Stability of Helical Vortex Structures Shed from Flexible Rotors
Steven N. Rodriguez, Justin W. Jaworski, John G. Michopoulos
SStability of Helical Vortex Structures Shed from Flexible Rotors
Steven N. Rodriguez a , Justin W. Jaworski b , John G. Michopoulos a a Computational Multiphysics Systems Laboratory, U. S. Naval Research Laboratory; Washington, DC, 20375, USA b Department of Mechanical Engineering and Mechanics, Lehigh University; Bethlehem, PA, 18015, USA
Abstract
The presented investigation is motivated by the need to uncover connections between underlying rotor fluid-structureinteractions and vortex dynamics to fatigue performance and characterization of flexible rotor blades, their hub, andtheir supporting superstructure. Towards this e ff ort, temporal stability characteristics of tip vortices shed from flexiblerotor blades are investigated numerically. An aeroelastic free-vortex wake method is employed to simulate the helicaltip vortices and the associated velocity field. A linear eigenvalue stability analysis is employed to quantify stabilitytrends (growth-rate v. perturbation wavenumber) and growth-rate temporal evolution of tip vortices. Simulations of acanonical rotor with rigid blades and its generation of tip vortices are first conducted to validate the stability analysisemployed herein. Next, a stationary wind turbine is emulated using the National Renewable Energy Laboratory 5MWreference wind turbine base design to investigate the impact rotor aeroelasticity has on tip vortex stability evolutionin time. Blade flexibility is shown to reduce the sensitivity of tip vortex destabilization to low wavenumber pertur-bations, also blade-pitch reduces growth-rate magnitude and alters the growth-rate peak dependence on perturbationwavenumber, all of which have in the past not been reported in the rotorcraft literature. The current investigationaims to develop insight into rotorcraft tip vortex kinematics and stability to work towards quantifying fatigue loadingand its e ff ects, minimizing adverse blade-vortex interaction e ff ects, such as excessive noise emission and large bladevibrations also a ff ecting fatigue performance. Keywords:
Tip vortices, helical vortex dynamics and stability, rotor near-wakes, rotor aeroelasticity
1. Introduction
Unlike the vortices shed from fixed-wing aircraft, the wake of a rotor consists of a helical vortex structure thatpersists within proximity of the rotor plane. The helical vortex structure is a byproduct of the lift distribution acrossthe span of the rotorblade and the mutual interaction of the vortex elements that rollup and generate strong tip androot vortices, as shown in Fig. 1. However, blu ff bodies at the center of rotors, such as nacelles on wind turbines, havehistorically been associated with the destabilization and dissipation of root vortices [42]. Therefore, tip vortices arecommonly the dominant aerodynamic structure in the wake of rotors and rotorcraft [4]. These tip vortices persist asstrong coherent structures that can impact rotor blade loading and their associated performance significantly [4, 17].As the wake ages and travels downstream of the rotor, the tip vortices destabilize and become part of a complexwake breakdown and transition into the far-wake region, in which inflow recovery begins, as shown in Fig. 2. Thesewake stages have been extensively studied for decades for a variety of applications that range from coaxial rotorcraftto wind farms [6, 8, 11, 16, 17, 20, 25, 27, 44]. The velocity field of the transition and far-wake regions have negligibleimpact on rotor blade loading and performance as compared to the near-wake region [17, 43]. The aerodynamics ofthe transition and far-wake regions is usually of interest for applications in which a collection of rotors must operatein the wake of others, such as in wind farms.Research concerned with the near-wake generally involves e ff orts aimed toward understanding the impact of near-by vortex structures on rotor performance and the vortex interactions with rotor blades [3, 7, 17, 29, 30, 37, 46]. Thesenear-wake investigations are mainly motivated by the need to understand, control and minimize adverse operationalimpact, such as excessive blade-vortex interaction noise, blade and supporting super-structure fatigue and materialdegradation, and poor rotor maneuverability. Furthermore, recent progress in additive manufacturing (AM) nowenables the manufacturing of components in the rotor blade-supporting superstructure, which further emphasizes the Preprint submitted to the Journal of Fluids and Structures
DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 1 a r X i v : . [ phy s i c s . f l u - dyn ] A ug V e r ti ca l ( z / D ) Tip VortexRoot VortexTrailing Vortex Sheet
Figure 1: Rotor near-wake aerodynamic vortex structures need to understand fatigue and material degradation concerns due to AM-induced porosity and surface roughness[14]. Consequently, this underlines the need for an e ff ort to understand how the aeroelastic loading is a ff ected by thevortex structures shed from flexible rotors.Exploration of the complex aerodynamics of rotor-wakes and associated temporal characteristics as the waketravels downstream of the rotor has generated a large field of active research problems that aim to improve rotortechnology and their physics involving inter alia turbulent rotor inflow modeling, near-wake and far-wake simulations,rotor-to-rotor near-wake interaction, rotor-wake instabilities, wake meandering, and wake stability [11, 17, 21, 38, 43,44]. Figure 2: Example of wake stages generated by the operation of a wind turbine
The current work is focused on the research of kinematics and stability of shed helical vortices in the near-wakeregion. Investigations into the stability of coherent helical vortex structures generally study the influence of twotypes of disturbances: short-wave and long-wave perturbations [28]. Short-wave perturbations are able to disturbvortex cores and are generally a byproduct of external strain fields or the curvature and torsion of the vortex itselfDISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 228]. The present study is concerned with long-wave perturbations, which are generated by external disturbances withcharacteristic lengths much larger than the vortex core, such as gusts and atmospheric turbulence [28].Fluid mechanical insight into the dynamics and stability of helical vortices has practical benefits in engineeringapplications, such as flow control, design and manipulation of early or prolonged wake breakdown, and the ability toanticipate uns and catastrophic wake behaviors in rotor operations, such as vortex-ring state in helicopter flight [18].The benefits associated with understanding helical vortex dynamics and stability have sustained research activityon this topic for almost a century. The earliest recorded e ff ort was conducted by Levy and Forsdyke [19], whichinvestigated the stability of a helical vortex filament subjected to long-wave sinusoidal perturbations. Widnall [45]rectified their incorrect treatment of the Biot-Savart singularity along the vortex line by using the cuto ff method andidentified three modes of helical vortex instability: short-wave, long-wave, and mutual-inductance modes. Guptaand Loewy [10] extended the work of Widnall to investigate the stability of multiple helical vortex filaments woundabout a common axis. Their work concluded that the magnitude of their stability trends (growth-rate as a functionof perturbation wavenumber) depends on the number of helical vortices in the domain. Specifically, an increase inthe number of helical vortex structures will result in an increase of growth rate levels. It was also found that thedivergence rates decreased as the perturbation wavenumbers increased. Decades later, following the advances innumerical and computational methods, Bhagwat and Leishman conducted a stability analysis of tip vortices generatednumerically by a helicopter rotor [3]. Bhagwat and Leishman determined that the stability trends depend on thenumber of intertwined helical vortices, as suggested previously by Gupta and Loewy. Bhagwat and Leishman alsospeculated that tip vortices are most unstable to perturbation wavenumbers equal to half-integer multiples of thenumber of helical vortex filaments, i.e. ω = N ( k − / ω is the perturbation wavenumber, N is the number oftip vortex filaments shed from rotor blades, and k = , , . . . , denotes any natural number. Similar findings as reportedby Bhagwat and Leishman were also found by Ivanell et al. [13] who used a large eddy simulation (LES) to study thestability of a wind turbine wake. More recent theoretical and numerical investigations into helical vortex dynamicsand their stability have led to similar conclusions that the near-wake is unconditionally unstable and that the mostunstable modes occur at wavenumber perturbations equal to half-integer multiples of the number of tip vortices in thedomain [24, 32, 33, 34].Experimental research of helical vortex dynamics and stability has historically lagged behind theoretical and nu-merical investigations due to the technological limitations of capturing the three-dimensional velocity field of thesevortex structures. However, advancements in flow visualization and tracking in the past decade have enabled experi-mental investigations to reach a state where they can examine the validity of the conclusions postulated by theoreticaland numerical studies. For example, Felli et al. [9] tracked the generation of the helical vortex structure of a two-bladed, three-bladed, and four-bladed propeller through velocity measurements and high-speed visualizations, andwere the first to observe the onsets of short-wave, long-wave, and mutual-inductance instabilities of helical vorticespredicted by Widnall [45]. Another recent study by Quaranta et al. [28] sought to conduct long-wave instability ex-periments of helical vortices for the purposes of comparing their results against classical theoretical works. Theirwork was generated by a single-bladed rotor, the velocity field and vorticity distributions were captured via particleimage velocimetry (PIV) measurements, and the vortex was visualized by applying fluorescent dye to the rotor-bladetip. Their works showed consistent agreement with classical stability analyses, i.e. the experimental helical vortexgrowth-rates caused by long-wave perturbations agreed well with the theoretical results of Widnall [45] and Guptaand Loewy [10]. It was also observed that helical vortices are extremely receptive to small-amplitude spatial-temporalperturbations, which further reinforces the theoretical conclusions that helical vortices are unconditionally unstable.The reader is referred to [9, 12, 22, 23, 28, 43] for more recent works on experimental helical vortex stability.Recent developments of helical-vortex dynamics and stability suggest that fundamental knowledge regarding sta-bility modes and mechanisms has begun to converge for simple, uniform, constant-pitch, and steady helical vortexstructures. However, near-wakes encountered in real rotor engineering applications are rarely simplistic in nature,and generally involve multiphysics phenomena, such as aerodynamic-elasticity (aeroelasticity), which introduce ad-ditional layers of complexity to tip-vortex dynamics and stability. The present investigation examines the motion andstability of more realistic vortex structures generated by the multiphysics behavior of flexible rotors. Specifically, thework aims to investigate the temporal stability characteristics of tip vortices that are captured in the near-wake regionthat deformed by rotor-blade aeroelasticity . Ultimately, the present work seeks to contribute fundamental knowledgeof realistic rotor near-wake dynamics in the e ff orts to improve rotor-wake applications and technology, such as tip-vortex flow control, and that can work towards minimizing adverse blade-vortex interaction e ff ects, such as excessiveDISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 3oise emission and large blade vibrations.The remainder of this paper is structured as follows. Section 2 reviews the free-vortex wake method that is used tosimulate rotor operation and to generate the helical wake. Section 3 introduces a linear-eigenvalue stability analysis,which is used to quantify temporal stability characteristics of tip vortices. This section also introduces a canonicalthree-bladed rotor configuration, with additional two- and four-bladed canonical rotor configurations presented inAppendix A, to validate the stability analysis presented in this paper against earlier investigations. Section 4 appliesthe stability analysis to tip vortices generated by a stationary symmetric (i.e., rotor-plane perpendicular to inflow)configuration of the National Renewable Energy Laboratory (NREL) 5MW reference wind turbine rotor with flexibleblades under a range of static rotor (non-rigid body rotor kinematics rotor) conditions, such as variable tip speed ratio,inflow speed, and blade pitch. Finally, the conclusions of this research paper are discussed in Section 6.
