2008-ahs-dual-plane Piv Paper -1

Turbulent Tip Vortex Measurements Using Dual-Plane Digital Particle Image Velocimetery Manikandan Ramasamy∗

Bradley Johnson†

J. Gordon Leishman‡

Department of Aerospace Engineering Glenn L. Martin Institute of Technology University of Maryland, College Park, MD 20742

Abstract The formation and evolutionary characteristics of the tip vortices trailing from a model-scale, hovering helicopter rotor was analyzed by performing comprehensive flow field measurements inside the rotor wake. The use of a dual-plane stereoscopic digital particle image velocimetry (DPS-DPIV) technique allowed measuring all the terms involved in Reynolds-averaged stress transport equations. These include, not only all the three components of velocity, but also all nine velocity gradient tensors simultaneously, a capability not possible with conventional PIV technique. High-resolution imaging of the vortex sheet trailing behind the rotor blade, and detailed turbulence measurements on the process of the roll up of the tip vortices revealed the presence of several micro-scale flow features that play a critical role in the overall evolution of the tip vortices. These measurements help provide a benchmark case for calibrating turbulence models, as well as for validating CFD predictions. Also, the measurements clearly confirmed that an isotropic assumption of turbulence properties is not valid, and that stress should not be represented as a linear function of strain for the development of future turbulence models. DPIV measurements were complemented with three-component, laser Doppler velocimeter (LDV) measurements made on same rotor. Good correlation was found between the two measurement techniques.

Nomenclature rotor disk area blade chord rotor thrust coefficient, = T /ρAΩ2 R2

A c CT

i, j, k Nb p R r, θ, z r0 rc Rev u, v, w u0 , v0 , w0 u0 v0 v0 w0 u0 w0 Vax Vr Vθ Vr0 , Vθ0 , Vz0 Vtip x, y, z xT , yT , zT X, Y, Z α Γv δ δi j ν ζ ρ ψ ψω Ω 2-C 3-C BSPCs

direction vectors number of blades static pressure radius of blade polar coordinate system initial core radius of the tip vortex core radius of the tip vortex vortex Reynolds number, = Γv /ν velocities in Cartesian coordinates normalized RMS velocities in DPIV coordinates normalized Reynolds shear stress in X,Y plane normalized Reynolds shear stress in Y, Z plane normalized Reynolds shear stress in X, Z plane axial velocity of the tip vortex radial velocity of the tip vortex swirl velocity of the tip vortex normalized RMS velocities in polar coordinates tip speed of blade tip vortex coordinate system rotor coordinate system DPIV coordinate system Lamb’s constant, = 1.25643 total vortex circulation, =2πrVθ ratio of apparent to actual kinematic viscosity Kronecker delta kinematic viscosity wake age air density azimuthal position of blade relative blade position rotational speed of the rotor two-component three-component beam splitting polarizing cubes

∗

Assistant Research Scientist. [email protected] Graduate Research Assistant. [email protected] ‡ Minta Martin Professor. [email protected] Presented at the 64th Annual National Forum of the American Helicopter Society International. Inc., April 29–May 1, 2008, c Montr´eal, Canada. 2008 M. Ramasamy et al. Published by the AHS International with permission. †

Introduction The thorough understanding of a helicopter rotor wake is a challenging one to say the least, even in hover where Page 1

the flow state relatively axisymmetric. Decades of research have been focused towards a general understanding of the complex vortical wake generated by a rotor, and its effect on vehicle performance, vibration, and noise levels (Refs. 1–11). Much of the research has been rightfully focused on the understanding of the blade tip vortices, which are the most important features of the rotor wake (Refs. 12–18). These vortices can remain in the proximity of the rotor disk for several rotor revolutions, and can lead to rotor vibration and unsteady airloads. It is, therefore, important to better understand and predict the physics that determine the formation, strength, and trajectories of these tip vortices to develop more consistent and reliable mathematical models to predict rotor aerodynamics. To this end, no model can be completely successful unless it is able to accurately capture the threedimensional, turbulent flow distribution present inside the vortex cores, and their changes as a function of wake age and other functions. Mathematical models of tip vortex evolution have been developed by making several assumptions to the N–S equations (e.g., the Lamb–Oseen model). These assumptions, such as incompressible, one-dimensional, inviscid flow do not completely represent the real flow field complexities, and can result in inaccurate predictions of the growth properties of real tip vortices. Nevertheless, these models have formed the basis for several empirical modifications, which are used today in many comprehensive rotor analyses (Refs. 7, 19, 20). Efforts towards resolving rotor wakes using Navier– Stokes (N–S) methods have been carried out, and are steadily on the rise. Because Direct Numerical Solution (DNS) of the N–S equations is presently unrealistic for the rotor wake problem because of the enormous computational expense more effort has to be focussed toward solving the Reynolds-Averaged Navier–Stokes (RANS) equations. RANS simplifies the process by time-averaging the N–S equations by representing the flow velocity at a point (ui ) as a combination of the mean component, ui , and a fluctuating component, u0i , as given by ui = ui + u0i

Time-averaging bypasses the need to computationally represent the high-frequency, small-scale flow fluctuations (i.e., u0i ) caused by the turbulent eddies. However, this advantage is countered by the presence of an additional unknown term (i.e., the Reynolds stress u0i u0j ), which makes the RANS equations unsolvable unless a closure model to rebalance the number of equations and unknowns. This stress term (or the correlation term) basically accounts for the effect of velocity fluctuations created by the presence of eddies of different length scales. It is vital, therefore, that any turbulence closure model adopted be consistent with the flow physics to correctly model the contributions of turbulent flow, and also considering any numerical stability limits inherent to the specific discretization scheme used. The basis for turbulence models is empirical evidence and measurements. Therefore, the objective of the present work was to explore the complex wake (both the vortex sheet and the tip vortex) of a rotating-wing, using high spatial and temporal resolution DPIV and LDV to measure the mean and turbulent flow quantities in all three flow directions. These objectives were accomplished by employing: (1) A conventional DPIV with an extremely high-resolution (11 mega pixel) camera that provided a great deal of quantitative information on the formation of the tip vortices, and the turbulence transport between the vortex sheet and the tip vortex, and (2) A dual plane DPIV (DPS-DPIV) technique, which can simultaneously measure the velocity across two adjacent, parallel planes. An advantage of DPS-DPIV is that it can determine not only the 6 in-plane velocity gradients, but also the 3 out of plane velocity gradients, which are needed for understanding the turbulence transport. Separate LDV measurements on the same rotor were also used to compare with the DPIV data. While typical experimental goals of measuring the tip vortex core size, peak swirl velocity, and their evolution with time were accomplished. The focus here was on understanding the turbulent production, transport, and diffusion present in the near- and far-wake of the rotor. This information can be used to validate and improve vortex models, as well as to validate turbulence models for RANS solvers.

