structure is formed due to the roll-up on the downstream side alone. This formation of hairpin vortices is similar to that observed in the experiments of Acarlar & Smith (987) at velocit ratios much smaller than those used in our simulation.... Large L/: hairpin vortex shedding For low velocit ratios and large stroke ratios, a series of hairpin vortices is shed downstream. Figure shows an iso-surface of pressure for L/ = and r =.. Note that once a hairpin vortex is discharged, then the next hairpin starts to form. This shedding of hairpin vortices is Renolds-number dependent. Figure shows vorticit contours in the smmetr plane at different Renolds numbers:, and. The stroke ratio L/ = so that the periodic shedding of the hairpins can be observed. Instantaneous contours of vorticit are shown at t = for each of the cases. Note that for Re =, the vortices are unstable. For lower Renolds number, periodic shedding of coherent hairpins is observed as shown. Also, the shedding frequenc depends upon the Renolds number.

12 R. Sau and K. Mahesh 8 8 (c) 8 x/ Figure. Vorticit contours in the smmetr plane at t = for r =. andl/ =. Re =, Re =, (c) Re =. Note that shedding of the hairpin vortices is unstable at Re =. The shedding is ver periodic for lower Renolds number. The evolution of hairpin vortices ma be explained as follows. Figure shows a hairpin structure and its image in the presence of the wall. Schematics of the hairpin cross-sections in spanwise and smmetr planes are also shown. The smmetr plane cuts the head off the hairpin structure. The spanwise plane consists of its counterrotating legs. The presence of the wall can be modelled b the image vortices as shown in the schematic. In the spanwise plane, the counter-rotating legs follow the path shown as a thick-dotted line in figure. The legs approach each other and also move upwards due to their mutual induction and interaction with the image vortices. On the other hand, the crossflow exerts an upward Kutta Joukowski lift force on the head of the hairpin and the image vortex induces a negative u-velocit. As a result, the head portion of the hairpin tilts as it propagates downstream and becomes normal to the crossflow direction as shown in figure. The hairpin structure detaches from the emerging shear laer due to the particular motion of the hairpin legs in the presence of the wall. While the hairpin detaches from the shear laer and is shed downstream, the next hairpin is formed. These results suggests that for ver low velocit ratio and low Renolds number, a stead jet in the presence of a crossflow results in a series of hairpin structures. And if the Renolds number is ver low then the horseshoe-like vortices are shed in a periodic manner. This suggests that at ver low velocit ratios, the structure of jets in crossflow is quite unlike the classical structure where the counter-rotating vortex

13 namics and mixing of vortex rings in crossflow Smmetr plane Smmetr plane u Wall z x Horseshoe vortex Image Wall Wall Spanwise plane Image vortex Image Spanwise plane Figure. Evolution of a hairpin vortex structure in the presence of a wall. The dotted curves in the spanwise plane show the trajector of the legs due to the wall. pair (CVP), a stationar horseshoe vortex, wake vortices and Kelvin Helmholtz instabilit vortices characterize the flow (Fric & Roshko 99). This observation is also consistent with the experiments of Gopalan, Abraham & Katz () on a turbulent jet in crossflow at r<. The report that a semi-clindrical vortical laer forms behind the jet and suggest that this vorticit starts from the jet shear laer.. Mixing and entrainment characteristics The mixing of nozzle fluid with the crossflow fluid is studied b the transport of a passive scalar. The Schmidt number Sc of the scalar is.. Fluid exiting from the nozzle has scalar concentration of. The crossflow fluid which initiall had scalar concentration of zero, mixes with the nozzle fluid to give intermediate scalar values between zero and one. When the ambient fluid is stationar, an axismmetric vortex ring entrains the ambient fluid radiall inward into the core when it forms. The entrained ambient fluid mixes with the nozzle fluid inside the core. In contrast, the trailing column is surrounded b quiescent ambient fluid and does not entrain the ambient flow. The trailing column is much less effective than a single vortex ring in entraining the ambient fluid and mixing b vortex rings in stationar fluid is optimal at a stroke ratio equal to the formation number (Sau & Mahesh 7). The presence of crossflow changes the mixing and entrainment characteristics completel. As shown above, three different flow regimes exist depending on the velocit ratio and stroke ratio. The different entrainment mechanisms of these three flow regimes are studied below.

