Far Wake Rotorcraft Vortex Tumbling

Transcription

1 5th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition 9-12 January 212, Nashville, Tennessee AIAA Far Wake Rotorcraft Vortex Tumbling James H. Stephenson, Swathi M. Mula, Charles E. Tinney, Jayant Sirohi Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin, Austin, TX 78712, USA The far wake trajectory of the tumbling tip vortex effect of a reduced-scale, 1 m diameter, four-bladed rotor during hover is studied using vortex methods combined with a center of mass analysis approach. Measurements of all three components of the velocity field are acquired using a stereo PIV system synchronized to capture up to 8 wake age of the vortex with 1 offsets during hover conditions. The nominal operating condition of the rotor is at a rotational speed of 152RPM, corresponding to Re c = 248, with a chord length of 58.5mm. The rotor was operated with a pitch of 7.2 ±.5 and a C T/σ of.29. The far wake vortex tumbling phenomenon is captured and described. It is shown that tip vortices from two blades tumble through approximately 18 of rotation before they coalesce. It is seen that the parent vortices are stronger than the daughter vortex, but due to vortex stretching the daughter vortex has a smaller radius with greater swirl strength. An accurate characterization and prediction of the trajectory of the far wake vortex tumbling can enhance the ability to predict and alleviate the resuspension of particles during brownout as well as provide a database for far wake validation of CFD codes. I. Introduction The aerodynamic wake of a helicopter in hover is highly complex, unsteady, three dimensional and dominated by the shed tip vortices. Flying near the ground or other obstacles, or making flight maneuvers further exacerbate the complexity of the rotor wake. Interactions between the shed tip vortices and the fueslage, empennage or tail rotor adds another level of complexity to the wake structure. As CFD and CAA codes increase in their capability and reliability to handle these circumstances, it becomes necessary for the experimentalist to focus on high-fidelity measurements of fundamental phenomena that have yet to be modeled or resolved. As such, there has recently been a great deal of research focused on the rotorcraft wake and its associated aeroacoustics. Despite recent progress, the dependence of tip vortex evolution on rotor parameters is still poorly understood. Additionally, numerical tools as well as experimental analysis techniques to track the statistical behavior of the rotor wake are currently lacking. The transition from the near-wake to the far-wake region of a rotorcraft is defined by the deterministic pairing of vortices. This region has proven a particular challenge for researchers, as vortex tumbling and coalescence is present and the ability to track a given vortex becomes even more complicated. In order to begin tracking the statistical behavior of vortices in the wake of a rotorcraft, a definition of what constitutes a vortex must first be decided on. Unfortunately, this is still a contentious topic in the field of vortex dynamics and as of yet there has not been an academically rigorous description of a vortex. 1,2,3,4,5,6,7 Starting in the late 7 s and early 8 s researchers began with intuitive definitions of vortices. Yule (1978) postulated that a vortex has a concentrated, continuous, coherent distribution of vorticity which is uniform in the direction of the vorticity vector and which grows in scale by viscous diffusion alone. 1 Lugt (1979) described a vortex as a group of particles rotating about a common axis, 2 while another description was proposed that a vortex is composed of closed or spiraling streamlines. The problem with the closed streamline description, is that it is highly frame dependent and can change under SMART Fellow, AIAA Member Graduate Assistant, AIAA Member Assistant Professor, AIAA Senior Member Assistant Professor, AIAA Member 1 of 11 Copyright 212 by James H Stephenson. Published by the, Inc., with permission.

