Measurement of ambient fluid entrainment during laminar vortex ring formation


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1 Exp Fluids (8) 44:35 47 DOI.7/s RESEARCH ARTICLE Measurement of ambient fluid entrainment during laminar vortex ring formation Ali B. Olcay Æ Paul S. Krueger Received: December 6 / Revised: 4 September 7 / Accepted: 4 September 7 / Published online: 4 October 7 Ó SpringerVerlag 7 Abstract Planar laser induced fluorescence (PLIF) and digital particle image velocimetry (DPIV) combined with Lagrangian coherent structure (LCS) techniques are utilized to measure ambient fluid entrainment during laminar vortex ring formation and relate it to the total entrained volume after formation is complete. Vortex rings are generated mechanically with a pistoncylinder mechanism for a jet Reynolds number of,, stroke ratios of.5,. and., and three velocity programs (Trapezoidal, triangular negative and positive sloping velocity programs). The quantitative observations of PLIF agree with both the total ring volume and entrainment rate measurements obtained from the DPIV/LCS hybrid method for the jet Reynolds number of,, trapezoidal velocity program and stroke ratio of. case. In addition to increased entrainment at smaller stroke ratios observed by others, the PLIF results also show that a velocity program utilizing rapid jet initiation and termination enhances ambient fluid entrainment. The observed trends in entrainment rate and final entrained fluid fraction are explained in terms of the vortex rollup process during vortex ring formation. A. B. Olcay Department of General Engineering, University of WisconsinPlatteville, University Plaza, Platteville, WI 5388, USA P. S. Krueger (&) Department of Mechanical Engineering, Southern Methodist University, P.O. Box 75337, Dallas, TX 7575, USA Introduction Transient ejection of a jet from a nozzle is a common flow configuration which engenders the formation of a vortex ring. During the jet ejection, the shear layer which separates at the nozzle lip rolls up and entrains some of the ambient fluid into the forming vortex ring as described by Didden (979). Consequently, both ejected fluid which comes from inside the cylinder and ambient fluid which is pulled from the vicinity of the nozzle must be accelerated as the ring forms. The convective nature of the entrainment process is directly relevant for a wide variety of problems including cooling of a CPU unit (Kercher et al. 3), extinguishing oil well fires at places where bringing manpower and technology can be very expensive (Akhmetov et al. 98), mixing two different fluids, and transferring mass from one location to another. Hence, a detailed understanding of the entrainment mechanics in transient jets could be applied to enhance or lessen entrainment effects in a variety of applications. Most work on entrainment in vortex rings to date has considered formed or steady vortex rings. In this state, a closed streamline encircles the vortex ring in a frame of reference moving at the vortex ring velocity. Thus, the fluid transported within the ring in the vortex bubble is clearly defined as described by Shariff and Leonard (99). Maxworthy (97) conducted experiments using dye visualization and hydrogen bubble techniques to study the diffusion of vorticity, which results in entrainment after the vortex bubble is formed. He concluded that while some of the vorticity diffuses by pulling irrotational fluid inside the vortex bubble making vortex bubble larger in size, some stays behind the traveling vortex ring forming a wake. Maxworthy (977) also performed some experiments to study the vortex ring formation as well as evolution of the 3
2 36 Exp Fluids (8) 44:35 47 vortex rings. He noted the effect of Reynolds number on the formation process, but he did not comment about its effect on entrainment. Fabris and Liepmann (997) analyzed vortex ring formation, and they concluded that there are three distinct regions in a steady vortex ring, namely the core region where rotational flow is present, an intermediate region where irrotational flow exists in the form of ejected and entrained fluid, and finally an external region where potential flow encloses the moving vortex bubble. Dabiri and Gharib (4) investigated entrainment using steady bulk counterflow to hold the rings in the field of view and streamlines obtained from DPIV to identify the bubble volume. They observed the rings contained up to 65% entrained fluid volume when they were completely formed. It was also observed that entrainment fraction could be increased by using a smaller stroke ratio. A diffusive fluid entrainment model was developed by relating the ratio of entrained fluid flux (i.e., time rate of change of entrained fluid volume) to the total fluid flux in the dissipation region behind the formed ring with the ratio of ambient fluid energy loss rate by viscous dissipation to ambient fluid energy entering the dissipation region. These ratios where expressed in terms of the vortex ring s governing parameters (e.g., velocity, volume of the vortex ring, and diameter of vortex ring generator). For formed rings experiencing periodic forcing, the unstable manifold of the forward stagnation point no longer coincides with the stable manifold of the rear stagnation point and intersections between these manifolds identify lobes of fluid which can cross the ring boundary (entrainment or detrainment) as the flow evolves. Shariff et al. (6) studied this process for numerically generated, timeperiodic vortex rings using dynamical systems theory. By