2. Free-Vortex Wake Method
The numerical model used to simulate rotor operation and generate the helical vortex system is based upon the free-vortex wake method (FVM) presented by Rodriguez [31] and Rodriguez and Jaworski [35, 36] developed to emulatethe NREL 5MW reference wind turbine and its aeroelastic performance. Their model coupled a linear-kinematic beamtheory for spinning structures to the free-vortex wake method code developed by Sebastian and Lackner, known asWInDS [39, 40]. To provide comprehensive context, the FVM framework employing both the Vatistas vortex model[17, 40] and vortex cut-o ff models [40] are now briefly reviewed. Note that in the context of this paper all vectors arenoted with a bold face, (i.e., x ) and matrices are noted as a bold faces inside square brackets (i.e., [ x ]).The flow physics modeled by the free-vortex wake method is governed by, d r dt = V ∞ + V induced + V rbm , (1)where the left hand side tracks the velocity of the discrete filaments in the wake, and V ∞ , V induced , and V rbm , are thefreestream velocity, the induced velocity, and the velocity generated by rigid body motions (rbm) of the rotor (suchas wave-induced motions of a floating o ff shore wind turbine), respectively. A uniform freestream velocity in theaxial direction is assigned and prescribed in the present study. The rigid-body motion velocity may be superimposedonto the rotor kinematics, as has been done in Rodriguez [31] and Rodriguez and Jaworski [35, 36] for wind turbineblades in an o ff shore environment. However, no such rotor motion is imposed in this investigation, i.e. V rbm = .The induced velocity is computed using the Biot-Savart law where the before-mentioned vortex core models areemployed. Two standard forms of the Biot-Savart law appear in the literature: the traditional form [17] and anotherthat is computationally more e ffi cient [26]. For the reader’s convenience we present both and provide the relationsused to arrive at either. Figure 3: Induced velocity relationship of an i th semi-infinite vortex filament The traditional form of the Biot-Savart law for semi-infinite filaments is V induced = Γ π (cid:90) l d l × r | r | (2)where r is the position along a semi-infinite vortex filament. For a straight-segment filament the Biot-Savart can beDISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 4ritten as V induced , i = Γ π h (cos θ − cos θ ) e , (3)where i indicates the i th filament, i.e. the induced velocity contribution of the i th vortex filament on a point in space.Geometric relationships of the i th filament configurations illustrated in Fig. 3 lead to the following expressions h = | l × r | l , cos θ = l · r l r , cos θ = l · r l r , e = l × r | l × r | , (4)where e is a column vector in R . Substitution of these expression into Eq. 2 and algebraic manipulation then yieldsEq. 5, as presented in [26], V induced , i = Γ π l × r | l × r | l · (cid:32) r r − r r (cid:33) . (5)Equation 5 can be rearranged to benefit numerical calculations by using the trigonometric relations l = r − r , r · r = r r cos θ , | r × r | = r r sin θ . (6)Substitution of these relations into Eq. 5 yields V induced , i = Γ π l × r | l × r | (cid:32) ( r − r ) · (cid:32) r r − r r (cid:33)(cid:33) = Γ π ( r + r ) ( l × r ) l r ( l r + l · r ) . (7)The form of the Biot-Savart law presented by Phillips and Snyder [26], Sebastian and Lackner [40], and Sebastian[39] includes the vortex cuto ff model [40] and the Vatistas vortex model [17, 40]: V induced , i = Γ π ( r + r )( l × r ) l r ( l r + l · r ) + ( δ l ) , if cuto ff model , C ν Γ π ( r + r )( l × r ) l r ( l r + l · r ) , if Vatistas model, (8)where C ν = ( l r ) − ( l · r ) l r nc + ( l r ) − ( l · r ) l n − / n . (9)The geometric relations in Eq. 4 enable Eq. 8 to be expressed in the traditional form as V induced , i = Γ π ( h + ( δ l ) ) (cos θ − cos θ ) e , if cuto ff model , Γ π h ( r nc + h n ) / n (cos θ − cos θ ) e , if Vatistas model. (10)The stability analysis presented in this investigation is derived from Bhagwat and Leishman [3], who used thetraditional form of the Biot-Savart law (Eq. 10 with the Vatistas model), which for consistency will be used for theremainder of this paper.
3. Methods for Stability and Dynamics Analysis
The stability of helical vortex structures generated by rotor systems is typically considered in the context of eithershort-wave perturbations or long-wave perturbations [28]. Short-wave perturbations disturb the vortex core structure,which may be generated by strain or torsion induced by a neighboring vortex. Long-wave perturbations consider thedisturbance of the local helical geometry as a whole, without perturbing the vortex core. Long-wave perturbationsmay arise from atmospheric turbulence or any wave disturbance much larger than the vortex core radius and are theperturbation type considered in this work.DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 5 .1. Linear-Eigenvalue Stability Analysis
The current investigation considers long-wave perturbations on the rotor wake of a stationary wind turbine rotor(i.e., no rigid body motion due to o ff shore wave-induced forcing). The analysis used in this work was originallydeveloped by Bhagwat and Leishman [3] to perturb the helicopter wake geometries harmonically in both space andtime. Their investigation evaluated the tip-vortex stability of a hovering rotor, in which the tip-vortex geometry wasgenerated by an FVM aerodynamic model. The current investigation also employs FVM. However, a major di ff erencebetween the FVM employed by Bhagwat and Leishman [3] and the FVM used herein is that the current frameworktakes consideration of the impact that the trailing vortex sheet and its temporal changes have on the production of thetip vortex and roll-up e ff ects. Hence, the wake geometry presented by this work is able produce additional physicalinsight into these e ff ects on the stability analysis.It is also important to note that Bhagwat and Leishman investigated perturbations of the tip vortices only, neglectedblade elasticity, and modeled a simple canonical rectangular rotor blade geometry. The work presented here will lookinto the stability of wakes generated by the flexible, non-uniform, and tapered rotor-blade geometry of the NREL rotorblade. However, the current stability analysis of the wake, like Bhagwat and Leishman, will only consider perturbingthe tip vortices in isolation, even though the wake geometry was generated by accounting for the presence of thetrailing vortex sheet.Due to the di ff erent modeling approach taken in this study to investigate tip-vortex stability, it can be anticipatedthat study will recover similar trends reported by Bhagwhat and Leishman, but will also bring to light any deviation ofthese stability trends due to geometric distortion of tip-vortices as a result of rotor-blade deformation. The derivationof the tip vortex stability analysis is now presented. The free-vortex wake method depends on computing the local velocity of Lagrangian markers cast into the wake.Hence, the induced velocity field is perturbed by displacing the wake geometry by a small quantity, δ r . To evaluatethe stability of the system, the governing equation are perturbed as follows: d ( r + δ r ) dt = V induced , i ( r + δ r ) → d r dt + d ( δ r ) dt = V induced , i ( r + δ r ) (11) δ ˙ r = V induced , i ( r + δ r ) − V induced , i ( r ) . (12)where the overdot in δ ˙ r represents the time derivative. The perturbed velocity can be expressed as an ordered seriesin δ r as follows, where quadratic and higher order terms can be neglected: V induced , i ( r + δ r ) = V induced , i ( r ) + δ V induced , i ( δ r ) + O (( δ r ) ) . (13)Substituting Eq. 13 into Eq. 12, yields the perturbed governing equation: δ ˙ r = δ V induced , i ( δ r ) . (14)The unperturbed velocity field is determined by the Biot-Savart law. Thus, substituting δ f = ∂ f ∂ x δ x , where f is somefunction of interest being perturbed that is dependent on a variable x , into the Biot-Savart law using the Vatistas orcuto ff model for semi-infinite straight filaments from Eq. 10 will yield the perturbed induced velocity field: V (cid:48) = ¯ Γ (cid:20) (cos θ − cos θ ) e + ( δ cos θ − δ cos θ ) e + h f δ h (cos θ − cos θ ) e + (cos θ − cos θ ) δ e (cid:21) (15)where for the Vatistas and cuto ff models h f = − ( h + ( δ l ) ) , if cuto ff model , h − − h n − r nc + h n , if Vatistas model , (16)DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 6nd ¯ Γ = Γ π ( h + ( δ l ) ) , if cuto ff model , Γ π h ( r nc + h n ) / n , if Vatistas model . (17)Note that V (cid:48) = V induced , i + δ V induced , i , and the spatial parameters, cos θ , cos θ , h , and e must all be perturbed to arriveat the governing perturbed induced velocity. The