2 Galilean transformations. Lugt s (1979) definition while somewhat better, suffers from the lack of a rigorous mathematical expression. More recently, researchers have tried to develop ways to identify vortices based on the idea that the developed method must be Galilean invariant. The gradient of velocity has become an important parameter for these methods. Chong, Perry and Cantwell (199) describe a method where complex eigenvalues of u identifies the location of a vortex, while Hunt, Wray and Moin (1988) developed the Q criterion which states that a vortex exists where both the second invariant of the velocity gradient is positive and the local pressure is less than the ambient pressure. 3,4 Jeong and Hussain (1995) proposed the λ 2 method, which identifies a vortex as being located in a region where two negative eigenvalues exist for the sum of S 2 and Ω 2. 6 S and Ω represent the symmetric and antisymmetric parts of u, respectively. By assuming that unsteady irrotational straining and viscous effects can be neglected, this method is equivalent to the Q criterion, but inherently incorporates the pressure minimum criteria into it. Other identification techniques exist which handle more complicated flows in which the previously described techniques do not work. Zhang and Choudhury (26), for instance, presented a new technique based on the eigenvalues of u that works exceptionally well for compressible, variable-density flows in which Q, λ 2, and helicity methods all break down. 7 The helicity method is a technique which is Galilean variant, and is the scalar product of vorticity and velocity. An alternative method was developed by Kolar (27) that does not rely on local parameters (like local velocity gradients), but instead depends on global properties. 5 In doing this, Kolar identified what must be covered by any comprehensive vortex definition/identification technique. Kolar states that a new vortex identification scheme must, at a minimum, contain validity for compressible and variable density flows, avoid subjective thresholds for vortex boundary identification, identify the vortex axis, and it has to be Galilean invariant. Kolar puts forth a triple decomposition method (TDM) in which pure shear motion can be separated from the actual swirling motion caused by a vortex. Kolar s method differs from the typical double decomposition ( u = S + Ω), which comes from the Cauchy-Stokes decomposition theorem, by explicitly including a pure shearing motion. While Kolar s method is able to identify vortices in a near-wall region without identifying vorticity due to shear stress, the ramifications that TDM has to 3D flows has yet to be determined. One thing is certain from the discussion, whole field measurements are required for correct eduction of vortical structures. The single point measurements of Bhagwat, Leishman and others were inherently limited as they were unable to spatially resolve the relevant instantaneous flow field. 8,9,1 This is certainly no fault of the experimentalist, but it has prompted demands for spatially resolved measurement techniques, which have become more sophisticated over the past decade with the advent of commercially available digital Particle Image Velocimetry (PIV) systems. Stereoscopic PIV allows for the instantaneous in-plane vorticity to be captured, thus making it a well suited technique for accurately identifying and characterizing the wake and slipstream generated by rotors. It has been shown in previous studies that the Q criterion and λ 2 criterion are equivalent for model scale helicopters in hover. 11 As such, in the present study, the Q criterion will be used to identify the region the vortex is located in, and a Center of Mass (CoM) approach will be used in conjunction with the Q identification criterion to identify the center of the vortex. Application of this Q/CoM method to problems in rotorcraft was previously described in Mula, et al. (211) and provides a consistent method for identifying vortex centers and thereby allowing for the statistics of vortex jitter to be tracked. In the current study, the far wake vortices are identified and tracked. A description of the vortex strength before and after vortex tumbling and coalescence is provided, as well as a statistical characterization of the vortex jitter. Previous studies have investigated the aperiodicity corrections and statistical characterizations of vortex jitter, but they were focused solely on the near field wake structure, before vortex tumbling commenced. 11,12 II. Experimental Setup The goal of this study is to capture the dynamical characteristics of the far wake of a rotor. A laboratory scale rotor stand was developed, which allows for high-fidelity PIV measurements to be acquired. The current study was conducted at The University of Texas at Austins J.J. Pickle Research Campus (UT-PRC), in a room measuring 6.5 x 8 x 6.5 rotor diameters. With the rotor hub situated 3 rotor diameters above the ground, this provided sufficient volume in which to perform where external surfaces could not sufficiently 2 of 11