monitoring evolution of the lobes and changing oscillation amplitude of the periodic disturbance, they showed that the exchanged volume can be increased when a higher oscillation amplitude is used. They also noted the quantitative similarity between their results and detrainment from experimentally generated turbulent rings. Shadden et al. (6) studied empirically generated vortex rings and observed entrainment and detrainment by lobe dynamics for nominally steady (i.e., quasisteady), aperiodic flows. In this case, the stable and unstable manifolds delineating the vortex boundary were identified as ridges in the finitetime Lyapunov exponent (FTLE) field obtained from digital particle image velocimetry (DPIV) data of the velocity field. They also computed the bubble volume identified by the ridges [called Lagrangian coherent structures (LCSs)] and compared it with that determined from streamlines. Although all of these studies highlight various features of entrainment during the steady (or quasisteady) phase of laminar vortex ring motion when a closed vortex bubble has formed, none of them address the process of entrainment during initial ring formation and rollup. Yet, the bulk of the entrained fluid in a steady vortex ring is acquired during the formation process (Dabiri and Gharib 4). Indeed, Auerbach (99) made the distinction between convective entrainment during shear layer rollup phase and diffusive entrainment after the vortex ring is formed. While he was unable to provide quantitative measure of ambient fluid entrainment during vortex ring formation, he concluded that depending on the formation details as much as 4% of the fluid carried with a steady ring can be ambient fluid. The objective of this study is to measure ambient fluid entrainment during laminar vortex ring formation, and evaluate the effect of vortex ring formation parameters on the entrainment process. Since there is no closed volume associated with the vortex during the formation process (i.e., prior to achieving a nearly constant translational velocity), we instead focus on the rate at which fluid is entrained into the forming vortex spiral. Explicit identification of the entrance to the spiral will be given in Sect The experimental observations are made using planar laser induced fluorescence (PLIF) and DPIV. Using dye as a Lagrangian marker of the ejected fluid, PLIF allows direct observation of the vortex spiral during the formation process. DPIV combined with LCS techniques gives analogous results, but also includes velocity information allowing a more detailed analysis. Both data sets (PLIF and DPIV) can be used to deduce the size of the vortex bubble once formed, indicating the overall entrainment. Using these techniques, the present investigation studies the evolution of entrainment during ring formation under the influence of different jet velocity programs and ejected jet lengthtodiameter ratios (L/D). Experimental setup and techniques A schematic of the experimental apparatus is given in Fig.. The experimental apparatus consisted of a piston cylinder mechanism for generating the vortex rings, a water tank, and a pressurized tank to drive the piston. The water tank was 6 cm wide, 6 cm deep and 44 cm long. The walls of the tank were.7 cm thick glass for flow visualization purposes. The piston and cylinder of the vortex ring generator were made from highimpact strength PVC rod and clear PVC schedule4 pipe, respectively. The cylinder s inner diameter (D) was 3.73 cm. A critical parameter for vortex ring formation was the lengthtodiameter ratio (L/D), defined as the ratio of the total piston displacement (during jet ejection) to the piston diameter. The outer surface of cylinder nozzle was machined to have a wedge with a tip angle (a) of7 to ensure clean flow 3
3 Exp Fluids (8) 44: separation at the nozzle exit plane. The piston cylinder mechanism was connected to a 3gallon tank which was pressurized to 5 psig (3 kpa gage) to actuate the piston during measurements. A proportional solenoid valve (SD8, ASCO Valve Inc.) and an inline ultrasonic flow rate probe (ME9PXN, Transonic Systems Inc.) were used to control and measure the volumetric flow rate, respectively. The piston velocity was determined from the ratio of volumetric flow rate to the piston area. An inhousecode programmed in Labview (National Instruments) provided feedback control of the flow rate allowing the piston to follow an arbitrary velocity program. To generate the vortex rings, the piston was commanded to execute finite duration jet pulses. Three different jet velocity programs were considered: trapezoidal, triangular positive sloping (PS), and triangular negative sloping (NS). Examples of all three are shown in Fig.. Jet Reynolds number (Re J ) is calculated based on the piston s maximum velocity (U M ), piston diameter (D) and the fluid viscosity (m), namely, Re J ¼ U MD : ðþ m In Fig. solid lines show the commanded velocity, and hollow triangles indicate the measured piston velocity. As seen from the plots, there is a very good agreement between commanded and measured velocities. Repeatability of velocity programs was better than % in Re J and 5% in L/D. While acceleration and deceleration periods were chosen to last.t p (t p being the pulse duration) for a trapezoidal velocity program, triangular PS and triangular NS velocity programs commanded the piston with.9t p and.t p in the acceleration phase,.t p and.9t p during the deceleration phase, respectively. The triangular PS and NS cases introduce the effects of nonimpulsive jet initiation and