final expression for the perturbed induced velocity field requiresfurther simplification of the perturbed parameters δ h , δ (cos θ ), δ (cos θ ), and δ e , in terms of the position vectors, r A , r B , and r P . Applying this simplification yields the final form of the perturbed induced velocity on the point ofinterest, P , δ V induced , i = ¯ Γ { [ M ] δ r A + [ N ] δ r B + [ O ] δ r P } , (18)where δ r T A = [ δ r Ax , δ r Ay , δ r Az ] , δ r T B = [ δ r Bx , δ r By , δ r Bz ] , δ r T P = [ δ r Px , δ r Py , δ r Pz ] . The coe ffi cient matrices [ M ], [ N ], and [ O ] are now defined. First, consider the second term in Eq. 15 as( δ cos θ − δ cos θ ) e = ¯ A e T δ r A + ¯ B e T δ + ¯ P e T δ r P . (19)where ¯ A T1 = (cid:32) ∂ (cos θ ) ∂ r Ax − ∂ (cos θ ) ∂ r Ax , ∂ (cos θ ) ∂ r Ay − ∂ (cos θ ) ∂ r Ay , ∂ (cos θ ) ∂ r Az − ∂ (cos θ ) ∂ r Az (cid:33) , (20)¯ B T1 = (cid:32) ∂ (cos θ ) ∂ r Bx − ∂ (cos θ ) ∂ r Bx , ∂ (cos θ ) ∂ r By − ∂ (cos θ ) ∂ r By , ∂ (cos θ ) ∂ r Bz − ∂ (cos θ ) ∂ r Bz (cid:33) , (21)¯ P T1 = (cid:32) ∂ (cos θ ) ∂ r Px − ∂ (cos θ ) ∂ r Px , ∂ (cos θ ) ∂ r Py − ∂ (cos θ ) ∂ r Py , ∂ (cos θ ) ∂ r Pz − ∂ (cos θ ) ∂ r Pz (cid:33) . (22)The coe ffi cient matrices are collected as [ M ] = ¯ A e T , [ N ] = ¯ B e T , [ O ] = ¯ P e T . (23)Next, the third term in Eq. 15 is defined as h f δ h (cos θ − cos θ ) e = h f (cos θ − cos θ ) (cid:18) ¯ A e T δ r A + ¯ B e T δ r B + ¯ P e T δ r P (cid:19) , (24)where the coe ffi cient matrices are collected and defined as[ M ] = h f (cos θ − cos θ ) ¯ A e T , [ N ] = h f (cos θ − cos θ ) ¯ B e T , [ O ] = h f (cos θ − cos θ ) ¯ P e T , (25)using ¯ A T2 = (cid:32) ∂ h ∂ r Ax , ∂ h ∂ r Ay , ∂ h ∂ r Az (cid:33) , (26)DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 7 B T2 = (cid:32) ∂ h ∂ r Bx , ∂ h ∂ r By , ∂ h ∂ r Bz (cid:33) , (27)¯ P T2 = (cid:32) ∂ h ∂ r Px , ∂ h ∂ r Py , ∂ h ∂ r Pz (cid:33) . (28)Finally, the fourth term in Eq. 15 is defined as(cos θ − cos θ ) δ e = (cos θ − cos θ ) [ ¯ A ] δ r A + (cos θ − cos θ ) [ ¯ B ] δ r B + (cos θ − cos θ ) [ ¯ P ] δ r P . (29)where [ ¯ A ] = ∂ e x ∂ r Ax ∂ e x ∂ r Ay ∂ e x ∂ r Az ∂ e y ∂ r Ax ∂ e y ∂ r Ay ∂ e y ∂ r Az ∂ e z ∂ r Ax ∂ e z ∂ r Ay ∂ e z ∂ r Az , [ ¯ B ] = ∂ e x ∂ r Bx ∂ e x ∂ r By ∂ e x ∂ r Bz ∂ e y ∂ r Bx ∂ e y ∂ r By ∂ e y ∂ r Bz ∂ e z ∂ r Bx ∂ e z ∂ r By ∂ e z ∂ r Bz , [ ¯ P ] = ∂ e x ∂ r Px ∂ e x ∂ r Py ∂ e x ∂ r Pz ∂ e y ∂ r Px ∂ e y ∂ r Py ∂ e y ∂ r Pz ∂ e z ∂ r Px ∂ e z ∂ r Py ∂ e z ∂ r Pz . (30)the coe ffi cient matrices are then defined as [ M ] = (cos θ − cos θ ) [ ¯ A ] , [ N ] = (cos θ − cos θ ) [ ¯ B ] , [ O ] = (cos θ − cos θ ) [ ¯ P ] . (31)Finally, the global coe ffi cient matrices of Eq. 18 are obtained by summing the appropriate coe ffi cient matrices:[ M ] = [ M ] + [ M ] + [ M ] , [ N ] = [ N ] + [ N ] + [ N ] , [ O ] = [ O ] + [ O ] + [ O ] . (32)Substituting Eq. 32 into Eq. 18, defines the perturbed induced velocity field completely. Now that the perturbed induced velocity is expressed by its independent parameters, the type of perturbation to beused on the Lagrangian marker velocity must be defined, i.e., the left hand side of Eq. 14. For this analysis the wakeis perturbed in a harmonic fashion. In other words, the form of the perturbation is assumed to be a travelling wavesuch that a series of normal mode perturbations could describe any arbitrary disturbance [3]. These perturbations areset in cylindrical coordinates for convenience, and their amplitudes are denoted by δ . The harmonic perturbation isdefined by the following expression: δ p k = δ x δ r δθ e α t + i ωζ k = δ e α t + i ωζ k , (33)where k = A , B , or P , which correspond to endpoints of the straight segment vortex filament and a point in space, asshown in Fig. 3. It is important to note here that i = √− T ] = θ − r sin θ θ r cos θ , (34)where the harmonic perturbation Eq. 33 is transformed to Cartesian coordinates as follows δ r A = [ T ] δ p A , δ r B = [ T ] δ p B , δ r P = [ T ] δ p P . (35)DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 8 igure 4: Illustration of a radial perturbation at ω = . − Figure 5: Illustration of an axial perturbation at ω = − Figure 6: Illustration of an azimuthal perturbation at ω = . − Also note the corresponding transformation matrix expressions:˙ r k = [ T ] ˙ p k , (36) δ ˙ r k = [ T ] δ ˙ p k + [ T ] δ p k , (37)DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 9 T ] = − ˙ θ sin θ − ˙ z θ cos θ ˙ y , (38)cos θ = y (cid:112) y + z , sin θ = z (cid:112) y + z , ˙ θ = y ˙ z − z ˙ yy + z . (39) A complete set of equations has now been obtained (Eqs. 18, 37) that is necessary to formulate an eigenvaluestability analysis. First, substitute Eq. 37 into the left hand side of the governing perturbed equation, Eq. 14, δ ˙ r k = [ T ] δ ˙ p k + [ T ] δ p k = ( α [ T ] + [ T ]) δ e α t + i ωζ k . (40)Next, substituting Eq. 35 into Eq. 18 will give δ V induced , i = ¯ Γ ([ M ] δ r A + [ N ] δ r B + [ O ] δ r P ) = ¯ Γ (cid:16) [ M ][ T ] δ e α t + i ωζ A + [ N ][ T ] δ e α t + i ωζ B + [ O ][ T ] δ e α t + i ωζ P (cid:17) = ¯ Γ (cid:16) [ M ] e i ω ( ζ A − ζ P ) + [ N ] e i ω ( ζ B − ζ P ) + [ O ] (cid:17) [ T ] δ e α t + i ωζ P = [ V i ][ T ] δ p P , (41)where by only considering the real contribution of e i ω ( ζ A − ζ P ) and e i ω ( ζ B − ζ P ) [ V i ] = ¯ Γ (cos ω ( ζ A − ζ P )[ M ] + cos ω ( ζ B − ζ P )[ N ] + [ O ]) . (42)The subscript i denotes the i th filament (the segment between r B and r A ). The superposition of each individual fil-ament’s perturbed induced velocity field on a point of interest is expressed as δ V = (cid:80) i δ V induced , i . Likewise, theexpression [ V ] = (cid:80) i V i is the total perturbed induced velocity field in cylindrical coordinates on the point of interest.Substituting Eqs. 40 and 41 into Eq. 14 yields, after some rearrangement, αδ r P = (cid:16) [ V ] − [ T ][ T ] − (cid:17) δ r P . (43)Setting [ W ] = (cid:16) [ V ] − [ T ][ T ] − (cid:17) and rearranging Eq. 43 further gives the final form of the eigenvalue problem foreach point of interest along the tip vortex: ([ W ] − α [ I ]) δ r P = , (44)where solving for the maximum α along the age of the tip vortex, at a specified radial wavenumber ω and instant intime t , produces the stability trend, ω v. α , of the tip vortices. Validation of the stability analysis is performed on a three-bladed rotor with rectangular blades. Further validationis performed on two and four-bladed rotors in which their data is available in Appendix A. The rotor specificationsare listed in Table 1.Simulations of the canonical rotor were generated using the aeroelastic free-vortex wake method presented byRodriguez and Jaworski [35, 36], but rotor blades are constrained to be rigid for this validation case. Simulationsfor each rotor configuration were performed over 10 seconds with vortex shedding frequencies f = , ,
30 Hzat an inflow condition of V ∞ =
18 m / s, tip-speed ratio (blade tip-speed / inflow speed) λ =
9, and blade pitch angle θ bl = ◦ . Snapshots at t =
10 s of these simulations at f =
20 Hz for the Vatistas models ( n = , , f = ff models ( δ = . , − , − ) are shown in Fig. 7, where axis labels X = x / D and Z = z / D arethe nondimensional distances, where x , z , and D are distances in the downstream direction, distance in the verticaldirection, and the rotor diameter, respectively. Finally, stability analyses were performed on the generated tip vortices.DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 10 able 1: Canonical rotor specifications modified from [15] Rotor orientation and configuration: Upwind; 2, 3, 4 BladesRotor and hub diameter: 126 m, 3 mAirfoil: NACA64-A17Chord length: 4.5 mBlade twist: 0 ◦ Rotor plane tilt: 0 ◦ It is important to note that previous work by Rodriguez and Jaworski [34] validated their stability analysis onlyagainst peak divergence trends of the flexible NREL 5MW reference wind turbine rotor. The current validationwork supersedes previous validation attempts by Rodriguez and Jaworski [34] such that the current analyses confirmsclassical stability trends and vortex behaviors of symmetric, multi-bladed configurations, with variable finite-corevortex modeling [3].