3 impact the experiment. A custom fabricated reduced-scale rotor test stand provided the necessary conditions for characterizing the vortex trajectory from a four-bladed rotor at various wake ages and collective pitch angles. At the heart of the test stand is a 9kW electric motor powered by a 1kW (max) Lambda TKE ESS 5-2 programmable DC power supply that outputs up to 5V at 2A. The motor is capable of a maximum rotational speed of 8RPM (133Hz) and a maximum torque of 1Nm. A custom fabricated optical encoder fixed to the motor allows both 1/rev and 6/rev positioning of the rotor to be phase aligned with other laboratory instruments (PIV, DIC, acoustics, etc). This is especially important for developing accurate statistical models of the wake age and far-field / near-field acoustic signatures as a function of rotor azimuth. For the current study, a custom fabricated, four-bladed, fully articulated hub assembly was mounted to the rotor shaft. The rotor was directly driven by the electric motor, sans transmission. Rotor system loads were unavailable for these particular measurements, although total thrust could be estimated from measurements of the inflow through the rotor. In order to capture the dynamical characteristics of the far wake, a 3-component (stereo) Particle Image Velocimetry (PIV) system was used. The PIV system, by LaVision, consists of two 2M pixel (1648x1214 pixel) CCD cameras and a 135mJ/pulse Nd-YAG laser. The cameras have 7.4µm square pixels with a 28mm focal length lens and the f-stop was set at 2.8 for this experiment. The PIV system recorded images at an approximate rate of 4Hz, dependent upon the phase timing as acquired by the once per revolution optical sensor that measures the motor revolutions directly. Interframe timing was set to 11µs, which corresponds to a maximum pixel displacement of 5 pixels, based on a magnification factor of.23. This interframe rate was chosen to accurately resolve more of the slower speed flow located outside of the slipstream. Vector maps were generated by DaVis v7.2 with interrogation windows that started at 32x32 pixels and iteratively reduced to 16x16 pixels with a 5% overlap region. The PIV cameras themselves were mounted on an 8/2 rail system with linear bearings that allowed for the calibration in one plane and translation to another. The cameras were calibrated in the near-field plane of the rotor and than subsequently lowered to the far-field recording plane. The cameras and recording planes are shown in figure 1, and the cameras were oriented in such a way that radial (r) and vertical (y) velocity gradients are measured, while out of plane gradients are not. 25 PIV images at every 1 of wake age (ψ) were acquired in both the near field and far field locations, while only the far field data will be discussed here. The total data set comprises 18 independent PIV images capturing the first 8 of wake age. This study used blades comprised of carbon fiber with a foam core, that had a NACA 12 profile with constant chord length of 58.5mm and no twist. Each blade was manufactured with square tips and had all surfaces painted with a matte-black finish (to reduce laser blooming effects). The rotor blades did not possess any artificial boundary layer tripping devices (sandpaper, carborundum) and were balanced to within 1% of the blade mass. The custom fabricated hub is fully articulated with flap, pitch, and lead-lag hinges. In order to accurately track the rotor blades, the lead-lag positions of the blades were fixed (rigid) and adjusted until the blades were tracked within 5% of the blade chord, with blade one being set slightly behind the others to enforce vortex tumbling. The blades and their associated vortices are named in the order they pass through the laser sheet. Blade one passes through at ψ =, with blade two coming 9 later, and so forth. For this study, the wake age refers to the azimuthal angle as measured after blade one passes through the laser sheet the first time, while the vortex age corresponds to how far the blade that spawned that vortex has traversed after it passes through the laser sheet on its first time. For instance, at a wake age of 1, vortex one will be 1 old, while vortex two is only 1 old. The diameter of the entire assembly including blades, rotor hub and blade grips is 1.1m with the blades being tracked using a two and four per revolution stroboscope. For the current experiment, a fixed collective pitch angle of 7.2 (±.5 per blade) was investigated at a rotor speed of ω = 25.33Hz (152 RPM). Given the diameter of the rotor assembly, this results in a rotor tip speed of 8.33m/s (Re c = 248,, M tip =.24). The parameters of the rotor and the test condition are given in Table 1. 3 of 11

4 Figure 1. Experimental setup with the near- (1) and far-field (2) PIV windows shown. III. Vortex Identification Methods Many vortex identification schemes have been proposed and can be shown to reduce to almost identical results for 3-component PIV measurements on model scale rotorcraft in hover. 11 As such, the Q/CoM method described by Mula et al (211) and mentioned previously is used to identify the vortex as well as its center. The Q criterion represents the balance of the rotation rate and strain rate in all directions and is described in equation 1. Q = 1 2 (u2 i,i u i,j u j,i ) = 1 2 (u i,ju j,i ) (1) = 1 2 ( Ω 2 S 2 ) ( ) y Q(y,r) r Q(y,r) (Y c,r c ) =, Q(y,r) Q(y,r) (2) In equation 1, u i,j represents the partial derivative of the i-th component of velocity with respect to the j-th direction. The center of mass technique is then applied to the Q contours via equation 2, where the vortex center is given by (Y c,r c ). This can be automated to find all vortices in a given PIV image by averaging around each peak in the Q contour levels that exist above a predetermined threshold value. Figure 2 demonstrates application of the Q/CoM technique to the far wake region of this rotor, at ψ = 39. Within the figure, the vortices are clearly identified by the Q-contour and the vortex centers have been marked. The PIV images of the near- and far-field wake region were overlapped by identifying the shared vortex core, and translating the far-field vector maps in such a way that the ensemble averaged vortex cores from the common vortex was collocated. Figure 3 shows the ensemble averaged vector fields corresponding to a 21 wake age (3 age for the top vortex), with the overlap region corresponding to a y/r of.15 to.25 resulting in a 2% overlap for the wholefield view. It should be noted that due to the increase in vortex 4 of 11