termination, respectively. In order to study the effects of piston velocity program and L/D ratio, a number of experiments were performed. A summary of tests used in this study along with the utilized techniques are given in Table. For PLIF measurements a cover plate similar to that used by Johari (995) was initially placed over the end of the cylinder (see Fig. ). A hole in the plate allowed dyed fluid to be drawn into the cylinder while the plate was in place ensuring the fluid in the cylinder and the ambient fluid were initially separate. Just before the test began the cover plate was drawn up vertically. Velocity of the cover plate was.49 ±. cm/s. This velocity was small enough to cause minimal dye disturbance prior to tests. When the plate was clear, the piston was actuated. Using this procedure, only dyed fluid was in the cylinder when the jet was initiated and therefore the dye may be considered as a Lagrangian marker for the ejected fluid. Fluorescein dye at M was used as the fluid marker. The Schmidt number for fluorescein was about, as suggested by Green (995), so the dye tracked the fluid motion, but not vorticity diffusion. An Argon ion laser (Innova 7, Coherent Inc.) was used to illuminate the dye. A.5 cm thick laser sheet was obtained using a cylindrical lens. A black and white 8bit digital CCD camera (UP83, Uniq Vision Inc.) was placed perpendicular to the laser sheet to record the flow evolution at 3 Hz. The recorded field had a,4,4 pixels spatial resolution with a 4.65D 4.65D field of view. Digital particle image velocimetry (DPIV) was used to obtain velocity field data for the vortex ring formation process. The DPIV system consisted of a pair of frequency doubled, pulsed Nd:YAG lasers (Vlite, LABest Inc.), Fig. Schematic of experimental apparatus Pressurized air Pressure regulator Pressurized tank Proportional solenoid valve Flowrate meter Cover plate and lifting mechanism Laser sheet Water tank Water Piston α U p (t) D Feedback control through Labview Vortex ring generator 3
4 38 Exp Fluids (8) 44:35 47 Fig. Typical piston velocity programs for Re J =,, L/D =.. a Trapezoidal velocity program (t P =.77 s), b triangular positive sloping (t P = 4.98 s), and c triangular negative sloping (t P = 4.98 s) velocity programs (a) U p (t) / U M (b). U p (t) / U M Commanded velocity Measured velocity. Commanded velocity Measured velocity (c). t/t p t/t p.8 U p (t) / U M.6.4. Commanded velocity Measured velocity t/t p Table Table of the tested cases L/D Re J Velocity program Flow analysis technique.5,., Trapezoidal PLIF., Trapezoidal PLIF, Streamline, and LCS.5,., Triangular NS PLIF and LCS., Triangular NS PLIF and LCS.5,., Triangular PS PLIF., Triangular PS PLIF and LCS optics to transform the laser beam into a. cm thick laser sheet, and a delay generator (555 Pulse Delay Generator, Berkeley Nucleonics Corporation) for synchronizing the laser and the camera. The particles used to seed the flow were 5 lm diameter neutrally buoyant silvercoated hollow glass spheres (SH4S, Potters Industries Inc.). The obtained,4,4 pixels particle images were recorded, paired and processed using Pixel Flow (FG Group LLC.), which uses a crosscorrelation algorithm similar to the one described by Willert and Gharib (99). A laser pulse separation of.7 ms was used, giving maximum particle displacements of 7 8 pixels for a 3.D 3.D field of view. With 3 3 pixel interrogation windows at 5% (6 pixels) overlap, the spatial resolution of the resulting vector fields was.94d.94d. To improve the accuracy, the data were processed a second time with a windowshifting algorithm as described by Westerwheel et al. (997). The uncertainty of velocity measurements was.4 pixels as stated by Westerwheel et al. (997), which was less than % for the majority of the measured velocity field. The DPIV data were also used to obtain Lagrangian information about the flow as expressed using the finite time Lyapunov exponent (FTLE). Details of this approach may be found in Shadden et al. (5); however, a brief overview will be presented here for completeness. The equation describing the trajectory of a fluid particle at position x at time t may be expressed as _xðt; t ; x Þ¼Vðxðt; t ; x Þ; tþ ðþ where xðt ; t ; x Þ¼x : The right side of Eq. () can be attained from the DPIV velocity field data. The solution to () is a flow map ð/ t þt t ðx ÞÞ describing the position at time t = t + T of the fluid particle initially at x at time t namely; / t þt t ðx Þ¼xðt þ T; t ; x Þ: ð3þ Then the finite time Lyapunov exponent (FTLE) is defined as 3
5 Exp Fluids (8) 44: Fig. 3 PLIF flow visualization vortex rings generated by trapezoidal, triangular NS and PS velocity programs for Re J =, and L/D =.. Gray and black pixels represent the ejected and ambient fluid, respectively Trapezoidal Velocity p r o g ra m t * =.5 t * =. t * =.5 t * =. Triangle (NS) V elocit y P r o g ra m t * =.5 t * =. t * =.5 t * =. Triangle (PS) V elocit y P r o g ra m t * =.5 t * =. t * =.5 t * =. r T t ðxþ jtj ln pffiffiffiffiffiffiffiffiffi k max where k max is the maximum eigenvalue of r/ t þt t ðxþ r/ t þt t ðxþ ð4þ ð5þ and ()* denotes the adjoint operation. It can be shown (Shadden 5) that the separation of particles advected by the flow is proportional to e rt t ðxþjt j to highest order. Hence, the FTLE is roughly a measure of the maximum expansion rate of particle pairs advected by the flow. Lagrangian coherent structures (LCS) are defined as the ridges in the FTLE field. Shadden et al. (5) show that the flux across a LCS scales like jtj and thus, for large T a LCS can be