14 R. Sau and K. Mahesh 8 x/ x/ Figure. Instantaneous scalar contours on the smmetr plane for a vortex ring in crossflow with r = and L/ of and. Note the asmmetr in the scalar concentration in the case of small stroke ratio, and the enhanced downstream mixing for the case of large stroke ratio Velocit ratio > Figures and show scalar contours in crossflow of r = for vortex rings with stroke ratio and respectivel. istinctl different mixing and entrainment behaviour can be observed. For the single vortex ring shown in figure, the scalar mixing is similar to that in the absence of crossflow. The vortex ring entrains the ambient fluid during its formation. The entrained fluid is mixed with the nozzle fluid inside the core and some of the fluid from the boundar of the vortex ring is deposited behind the ring as the ring propagates. The crossflow introduces an asmmetr between the cores of the ring. Figure suggests that the core on the upstream side is more mixed than on the downstream side of the ring. This asmmetr in scalar mixing can be attributed to the deformation of the ring as explained in... For large stroke ratios, scalar contours in figure clearl show that the mixing on the downstream side is enhanced significantl. This behaviour is similar to that observed in jets in crossflow. As suggested b Muppidi & Mahesh (8), when a jet encounters a crossflow, the jet bends and the cross-section of the jet deforms. This creates a high pressure gradient directed towards the jet on the downstream side. This pressure gradient drives the flow toward the jet and causes the jet to entrain more fluid on the downstream side rather than the upstream side. Similar flow features are observed in the present case. Figure shows an iso-surface of vorticit along with three-dimensional streamlines for L/ = and r =. The iso-surface shows the structure of the ring along with the trailing column. It is interesting to observe the streamlines and the deformation of the trailing column cross-section on the downstream side. The streamlines clearl show that crossflow fluid goes around the trailing column and is entrained from the downstream side into the trailing column and ring. This downstream entrainment is further enhanced b the low pressure in the core of the vortex ring. This behaviour is in contrast to that in the absence of crossflow where the trailing column does not significantl contribute to overall mixing. Is there an optimal length of trailing column? For low stroke ratios (no trailing column or a ver small one), enhancement due to downstream entrainment is absent. On the other hand, for ver large stroke ratios, the leading ring would be far awa

15 namics and mixing of vortex rings in crossflow z x Figure. Iso-surface of vorticit along with some three-dimensional streamlines for L/ = and r =. The streamlines show the downstream entrainment of the crossflow fluid. (c) x/ x/ x/ Figure. Contours of pressure gradient magnitude ( p ) at different times: t =.7, t =., (c) t =7.. Note that the pressure gradient magnitude around the downstream of the trailing column is maximum around t =.. from the high-pressure region downstream of the trailing column and would not contribute to enhance the downstream pressure gradient.... Optimal downstream pressure gradient In order to obtain the optimal length of trailing column, the variation of pressure gradient magnitude with the length of the trailing column is computed for velocit ratio. Figure plots the contours of pressure gradient magnitude ( p ) inthe smmetr plane at different time instants. The figure shows that a high pressure gradient magnitude (denoted b p tr ) is created on the downstream side of the trailing column. Note that p tr increases till approximatel t =., following which it decreases. So, for optimal downstream entrainment, the length of the trailing column

16 R. Sau and K. Mahesh x/ Figure 7. Pressure gradient vectors on the smmetr plane near the downstream pressure gradient region at two different time instants:, t =.7;, t =.. Note that the vectors rotate towards the right in the direction of the vortex ring core x/ x/ Figure 8. Instantaneous scalar contours on the smmetr plane for r = and L/ of and. should be at stroke ratio equal to. The effect of the leading ring on the downstream pressure gradient is shown in figure 7. The downstream side of the trailing column is enlarged to show the pressure gradient vectors at two different time instants, chosen when p is increasing. Note that the pressure gradient vectors rotate towards the right, in the direction of the leading vortex ring... Velocit ratio < Recall that for ver low velocit ratio, a hairpin structure forms instead of a vortex ring, and that a series of hairpins is shed downstream for large stroke ratios. Figures 8(c) and 8(d) show the scalar contours in the smmetr plane for hairpin structures at r = for stroke ratios of and respectivel. It is immediatel apparent