(1)

Applying Eq. 1 to the incompressible N–S equations results in the RANS equation that is given by.     pδi j Dui ∂ ∂ui ∂u j 0 0 = − +ν + − ui u j Dt ∂x j ρ ∂x j ∂xi

Turbulence Modeling Generally, the turbulence characteristics of any given flow field depends on the length scales of turbulent eddies present in the flow, the time history effects, and the influence non-local effects, such as the presence of a solid boundary. Developing a turbulence model to address all these effects for general flow conditions is not possible. This can be appreciated from the fact that the turbulence fluctuations observed in a free shear flow, for example, are

(2)

where, D/Dt is the substantial derivative, p is the static pressure, δi j is the Kronecker delta function, ρ is the density of the fluid medium, ui is the instantaneous velocity, and ui u j is the correlation (or shear stress) term. The overbar represents the time averaged mean values. Page 2

Figure 1: Laser light sheet flow visualization of a rotor tip vortex, displaying 3 distinct flow regions

Figure 2: Variation of Richardson number across the vortex.

completely different from those found in bounded flow. As a result, numerous flow-dependent turbulence models have been tailor-made for a given set of flow conditions. The Johnson–King model, for example, is often used for separated flows (Ref. 21). Similarly, the V 2 –F model is more suitable for low Reynolds number flows (Ref. 22). Because different turbulent models must be developed for different problems and flows, they will understandably vary in complexity, and in the number of equations and coefficients used in the model. Regardless, every model requires a certain number of closure coefficients (or constants), and damping functions. The simple Prandtl’s mixing length model, for example, requires just one constant, while the modified Baldwin–Lomax model requires six constants (Ref. 23). The frequently used Spalart– Allmaras (Ref. 24) model has 8 constants and 3 damping functions, while Baldwin–Barth model requires 7 constants, 2 damping functions, and 1 additional function for the length scale (Ref. 25). These closure coefficients and damping functions are derived from free shear flows or homogenous flows, and are based on experimental measurements. A problem arises, however, when these models, whose coefficients are derived from simple measurements of a particular flowfield, are applied to a more complicated flow that may involve shock waves or streamline curvatures, (such as those present in tip vortices at the blade tip). It is known, for example, that tip vortices can produce relaminarization or rotational stratification of turbulence near the vortex core axis, which will not be accounted for if turbulence models derived from non-rotational flows are used without modification (Refs. 26, 27).

laser light sheet visualization performed in the flow field of a hovering model-scale rotor. It is apparent that the vortex comprises of three regions; (1) an inner laminar region, where the vortex behaves similar to a solid-body with little to no interaction between adjacent layers of fluid, (2) a transitional region with eddies of different scales, and (3) an outer potential region. This is similar to the make-up of a turbulent boundary layer, which is composed of (1) a viscous sub-layer, (2) a logarithmic layer, and (3) an outer free-stream flow. While Van Driest (Refs. 28, 29) used a damping function to model the logarithmic layer, Klebanoff (Ref. 30) used an intermittency function to represent the turbulent fluctuations from near the wall to the outer free stream flow. Based on a similar idea, Ramasamy & Leishman developed an intermittency function to represent the eddy viscosity variation across the tip vortex, and derived a semi-empirical solution for the evolutionary characteristics of tip vortices. This eddy viscosity model is given by νT = ν + VIF α2RL |σ|

(3)

where νT is the total viscosity, ν is the kinematic viscosity, VIF a vortex intermittency function, αRL is an empirical constant, and σ is the strain. The model is based on Prandtl’s mixing length hypothesis, and addresses the effects of rotational stratification through a Richardson number (Ref. 31). The Richardson number can be defined as the ratio of turbulence produced or consumed inside the vortex to the turbulence produced by shear. Figure2 shows the distribution of Richardson number measured inside a tip vortex along with predictions from two vortex models. It can be seen that the Ri number decreases quickly from infinity at the center of the vortex, and approaches a threshold value. It has been argued (Ref. 32) that turbulence cannot develop until the value of Ri number falls below

Applications to the Rotor Wake There are many challenges in developing a turbulence model for rotor wakes. Figure 1 shows a stroboscopic Page 3

this threshold. The R − L eddy viscosity model was based on this idea, and the closure coefficients associated with the model were derived by first finding a mathematical solution to the one-dimensional N–S equations, followed by comparing the Reynolds number dependent, similarity solution to the experimental measurements of the evolutionary characteristics of tip vortices (such as the swirl velocity distribution) obtained from various sources. Understanding the turbulence activity inside the tip vortices also requires an understanding of the turbulence in the inboard vortex sheet, especially at early wake ages, where the sheet becomes partly entrained into the vortex. However, understanding the roll-up process is difficult because the flow is turbulent, three-dimensional, and involves high pressure and velocity gradients. The pressure difference between the lower and upper surfaces of the blade tip accelerates the flow from the lower surface, which when combined with the free-stream flow results in the formation of a tip vortex. However, this is only an inviscid description, and it masks the intricate details of the flow physics. Secondary and tertiary vortices are known to form when the viscous nature of the flow couples with the local pressure gradients, which results in flow separations near the tip surfaces. These structures continue to evolve on the upper surface, and ultimately merge into a coherent vortex downstream of the trailing edge of the blade. Furthermore, part of the shear layer from the trailing edge is also entrained into the tip vortex flow. Although the tip vortex is fully rolled-up and largely axisymmetric in the far-field (Ref. 33), modeling the turbulence inside the vortex cores (and their evolution) depend directly upon the initial strength and its roll-up behavior.