treated as a material line or transport barrier in the flow. Additionally, LCS obtained by forward (T [ ) and backward (T \ ) time integration recovers the stable (also called repelling LCS) and unstable manifolds (also called attracting LCS), respectively, surrounding a vortex ring. Since LCSs behave like material lines, they can identify the vortex bubble boundaries. This was illustrated by Shadden et al. (6), who combined the attracting and repelling LCS to study entrainment of a formed ring. Since the formulation applies equally well to unsteady flows, it can be used to identify the vortex boundary during ring formation and hence, is a useful tool for studying entrainment during this phase of ring evolution as well. LCSs in this study were calculated by using a software package called ManGen developed by Francois Lekien and Chad Coulliette in (http://www.lekien.com/* francois/software/mangen/). ManGen provided the FTLE field by computing Eq. (4) for a grid of massless particles placed in the domain and advected using the given velocity field. For the present investigation, computation of r was performed with a uniform grid of.67d resolution to produce clear ridges for both attracting and repelling LCS. 3 Results 3. Qualitative observations Figure 3 illustrates vortex ring formation from a trapezoidal, triangular NS and PS velocity programs for Re J of, and L/D =.. All the rings in Fig. 3 travel from left to right, and t* in the figures is defined as t t p : In these figures, gray pixels represent the ejected fluid coming from inside the cylinder, and black pixels represent ambient fluid which was initially outside the nozzle. During jet ejection ( t* ), entrainment is apparent through the growing black spiral, but the volume of entrained fluid is clearly much less than the ejected fluid. Comparison of the spiral formation among the velocity programs shows distinct differences. Since trapezoidal and triangular NS velocity programs have a similar start up acceleration, both produce a long tightly wound spiral during the initial jet ejection. In contrast, the initial spiral for the PS case is not wound tightly because the slow jet initiation provides much less vorticity initially yielding fewer spiral loops, and the width of each loop is larger. In general, trapezoidal and triangular NS velocity programs have steeper start up accelerations than the triangular PS velocity programs. This steep start up acceleration is responsible for high velocity gradients at the inner nozzle wall, which produce stronger vorticity at start up. The 3
6 4 Exp Fluids (8) 44:35 47 (a) r / D x / D ω (/s) (b) r / D x / D ω (/s) (c) r / D x / D ω (/s) Fig. 4 Contour plots of vorticity with the stagnation streamline indicated by dashed lines. a c are at t* =.7,.8, and.8, respectively result is strong BiotSavart induction, a tightly wound spiral and, as will be shown later, high ambient fluid entrainment. Additionally, while the vortex rings obtained with trapezoidal and triangular NS velocity programs leave behind a noticeable quantity of ejected fluid, rings generated by triangular PS velocity program pull in nearly all the ejected fluid. This results in proportionally more ambient fluid within the ring for the former case. Finally, it is noted that most of the entrainment occurs after the piston has stopped (see Fig. 3 for t*[.). Once the piston stops, the ejected fluid boundary is no longer held out by the jet and it may contract under the influence of the ring vorticity, making a larger area available for entrainment of ambient fluid. The vorticity created during shear layer roll up drives the ambient fluid entrainment not only during formation phase but also after piston has stopped. These observations agree with Didden s (979) results. Qualitative observations of the ring formation are also obtained using DPIV by computing the streamfunction in a frame of reference moving with the vortex ring. The velocity field is converted to this reference frame by subtracting the ring velocity based on the position of the peak ring vorticity. To obtain accurate ring velocity measurements, a Gaussian fit of the vortex core is used to obtain subgrid estimates of vortex location and a third order polynomial fit of the results is used for computing velocity. Then, Stoke s streamfunction is obtained by solving the governing equation r o W ox þ o or r ow or ¼ x h ð6þ with a second order accurate finite difference method using the vorticity field obtained from DPIV data and the velocity data as boundary conditions. Figure 4 displays DPIV measurements of vortex ring evolution for Re J =, and L/D =.. The dashed line in Fig. 4 represents the stagnation streamline which identifies the boundary of the vortex bubble. Figure 4a shows that, in the field of view, the W = streamline does not reach r = on the backside of the ring while the jet is on. Although data was collected only for x [, we can conclude that the W = streamline does not reach r = for x \ while the jet is on since instantaneous streamlines cannot intersect. Thus, mass is still entering the ring in Fig. 4a. The W = streamline may, however, close on the outer annulus of the nozzle. Once the piston has stopped moving, a stagnation point develops behind the ring, forming a closed bubble as shown in Fig. 4b, c. The vorticity distribution is also given in Fig. 4. During jet ejection the vortex core is small and moves slightly outward, above the piston radius (i.e.,.5d). As the ring continues to form (Fig. 4b), the bounding streamline obtains foreaft symmetry. During further