5 Number of blades (b) 4 Diameter (m) 1.1 Chord (m).585 Twist [θ 1deg ](deg) Blade airfoil NACA 12 Collective (deg) 7.2 ±.5 Tip speed (m/s) 8.33 Re c 248, M tip.24 Coefficient of Thrust (C T ).3982 Rotor Solidity (σ).139 Hub flap Hinged Hub Lead/Lag Rigid Blade tip shape Square Table 1. Parameters and test condition of the rotor system Figure 2. Q criterion identifying 2 rotating vortices in an instantaneous PIV image, with Center of Mass deducing the vortex center. jitter, the ensemble averaged vector fields do not accurately depict the shape, strength, nor size of the older vortices. Since focus was paid on aligning the tip vortices, a minor translation offest can be seen in the out of plane velocity component (gray-scaled background) closer to the hub. The average position of four of the vortices can clearly be seen in figure 3, as well as the wake from two passing blades, as denoted by the large (white) out of plane component of velocity. Since there is little to no counter-rotating flow for a rotor in hover, black denotes a small but positive out of plane velocity and white denotes a large and positive velocity. By conditionally averaging the vortex fields, that is, averaging based on the individual vortex centers determined by the Q/CoM method, a characteristic vortex can be recovered that will have the correct characteristics. This form of conditional averaging was first described by van der Wall and Richard (26) when they applied it to the HART-II data set and has also been explored by Ramasamy et al (29), who renames it conditional ensemble phase averaging. 12,13 An example of a characteristic vortex can be seen in Figure 7(a). 5 of 11

6 Figure 3. Overlayed ensemble averaged vector fields corresponding to a 21 degree wake age. The W (out of plane) velocity is the gray-scaled background, while U (radial) and V (vertical) velocities are depicted by the vectors. After the identification of the vortex cores, the slipstream of the rotorcraft, as given by the average position of the vortex cores, can be compared with the empirical trend seen by Landgrebe. 14 Landgrebe used the approximate relations shown in equations 3-6, where A =.78 and λ = C T. { y R = k 1 ψ; ψ 2π b ( y R ) ψ= 2π +k 2(ψ 2π b b ); ψ 2π (3) b r R = A+(1 A)e λψ (4) k 1 =.25(C T /σ +.1θ 1deg ) (5) k 2 = ( θ 1deg ) (C T /2) (6) Direct thrust measurements were not acquired, so the blade loading was calculated by integrating the measured inflow velocity and was concurrently compared to blade element momentum theory, to yield the approximate C T given in table 1. Since velocity measurements were only available for r/r values from.78 to past the blade tips, the velocity inflow was curve fit to give an approximate inflow for the entire rotor. The coefficient of thrust calculated by the curve fit induced velocity compared very favorably to the blade element momentum theory result, giving confidence in the assumptions. Figure 4 shows the slipstream position outlined by vortex 3 spawned by blade 3. Both the vertical and radial positions compare well with Landgrebe s model, the minor variance seen at a wake age of 23 is a remnant of the overlapping process. Vortex 3 was chosen specifically because it did not undergo vortex tumbling, which better emulates Landgrebe s experiments. Once the vortices are tracked using the Q/CoM method, the predominant jitter can be identified. By calculating the ensemble averaged vortex center, a 95% confidence ellipse can be fit to encompass the individual vortex positions, centered around the average position. Figure 5 shows the slipstream generated by vortex 3 and its 95% confidence ellipse. It can be seen that a vortex jitters predominantly perpendicular to the slope of the slipstream boundary. The amount of vortex jitter also grows with age, as the vortices have 6 of 11

