Assessment of a State-Space free wake model

Transcription

1 Assessment of a State-Space free wake model Master of Science Thesis DELFT UNIVERSITY OF TECHNOLOGY Faculty of Aerospace Engineering Flight Performance and Propulsion Diego Hidalgo López I.A.

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3 Assessment of a State-Space free wake model Master of Science Thesis For obtaining the degree of Master of Science in Aerospace Engineering at Delft University of Technology Thesis number 248#14#MT#SEAD-FPP D. F. Hidalgo I.A December 9, 2014 Faculty of Aerospace Engineering Delft University of Technology

4 Copyright D.F. Hidalgo All rights reserved

5 Delft University of Technology Department of Flight Performance & Propulsion The undersigned hereby certify that they have read and recommend to the Faculty of Aerospace Engineering for acceptance a thesis entitled Assessment of a State-Space Free Wake Model for Rotorcraft Flight Mechanics Applications by D. F. Hidalgo in partial fulfillment of the requirements for the de degree of Master of Science. For obtaining the degree of Master of Science in Aerospace Engineering at Delft University of Technology Dated: December 9, 2014 Chairman of the graduation committee Prof. dr.ir. L.L.M. Veldhuis Supervisor(s): Reader(s): Dr.ir.Marilena Pavel Dr. ir. Mark Gerritsma Dr. ir. Mark Voskuijl

6 Abstract The goal of this work to provide an initial insight into the potential of a rotorcraft aerodynamic wake model in state space form and its impact on rotor performance and the capability of current analytical and numerical models to consider such potential into a basic flight mechanics environment. First a brief overview of the physical phenomena is included to highlight the origin of this study and the relevance of the physics fundamentals. Then, with the presence of the wake already established, a review of diverse literature is shown containing the different approaches to rotor wake modeling. The free wake model in state space is introduced and its definition is presented as well as problem setting solution approach trough the Method of Lines, linkage to blade motion is also presented. A dissection of the different model components is presented and individual validation for every component is included. The integrated model is tested for an isolated rotor in different circumstances and compared against the behavior of experiments in order to check for suitability of the model into potential more complex flight mechanics evaluations and analysis via simulation. Conclusions in this regard are finally presented. I

7 Acknowledgments The author would like to acknowledge the kind help, patience and guidance of Prof. Marilena Pavel, her consideration was definitive for the completion of this thesis assignment, my sincere appreciation for her help. Prof. Schiesser at Lehigh University who took time from his busy schedule to review my progress and who generously provided advice and insight to the method of lines. Dr. Wim van Hoydonck, his help and expertise in the particular topic of rotorcraft free wake modelling were the drivers that allowed the progress of this work in that regard, his Ph.D. thesis and previous work were a key reference. To Prof. Cesar Nieto at UPB for the great amount of patience and perhaps undeserved confidence on my capabilities and ideas, to him my sincere appreciation and respect, as researcher and teacher. Finally, to my family, to my mother who was the will behind the possible and the faith behind the impossible, to my father, whose silent patience and careful teachings are always in my heart and memory. II

8 Contents Abstract... I Acknowledgments... II Contents...III List of Figures... VI Nomenclature... XI Chapter 1: Introduction Helicopter Rotor Wake, a Context Thesis Objectives General Objective Specific Objectives Thesis Scope Thesis Outline... 4 Chapter 2: Background Rotor wake features Tip Vortex Inboard vortex sheet Root Vortex Near wake Mid-wake Far wake Rotor wake models Dynamic Inflow Methods Boundary Element Methods (BEM) wake models III

9 2.2.3 Vortex filaments for rotor wake model Rigid/Prescribed wake models Free wake models Vortex ring / Vortex tube wake models Fast vortex/panel methods State-space free wake Rotor Wake models issues and considerations Blade Aerodynamics Maneuvering Flight Ground effect Compressibility Wake/Body interaction The real time problem Chapter 3: State-space rotor - wake formulation Assumptions and constraints A Lagrangian approach for wake Induced velocity field Blade model Considerations and assumptions Airfoil velocity sources The Theodorsen model The indicial function approach, Wagner problem The Sharp-edged gust, Kussner problem State space unsteady airfoil model Blade flapping model Formulation of velocities for rotating blade Induced velocity transformation IV

10 3.5 The Method of Lines (MOL) for rotor wake solution Problem statement Auxiliary conditions Spatial differentiation of LHS (5PBU4) Chapter 4: Components Validation and vorticity coupling Biot-Savart law MOL wake geometry representation Unsteady blade section Pure Angle of attack oscillations Harmonic plunge oscillations Blade load integration Uniform inflow distribution - hover Linear inflow distribution - forward flight Free wake periodicity Bound vorticity coupling Chapter 5: Integrated Model Cases Steady state wake shape -forward flight Rotor controls setting evolution with advance ratio Time dependent collective pitch inputs Chapter 6: Conclusions Bibliography Appendix A. Velocity-vorticity Navier-Stokes Equation V