evolution of the vortex ring, vorticity starts to diffuse out of the vortex bubble (Fig. 4c) causing enlargement of the vortex bubble volume as mentioned by Maxworthy (97). The movement of the vorticity core during ring formation determines the area where induced ambient fluid entrainment takes place. Once the DPIV velocity vector fields are obtained from experiments, FTLE fields are calculated using ManGen (see Fig. 5a). It is noted that high FTLE values (i.e., ridges) illustrate the LCS (also called attracting LCS since T \ ) as stated by Shadden et al. (6). During postprocessing, a threshold is applied to FTLE fields to identify the LCS and locate Dr to be used for volume and entrainment calculations, respectively (see Fig. 5b). This is discussed further in Sect The time evolution of the LCS for a trapezoidal velocity program for Re J =, and L/D =. is shown in Fig. 6. The repelling and attracting LCSs in Fig. 6a c were obtained with forward integration (i.e., T [ ) and backward integration (i.e., T \ ), respectively. Combining these repelling and attracting LCSs generate a closed transport barrier defining the vortex bubble as described by Shadden et al. (6). The extended back side can be observed in Fig. 6a, b representing the mass that will comprise the final vortex bubble. This unique property of LCS determines the volume associated with the vortex ring 3
7 Exp Fluids (8) 44: Fig. 5 a shows the color contour plots of FTLE field at T = 5. s( T /t p =.8); b illustrates the LCS after thresholding. It is noted that ambient fluid entrainment occurs through Dr r / D.5.75 (a) x / D ω (/s) r / D Primary vortex (b)  Stopping vortex x / D ω (/s) r / D.5.75 (c) x / D ω (/s) Fig. 6 Vortex evolution observed from LCS. The solid line is the repelling LCS and the dashed line is the attracting LCS. a c illustrates evolution of the vortex bubble at t* of.7 (T =.99 s),.8 (T =.99 s) and.8 (T = 5. s), respectively before it is completely formed. It is also noted that the LCS in Fig. 6b delineates the primary vortex and the stopping vortex as observed by Didden (979). Since the stopping vortex does not enter into the forming ring, the LCS verifies this flow is outside of the vortex bubble. Lastly, Fig. 6c illustrates the LCS after the ring is completely formed. The LCS representing boundary of the vortex bubble demonstrates near foreaft symmetry once ring formation is complete (in agreement with Fig. 4c). 3. Entrainment for a trapezoidal velocity program For a quantitative assessment of the entrainment, we first focus on the trapezoidal velocity program as a canonical case frequently studied in the literature. Figure 7 illustrates PLIF of a vortex ring just after formation is complete for Re J =, and L/D =.. To find the boundary of the vortex bubble ðoxþ from such data, an edge detection algorithm was applied to the images after thresholding. While the front edge of the moving vortex bubble can be clearly observed with the PLIF technique, the rear edge of the vortex bubble cannot be identified by this technique. Therefore, frontback symmetry is assumed using the upper and lower boundary of the vortex bubble to define the mirror axis as indicated in Fig. 7. As noted earlier, this assumption applies well for a formed ring (Fig. 4b, c). Integrating over the hatched region in Fig. 7 and assuming axisymmetry gives the vortex bubble volume (V B ). Although the uncertainty of the volume calculation depends on the location of mirror axis, threshold value for edge detection, and image quality, uncertainty analysis shows that overall uncertainty in V B falls below 7%. Once the vortex bubble boundary is determined, the volume of ejected fluid within this volume (V EJ ) is calculated by integrating the volume of the gray pixels within ox (again assuming axisymmetry). This approach accurately obtains the amount of ejected fluid which remains in the vortex bubble at the end of a piston stroke since it does not consider ejected fluid not entrained into the vortex 3
8 4 Exp Fluids (8) 44: V B /V EJ.8 Fig. 7 The components of a moving vortex bubble obtained by PLIF for Re J =, and L/D = Volume by PLIF Volume by streamlines Volume by LCS Ejected Volumeinring (PLIF) A Mirror axis t / t P Fig. 9 Vortex bubble volume calculation using PLIF, DPIV and LCS data (Re J =,, L/D =.) B Fig. 8 Illustration of the rear edge obtained in PLIF images assuming foreaft symmetry bubble (see PLIF data for trapezoidal and triangular NS when t*.5 in Fig. 3). Therefore, once the ejected fluid in the vortex bubble is known, volume of entrained ambient fluid (V E ) can be calculated as V E ðtþ ¼V B ðtþ V EJ ðtþ: ð7þ Dye diffusion in the vortex core, however, leads to ambiguity in identification of ejected versus entrained fluid in the core. The related uncertainty in V EJ is less than 4.