Figure 3: Schematic and photograph of DPS-DPIV advanced optical setup

The present study involved the application of both LDV and DPIV techniques. In the case of DPIV, a simple 2component configuration was first used to analyze the vortex sheet using an ultra-high resolution (11 mega pixel) camera. This was followed by a DPS-DPIV technique, where a pair of 4 megapixel cameras and a single 2 mega pixel camera were used. Because a 2-component DPIV configuration is relatively simple to set up, the discussion in this paper is focussed on the DPS-DPIV technique. All of the measurement techniques used the same seed particles, whose average size was approximately 0.25 µm in diameter. This mean seed particle size was small enough to minimize particle tracking errors (Ref. 34).

gle blade rotor have been addressed before (Refs. 35, 36). This includes the ability to create and study a helicoidal vortex filament without interference from other vortices generated by other blades (Ref. 35). Also, a single helicoidal vortex is much more spatially and temporally stable than with multiple vortices (Ref. 36), thereby allowing the vortex structure to be studied to much older wake ages and also relatively free of the high aperiodicity typical of multi-bladed rotor experiments. The blade was of rectangular planform, untwisted, with a radius of 406 mm (16 inches) and chord of 44.5 mm (1.752 inches), and was balanced with a counterweight. The blade airfoil section was the NACA 2415 throughout. The rotor tip speed was 89.28 m/s (292.91 ft/s), giving a tip Mach number and chord Reynolds number of 0.26 and 272,000, respectively. The zero-lift angle of the NACA 2415 airfoil is approximately -2− circ at the tip Reynolds number. All the tests were made at an effective blade loading of CT /σ ≈ 0.064 using a collective pitch of 4.5◦ (measured from the chord line). During these tests, the rotor rotational frequency was set to 35 Hz (Ω = 70π rad/s).

Rotor System

DPS-DPIV

A single bladed rotor operated in the hovering state was used for the measurements. The advantages of the sin-

DPS-DPIV differs from conventional DPIV because it can measure all nine components of the velocity gradi-

Description of Experiment

Page 4

ent tensor. For example, a conventional, stereoscopic (3component) DPIV system is capable of measuring all the three components of velocity in a given plane (Refs. 37– 40). These instantaneous velocities can then allow for velocity gradient calculations to be estimated in the in-plane x- and y- directions. However, estimating the velocity gradient in the out-of-plane direction (i.e., ∂/∂z) requires the measurement of all three components of velocity in at least two planes that are parallel to each other, and are separated by a small distance in the z direction. The optical set up of the DPS-DPIV technique is shown in Fig. 3. For dual-plane measurements, two separate DPIV systems are required to simultaneously measure the flow velocities in both the planes, independently. It should be noted that DPS-DPIV can be arranged as a combination of two stereoscopic PIV systems, or as a combination of a stereoscopic PIV system and a 2-component PIV system (Refs. 41, 42). While the former combination provides all the three components of velocity in both of the two parallel planes, the latter provides all the three components of velocity in one plane and only the in-plane velocities (i.e., two components) in the other plane. The out-ofplane velocity is usually calculated using continuity assumptions. This latter method provides some advantages over the stereoscopic setups, mainly because of its simpler configuration. The present set up is shown in Fig. 3. A conventional DPIV setup (the 2 mega-pixel camera is labeled as C2 in Fig. 3) is used to measure two components of velocity in one plane, while a stereo setup (a pair of 4 mega-pixel cameras labeled C1 and C3 in Fig. 3) is used to measure three velocity components in the second plane. Continuity in the form of Eq. 4 is then applied to estimate the third component of velocity in the 2-C measurement plane (shown in green), i.e., w1 = −(

∆u1 ∆v1 + )∆z + w2 ∆x ∆y

Figure 4: Schematic of DPS-DPIV timing. with the lasers (i.e., the first image in each plane is captured upon the firing of lasers 1 & 3 and the second image in each plane is captured during the firing of lasers 2 & 4). There are several challenges with simultaneous measurement in adjacent, parallel, laser planes, however, mainly resulting from crosstalk between cameras. This occurs despite each camera having a finite depth of field, and would record seed particle reflections from both illuminated laser planes (because the laser planes are apart by only a few millimeters). If any camera captures reflection, from seed particles in both the planes, not only will its planar velocity map be erroneous after DPIV processing, but also the comparison between the velocity map in the first plane with that of the second plane (which is needed to calculate velocity gradients in the z direction) would be meaningless. This problem is heightened by the need to have the intensity of each laser each laser set to high levels so that the individual seed reflections can be captured by the cameras. To guarantee that each respective set of cameras is only seeing the flow in its designated laser plane, a special optical setup was used, as shown in Fig 3.

(4)

The resulting velocity fields in the two planes can then be compared to determine all nine components of velocity gradient tensor. To ensure the correctness of these gradient calculations, however, several precautions have to be taken. In terms of the setup procedures, the two laser light sheet planes must be both parallel, and adjacent (ideally, just a small distance apart) to each other. Additionally, the lasers must be synchronized not only with each other, but also with both sets of cameras and to the rotor rotational frequency. For the present experiment, each laser pair (i.e., lasers 1 & 2, and lasers 3 & 4 in Fig. 1) delivers two pulses of laser light with a pulse separation time of 2 µs. The first laser pulse from the green pair (laser 1) must be synchronized with the first laser pulse from the blue pair (laser 3), and the same for the second laser pulse from each laser pair (lasers 2 & 4). Each of the three cameras must then be synchronized