Before proceeding to the validation of the stability analysis, a few notes on the numerical simulations must be pre-sented. Rodriguez and Jaworski [35, 36] showed that the FVM aeroelastic framework employed herein is capable ofreproducing accurate, stable, and robust rotor-blade and rotor-wake performance metrics of the NREL 5MW referencewind turbine rotor reported by Jonkman et al. [15], such as blade forces, blade deflection, and rotor thrust. The aero-dynamic framework used in Rodriguez and Jaworski [35, 36], initially developed by Sebastian [39], has capabilitiesof faithfully generating the near-wake geometry of rotor wakes as validated in Sebastian [39]. However, convectionof Lagrangian markers can become numerically unstable as discussed in Bagai and Leishman [2] and Rodriguez [31]especially as the number of rotor-blades increase which can be seen qualitatively in Figs. 7, A.23, and A.27. This nu-merical issue is associated with numerical instabilities that have been studied by Leishman et al . [1, 2, 3]. Specifically,depending on the numerical method employed, the discretization of the induced velocity field can result in the additionof anti-dissipative terms, which can lead to exponential growth of non-physical disturbances caused by roundo ff error[18]. To circumvent non-physical disturbances caused by numerical artifacts, sophisticated FVM-specific numericaltechniques, such as those presented in [1, 2, 3, 5], can be implemented to numerically stabilize the wake geometry.The present investigation takes an alternate approach introduced by Rodriguez [31] to avoid artificial disturbancesthat may corrupt the stability analysis of tip vortices. Rodriguez [31] has shown that there are regions along thetip vortex where a stability analysis can be performed where the e ff ects of artificial numerical instabilities can beminimized by truncating the wake downstream of the rotor. These regions are defined by truncating segments of thetip vortex which have already been dominated by artificial numerical instabilities. Rodriguez [31] also showed thatusing the stability analysis presented herein, one can quantitatively and qualitatively find the location of the onset ofthe numerical wake breakdown. The current validation analysis is conducted on one rotation (a wake age of ζ = π )of the tip vortices as they are shed o ff of the rotor blades, where ages longer than one full rotation are truncated. Three conditions must be met to validate the stability analysis as presented. The stability trends (divergencerate versus perturbation wavenumber, α v. ω ) must obey the following classical results: 1) peak divergence ratesoccur at perturbation wavenumbers ω = N b (cid:16) k − (cid:17) , 2) divergence rates must converge to a constant value as thewavenumber goes to infinity, 3) because the canonical rotor configuration generates symmetric results, it is requiredthat the stability trends of each individual tip vortex be identical. In addition to these requirements, it can also beanticipated that stability trends have larger divergence rates as the number of blades increase, which was observed byBhagwat and Leishman [3].DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 11 lade 1Blade 2Blade 3 Figure 7: Three-bladed rotor wakes modeled by employing the Vatistas core model (first row) and the cuto ff model (second row). Sheddingfrequency is f =
20 Hz.
Stability results are now presented for the tip vortices produced by the canonical rotor. Figure 7 presents sideviews of the three-bladed rotor and its wake, where the Vatistas core modeling and cuto ff core modeling have beenused where indicated. The two-bladed and four-bladed rotor wakes are presented and briefly discussed as auxiliarystudies in Appendix A. First, the stability trends ( α v. ω ) are presented in Figure 8 for the Vatistas model and cuto ff model at variable vortex shedding frequencies. Perhaps the most notable feature of Fig. 8 is the periodic behaviorin divergence rate oscillations as a function of wavenumber. Classical studies [3, 10, 45] have traditionally lookedat low wavenumbers that are consistent with the underlying long-wave perturbation qualitative assumption, i.e. long-wave perturbations are defined as a perturbations much larger than the vortex core. However, because of the long-wave assumption, no investigation, to our knowledge, has presented stability trends for a high range of wavenumberperturbations to visualize the limits of long-wave perturbation analyses.It is believed that this periodic behavior is caused by both the long-wave perturbation analysis and a numericalartifact from the time-marching scheme (second-order Runge-Kutta) of the free-vortex wake method currently usedbased on the following observations. Notice that for a relatively low vortex shedding frequency, such as f =
10 Hz(blue trends in Fig. 8), the periodic behavior has lower amplitude and lower period than at shedding frequencies of f =
20 and 30 Hz. Higher shedding frequencies of f =
20 and 30 Hz show a high bell-shaped divergence rate trend.It is also observed that the width of this “bell” increases with the shedding frequency. This result suggests that ifthe shedding frequency approaches infinity then the rotor-wakes simulated by the current free-vortex wake methodwould be most sensitive to high wavenumbers, i.e. short-wave perturbations. Finally, the bell curve maxima occur atapproximately the same location ( ω ≈
25 (1 / rad) at f =
20 Hz; and ω ≈
37 (1 / rad) at f =
30 Hz) for rotor wakegeometries employing the Vatistas model, which further reinforces the conjecture that the periodic bell-curve is anartifact of the long-wave perturbation analysis, the numerical integration, and their relationship to the Vatistas finitecore modeling.At wavenumbers between 0 ≤ ω ≤ . − , the stability trends generated by the Vatistas models ( n = , , and3)with shedding frequencies f =
10 and f =
20 Hz show very good agreement with reported classical stability trends,DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 12 a) Vatistas vortex core model, n = n = n = ff vortex core model, δ = − Figure 8: Stability trends for one tip vortex shed from a three-bladed rotor at variable vortex shedding frequencies (10, 20, 30 Hz) with a one-rotation(2 π ) tip vortex window. Vatistas models for n = , , and 3 are employed, and the cuto ff model is presented with a radius of δ = . i.e., divergence-rate peaks occur at ω = N b ( k − / k is any natural number. However, at f =
20 and 30Hz, the stability trends do not exhibit the correct behavior beyond the second divergence rate peak. In fact, thestability trend becomes erratic as wavenumbers increase and approach the bell. This result may indicate that theshedding frequency has neighboring vortices spaced too closely to remain numerically stable to record the vortexstability trends reported by classical studies with the Vatistas model. Limitations of the Vatistas model with regardto capturing the classical stability results at higher vortex shedding frequencies is further highlighted by the cuto ff vortex model results in Fig. A.28d. The cuto ff model shows very consistent oscillations of the divergence rates for allvortex shedding frequencies. It appears that as the vortex frequency increases the period of the bell behavior tends toinfinity. This dependence on vortex shedding frequency suggests that an infinite vortex shedding frequency is requiredto achieve classical stability trend results for the cuto ff model. The absence of the bell as seen for the Vatistas modelsalso implies that the cuto ff model is not susceptible to short-wave (high wavenumber) numerical instabilities.The di ff erences between the Vatistas and cuto ff models motivated further investigation into reducing the cuto ff radius, which becomes the Rankine vortex as δ →
0. The cuto ff radius reduction results are presented in Fig. A.25for a vortex shedding frequency of f =
30 Hz. Results show marginal di ff erences between cuto ff radii as δ → ff radii further reinforces the notion that the Vatistas core modelmay be more sensitive to short-wave numerical instabilities.The above stability trends are for single tip vortex shed from one blade. Recall that the stability trends for each tipDISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 13 igure 9: Stability trends for the three-bladed rotor wake with shedding frequency set at f =