% for t*.5. The preceding analysis employing foreaft symmetry to identify the boundary of a formed ring may be readily extended to a forming ring. During jet ejection, foreaft symmetry is clearly violated near the centerline because there is no stagnation point on the back side of the ring, but near the forming spiral such symmetry holds approximately. This is illustrated in Fig. 8 where the boundary obtained from the front of the ring is reflected about the mirror axis to compare with the back side of the ring. The lack of symmetry on the back side near the axis is of no consequence for computing V E, however, since the ejected fluid in this region is subtracted out. Thus, V E (t) computed with this procedure, in essence, identifies the fluid entrained into the spiral across the arc AB in Fig. 8. Specifically, dv E dt computed from these results should give an accurate measurement of the rate at which ambient fluid is entrained into the ring during formation. This statement will be justified experimentally in Sect Results of the PLIF volume calculations for Re J =,, L/D =., and a trapezoidal velocity program are shown in Fig. 9. Volume by PLIF refers the volume calculation performed on hatched region given in Fig. 7, and ejected volume in the ring is the volume of gray pixels (i.e., ejected fluid) in the hatched region of Fig. 7. The results show that the computed V B and V EJ increase at nearly identical rates during formation so that V E is small during this phase. Once the piston stops, ejected fluid entry slows dramatically since any ejected fluid left at the vicinity of the nozzle does not have enough momentum to catch the vortex bubble. Nevertheless, the vortex bubble volume continues to increase after the jet stops until it reaches an asymptotic value of V B /V EJ =.5. This final increase in the volume is mostly due to the ambient fluid entrainment. Therefore, the vortex ring velocity (see Fig. ) decelerates in this region (between t* of. and.5) since momentum initially supplied by the piston needs to be shared with this additional mass, namely the entrained ambient fluid. This indicates most of the ambient fluid is entrained during the impulsepreserving phase of motion after the jet stops (as observed qualitatively in Sect. 3.). The PLIF data shows a nearly constant bubble volume after ring formation is complete, verifying that the ring is formed since its shape and volume remain unchanged. In reality, the vortex bubble continues to increase slowly due 3
9 Exp Fluids (8) 44: W r /U M bubble volume for.7 t*.4, even though the PLIF data indicate the ring is not formed until t* [.5. This is because the nature of the LCS as transport barriers allows them to identify the fluid volume eventually to appear in the ellipsoidal volume of the formed vortex ring bubble, even before the bubble is formed (see Fig. 6a, b). The LCS data for t* [.4 illustrates an increase in vortex bubble volume in agreement with the streamfunction volume calculation to within experimental uncertainty. As with the streamfunction data, the LCS data agree with the volume obtained from the PLIF results to within experimental uncertainty, confirming the validity of the foreaft symmetry assumption used in computing the bubble volume from the PLIF images, at least in the case of a completely formed ring. t/t P Fig. Vortex ring velocity obtained from polynomial fit of vortex peak locations (Re J =,, L/D =.) to the vorticity diffusion as mentioned by Dabiri and Gharib (4) and Maxworthy (97). This is not apparent in the PLIF data due to slow dye diffusion (high Schmidt number). As a consequence, the PLIF data indicates volume obtained during formation only and not as a result of subsequent vorticity diffusion. The streamfunction volume calculation is also given in Fig. 9 and verifies the volume increase due to vorticity diffusion. Streamfunction volume calculations are only obtained after a vortex bubble is formed so that a closed streamline (see Fig. 4b, c) is obtained. Using the stagnation streamline defining the vortex bubble as shown in Fig. 4, V B can also be computed once the ring is formed. As before, axisymmetry is assumed in the volume calculation. The major source of uncertainty comes from the vortex ring velocity calculation and is less than %. The streamfuction volume calculation agrees with the PLIF results to within the experimental error just after ring formation is complete (.35 t*.). As time proceeds, the vortex bubble volume obtained from the streamfunction starts to increase, reflecting the effect of vorticity diffusion as the ring advects downstream. Since vortex bubble volume increases by this additional mass, we can see that vortex ring slows down after t* of.75 (see Fig. ). The filled diamonds in Fig. 9 represent the vortex bubble volume calculation from the LCS technique. These values were obtained by integrating the volume inside the LCS boundaries (see Fig. 6) assuming axisymmetry. The uncertainty in these volume calculations comes mainly from threshold values applied for repelling and attracting LCSs and is less than 3%. The LCS data indicate a constant 3.3 Entrainment rate The rate of fluid entrainment into the vortex ring (Q E )is defined as Q E dv E ð8þ dt where V E is the volume of ambient fluid in the vortex ring spiral. This can be estimated from the PLIF data, but the PLIF data relies on the assumption of frontback symmetry during ring formation. The LCS data, on the other hand, can obtain Q E directly. Specifically, we consider the volume flow rate into the entrance gap of the vortex spiral as identified by the attracting LCS. For convenience, we consider the gap Dr = r o r i identified along the line connecting vortex cores as shown in Fig. a. This is analogous to entrance of the vortex spiral identified in the PLIF data as can be seen by comparing Fig. a, b. Although taken from different runs, the location and the magnitude of Dr for the LCS and PLIF data show reasonable agreement. In particular, the entrance surface area (p(r o r i )) in LCS and PLIF is calculated to be 3. and 3.75 cm, respectively. The percent difference is comparable to the repeatability of Reynolds number and L/D. Both Shariff et al. (6) and Shadden et al. (6) used attracting and repelling LCSs for an already formed vortex ring to investigate the entrainment/detrainment between irrotational fluid outside the vortex ring and the fluid in the vortex ring. Here, however, we are interested in the flow of fluid into the developing spiral. While the attracting LCS identifies the spiral (i.e., unstable manifold which separates ejected fluid from entrained fluid in the vortex ring), the repelling LCS does not. Thus, we only use the attracting LCS in this analysis. With the spiral entrance identified, Q E is computed as 3