The purpose was to split the polarizations of the two respective laser pairs (lasers 1 & 2 are s-polarized, and lasers 3 &4 are p-polarized), and then to use appropriate filters and beam-splitting optical cubes placed in front of each camera to guarantee that they only see one type of polarizedlight. In the present setup, the middle 2-C camera (C2) was tuned to the s-polarization of lasers 1 & 2, Page 5

and the stereo cameras (C1 & C3) were tuned to the ppolarized light lasers from 3 &4. Figure 3 shows a diagram explaining the setup, where the blue (p-polarized) and green (s-polarized) light rays trace the paths of the light reflections from each respective laser sheet. One beam splitting cube in front of the 2-C camera initially sees both sets of laser reflections, and allows the ppolarized blue light to pass directly through and re-directs the s-polarized green light to a second beam splitting cube. The second cube simply acts as 45◦ mirror by re-directing the s-polarized light to the camera. A linear filter is placed over the lens to act as a final buffer against any p-polarized light. Each stereo camera also has one beam splitting cube in front of it, which re-directs the s-polarized light into separate light dumps adjacent to the cubes, and allows the p-polarized blue light to pass through to the camera lens. Each stereo camera also has a linear filter over the lens (oriented at a different angle than that over the 2-C camera) to act as a final buffer against any s-polarized light. Final verification of the working condition of the optical set up was made before measurements were started. It was ensured that the cameras see only their designated lasers and not the reflections from the other lasers, i.e., there was no crosstalk. The second challenge in the experiment involves the need for simultaneous measurement in each plane. Even after optically separating the two DPIV systems, care has to be taken to ensure that both systems are synchronized with each other so that the flow is measured in each plane at exactly the same time. This will guarantee that the turbulence estimates will be derived from measurements of the same exact flow features. Figure. 4 shows the timing setup of the experiment, which takes a 1/revolution TTL signal from the rotor, and uses it to synchronize both ND:Yag laser pairs, with each other, and their respective cameras. The processing of the acquired images from the DPSDPIV technique used a deformation grid correlation algorithm (see Ref. 43), which is optimized for the high velocity gradient flows found in rotor tip vortices. The interrogation window size was chosen in such a way that the images from both the cameras were resolved to approximately the same spatial resolution to allow measurements to be made of the velocity gradients in the out-of-plane direction. The steps involved in this correlation algorithm are shown in Fig. 5. The procedure begins with the correlation of an interrogation window of a defined size (say, 64-by-64), which is the first iteration. Once the mean displacement of that region is estimated, the interrogation window of the displaced image is moved by integer pixel values for better correlation in the second iteration. This third iteration starts by moving the interrogation window of the displaced image by sub-pixel values based on the displace-

Figure 5: Schematic of the steps involved in the deformation grid correlation.. ment estimated from second iteration. Following this, the interrogation window is sheared twice (for integer and sub-pixel values) based on the velocity magnitudes from the neighboring nodes, before performing the fourth and fifth iteration, respectively. Once the velocity is estimated after these five iterations, the window is split into four equal windows (of size 32 × 32). These windows are moved by the average displacement estimated from the final iteration (using a window size of 64 × 64) before starting the first iteration at this resolution. This procedure can be continued until the resolution required to resolve the flow field is reached. The second interrogation window is deformed until the particles remain at the same location after the correlation.

LDV System A fiber-optic based 3-C LDV system was used to make three-component velocity measurements. To reduce the effective size of the probe volume visible to the receiving optics, the off-axis backscatter technique was used, as described in Martin et al. (Ref. 44). This technique spatially filters the effective length of the LDV probe volume on all three channels. Spatial coincidence of the three probe volumes (six beams) and two receiving fibers was ensured to within a 15 µm radius using an alignment technique (Ref. 44) based on a laser beam profiler. Alignment is critical for 3-component LDV systems because it is geometric coincidence that determines the spatial resolution of the LDV probe volume. In the present case, the resulting LDV probe volume was measured to be an ellipsoid of dimensions 80 µm by 150 µm, which for reference was about 3% of the maximum blade thickness or 0.5% of the blade chord. A coincident window of 80µs was used to ensure that the same set of particles provide all the three components of velocity. The flow velocities were then converted into three orthogonal components based on measurements of the beam crossing angles. Each meaPage 6

Figure 6: Experimental set up for 2-D DPIV.

Figure 9: Close-up view of the tip vortex and various segments of the shed vortex sheet

Figure 7: Presence of Taylor–Gortler vortices in the vortex sheet trailing behind a rotor blade.

Results The observed results have been classified into three categories, in which they will be analyzed: (1) Highresolution imaging of the vortex sheet and the formation of tip vortices, (2) The mean characteristics of the rotor tip vortices, and (3) The turbulent characteristics of the rotor tip vortices. It should be noted that the measurements of the vortex sheet were carried out using 11 mega pixel camera, while the remainder of the results were obtained using the DPS-DPIV and 3-C LDV techniques.

Vortex Sheet and Tip Vortex Formation Figure 8: Close up view of the trailing tip vortex.

Figure 6 shows the schematic of the experimental set up used for the high resolution 2-C DPIV measurements. The laser was fired along the span of the rotor blade and the 11 mega pixel CCD camera was placed orthogonal to the laser light sheet. An instantaneous DPIV velocity vector map obtained at 2◦ wake age is shown in Fig. 7, where the color contour is stream wise vorticity (only every 4th vector is shown to

surement was phase-resolved with respect to the rotating blade by using a rotary encoder, which tagged each data point with a time stamp. The temporal phase-resolution of the encoder was 0.1 deg, but the measurements were averaged into one-degree bins (Ref. 44). Page 7

smeared through the averaging process. Nevertheless, it is these type of small scale (or even smaller), high frequency aerodynamic structures that contribute primarily to the initial turbulence fluctuations in the wake flow. These fluctuations, when combined with the high velocity gradients found in these tip vortices, play a substantial role in the momentum transfer process in the boundary layer.

Tip Vortex Measurements While the small-scale turbulent vortices in the inboard vortex sheet must play a significant role in defining the turbulence-scale in the near-wake of the blade, the most dominant coherent structure in both the near- and far-wake of the rotor blade is still the tip vortex. Therefore, it is important to study and measure the mean and turbulent quantities inside the tip vortices, and to do so as a function of wake age.