17 namics and mixing of vortex rings in crossflow Streamline z x Streamline z/.7. (c)..... w/ C/C Figure 9. Entrainment b hairpin vortex legs. Scalar contour lines on a spanwise cross-section of the hairpin legs at x =.9. The corresponding scalar contours on the spanwise plane along with some in-plane streamlines. (c) Spanwise velocit (w/) and scalar concentration (C/C ) profiles along a line passing through one of the cores of the hairpin legs on the spanwise plane. that the mixing behaviour is quite different from that observed at high velocit ratios. These differences are discussed in detail below.... Entrainment b hairpin legs Figure 9 shows the hairpin structures for L/ = at r = along with scalar contour lines on a spanwise plane at x/ =.9. A streamline emanating from near the nozzle exit is also shown. Figure 9 shows scalar contours in the spanwise plane (x/ =.9) which cuts the counter-rotating legs of the hairpin, and some instantaneous streamlines around one of the hairpin-leg cores. A ver low-pressure region is created at the core. The resulting pressure gradient causes the crossflow fluid around the legs to be entrained. This entrainment could also be explained as Biot-Savart induction b the vortex cores in the hairpin legs. The streamlines in figure 9 show this entrainment of crossflow fluid b the hairpin legs. To further illustrate this behaviour, w-velocit and scalar concentration profiles are plotted in figure 9(c) along a line passing through one of the cores of the hairpin legs on the spanwise plane at x/ =.9. The figure shows large levels of w-velocit near the vortex core close to the wall. The scalar concentration of the fluid with strong w-velocit is close to zero, indicating crossflow fluid. An interesting point to note here is that the scalar contours due to the counter-rotating hairpin legs in the spanwise plane appear similar to those of the counter-rotating vortex pair (CVP), observed in a jet in crossflow, although in the present case, a CVP does not exist.

18 R. Sau and K. Mahesh.. V sc dv sc dt *... 8 t * % Increase in dv sc dt * 8 7 (c) 7 L/ 7 8 L/ Figure. Variation of volume of scalar-carring fluid with time for different stroke ratios and r =:, L/ =.;, L/ =.;, L/ =.. The rate of change of scalar volume after formation against the stroke ratio for r = ( ). The results for the vortex rings in the absence of crossflow are also shown ( ). The vertical line denotes the formation number. The dot-dashed line shows the linear dependence for stroke ratios less than the formation number in the absence of crossflow. (c) Percentage increase in the rates due to crossflow are plotted against the stroke ratio. Note that the percentage increment is maximum around L/ =... Rate of Mixing The mixing is quantified b computing the rate of change of total volume of scalar after formation. The procedure is similar to that followed b Sau & Mahesh (7) for vortex rings without crossflow. Figure shows the total volume of scalarcarring fluid (V sc ) plotted against non-dimensional time (t ) for different stroke ratios and a velocit ratio of. V sc is computed as the sum of all volume elements which have scalar concentrations above a threshold value (set to.) in the domain above the nozzle exit plane (/ >). The slopes of these volume curves after the formation of the ring ield the rate of change of total volume of scalar-carring fluid. The rates are shown in Figure along with the results for rings without crossflow. The figure suggests that the crossflow enhances the mixing in each of the cases, but the enhancement is much higher for larger stroke ratios. This is due to the enhanced downstream entrainment b the trailing column. The percentage increase in entrainment rate for each stroke ratio is shown in figure (c). For a stroke ratio of, the rate of volume change increases onl about %. The increase is about %

19 namics and mixing of vortex rings in crossflow 7 V sc t * 8 Figure. Variation of volume of scalar-carring fluid with time for different stroke ratios and r =:, L/ =.;, L/ =.;, L/ =.;, L/ =.;, L/ =7.;, L/ =8.. Note that the slope of the volume curve after formation is almost the same for all stroke ratios. for stroke ratio of and about 7% for stroke ratio.. The percentage increase in rate appears optimal around a stroke ratio of., which is consistent with optimal downstream pressure gradient. The behaviour for low velocit ratios is different. Figure shows the total volume of scalar-carring fluid for different stroke ratios and r =. Note that the slopes of the curves after formation are nearl equal. The rate of change of V sc in this case is therefore nearl the same at all stroke ratios. This behaviour is due to the formation of hairpin structures at low velocit ratios. As shown earlier, periodic hairpin vortices are generated at all stroke ratios. Therefore this regime does not have an optimal stroke ratio which maximizes entrainment of crossflow fluid.. Summar: a global classification map Vortex rings in crossflow can be classified into three different regimes with differing flow structures, mixing and entrainment characteristics. Figure shows these different regimes in the space of stroke ratio (L/) and velocit ratio (r). This global classification map includes the transition curve for r>. The transition curve defines the transition from a discrete vortex ring (upstream tilting) to a vortex ring with a trailing column (downstream tilting). As the velocit ratio approaches ver high values, the flow field approaches a vortex ring in stationar fluid, and consequentl, the transition stroke ratio approaches the formation number. Again, as the stroke ratio increases toward ver large values, the flow field approaches a jet in crossflow. The regime to the right of the transition curve is characterized b enhanced downstream entrainment due to the trailing column structure, and, for a particular velocit ratio, there is an optimal length of the trailing column. For velocit ratio less than approximatel, complete vortex ring structures do not form. This regime is characterized b the formation of hairpin-like vortex structures. The global classification map has important implications. It categorizes the complete space of velocit ratio and stroke ratio into three different regimes. A desired flow feature can be attained b choosing the corresponding parameter from the map.