10 44 Exp Fluids (8) 44:35 47 Fig. Entrance surface area through which ambient fluid is entrained is shown for both LCS data in a with T = 5. s ( T /t p =.8) and PLIF data in b with t* =.8. Reference horizontal lines based on the LCS data are given for comparison purposes Q E ðtþ ¼p Z r o r i ðuðrþ W r Þrdr ð9þ where u(r) is obtained from DPIV and W r is the ring velocity. Although Dr considered here is different from the spiral entrance identified in the PLIF results (AB in Fig. 8), that should not affect the comparison of Q E between the methods since the fluid entering AB also passes through Dr. The comparison of Q E obtained from LCS/DPIV and PLIF is given in Fig.. The average slope obtained from PLIF data for t*.5 and t*. demonstrates reasonable agreement with LCS/DPIV results for the same interval. However, PLIF results during.5 \ t* \. were not able to capture gradual increase of entrainment obtained from LCS/DPIV. The disagreement is a combination of the fact that the PLIF result is an average value and uncertainty in V E obtained by PLIF due to dye diffusion in the vortex core. Nevertheless, the simplistic approach used in PLIF results during ring formation give good quantitative measurements of entrainment rates during and after jet ejection, even though the transition between the two rates appears more abrupt in the PLIF data than is actually indicated by the LCS/DPIV data. The high entrainment rate after the jet stops agrees with the previous observation that most of the ambient fluid is entrained during the impulse preserving phase of motion. As indicated in Fig. 3, a key factor in the increased Q E is the dramatic increase in Dr after jet termination. From Q E / (.5pD P UM ) t / t p Fig. Entrainment rate comparison between PLIF and LCS/DPIV results (Re J =, and L/D =.) Eq. (9), however, the flow velocity plays a role as well. In particular, note that timeaccurate LCS results show a descending trend for.9 t*.5. The peak near t* =.9 is reasonable since piston velocity program starts to slow down after this point. This causes reduction in velocity, but the Dr is almost constant between t* of and.5. Consequently, the entrainment rate starts to diminish until Dr starts to rise after t* of.5. This is observed as the LCS entrainment rate starts to increase again at late time (i.e., t* [.5). 3
11 Exp Fluids (8) 44: Dr /D. Γ/Γ Μ t/t P.4. Trapezoidal Triangle (NS) Triangle (PS) Fig. 3 Variation of Dr D for Re J =, and L/D =. (The uncertainty of Dr D is less than 5%) Q E / (.5pD P UM ) t/t P 3.4 Quantitative comparison between trapezoidal and triangular velocity programs Trapezoidal Triangular NS Triangular PS Fig. 4 PLIF entrainment rate (calculated from the average slope of PLIF entrained volume data) for trapezoidal, triangular NS and PS velocity programs at Re J =, and L/D =. The quantitative validity of the PLIF results having been confirmed, only PLIF entrainment results will be used here for brevity. The effect of velocity program on entrainment rate obtained from PLIF data can be seen in Fig. 4. Since trapezoidal and triangular NS velocity programs exhibit similar acceleration behavior at the start up, nearly the same entrainment rates are obtained during jet initiation ( t*.). Similarly, during the momentum conserving period (. t*.5), near identical entrainment rates are calculated for trapezoidal and triangle PS velocity t/t p Fig. 5 Vortex ring circulation comparison for trapezoidal, triangular NS, and PS velocity programs for Re J =, and L/D =.. Circulation is calculated from C ¼ R R x h dxdr and C M refers the maximum circulation obtained from trapezoidal velocity program. Uncertainty of the circulation calculations is less than % of the maximum circulation programs since these programs behave similarly at the deceleration phase. For rapidly accelerating velocity programs, both the vorticity in the forming spiral and the entrainment area are key parameters. First, Fig. 5 shows that the fast acceleration cases (i.e., trapezoidal and triangular NS) have higher circulation during jet ejection, which produces a higher entrainment velocity (due to the BiotSavart induction) compared to the triangular PS velocity program. Second, since rapidly accelerated velocity programs cause tighter spirals for t*., a larger Dr is available for entrainment as shown in Fig. 6. For rapidly decelerating velocity programs, on the other hand, the primary effect is on the area through which fluid is entrained (Dr), as determined by the effect of the stopping vortex. Figure 6 shows that Dr rapidly increases for both the trapezoidal and triangular PS cases following jet termination, arriving at a final value of nearly twice that of the triangular NS case. This is contrasted with the total circulation, C M, which differs by only about 3% between these three cases. Finally, to understand the effect of L/D on total entrained fluid volume, the entrainment fraction ði.e., g ent V E V B Þ is plotted in Fig. 7 below. When Re J is fixed, the piston is required to reach the commanded velocity in a shorter time as L/D is decreased. This results in a higher g ent as L/D is decreased for all the velocity programs given in Fig. 7 as a more compact vortex core is generated at initiation for higher initial jet acceleration. On 3