7 1.9 W ake Displacements Radial displacements (r/r) Vertical displacements (y/r) ψ[deg.] Figure 4. Estimates of the radial and vertical wake displacements of vortex 3 compared with the predictions from Landgrebe (1972). had a chance to dissipate and become more susceptible to turbulence and the effects of neighboring vortices. The primary direction of vortex jitter and the growth of jitter has been seen in similar work by Mula et al (211) and Kindler et al (21), both of which were limited to a smaller range of wake ages. 11,15 y/r r/r Figure 5. Slipstream boundary from vortex 3 with its 95% confidence region at each wake age The vortex tumbling for this rotor wake takes place at a wake age of 35 degrees. Figure 6 shows the tumbling between vortex one and two, spawned by blades one and two, respectively. Tumbling occurs only 7 of 11

8 between these two vortices, because blade one was left slightly lagging the other blades, when they were initially tracked. This resulted in vortex one being nearer vortex two, and therefore their resulting induced velocities were greater, instigating a vortex tumbling episode. Figure 6 also suggests that in the absence of external factors, if a rotorcraft s blades are perfectly tracked, vortex tumbling will not occur, or will occur only at very large wake ages. This can be seen by the fact that vortices 3 and 4 are not involved in a vortex tumbling episode. Figure 6 also shows that vortex one and two rotate around each other for approximately 18 degrees before they coalesce. Vortex one begins at a wake age of 3 below vortex two, and ends up above vortex two by 4 of wake age. Vortex one by this point is very weak and begins to dissipate and is gone by a wake age of 45 degrees. Vortex two is then considered to be the daughter vortex generated by both vortices two and one. It is interesting to note that there is no extended vortex tumbling period, with multiple rotations, for this wake structure. Figure 7 shows characteristics of vortex one at an age of 23 (ψ = 23, figure 7(a)), vortex two at an age of 18 (ψ = 27, figure 7(b)), and their combined daughter vortex (figure 7(c)) at a wake age of 46. Figure 7 provides the vortex velocity components in radial and vertical directions, as well as their calculated Q-contours. Note that the Q-contours are centered to the right of the center of swirl and is due to the Q-criterion taking into account the out of plane gradients as well as in-plane gradients. The radial and vertical components in figure 7 have been renormalized by R 1 =.326 R, which is a more appropriate vortex relative scale. Figure 8 shows a horizontal cross-section of Q magnitude(8(a)) and velocity magnitude(8(b)) for all three vortices, both of which are normalized by their respective maximas. Figure 8(a) shows that the combined vortex core has a tighter structure, indicated by a stronger swirling motion than is seen in either of its parent vortices. This can be seen by the steeper gradient of Q-criterion and higher magnitude in the r/r 1 direction. Figure 8(a) also identifies a smaller core region for the daughter vortex than for the parent vortices. This suggests that the vortices were stretched during the vortex tumbling episode, and that this stretching phenomenon was more powerful than viscous diffusion. The velocity magnitude cross-section in figure 8(b) on the other hand, shows that the vortex is weaker than either of its parent vortices. This conclusion is reached because the velocity differential across the core of the parent vortices is greater than that of the daughter vortex. The gradient of velocity is also greater for the parent vortices than the daughter vortex, although the daughter vortex does possess higher velocity magnitudes. The higher velocity magnitudes in the daughter vortex are expected since the slipstream is still contracting causing convective acceleration within the flow. The steeper Q-criterion gradients, as shown in figure 8(a), are an unexpected finding as it was assumed that viscous diffusion would be dominant in the far wake leading to a larger vortex core with lessor gradients. IV. Conclusions The tip vortices from a 4-bladed model helicopter rotor during hover was investigated experimentally in order to research the far-field vortex tumbling phenomenon. The tip vortices were tracked through the use of a three-component, stereo-piv system and the data was further processed with the help of the described Q/CoM vortex tracking technique. The vortices were tracked to a wake age of ψ = 8, encompassing both the near and far wake regions. The far-wake began around a wake age of 27, when vortex one and two were shown to begin tumbling. The vortices had coalesced into a daughter vortex by a wake age of 46 ; the strength and radius between the characteristic parent and daughter vortices was accessed. It was shown that the daughter vortex possessed a tighter core as denoted by a stronger swirling motion, while it was overall weaker than the parent vortices. The vortex jitter motion was also investigated by fitting a 95% confidence ellipse to the individual vortex locations, centered on the average core location. It was shown that the major axis of the ellipses aligned approximately perpendicular to the slipstream. This result was seen in previous studies, which were confined to a much smaller wake age. It was also seen that vortex tumbling either does not occur in hover, or occurs at much larger wake ages than those investigated, if the rotor blades are perfectly tracked and there are no external forces to disrupt the flow. Future work will be focused on two separate tracks. The first track will be to attempt to show that vortex tumbling does not need to occur before viscous diffusion dissipates the tip vortices; while the second track 8 of 11