30 Hz and reducing the cuto ff radii to δ → vortex on a multi-bladed rotor must be identical for the analysis to be valid. The stability trends for each individualtip vortex for a three-bladed rotor are presented in Fig. 10. Here it is demonstrated that the stability trends for the tipvortex from each blade are identical, as expected due to the symmetric rotor configuration. For both the Vatistas andthe cuto ff models, the divergence rates peak at perturbation wavenumbers ω = N b ( k − / ω = . − . After ω = . − ) peak divergence ratescorrespond to wavenumbers that are shifted forward by some value (cid:15) (i.e., peaks occur at some ω = N b ( k − / + (cid:15) ),which deviates from the classical stability trend. The resulting stability trends for the three-bladed canonical rotor (and corresponding two- and four-bladed rotorsseen in the Appendix A) satisfy generally the criteria established at the beginning of Section 3.1.6 to validate thestability analysis presented. However, it is important to remember a few numerical caveats: First, vortex sheddingfrequencies of the free-vortex wake method impact stability trends. For the Vatistas model, too high of a vortexshedding frequency will deconstruct the classical stability trend. However, for the vortex cuto ff model, no such de-construction of the classical stability trend occurs. Second, the present stability analysis employing a wake generatedby the Vatistas model or the cuto ff model and the current time integration scheme only satisfies the classical stabilitytrend for low wavenumbers (low wavenumbers will be identified after this discussion). Third, stability trends obtainedfrom wakes employing the Vatistas model experience large periodic divergence (the bell behavior) as the perturbationwavenumber increases. Stability trends from di ff erent rotor configuration are overlayed in Fig. A.31. It is made clearfrom Fig. A.31 that the Vatistas model would best reconstruct the classical stability trends as n → ∞ , which wouldgenerate the Rankine model and would be identical to the cuto ff model for δ →
0. However, for consistency withprevious aeroelastic works of Rodriguez and Jaworski [35] and Rodriguez [31], Rodriguez and Jaworski [36], thestability analyses from now on will employ the Vatistas finite core model with n =
4. Stability and Dynamics of Tip Vortices Shed from Flexible Rotors
Tip vortices shed from the NREL 5MW Reference Wind Turbine rotor, with a rigid rotor, and a flexible rotor, arenow investigated. Three cases spanning low-level and high-level operating conditions of the NREL 5MW referencewind turbine are considered to evaluate a range of rotor aeroelasticity and its impact on tip vortex dynamics andstability. The operational conditions considered in the current work were first presented in [41]. The tip-speed ratio λ (defined as blade-tip speed V ∞ ), blade pitch θ bl , inflow velocity V ∞ , and rotor diameter-based Reynolds number Re D = ρ DV ∞ µ ,where for all cases ρ = .
23 kg / m , µ = . × − kg / (m · s), D =
126 m [15], are used to describe case conditions.Table 2 lists the parameters for all cases. In addition, tip vortices shed from a symmetric rotor configuration (rotor-plane is perpendicular to the horizontal uniform inflow) and the rotor operational configuration as specified by NRELDISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 14 a) Vatistas vortex core model, n = n = n = ff vortex core model, δ = − Figure 10: Stability trends for all the individual tip vortices shed from a three-bladed rotor from Vatistas and cuto ff finite core models (rotor-plane at 5 ◦ tilt) will be investigated. All simulations were run for 60 s with a vortex shedding frequency of f =
12 Hz, using the Vatistas finite core model with n =
2. As in the canonical rotor case, simulation snapshotsare presented with X = x / D and Z = z / D axis labels, where x , z , and D are distances in the downstream direction,distance in the vertical direction, and the rotor diameter, respectively. The stability analyses employed the windowcuto ff specified by Rodriguez [31], such that case 1 employs a 2 π window due to small tip vortex pitch rate, and case2 and 3 employ a 4 π window due to a larger tip vortex pitch rate. Thus, it is important to remember that by performingthe stability analysis on a windowed portion of the wake, conclusions to be drawn apply only to early ages of tipvortices and not as a whole vortex structure that includes far-wake regions. To satisfy the window specifications, thestability analyses are conducted on tip vortices between simulation instants t =
20 s and t = First, case 1 the NREL 5MW wind turbine is investigated. Snapshots of the symmetric and operational configura-tions at t =
60 s are shown in Fig. 11. The di ff erence in rigid symmetric and operational configurations is highlightedin the location at which the numerical onset of qualitative wake breakdown occurs. The symmetric configurationappears coherent approximately between 0 ≤ X ≤
1, whereas the operational configuration maintains coherence ap-proximately between 0 ≤ X ≤ able 2: Parametric conditions considered λ V ∞ m / s θ bl ◦ Re D Case 1: 9.63 6 0 7.75 × Case 2: 7 11.4 0 14.7 × Case 3: 4.43 18 15 23.2 × blade deflection and its subsequent influence on the formation of the tip vortices. However, for both flexible rotor con-figurations (symmetric and tilted) the location at which the numerical wake break-down begins is approximately thesame, which indicates that rotor-configuration is not a major contribution to wake-breakdown at relatively low-inflowconditions. This result is likely due to the wake breakdown being largely dominated by the computational start-uptransients of the blade deflection rather than being dominated by periodic aerodynamic loading, as discussed in [35]. Rigid rotorFlexible rotor
Blade 1Blade 2Blade 3 (a) Symmetric rotor configuration
Rigid rotorFlexible rotor
Blade 1Blade 2Blade 3 (b) Operational rotor configuration
Figure 11: Case 1 tip-vortex snapshot at t =
60 s for rigid and flexible rotor operation
The qualitative evaluation of the tip vortices above is reinforced by the stability analysis performed on the snapshot t =
40 s in Fig. 12. As expected, via the classical stability trend criteria, the maximum eigenvalue for all configura-tions as shown in Fig. 12 corresponds to a perturbation wavenumber of ω = . − . The operational configurationsexhibit slightly higher peaks than the symmetric rotor configuration counterparts, which bolsters the notion that oper-ational configurations generate more unstable tip vortices for rigid rotors. However, it is interesting to note in Fig. 12that wake breakdown appears to occur at similar locations downstream, despite the flexible rotor operational config-uration having larger maximum eigenvalues than the flexible rotor symmetric configuration. It is important to note,though, that similar locations of wake breakdown may be a result from the initial transient perturbations from bladedeformation that dominate the wake formation downstream. However, as time progresses the “windowed” stabilityanalysis captures the inherent tip vortex stability of each corresponding rotor configuration, which results in highermaximum eigenvalues for operational rotor configurations in the 4 π window.To further highlight the impact blade flexibility has on tip vortex stability, eigenvalues corresponding to pertur-bation wavenumbers ω = . , . . − , i.e. N b = k = , , ω = N b ( k − / (rad -1 ) ( s - ) Rigid rotor
Blade 1Blade 2Blade 3 (rad -1 ) ( s - ) Flexible rotor
Blade 1Blade 2Blade 3 (a) (rad -1 ) ( s - ) Rigid rotor
Blade 1Blade 2Blade 3 (rad -1 ) ( s - ) Flexible rotor
Blade 1Blade 2Blade 3 (b)
Figure 12: Case 1 stability trend snapshots at t =
40 s of tip vortices shed from rigid and flexible (a) symmetric rotor and (b) operational rotorconfigurations. configuration, it is shown in Fig. 20a that within the 2 π window of tip vortices under consideration, blade flexibilityinitially generates less unstable (lower positive eigenvalues) tip vortices than rigid rotors. As time progresses, theeigenvalues generated from the stability of rigid rotor tip vortices tend to approach the temporal eigenvalue character-istics of tip vortices generated by flexible rotors. The impact of blade flexibility on the stability of tip vortices is likelyan indication that aeroelastic interactions allow the wake to more quickly approach its equilibrium state.The conjecture presented earlier that operational rotor configurations break down earlier than symmetric rotor con-figurations is supported by Fig. 20a, which shows that operational configurations periodically reach higher maximumeigenvalues ( α corresponding to ω = . − ) than symmetric operational configurations. The periodic behaviorseen in Fig. 20a, and is not present in the symmetric configuration, corresponds to a change in the angle-of-attackof individual blades as they pass through a full rotation in the tilted rotor plane. Figure 20b further illustrates theout-of-phase behavior of the maximum eigenvalues (corresponding to perturbation wavenumber ω = . − ) ofeach tip vortex shed o ff of individual blades. It is seen that the each individual tip vortex contains the same stabilitytrend but with a phase shift that corresponds to the periodic change in angle of attack of each rotor blade as it movesaround the tilted rotor plane.The impact that the operational configuration has on stability trends is highlighted by the fast-Fourier transform(FFT) of eigenvalues corresponding to ω = . − , in Fig. 14. Both symmetric (14a) and operational (14b) config-urations are compared to evaluate rotor-plane tilt impact on tip vortex stability. The FFT for the rigid symmetric rotorconfiguration reflects a low-frequency contribution of f ≈ .