12 46 Exp Fluids (8) 44: Triangular NS Triangular PS.6.5 Dr /D Dr F /D the other hand, the proximity of the ring to the nozzle when the jet terminates and the strength of the stopping vortex determine the final size of Dr (Dr F ) following jet termination. When L/D is decreased causing a stronger stopping vortex, Dr F /D increases as illustrated in Fig. 8. This also contributes to the larger g ent at small L/D. 4 Conclusions t/t P Fig. 6 Dr comparison between triangular NS and PS velocity programs for Re J =, and L/D =.. The dashed line shows the Dr D obtained from trapezoidal velocity program as a reference η ent L/D Trapezoidal Triangle (NS) Triangle (PS) Fig. 7 Entrainment fraction ði.e.; g ent VE V B Þ comparison for vortex rings at Re J =, Ambient fluid entrainment during vortex ring formation due to BiotSavart induction was investigated from a L/D Fig. 8 Dr F /D variation with respect to L/D for Re J =, and the Triangular NS velocity program piston cylinder mechanism. PLIF and DPIV combined with LCS methods were utilized first to identify the vortex bubble and then to compute the ambient fluid entrainment. The piston velocity programs and L/D ratio were changed in order to study the effects of these parameters on entrainment in the forming vortex ring. PLIF method gives entrainment during ring formation using the assumption of the foreaft symmetry. This assumption is justified for a formed ring. During ring formation it is also justified for calculation of V E due to approximate foreaft symmetry of the vortex spiral. The DPIV streamline method accurately provides the vortex bubble shape via the stagnation streamline once the vortex is formed. Indeed, the streamfunction shows bubble growth by diffusion; however, it can only be used after the ring is formed. The DPIV/LCS method provides more detailed information than either method, but requires significantly more data processing. The LCS results confirmed the conclusions drawn from PLIF using the assumption of foreart symmetry. Studying the PLIF results in detail revealed several key factors affecting entrainment during vortex ring formation. First, the effect of the velocity program on entrainment rate is determined primarily by the magnitude of jet acceleration during initiation and termination as shown in Fig. 4. While high initial accelerations such as a trapezoidal or a triangular NS velocity program can enhance the initial entrainment rate by as much as 5% compared to a triangular PS velocity program during jet ejection, high terminal deceleration like a trapezoidal velocity program or a triangular PS velocity program can increase the final entrainment rate up to only % compared to a triangular 3
13 Exp Fluids (8) 44: NS velocity program. Second, L/D has a strong effect on entrainment as well. As L/D is lowered from. to.5, the piston must be accelerated at a higher rate causing higher vorticity in the forming spiral. Also, when L/D is reduced from. to.5, Dr F /D increases by as much as 8%. This is due to the rapid stop of the piston which generates a stronger stopping vortex and creates a larger area for ambient fluid entrainment. The net effect of these trends is an increase of up to 67% in the final entrainment fraction as L/D is reduced from. to.5. These observations provide insight into enhancing ambient fluid entrainment during vortex ring formation. Specifically, a trapezoidal velocity program with low L/D ratio is an ideal candidate since this program benefits from an impulsive jet initiation as well as a rapid jet termination. Acknowledgments This material is based upon work supported by the National Science Foundation under Grant No References Akhmetov DG, Lugovtsov BA, Tarasov VF (98) Extinguishing gas and oil well fires by means of vortex rings. Combus Explos Shock Wave 6: Auerbach D (99) Stirring properties of vortex rings. Phys Fluids A3: Dabiri JO, Gharib M (4) Fluid entrainment by isolated vortex rings. J Fluid Mech 5:3 33 Didden N (979) On the formation of vortex rings: rollingup and production of circulation. Z Angew Math Phys 3: 6 Fabris D, Liepmann D (997) Vortex ring structure at late stages of formation. Phys Fluids 9:8 83 Green S (995) Fluid vortices. Kluwer, Dordrecht Johari H (995) Chemically reactive turbulent vortex rings. Phys Fluids 7:4 47 Kercher DS, Lee JB, Brand O, Allen MG, Glezer A (3) Microjet cooling devices for thermal management of electronics. IEEE Trans Components Packaging Technol 6: Maxworthy T (97) The structure and stability of vortex rings. J Fluid Mech 5(part ):5 3 Maxworthy T (977) Some experimental studies of vortex rings. J Fluid Mech 8(part 3): Shadden SC, Lekien F, Marsden JE (5) Definition and properties of Lagrangian coherent structures from finitetime Lyapunov exponents in twodimensional aperiodic flows. Physica D :7 34 Shadden SC, Dabiri JO, Marsden JE (6) Lagrangian analysis of fluid transport in empirical vortex ring flows. Phys Fluids 8: Shariff K, Leonard A (99) Vortex rings. Annu Rev Fluid Mech 4:35 79 Shariff K, Leonard A, Ferziger JH (6) Dynamical systems analysis of fluid transport in timeperiodic vortex ring flows. Phys Fluids 8: Westerwheel J, Dabiri D, Gharib M (997) The effect of a discrete window offset on the accuracy of crosscorrelation analysis of digital PIV recordings. Exp Fluids 3: 8 Willert CE, Gharib M (99) Digital particle image velocimetry. Exp Fluids :8 93 3
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