Figure 10: Signature of the Taylor-Gotler vortex pairs seen in the instantaneous flow field vector maps. avoid image congestion). This image reveals the complex turbulent flow pattern in the near wake, and shows the interplay of the tip vortex with the turbulent inboard vortex sheet. Several observations can be made. First, the instantaneous vector map (Fig. 7) shows not only a clear tip vortex forming behind the blade, but also a chain of counter-rotating vortices clearly intertwined with one another, and with blade tip vortex (note the interchange between red and blue vorticity contours which represent clockwise and counter-clockwise rotation respectively). Commonly known as Taylor–Gortler vortices (Ref. 45), these counter-rotating structures are present along the entire span of the shed vortex sheet. Various sections of the flow field (along the span of rotor blade) are shown in a close-up view in Figs. 8 and 9, which evidently show the tip vortex and the presence of these T-G vortices, respectively. These are highly unsteady, making the vortical sheet behind the blade, and its roll-up into the tip vortex at early wake ages a highly aperiodic and turbulent process. The presence of these vortical pairs is directly attributed to the streamline curvature of the boundary layer (Ref. 46). This aperiodicity is seen using Fig. 10, which is obtained by making a horizontal cut through the center of the tip vortex along the entire span of vortex sheet. The results include measurements from both the instantaneous vector map, as well as from a phase averaged vector map, which was obtained by simple averaging of 1,000 such instantaneous velocity vector maps. It can be seen that the instantaneous swirl velocity profile shows velocity fluctuations that arise from the presence of the Taylor–Gortler vortex pairs, while the phase average vector field essentially eliminates many of the intrinsic turbulent structures inboard. This is understandably a result of the aperiodic nature of these vortex pairs that do not maintain the same spatial location from one measurement to the next. Consequently, the signature of these coherent structures are

Aperiodicity Correction This distinction between mean and turbulent velocities in the tip vortex, however, is complicated by the fact that the wake is aperiodic. Even in hover, it is well known that the convecting vortex filaments develop various types of selfand mutually-induced instabilities and modes (Refs. 36, 47) from their interaction with each other. This causes their spatial location to change slightly from one revolution to the next. This leads to the vortex center wandering about a mean position in each instantaneous measurement plane, and so this effect poses a problem in finding the correct mean flow measurements. Unless corrected for, this manifests in inaccurate estimation of turbulent flow components based on Eq. 1. To achieve accurate mean flow velocities, the vortices first have to be co-located for the averaging process, such that the center of each vortex is aligned with one another. In essence, this guarantees that individual mean velocities at a point in the flow are calculated based on that points location with respect to a defined tip vortex center, not based on its unadjusted location with respect to the image boundaries. The conditional aperiodicity bias correction procedure used in the present study (helicity-based centering) was successfully analyzed in detail in (Ref. 37). Mean and turbulence measurements made from 1,000 instantaneous velocity vector maps, and co-locating them such that the point of maximum helicity (ωz w) in each of the instantaneous vector maps coincided (before phaseaveraging) were found to result in accurate estimates of the core properties. Mean Flow Characteristics of Tip Vortices Only after applying the conditional helicity phaseaveraging technique can accurate mean flow properties to Page 8

Figure 11: Growth characteristics of tip vortices trailing a helicopter rotor blade

Figure 13: Comparison of LDV and DPIV to axial velocity the swirl velocity reached a maximum value on these cuts were then averaged to estimate the core radius. This procedure has been proven to result in accurate estimates of the size of the tip vortex cores (Ref. 37). The measured core sizes obtained at various wake ages are shown in Fig. 11, along with measurements made on the same rotor using LDV. All of the measurements were normalized using the rotor blade chord. The figure also includes core growth estimated from Squire’s model, as modified by Bhagwat and Leishman (Ref. 48) given by s   ζ 2 rc = r0 + 4ανδ (5) Ω When δ = 1, the model reduces to the laminar Lamb– Oseen model. Increasing the value of δ basically means that the average turbulence inside the tip vortex is increased, which can be expected to result in a higher core growth rate. It can be seen that the present measurements follow the δ = 8 curve, suggesting that the momentum transfer occurs eight times faster than for laminar flow. This was the case for both the present measurements, as well as for previous measurements made on the same rotor. The swirl and axial velocity distributions measured using PIV (by making a horizontal cut across the vortex) are shown in Fig. 12. The classical signature of the swirl velocity distribution can be seen, with the peak swirl velocity continuously decreasing with an increase in wake age. This, combined with the increase in vortex core size for increasing wake age, suggest that viscous and turbulent diffusion are important flow mechanisms. For the mean axial velocity, it can be seen that the earliest wake age (2◦ ) exhibited an axial velocity deficit of 75% of the tip speed of the rotor blade. This immediately reduces to about 45% of tip speed within 2◦ of wake age. However, any further reduction in the mean axial velocity occurred relatively slowly, as it remained near 30%, even

Figure 12: Normalized swirl and axial velocity distribution at various wake ages be estimated. One important derived parameter is the core radius of the vortex, which is usually assumed as the distance between its center (maximum point of helicity) and the radial location at which the maximum value of swirl velocity occurs. In the present study, the core size was determined using a two step process. First, a horizontal cut (along the span of the blade), and a vertical cut (normal to the span) were made across the tip vortex. The radial distances at which Page 9

Figure 14: Schematic showing the coordinates followed in the present experiment. after 60◦ of wake age. Such high values of axial velocity deficits at the centerline of the tip vortices from trailing from micro-rotor blades were reported in Ref. 49. LDV measurements made on the same rotor at 45◦ wake age, as reported in Ref. 50, are compared to the current results in Fig. 13. It can be seen that the axial velocities estimated from LDV are lower in magnitude compared to those estimated using DPIV. The difference can be directly attributed to the lack of any correction procedure for aperiodicity bias for the LDV measurements. The DPIV measurements are corrected using helicity based aperiodicity correction technique. It was shown in Ref. 49 that axial velocity is one of the most sensitive parameters to the aperiodicity in the rotor wake. The result in this figure clearly shows the need to correct the velocity measurements for aperiodicity bias, otherwise the estimated peak values will be flawed. It should be noted that estimating accurately the peak value of axial velocity and its gradient is critical to understand the evolutionary characteristics of tip vortices. Velocity Gradients Corrected mean measurements using DPS-DPIV allow for accurate estimations of all 9 velocity gradients in 3 flow directions. Figure 15 shows the nine gradients for a wake age of 12◦ wake age. The solid circles represent the average core size of the tip vortex. As mentioned previously, ∂w/∂z is obtained using the continuity equation. The coordinates (and the sign convention) used in the present experiment can be understood from Fig. 14. It should be noted here that not only do all these gradients have different orders of magnitude, but their distributions throughout the vortex cores are also different.