9 .2.25 Vortex 1 Vortex 2 Vortex 3 Vortex 4 y/r ψ[deg] Figure 6. Vortex tumbling phenomenon between the vortices created by blade 1 (black square) and blade 2 (red circle). No tumbling happens between blades 3 (blue diamond) and 4 (cyan star). The combined, daughter vortex after tumbling is denoted by the green squares. will be to investigate further the vortex tumbling phenomenon by linking it to Biot-Savart s Law as well as looking deeper into the statistical characteristics of said phenomenon. Investigations into the acoustic signature produced by both tracks will also be explored. V. Acknowledgments The authors are grateful for the support provided by the DoD SMART Fellowship program. Financial support for this study was graciously provided by The University of Texas at Austin. References 1 Yule, A. J., Large-scale structure in the mixing layer of a round jet, Journal of Fluid Mechanics, Vol. 89, No. 3, 1978, pp Lugt, H. J., The dilemma of defining a vortex, 1979, pp Chong, M., Perry, A., and Cantwell, B., A general classification of three-dimensional flow field, Physics of Fluids A, Vol. 2, No. 765, Hunt, J. C. R., Wray, A. A., and Moin, P., Eddies, stream, and convergence zones in turbulent flows. Center for Turbulence Research Report: CTR-S88, 1988, pp Kolar, V., Vortex identification: New requirements and limitations, International Journal of Heat and Fluid Flow, Vol. 28, 27, pp Jeong, J. and Hussain, F., On the identification of a vortex, Journal of Fluid Mechanics, Vol. 285, 1995, pp Zhang, S. and Choudhury, D., Eigen helicity density: A new vortex identification scheme and its application in accelerated inhomogeneous flows, Physics of Fluids, Vol. 18, No. 5, Bhagwat, M. J. and Leishman, J. G., Correlation of Helicopter Rotor Tip Vortex Measurements, AIAA Journal, Vol. 38, No. 2, 2, pp Leighman, J. G., Measurements of Aperiodic Wake of a Hovering Rotor, Experiments in Fluids, Vol. 25, 1999, pp Caradonna, F. and Tung, C., Experimental and Analytical Studies of a Model Helicopter Rotor in Hover, Tech. rep., Moffett Field, Mula, S. M., Stephenson, J. H., Tinney, C. E., and Sirohi, J., Vortex Jitter in Hover, American Helicopter Society Southwest Region Technical Specialists Meeting, Fort Worth, February, Ramasamy, M., Johnson, B., Huismann, T., and Leishman, J. G., Digital Particle Image Velocimetry Measurements of Tip Vortex Characteristics Using an Improved Aperiodicity Correction, Journal of the AHS,, No. 54, 29, Paper 124, pp of 11

10 y/r1 y/r r/r 1 (a) r/r 1 (b) 1.5 y/r r/r 1 (c) Figure 7. Velocity with overlapped Q contours for vortices spawned by blades 1 (a) and 2 (b), as well as their combined vortex (c) after tumbling. 13 van der Wall, B. G. and Richard, H., Analysis methodology for 3C-PIV data of rotary wing vortices, Experiments in Fluids, Vol. 4, 26, pp Landgrebe, A. J., The wake geometry of a hovering helicopter rotor and its influence on rotor performance. Journal of the American Helicopter Society, Vol. 17, No. 4, 1972, pp Kindler, K., Mulleners, K., Richard, H., Van der Wall, B., and Raffel, M., Aperiodicity in the near field of full-scale rotor blade tip vortices, Experiments in Fluids, 21, pp of 11

11 1.8.6 Q r/r 1 (a) 1.8 U r/r 1 (b) Figure 8. (A) corresponds to a horizontal cross-section of Q-level through the vortex center, while (b) is the same for velocity magnitude. Both (a) and (b) are normalized by their respective maximas. 11 of 11