02 Hz that is not trivial to interpret. This contribution isseen in Fig. 20a where the time-history of the eigenvalue shows a slight decay and growth, which may likely be causedby the meandering of the initial transient e ff ects of the simulation, i.e., the dynamics of the agglomeration of filamentsdownstream in Fig. 11. The flexible symmetric rotor configuration also exhibits the low-frequency contribution of f ≈ .
02 Hz but also exhibits a significant contribution of f ≈ . f = . f = Ω / π ≈ .
14 Hz. This frequency was first presented by Rodriguez and Jaworski [34]and then by Rodriguez [31], where it was determined that stability trends fluctuate at a period
Λ = π/ Ω . However,DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 17 a)
10 20 30 40 50 60 70time (s)0.020.0250.030.0350.040.0450.05 ( / s ) Rigid Blade 1Rigid Blade 2Rigid Blade 3 Flexible Blade 1Flexible Blade 2Flexible Blade 3 (b)
Figure 13: Case 1 time history growth rates: a) eigenvalues corresponding to perturbation wavenumbers ω = − for symmetric andoperational rotor configurations(legend provides color scheme of rotor configuration and dashed lines represent time-history of growth-rates fromflexible blades)); and b) operational configuration eigenvalues corresponding to ω = − for all three blades (solid lines = rigid rotors, dashedlines = flexible rotors). Rodriguez and Jaworski [34] and Rodriguez [31] concluded the stability trend fluctuation was a byproduct of the bladepassing frequency, and not the rotor plane tilt. Figure 14 shows that the fluctuating frequency, f = Ω / π , correspondsto the impact of the operational configuration (rotor plane tilt) on the rotor blade angle of attack and stability trends,and does not correspond to the rotation rate alone. || (r a d - ) -4 Flexible rotorRigid rotor (a) || (r a d - ) -3 Flexible rotorRigid rotor (b)
Figure 14: FFT-derived frequency spectra of eigenvalue time history signal for perturbation wavenumber ω = . -1 : a) symmetric rotorconfiguration; b) operational rotor configuration DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 18 .2. Case 2
Case 2 conditions are now investigated. Snapshots of the symmetric and operational configurations at t =
60 sare shown in Fig. 15. Di ff erences in wake breakdown location between rigid symmetric and operational rotors areless visible qualitatively than case 1 rotor-wakes. Di ff erences in wake breakdown location are di ffi cult to identifylikely because of the decrease in tip-speed ratio, which increases tip-vortex to tip-vortex spacing (pitch), as therebygenerating less unstable vortices and mitigating the e ff ect of perturbation propagation across tip vortices comparedto case 1 operation. However, one major di ff erence between symmetric and operational wakes is the transition intothe agglomeration of tip vortices in the far wake. The symmetric configuration exhibits a smooth transition intothe far wake as opposed to the operational configuration which exhibits distortion in tip vortices as early as X ≈ . Rigid rotorFlexible rotor
Blade 1Blade 2Blade 3 (a) Symmetric rotor configuration
Rigid rotorFlexible rotor
Blade 1Blade 2Blade 3 (b) Operational rotor configuration
Figure 15: Case 2 tip-vortex snapshot at t =
60 s for rigid and flexible rotor operation
Snapshots at t =
40 s of the stability analysis performed on tip vortices for two rotations (4 π window) are presentedin Fig. 16. Maximum eigenvalues occur at the expected perturbation wavenumber of ω = . − . The rigid rotoroperational configuration exhibits higher eigenvalue peaks than its symmetric rotor counterpart, which again supportsthe notion that operational configurations tend to further destabilize tip vortices. However, it is interesting to notein Fig. 16 that wake breakdown appears to occur at similar locations downstream in Fig. 15, despite the flexiblerotor operational configuration having larger maximum eigenvalues than the flexible rotor symmetric configuration.It is important to note, though, that similar locations of wake breakdowns may be a result from the initial transientperturbations from blade deformation that dominate the wake formation downstream. However, as time progressesthe “windowed” stability analysis captures the inherent tip vortex stability of each corresponding rotor configuration,which results in higher maximum eigenvalues for operational rotor configurations in the 4 π window. Furthermore,the stability analysis for flexible rotors result in distorted (non-smooth) stability trends as perturbation wavenumbersincrease, which was not as prevalent in case 1 and is likely due to blade vibrations. The level may be an indicator thatblade flexibility may reduce, shift, or distort tip vortex instability at specified perturbation frequencies.The time histories of eigenvalues corresponding to perturbation wavenumbers ω = . , . . − , i.e., N b = k = , , ω = N b ( k − /
2) are now tracked in time for symmetric (sym.) and operational (op.)configurations in Fig. 17. Independent of the configuration, Fig. 17a shows that within the 4 π window of tip vorticesunder consideration, blade flexibility enables less unstable (lower positive eigenvalues) tip vortices at ω = . − than rigid rotors. As time progresses, the presence of blade flexibility may generate tip vortices that are more sensitiveDISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 19 (rad -1 ) ( s - ) Rigid rotor
Blade 1Blade 2Blade 3 (rad -1 ) ( s - ) Flexible rotor
Blade 1Blade 2Blade 3 (a) (rad -1 ) ( s - ) Rigid rotor
Blade 1Blade 2Blade 3 (rad -1 ) ( s - ) Flexible rotor
Blade 1Blade 2Blade 3 (b)
Figure 16: Stability trend snapshots at t =
40 s of tip vortices shed from rigid and flexible symmetric rotor (left column) and operational rotorconfigurations (right column).
DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 20o higher wavenumber perturbations as shown in the time histories for stability trends for ω = . . − ). Thesame higher wavenumber sensitivity was seen in tip vortices for conditions in case 1, but for case 2 this sensitivity ismore pronounced in Fig. 17.Figure 17a quantitatively supports the qualitative observation stated earlier that operational rotor configurationsbreakdown earlier than symmetric rotor configurations. As in case 1, it is shown in Fig. 17a that operational configura-tions periodically reach higher maximum eigenvalues ( α corresponding to ω = . − ) than symmetric operationalconfigurations. This periodic behavior seen in Fig. 17a, not present in the symmetric configuration, corresponds to achange in the angle-of-attack of individual blades as they pass through a full rotation in the tilted rotor plane, as wasalso observed in case 1 operation. Figure 17b further illustrates the out-of-phase behavior of the maximum eigenvalues(corresponding to perturbation wavenumber ω = . − ) of each tip vortex shed o ff of individual blades.