The presence of the lobed-patterns in Fig. 15 are a result of analyzing a rotational coherent structure in terms of Cartesian coordinates. Comparing the gradients, both the ∂u/∂y and ∂v/∂x components were to be at a maximum near the vortex center, albeit with opposite signs. Their peak values were also significantly higher than all other velocity gradients, followed by ∂w/∂x and ∂w/∂y whose higher magnitude can be explained by the steep rise in the axial velocity deficit within the viscous vortex cores. The ∂w/∂y and ∂w/∂x gradients predictably form a two lobe pattern of opposite sign about the vortex center because the axial velocity deficit should increase going radially inwards towards the vortex center, and decrease going radially outwards from the vortex center. The other in-plane gradients, i.e., ∂u/∂x and ∂v/∂y, developed a four-lobed pattern, that were approximately 45◦ to the x-y coordinate axes. Specifically, the ∂u/∂x component displays negative lobes at 45◦ and 225◦ , and positive lobes at 135◦ and 315◦ . The pattern developed in the ∂v/∂y gradient is offset from that in ∂u/∂x by 90◦ . As a result, their sum (which is ∂w/∂z, based on Eq. 4) will be relatively small. Notice, the positive lobes in ∂u/∂x will be added to the negative lobes in ∂u/∂x, and vice-versa. For the other streamwise gradients of in-plane velocities (∂u/∂z and ∂v/∂z), a two-lobed pattern should be expected. Based on the coordinate system followed in this work, ∂u/∂z is negative on the lobe aligned with the positive y-axis, and negative along the y-axis. In turn, the swirl velocity gradient, ∂Vθ /∂z, will be negative (z is positive streamwise) at all points inside the vortex core, indicating a reduction in the swirl strength of the tip vortices. This gradient can be expected to be positive on top of the blade when the tip vortex is still undergoing its roll-up process.

Turbulence Characteristics A detailed analysis was performed on the measured turbulence characteristics to help in understanding the evolutionary behavior exhibited by the tip vortices. 1,000 velocity vector maps were used to estimate the fluctuating velocity components. All the first and second order velocity fluctuations were normalized by Vtip , and Vtip 2 , respectively, and the length scale was normalized by the rotor blade radius. Turbulence Intensities Figure 16 shows the distribution of normalized turbulence intensities u0 , and v0 from ζ = −4◦ (z = −0.66 c) to 4◦ wake age (z = 0.33 c). The solid black circle represents the blade. It can be seen from Figs. 16(d) and (e) that u0 and v0 are biased along x and y axes, respectively. This bias also occurs in the initial stages of the roll up (i.e., ζ = −2◦ ), and correlates with previous measurements made in

Page 10

dudy: -2.000

dudx: -6.556 -4.635 -2.714 -0.793 1.128 3.050 4.971

0

-0.005

-0.005 0 0.005 Distance from the vortex center, Y/R

0.005

0

-0.005

-0.01 -0.01

0.01

-6.000

-3.000

Distance from the vortex center, X/R

Distance from the vortex center, X/R

0

-0.005

-0.005 0 0.005 Distance from the vortex center, Y/R

0.01

0.005

0

-0.005

-0.01 -0.01

0.01

Distance from the vortex center, Y/R

0.01

0.005

0

-0.005

0.01

-0.005

0.01

0

-0.005

(h) ∂w/∂y

-0.005 0 0.005 Distance from the vortex center, Y/R

0.01

(f) ∂v/∂z

0.005

-0.005 0 0.005 Distance from the vortex center, X/R

dvdz: -1.20 -0.80 -0.40 0.00 0.40 0.80 1.20

0

-0.01 -0.01

0.01

dwdy: -9.000 -6.000 -3.000 0.000 3.000 6.000

-0.01 -0.01

0.01

0.005

(e) ∂v/∂y

dwdx: -7.000 -4.000 -1.000 2.000 5.000 8.000

-0.005 0 0.005 Distance from the vortex center, X/R

-0.005 0 0.005 Distance from the vortex center, Y/R

-0.005 0 0.005 Distance from the vortex center, Y/R

(c) ∂u/∂z

0.01

(d) ∂v/∂x

Distance from the vortex center, Y/R

-0.005

dvdy: -6.500 -4.500 -2.500 -0.500 1.500 3.500 5.500

0.000

0.005

(g) ∂w/∂x

0

(b) ∂u/∂y

0.01

-0.01 -0.01

0.005

-0.01 -0.01

0.01

Distance from the vortex center, X/R

-9.000

-0.005 0 0.005 Distance from the vortex center, Y/R

Distance from the vortex center, X/R

dvdx: -15.000 -12.000

0.01

dudz: -0.369 -0.228 -0.087 0.053 0.194 0.334 0.475

7.000 10.000 13.000 16.000

0.01

(a) ∂u/∂x

-0.01 -0.01

4.000

Distance from the vortex center, X/R

Distance from the vortex center, X/R

Distance from the vortex center, X/R

0.005

-0.01 -0.01

1.000

0.01

0.01

0.01

dwdz: -2.500 -0.500 1.500 3.500

0.005

0

-0.005

-0.01 -0.01

-0.005 0 0.005 Distance from the vortex center, Y/R

0.01

(i) ∂w/∂z

Figure 15: DPIV measurements of the nine velocity gradient tensors inside the tip vortex core the flow field of a micro-rotor using DPIV (Ref. 49), and those made behind a fixed-wing (Refs. 51, 52). While the bias pattern remains the same at all wake ages, the magnitude of the bias varies with wake age. Figure 17 shows the distribution of turbulence intensities obtained by making a horizontal cut across the centerline of the tip vortex at 2◦ wake age. Along the horizontal cut, the Cartesian components of the velocity u0 and v0 will be equivalent to Vr0 and Vθ0 , respectively, in polar coordi-

nates. Because of the bias of u0 (i.e., Vr0 in this case) along the horizontal cut, the Vr0 distribution is noticeably wider than the Vθ0 distribution. Also, both turbulence intensities reach a maximum value at the center of the vortex. This can be seen from the contour shown in Fig. 18. Furthermore, it can be seen that Vr0 > Vθ0 . This observation is of particular significance, and has been used by Chow et al. (Ref. 52) to explain the turbulence intensity bias observed in Fig. 16.