20 8 R. Sau and K. Mahesh 9 8 Formationnumber Towards vortex ring in stationar fluid Velocit ratio, r 7 Towards jet in crossflow 7 8 Stroke ratio, L/ Figure. A classification map showing three regimes with different flow structure and entrainment characteristics. Consider pulsation of the jet as a means to control jets in crossflow. A pulsed jet generates a series of vortex rings. The equivalent stroke ratio of each vortex ring can be easil derived from the pulsing parameters (e.g. Strouhal number, dut ccle, waveform). For example, the equivalent stroke ratio is the inverse of Strouhal number for a full modulated pulsed jet with % dut ccle and square waveform. Thus, for a fixed crossflow, the pulsing parameters can be chosen from the map according to the desired flow structure and entrainment characteristics. This work was supported b the Air Force Office of Scientific Research (AFOSR) under grant FA Computer time was provided b the National Center for Supercomputing Applications (NCSA), Minnesota Supercomputing Institute (MSI) and the San iego Supercomputer Center (SSC). We thank r Suman Muppidi for helpful discussions. REFERENCES Acarlar, M.S.&Smith, C. R. 987 A stud of hairpin vortices in a laminar boundar laer. Part. Hairpin vortices generated b fluid injection. J. Fluid Mech. 7, 8. Blosse, P., Naraanan, S.& Bewle, T. R. namics and control of a jet in crossflow: direct numerical simulation and experiments. Proc. IUTAM Smp. Turbulent Mixing Combustion (ed. A. Pollard & S. Candel), Kluwer., pp.. Chang, Y.K.&Vakili, A.. 99 namics of vortex rings in crossflow. Phs. Fluids 7, Eroglu,A.&Briedenthal, R. E. Structure, penetration and mixing of pulsed jets in crossflow. AIAA J. 9, 7. Fric, T.F.&Roshko, A. 99 Vortical structure in the wake of a transverse jet. J. Fluid Mech. 79, 7. Gharib, M., Rambod, E.& Shariff, K. 998 A universal time scale for vortex ring formation. J. Fluid Mech.,.

21 namics and mixing of vortex rings in crossflow 9 Gopalan, S.,Abraham, B.M.&Katz, J. The structure of a jet in cross flow at low velocit ratios. Phs. Fluids, Johari, H. Scaling of full pulsed jets in crossflow. AIAA J., Karagozian, A.R.,Cortelezi, L.&Soldati, A. Manipulation and control of jets in crossflow. CISM Courses and Lectures No. 9, International Center for Mechanical Sciences, Springer Wien, New York. Mahesh, K., Constantinescu, G.& Moin, P. A numerical method for large edd simulation in complex geometries. J. Comput. Phs. 97,. M Closke, R. T., King, J. M., Cortelezzi, L.& Karagozian, A. R. The activel controlled jet in crossflow. J. Fluid Mech.. Muppidi, S. irect numerical simulations and modeling of jets in crossflow. Ph Thesis, Universit of Minnesota. Muppidi, S.& Mahesh, K. 8 irect numerical simulation of passive scalar transport in transverse jets. J. Fluid Mech. 98,. Sau, R.&Mahesh, K. 7 Passive scalar mixing in vortex rings. J. Fluid Mech. 8, 9 9. Shapiro,S.R.,King,J.M.,Karagozian,A.R.&M Closke, R. T. Optimization of controlled jets in crossflow. AIAA J., Ting, L.&Tung, C. 9 Motion and deca of a vortex in a nonuniform stream. Phs. Fluids 8, 9. Wu, J. M, Vakili, A..& Yu, F. M. 988 Investigation of interacting flow of nonsmmetric jets in crossflow. AIAA J., 9 97.