10 20 30 40 50 60 70time (s)0.020.030.040.050.060.070.08 ( / s ) (a)
10 20 30 40 50 60 70time (s)0.030.040.050.060.070.08 ( / s ) Rigid Blade 1Rigid Blade 2Rigid Blade 3 Flexible Blade 1Flexible Blade 2Flexible Blade 3 (b)
Figure 17: Case 2 time history operation growth rates: a) eigenvalues corresponding to perturbation wave numbers ω = -1 forsymmetric and operational rotor configurations (legend provides color scheme of rotor configuration and dashed lines represent time-history ofgrowth-rates from flexible blades); b) Operational configuration eigenvalues corresponding to ω = -1 for all three blades The frequency spectra of the eigenvalue time-histories for case 2 symmetric and operational rotor configurationsis shown in Fig. 18. For the symmetric rotor configuration, the FFT shows a dominant low-frequency contribution thatwas also seen in case 1. As mentioned previously, identifying this low-frequency contribution is not trivial but seemsto be related to the agglomeration of tip vortices downstream. This low frequency contribution is seen in Fig. 20athrough a small and almost negligible eigenvalue fluctuation in time. For the symmetric flexible rotor configuration,the FFT is showing a secondary frequency of f ≈ . f ≈ . f = Ω / π ≈ . Case 3 of the NREL 5MW wind turbine is now investigated. Snapshots of the symmetric and operational configu-rations at t =
60 s are shown in Fig. 19. The generation of tip vortices presented in Fig. 19 exhibit much more coherentand smooth helical vortex geometry than any of the other operational conditions (case 1 and 2) for both symmetric andtilted configurations. The coherence of the tip vortex geometry is attributed to the decrease in tip-speed ratio, whichincreases the distance between adjacent vortices (helical pitch) that mitigates the influence of perturbation propaga-tion. Unlike prior cases 1 and 2, it is di ffi cult to qualitatively highlight in case 3 the influence of rotor-plane tilt onthe stability of tip vortices as both symmetric and tilted configurations maintain a very coherent tip vortex structure.Therefore, no clear qualitative conjecture can be made about the influence of operational configurations on numericalDISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 21 || (r a d - ) -3 Flexible rotorRigid rotor (a) Vatistas core model, n = || (r a d - ) -3 Flexible rotorRigid rotor (b) Vatistas core model, n = Figure 18: FFT-derived frequency spectra of eigenvalue time history signal for perturbation wave number ω = . -1 : a) symmetric rotorconfiguration; b) operational rotor configuration and qualitative wake break down. Similarly, the influence of blade flexibility on the stability of tip vortices generatedin symmetric and operational configurations is not clearly observed qualitatively, as the 15 ◦ blade pitch has reducedaerodynamic loading of the blades and blade deformation. Thus, unlike prior rotor operational conditions, no clearqualitative hypothesis can be made on the influence of blade elasticity on the stability of tip vortices. Rigid rotorFlexible rotor
Blade 1Blade 2Blade 3 (a) Symmetric rotor configuration
Rigid rotorFlexible rotor
Blade 1Blade 2Blade 3 (b) Operational rotor configuration
Figure 19: Case 3 tip-vortex snapshot at t =
60 s for rigid and flexible rotor operation
Snapshots of the stability analysis performed at t =
40 s for tip vortices in a two rotation window (4 π window) arepresented in Fig. 20. Snapshots of the stability analyses qualitatively show that eigenvalues show less variance across arange of perturbation wavenumbers than in other cases. In fact, the current case 3 appears not to adhere to the classicalstability criteria, namely that eigenvalue peaks of stability trends do not correspond to wavenumber perturbations equalto ω = N b ( k − / (cid:15) amount, i.e. ω = N b ( k − / + (cid:15) . Figure 20shows that at t =
40 s the stability for operational rotor configurations reaches higher eigenvalues than symmetricDISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 22 (rad -1 ) ( s - ) Rigid rotor
Blade 1Blade 2Blade 3 (rad -1 ) ( s - ) Flexible rotor
Blade 1Blade 2Blade 3 (a) (rad -1 ) ( s - ) Rigid rotor
Blade 1Blade 2Blade 3 (rad -1 ) ( s - ) Flexible rotor
Blade 1Blade 2Blade 3 (b)
Figure 20: Stability trend snapshots at t =
40 s of tip vortices shed from rigid and flexible symmetric rotor (left column) and operational rotorconfigurations (right column). rotor configurations.The time histories of eigenvalues corresponding to perturbation wavenumbers ω = . , . . − , i.e., N b = k = , , ω = N b ( k − /
2) are now tracked in time for symmetric (sym.) and operational (op.)configurations in Fig. 21. Unlike cases 1 and 2 shown in Figs. 13 and 17, respectively, the time histories of eigenvaluesin case 3 for perturbation wavenumbers ω = . , . . ff erences betweentip vortices generated by symmetric and operational rotor in Fig. 19 are minor, the underlying physical di ff erences arehighlighted by the temporal stability characteristics shown in Fig. 21, which demonstrates that the periodic change inangle-of-attack generated by the rotor-plane tilt results in higher eigenvalues of the operational rotor configuration,thereby generating more unstable tip vortices. Figure 21b also shows the out-of-phase behavior of tip vortex stabilitytrends generated by the out-of-phase period change in angle-of-attack for individual rotor blades.The frequency spectra of the eigenvalue time-histories for case 3 symmetric and operational rotor configurationsis shown in Fig. 22. For the symmetric rotor configuration, the FFT shows a dominant low frequency contribution thatwas also seen previously in cases 1 and 2. It is also interesting to note that for the tip vortices generated by symmetricflexible rotors, this low frequency contribution is much more pronounced than a rigid rotor. As mentioned previously,identification of this low-frequency contribution is not trivial but seems to be related to the agglomeration of tipvortices downstream, and it may be that blade flexibility amplifies this low-frequency contribution in the amplitudespectrum. For the flexible rotor symmetric configuration, the FFT is showing a secondary frequency near the firstnatural frequency of the NREL rotor blade ( f ≈ . f = Ω / π ≈ . ( / s ) (a)
10 20 30 40 50 60 70time (s)0.0120.0130.0140.0150.016 ( / s ) Rigid Blade 1Rigid Blade 2Rigid Blade 3 Flexible Blade 1Flexible Blade 2Flexible Blade 3 (b)
Figure 21: Case 3 time history operation growth rates tip vortex 1: a) eigenvalues corresponding to perturbation wavenumbers ω = -1 for symmetric and operational rotor configurations (legend provides color scheme of rotor configuration and dashed lines represent time-history ofgrowth-rates from flexible blades); b) operational configuration eigenvalues corresponding to ω = -1 for all three blades. || (r a d - ) -4 Flexible rotorRigid rotor (a) || (r a d - ) -3 Flexible rotorRigid rotor (b)
Figure 22: FFT-derived frequency spectra of eigenvalue time history signal for perturbation wave number ω = . -1 : a) symmetric rotorconfiguration, and b) operational rotor configuration
5. Discussion
The tip vortex analyses presented show that a rotor-plane tilt and blade flexibility alters the time-dependent contentof tip vortex stability. By introducing the rotor-plane tilt, the time-history of eigenvalues corresponding to perturbationDISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 24ave numbers ω = . , . . − ) fluctuate at a frequency of f = Ω / π . Rodriguez and Jaworski [34]advanced the claim that stability trends are dependent on the blade passing frequency corresponding to f = Ω / π .However, their investigation incorrectly identified time-varying stability trends for tip vortices shed from di ff erentblades despite their base simulation being a symmetric rotor configuration. Rodriguez [31] also identified the samefrequency content in time-history trends for cases 1-3, but it was also assumed that the stability trend fluctuationwas a byproduct of the blade passing frequency. The current investigation shows that the stability trend fluctuationis not a byproduct of the blade passing frequency, but a result of the periodic change in angle-of-attack as the rotorblade travels around the tilted rotor-plane. It was also found that blade flexibility adds higher frequency contentto the stability trends equal to or about the first natural frequency of the rotor blade. The impact that both bladeflexibility and rotor-plane tilt have had on the stability trends indicate that tip vortices are sensitive to the dominantdynamics of rotor operation. This conclusion may be an important aspect to further investigate tip vortex stability offloating o ff shore wind turbines, where the dominant operational dynamics may interchange from rotor dynamics toenvironmental dynamics (wave-induced loading and rigid body motion).The analyses presented herein have also shown the impact that blade flexibility has on tip vortex stability for anaeroelastic rotor based on the NREL 5MW reference wind turbine rotor. Rodriguez and Jaworski [34] investigatedinitially the role of flexibility on the stability of tip vortices and concluded that blade deformation destabilizes tipvortices. However, their stability analyses were conducted on the entire tip vortex geometry, which included perturbingtip vortex locations that were already initially perturbed by artificial transients due to the impulse loading of the inflowconditions and numerical instabilities. To avoid performing stability analyses on tip vortices perturbed by numericalartifacts, Rodriguez [31] performed a windowed analysis to identify regions where numerical instabilities would notcorrupt stability analyses. Rodriguez [31] employed the windowed stability analysis and found that by evaluatingtime-history of eigenvalues corresponding to ω = . − wavenumbers, blade flexibility can generate less unstable(lower positive eigenvalues) tip vortices than rigid rotors. However, no time history analyses were performed onhigher wavenumber pertubations. The present analyses align well with results presented in [31] for wavenumbers ω = . − . Furthermore, it was also observed that for cases 1 and 2, where no rotor blade pitch is present, bladeflexibility can reduce sensitivity to low wavenumber perturbations and increase sensitivity to higher wavenumbers.Stability analyses of tip vortices for case 3 showed the lowest eigenvalues across a range of wavenumber per-turbations compared to cases 1 and 2. These relatively low eigenvalues reinforce the observation that the coherentand long tip vortex structures in Fig. 19 show minimal signs of wake break-down. Case 3 configurations (i.e. rotorconfigurations with a θ bl = ◦ blade pitch) also showed no agreement with the classical stability trend for α v. ω . Infact, it was seen that peak eigenvalues were located at forward shifted pertubation wavenumbers. Furthermore, eigen-values demonstrated monotonic values across the range of perturbation wavenumbers relative to case 1 and 2 stabilitytrends. The monotonic behavior of tip vortex stability was highlighted in the time history analysis in Fig. 21a thatdemonstrated tip vortices perturbed at ω = . , . . − showed almost identical eigenvalues. An overviewof the mathematical framework of the stability analysis highlights that by introducing blade pitch the local inducedvelocity field may introduce stability characteristics that are vastly di ff erent than the classical stability trends of thetip vortices. However, further investigations are required to identify the underlying mechanisms at play that cause thechange in stability trends due to blade pitch.