Page 11

(a) ζ = −4◦

(b) ζ = −2◦

(c) ζ = 0◦

Figure 16: In-plane measurements of turbulence intensities during the tip vortex roll up (Every third vector has been plotted to prevent image congestion)

Page 12

(d) ζ = 2◦

(e) ζ = 4◦

Figure 16: In-plane measurements of turbulence intensities during the tip vortex roll up (Cont.) Vθ0 transport shows that   ∂Vr Vθ 0 0 ∂Vr 0 0 02 0 +Vz Vr − V V Vr(prod.) = −2 Vr ∂r ∂z r r θ

(6)

and 0 Vθ(prod.)

Figure 17: Turbulence intensity distribution across the rotor tip vortex at ζ = 2” ◦ wake age.

Examining the turbulence production terms for Vr0 and

  ∂Vr ∂Vθ ∂Vθ 0 0 02 0 0 = −2 Vθ +Vz Vθ + V V (7) ∂r ∂z ∂r r θ

Comparing the two equations, the second term is the streamwise gradient of the in-plane velocities. This term is relatively small, and becomes even smaller when multiplied by shear stress. The velocity gradient in the first term is very small, mainly because the radial velocity is small, so the gradient of radial velocity becomes even smaller. However, the presence of a normal stress term (which is usually significantly larger than the shear stresses) does tend to compensate for the small gradient in radial velocity. The last term in both equations involves the shear

Page 13

(b) Vθ0

(a) V r0

Figure 18: Turbulence intensity distribution (in polar coordinates) across the rotor tip vortex at ζ = 4◦ .

(b) Vθ0

(a) V r0

Figure 19: DPIV measurements of the nine velocity gradient tensors inside the tip vortex core stress and swirl velocity gradient. Inside the vortex cores, the components Vθ /r, and ∂Vθ /∂r are very similar. The difference here, however, is the sign of the last term. While, this term is negative for Vr0 , it is positive for Vθ0 . While Vr0Vθ0 is usually assumed to be zero for solid body rotation, the present results show it to be predominantly negative. A non-zero Vr0Vθ0 will therefore increase the production of Vr0 and reduce Vθ0 , resulting in Vr0 > Vθ0 . This, in turn, explains the reason behind the turbulence intensity bias observed in u0 and v0 , as shown in Fig. 16. Figures 19(a) and (b) compare the turbulence intensities measured across the tip vortex at 30◦ obtained using DPIV and LDV. Good correlation can be seen between these two measurement techniques, even though the LDV measurements are not corrected for aperiodicity bias. Although procedures are available to correct the core size and peak swirl velocity in LDV, no such procedure is available for

turbulence measurements. However, at early wake ages, the magnitude of aperiodicity is relatively low and allows such a comparison to be readily made. Reynolds Stresses Figure 20 shows the distribution of Reynolds shear stress (u0 v0 ) and its associated strain (∂u/∂y + ∂v/∂x) from ζ = −4◦ (on the blade) to ζ = 4◦ . The reason to plot shear stress along with strain directly stems from the basic assumption in linear eddy viscosity-based turbulence models, where stress is represented as a linear function of strain. However, examination of the contours in Fig. 20 clearly suggests that this assumption is not valid, as has already been shown for curved stream lines (Refs. 32, 53). This is true regardless of the wake age. The magnitude of both the shear stress and strain continue to change with wake age, and eventually form a clear four-lobbed pat-

Page 14

___ u’v’ UV: -0.006 0.001 0.008

-0.02

Non-dimensional distance, Y/R

Non-dimensional distance, Y/R

0.02

0.01

0

-0.01

-0.02 -0.01

Strain rate: -15000

-7500

0

7500

15000

-0.01

0

0.01

0.02

0 0.01 0.02 0.03 Non-dimensional distance, X/R

-0.01

0 0.01 0.02 0.03 Non-dimensional distance, X/R

(a) ζ = −4◦ ___

UV: -0.006 0.001 0.008 u’v’

0.01

0

-0.01

-7500

0

7500

15000

-0.01

0

0.01

0.02

-0.02 -0.01

Strain rate: -15000

-0.02

Non-dimensional distance, Y/R

Non-dimensional distance, Y/R

0.02

-0.01

0 0.01 0.02 0.03 Non-dimensional distance, X/R

0 0.01 0.02 0.03 Non-dimensional distance, X/R

(b) ζ = −2◦ ___ u’v’ -0.017 -0.009 -0.002 0.006 0.013 UV:

-0.02

Non-dimensional distance, Y/R

Non-dimensional distance, Y/R

-0.02

-0.01

0

0.01

0.02 -0.01

Strain rate: -15000

-8182

-1364

5455

12273

-0.01

0

0.01

0.02

0 0.01 0.02 0.03 Non-dimensional distance, X/R

-0.01

0 0.01 0.02 0.03 Non-dimensional distance, X/R

(c) ζ = 0◦

Figure 20: Reynolds shear stress and strain at various wake ages.