6. Conclusions
The stability of tip vortices shed from flexible rotors has been investigated numerically. An aeroelastic free-vortexwake method was used to generate tip vortex structures and a linear-eigenvalue stability analysis was employed tohighlight the underlying tip vortex physics. Towards validation of the stability analysis, it was found that the Vatistasmodel used to desingularize the Biot-Savart was susceptible to numerical artifacts not present in the cuto ff modeling.Further investigation into the impact of numerical time integration with Vatistas and cuto ff finite core modeling arenecessary to understand the numerical issues at hand. Nevertheless, it was found that by employing a windowedstability analysis, whereby only the early-aged segments of tip vortices are considered, classical stability trends wererecovered on a three-bladed canonical rotor (two- and four-bladed validation is also presented in the Appendix A).The validated stability analysis was then employed on the tip vortices generated by the NREL 5MW referencewind turbine for rigid and flexible blades under symmetric (no tilt) and operational (5 ◦ rotor-plane tilt) configurations.The tip vortex stability analyses and corresponding time-history analyses demonstrated three main findings: 1) tipDISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 25ortex stability is sensitive to dominant dynamics of rotor operation; 2) blade flexibility may generate tip vortices lesssensitive to low wavenumber perturbations but more sensitive to higher wavenumber perturbations; and 3) introducingblade pitch alters the local induced velocity field and alters tip vortex stability such that peak eigenvalue trends do notadhere to classical stability criteria. Future work will include performing wide parametric investigations across oper-ational conditions and rotor configurations to more conclusively determine the mechanism rotor blade aeroelasticityplays on tip vortex stability.
7. Acknowledgments
Most of this research was performed while the first author was at Lehigh University. The first and second authorswish to acknowledge the support of the Air Force O ffi ce of Scientific Research under Award No. FA9550-15-1-0148monitored by Dr. Gregg Abate. The first author also wishes to acknowledge the support from the Naval Research Lab-oratory’s Karles Fellowship. The third author acknowledges the support from Technical Data Analysis Inc. throughthe Small Business Innovation Research Topics N171-027 and N171-0416 under Phase II agreements NRL-2019-051and NRL-2019-024 and the support from the O ffi ce of Naval Research through the Naval Research Laboratorys corefunding. Appendix A. Additional validation cases
Appendix A.1. Two-bladed rotor wake stability
First, the stability trends for the two bladed rotor wake are discussed. Figure A.24 presents the stability trendresults for the Vatistas model and cuto ff model at variable vortex shedding frequencies. Perhaps the most notablefeature of Fig. A.24 is the periodic behavior in divergence rate oscillations as the wavenumbers increase. Traditionally,classical studies [3, 10, 45] have looked at lower wavenumbers as classical stability analyses (the analysis presentedhere included) are based on long-wave perturbation assumptions, which are usually defined as a perturbations muchlarger than the vortex core, and recently a quantitative definition of long-wave perturbations was presented by Quarantaet al. [28], which will be discussed later. Because of these long-wave perturbation assumptions no investigation, to thebest of the authors’ knowledge, has presented stability trends for high wavenumber perturbations and have observedsuch periodic behavior in their trends as presented in Fig. A.24.It is believed that this periodic behavior is a numerical artifact from the time-marching vortex shedding frequencyof the free-vortex wake method. Notice that for a relatively low vortex shedding frequency, such as f =
10 Hzin Fig. A.24, the periodic behavior is more compact and occurs more often than at shedding frequencies of f =
20 and 30 Hz. Higher shedding frequencies of f =
20 and 30 Hz show a high bell-shaped divergence rate behavior.It appears that as the shedding frequency increases the width of the “bell” also increases. The relationship andbehavior between shedding frequency and bell width suggests that if the shedding frequency approaches infinitythat rotor-wakes simulated by the current free-vortex wake method would be most sensitive to high wavenumbers,i.e. short-wave perturbations.At wavenumbers between 0 ≤ ω ≤
13 rad − , the stability trends generated by the Vatistas models ( n = , , and 3)with shedding frequencies f =
10 Hz show very good agreement with reported classical stability trends, i. e. divergence-rate peaks occur at ω = N b ( k − / k is any natural number. However, at f = ff vortex model resultsin Fig. A.28d. The cuto ff model shows very consistent oscillations of the divergence-rates for all vortex sheddingfrequencies. It appears that as the vortex frequency increases the period of the oscillatory periodic behavior is pushedtowards infinity. This dependency on vortex shedding frequency suggests that an infinite vortex shedding frequency isrequired to achieve classical stability trend results for the cuto ff model. The absence of the bell as seen for the Vatistasmodels also implies that the cuto ff model is not subjected to short-wave (high wavenumber) numerical instabilities asthe Vatistas model.DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 26 lade 1Blade 2 Figure A.23: Two-bladed rotor wakes modeled by employing the Vatistas core model (first row) and the cuto ff model (second row). Sheddingfrequency is f =
20 Hz.
The di ff erences between the Vatistas and cuto ff model motivated further investigation into reducing the cuto ff radius, which when δ → ff radius reduction results are presented inFig. A.25 for a vortex shedding frequency of f =
30 Hz. Results show marginal di ff erences between cuto ff radii as δ →
0. Similarity between stability trends with variable cuto ff radii further reinforces the notion that the Vatistascore model may be subjected to short-wave numerical instabilities.The above presented stability trends all belong to the tip vortex shed from one blade. Recall that for the analysisto be validated the stability trends for each tip vortex must all be identical. The stability trends for each individual tipvortex are presented in Fig. A.26 over a smaller range of wavenumbers to highlight the details. From Fig. A.26, it canbe seen that all stability trends are coincident, as expected for a symmetric rotor configuration. Appendix A.2. Four-bladed rotor wake stability
Finally, the four-bladed rotor wake stability trends are presented in Fig. A.28 for variable vortex shedding fre-quency. The same features exist in these stability trends that were seen in the two and three-bladed rotor wake, suchas the location and widths of the bell. Also, the frequency content of the stability trends has lowered as the additionaltip vortex in the wake has lowered the number of periods in the divergence rate fluctuation but has overall increasedthe instability (has increased the divergence rate α ).The stability trend investigation as δ → f =
30 Hz vortex shedding frequency also shows the same overallbehavior as seen in the two and three-bladed cases, shown in Fig. A.29. However, because of the additional tip-vortexthe divergence rates have been amplified to visualize the stability trends in more detail. As δ →
0, divergence ratesconverge to a unique stability trend. However, the vortex shedding frequency of f =
30 Hz is not high enough tovisualize a complete amplitude decay in Fig. A.29 for wavenumbers less than ω ≤
60, as was seen for the two- andthree-bladed cuto ff model results. Prior stability trends of the two-bladed and three-bladed rotor would indicate thatas the shedding frequency approaches infinity a unique divergence rate would be reached for higher wavenumbers.DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 27 a) Vatistas vortex core model, n = n = n = ff vortex core model, δ = − Figure A.24: Stability trends for a single tip vortex shed from a two-bladed rotor at variable vortex shedding frequencies (10, 20, 30 Hz) with aone-rotation (2 π ) tip vortex window. Vatistas models for n = , , and 3 are employed, and the cuto ff model is presented with δ = . Finally, stability trends from individual tip vortices is shown in Fig. A.30. It is seen that all stability trends from theVatistas model show identical behavior for each individual tip vortex. The cuto ff model also exhibits nearly identicalresults, shown in Fig. A.30d. However, there are very minor deviations at the early peaks of the stability trend whichare likely also a numerical error from the wake simulations that has propagated through the stability analysis.DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 28 igure A.25: Stability trends for the two-bladed rotor wake with shedding frequency set at f =
30 Hz and reducing the cuto ff radii to δ → (a) Vatistas vortex core model, n = n = n = ff vortex core model, δ = − Figure A.26: Stability trends for all the individual tip vortices shed from a two-bladed rotor from Vatistas and cuto ff finite core models DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 29 =1 n=2 n=3 =10 -1 2 =10 -2 2 =10 -6 Blade 1Blade 2Blade 3Blade 4
Figure A.27: Four-bladed rotor wakes modeled by employing the Vatistas core model (first row) and the cuto ff model (second row). Sheddingfrequency is f =
20 Hz.
DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 30 a) Vatistas vortex core model, n = n = n = ff vortex core model, δ = − Figure A.28: Stability trends for a single tip vortex shed from a four-bladed rotor at variable vortex shedding frequencies (10, 20, 30 Hz) with aone-rotation (2 π ) tip vortex window. Vatistas models for n = , , and 3 are employed, and the cuto ff model is presented with δ = . DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 31 igure A.29: Stability trends for the four-bladed rotor wake with shedding frequency set at f =
30 Hz and reducing the cuto ff radii to δ → DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 32 a) Vatistas vortex core model, n = n = n = ff vortex core model, δ = − Figure A.30: Stability trends for all the individual tip vortices shed from a four-bladed rotor from Vatistas and cuto ff finite core models DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 33 (rad -1 ) ( s - ) (a) Vatistas core model, n = (rad -1 ) ( s - ) (b) Vatistas core model, n = (rad -1 ) ( s - ) (c) Vatistas core model, n = (rad -1 ) ( s - ) (d) Cut-o ff model, δ = − Figure A.31: Stability trends of 2, 3, and 4 bladed rotor tip vortices employing the Vatistas finite core model at, n = , , and 3, and the cuto ff model with a cuto ff radius of δ = − DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. 34 eferences [1] Bagai, A., Leishman, J.G., 1995a. 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