Page 15

___ u’v’ UV: -0.006 0.0005 0.008

-0.02 Distance from the vortex center, Y/R

Distance from the vortex center, Y/R

0.02

0.01

0

-0.01

-0.02 -0.01

0

0.01

0.02

Strain-rate: -15601.7

-4156

7936.85

26767.7

-0.01

0

0.01

0.02 -0.01

0.03

0

0.01

0.02

0.03

Distance from the vortex center, X/R

Distance from the vortex center, X/R

(d) ζ = 2◦

0.01

0

-0.01

-0.02 -0.01

Strain rate: -19819

-0.02

Distance from the vortex center, Y/R

Distance from the vortex center, Y/R

0.02

__ u’v’ UV: -0.006 0.001 0.008

-1301

12588

-0.01

0

0.01

0.02

0 0.01 0.02 0.03 Distance from the vortex center, X/R

-0.01

0

0.01

0.02

0.03

Distance from the vortex center, X/R

(e) ζ = 4◦

Figure 20: Reynolds shear stress and strain at various wake ages (Cont.). tern, as early as ζ = 2◦ . These lobes, whose magnitudes alternate in sign, are aligned along the Cartesian coordinate axes for strain, and 45◦ with respect to the coordinate axes for shear stress. The contours also suggest the presence of significantly high levels of shear stress inside the vortex sheet at early wake ages, which can be expected based on the instantaneous turbulent activity seen previously in Fig. 7(b). Figures 21 and 22 show the Reynolds shear stress component (u0 v0 ) and its associated strain (∂u/∂y + ∂v/∂x) for a fully developed tip vortex at ζ = 60◦ wake age. The figure also includes stress and strain contours from two other experiments. While Figs 21(a) and 22(a) are from a fixedwing experiment performed in a wind tunnel (Ref. 52), Figs 21(b) and 22b show DPIV measurements on a microrotor. Qualitatively at least, it can be seen that the correlation between all the three experiments is excellent. Figure 23 shows a similar plot for another component

of Reynolds shear stress (v0 w0 ), and its associated strain. Here, the current measurements again correlate well with the fixed-wing measurements, qualitatively. Unlike u0 v0 , the v0 w0 term has only two lobes. However, the alignment of the lobes are still 45◦ offset from the coordinate axes. The associated strain also shows only two lobes, which are aligned with the y axis. This suggests that the orientation of all the shear stress distributions are 45◦ offset from the shear strain distribution, regardless of the the vortex Reynolds number or the type of lifting surface used. The importance of v0 w0 (the correlation term between the streamwise and the cross flow directions) can be understood from the momentum equation in the z direction given by,

Page 16

U

∂U ∂V ∂W +V +W ∂z ∂z ∂z

= −

1 ∂p ∂u0 w0 + ν∇2W − ρ ∂z ∂x

−

∂v0 w0 ∂w02 − ∂y ∂z

(which is typically assumed to be zero for solid body rotation) can be concluded to be the primary reason for the anisotropy.

(8)

The pressure gradient, which is positive during the tip vortex roll up (resulting in increased axial velocity deficit as the wake age increases) can be assumed negligible at older wake ages. Similarly, the effects of molecular viscosity (ν) can be considered negligible compared with the effects of eddy viscosity. This leaves the gradients of the stress terms u0 w0 , v0 w0 , and w02 to play a significant role in defining the axial momentum at any wake age. This is especially true at the centerline of the vortex, where the axial velocity deficit is maximum. Figure 23(b) evidently suggests that the gradient ∂v0 w0 /∂y is very high inside the vortex core. Such high gradients directly transfer momentum from the streamwise direction to the cross flow direction, resulting in the reduction of the peak axial velocity, as shown in Fig. 12(b).

3. Qualitatively, good correlation was found in the Reynolds shear stress distribution and strain rate between the current measurement and other measurements made on tip vortices trailing behind a microrotor, and a fixed- wing. This suggests that the turbulence pattern remains the same for all tip vortices, independent of the operating Reynolds number or the type of lifting surface on which they are measured. The results conclude that shear stresses cannot be written as a linear function of strain, as assumed in most of the existing linear eddy viscosity based turbulence models. 4. Excellent quantitative correlation found between LDV and DPIV measurements of turbulent intensities clearly suggest that DPIV can be confidently applied for turbulence measurements. Additionally, the use of 1000 samples in the case of DPIV was found to be sufficient, based on a comparison with 200,000 samples used in the case of LDV. This establishes the sample size requirement for statistical convergence of turbulent properties of tip vortices using DPIV.

Conclusions Comprehensive measurements in the flowfield of a subscale rotor operating in hover were performed using dualplane digital particle image velocimetry. This method allowed for the successful, simultaneous measurement of all nine velocity gradient tensors. The measurements concentrated on the shed vortex system behind the blade, and studied the tip vortex evolution from as early as ζ = −4◦ (on top of the blade) to 270◦ wake age. These dual plane measurements were complemented by high resolution, 2D DPIV measurements of the near-wake, and LDV wake measurement on the same rotor. The present measurements can be used to help validating CFD predictions as well as to calibrate new turbulence models. The following are the specific conclusions derived from this study: 1. High-resolution imaging of the vortex sheet trailing behind the rotor blade revealed the presence of several micro-scale, high frequency, counter-rotating Taylor-Gortler vortex pairs, which produce substantial fluctuations in the flow velocity. These fluctuations, combined with the high velocity gradients found in these vortices play a significant role in the momentum transfer properties of the boundary layer. Consequently, this affects the roll-up process of the tip vortices. 2. Turbulence intensity measurements clearly showed anisotropy. Specifically, Vr0 was greater in magnitude than Vθ0 in both the near- and far-wake. This is consistent with previous observations made on the tip vortices trailing a micro-rotor at very low vortex Reynolds numbers. The very presence of Vr0Vθ0

5. The measured mean characteristics of the tip vortices, such as their core size, peak swirl velocity, and their variation with time were found to correlate well with previous measurements made using LDV. The turbulent diffusion, especially, was found to be eight times that of molecular diffusion. The measured peak axial velocity deficit (corrected for apreriodicity bias effects) was found to be about 75% of the tip speed at the earliest wake age behind the blade. This reduced to 40% at 60◦ wake age. Such high values were not found in any previous literature because of the unavailability of any method to correct for the effects of aperiodicity. These high deficit values clearly suggest that aperiodicity plays a substantial role in axial velocity estimation.

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(c) Current

Figure 21: Reynolds shear stresses (u0 v0 ) at 60◦ wake age.

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Distance from the vortex center, Y/R

___ VW: v’w’ -0.004 -0.002 0.000 0.002

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-0.005

0.005 0 -0.005 Distance from the vortex center, X/R

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