11 List of Figures Figure 1.1 Figure of merit for different types of helicopters ref [2][3]... 2 Figure 1.2 Rotorcraft aerodynamics environment ref [6]... 3 Figure 2.1 Helicopter rotor wake scheme ref [10]... 7 Figure 2.2 Wake major components... 8 Figure 2.3 Wake influence regions. Near field : red, Mid field: green, Far field: blue (original from ref [14])... 9 Figure 2.4 Wake inflow model families [21] Figure 2.5 Eulerian methods [9] Figure 2.6 Aircraft configuration discretization with panel methods definition [36] Figure 2.7 Wake potential strength and Kutta condition reinforcement for an airfoil and finite wing [36] Figure 2.8 Wake shape in hover, panel method doublet based solution [38] Figure 2.9 Rotor tip vortex filament as observed by shadowgraphy [44] Figure 2.10 Velocity induced by a straight vortex segment [45] Figure 2.11 rotor coordinate system and Lagrangian vortex filament discretization [11] Figure 2.12 Geometry for the Biot-Savart definition of curved vortex segments [46] Figure 2.13 Prescribed distorted wake (left) and undistorted rigid wake (right) [47] Figure 2.14 Location of equivalent vortex lines on critical trailing vortex system (top) and perspective view of the distorted vortex system (bottom) [51] Figure 2.15 Free wake development for a rotor in forward flight by using a relaxation method [39] Figure 2.16 Free wake development for a rotor in forward flight by using a relaxation method [64] Figure 2.17 NURBS defined rotor wake vortex tube [69] Figure 2.18 Wake trimming iteration process [70] VI

12 Figure 2.19 Source panels flight deck representation and interaction with rotor wake IGE [70] Figure 2.20 Octree structure and vortex interactions for fast vortex/panel methods [74] Figure 2.21 Geometry discretization for panel method for a blade [80] Figure 2.22 Systems of lifting surfaces using Phillips lifting line adaptation [82] Figure 2.23 Lifting line for rotary wing [11] Figure 2.24 Blade Velocity build-up [11] Figure 2.25 Velocity field decomposition [11] Figure 2.26 Evolution of effects of maneuver flight on wake [90] Figure 2.27 Ground effect nodel by method of images [12] Figure 2.28 Fully interacting rotor- wake-surface system for complex configurations [74] Figure 3.1 Rotor wake filament scheme (original from [12]) Figure 3.2 Biot-Savart influenced velocity by a vortex straight segment on a point in space Figure 3.3 Core swirl velocity models Figure 3.4 Schematics of interactions of induced velocities and blade loads Figure 3.5 Unsteady blade section velocity components [11] Figure 3.6 Wagner function comparison for analytical and approximated function Figure 3.7 Effect of airfoil penetration in a gust field (top), equivalent effect for blade section under influence of a nearby vortex filament (bottom). Original from [11] Figure 3.8 Kussner function, analytical and approximate behavior Figure 3.9 Schematics for treatment of Blade 2D sections for unsteady conditions Figure 3.10 Local velocity components for forward flight Figure 3.11 Basic typical arrangement of reference frames for blade motion Figure 3.12 Shaft plane rotations for induced velocity evaluation Figure 3.13 Linearized system poles for different discretization systems [77] Figure 4.1 Circular vortex ring influencing a point P in space Figure 4.2 Comparison of induced velocity solutions for a circular vortex ring Figure 4.3 Influence of discretization on induced velocity for a circular vortex ring Figure 4.4 Relative error for different number of elements for Biot-Savart numeric and analytic solutions VII

13 Figure 4.5 Induced velocity for a vortex ring, analytic solution vs regularized Biot_Savaart Figure 4.6 RMS and Number of evaluations(ncall) evolution for different wake angular spacing Figure 4.7 Comparison of wake solution, analytic solution of eq (blue) and MOL solution (red) for advance ratio of Figure 4.8 Comparison of wake solution, analytic solution of eq (blue) and MOL solution (red) for advance ratio of Figure 4.9Lift coefficient for unsteady harmonic AoA oscillation Figure 4.10 Load discrimination for harmonic AoA oscillation Figure 4.11 Lift coefficient output for harmonic plunging motion Figure 4.12 Angle of attack due to plunging Figure 4.13 Flapping moment integration for hover Figure Blade-wake configuration for hover case under uniform inflow Figure 4.15 Evaluation points along the blade radius (blue) and wake influencing elements (red) Figure 4.16 Induced velocity along the rotor radius. Comparison between uniform inflow and vortex filament analytic wake Figure 4.17 Induced velocity along the rotor radius. Regularization OFF Figure 4.18 Rotor wake system for forward flight Figure 4.19 Instantaneous induced velocity, longitudinal distribution Figure 4.20 Instantaneous induced velocity distribution at different azimuth stations Figure 4.21 Time averaged induced velocity, longitudinal distribution Figure 4.22 Time averaged induced velocity distribution at different azimuth stations Figure 4.23 Instantaneous induced velocity, azimuth distribution Figure 4.24 RMS behavior for wake periodicity study case Figure 4.25 Superposition of wake markers for different revolutions after periodic behavior is achieved (7 th rev.- solid blue line, 9 th rev.- red dots and 11 th rev. yellow stars) Figure 4.26 Time required for iteration convergence at a given time step Figure 4.27 Number of iteration required for convergence at a given time step VIII

14 Figure 5.1 Top view comparison for wake evolution against the experimental results and CAMRAD/JA as reported by Ghee and Elliot Figure 5.2 Z position for wake compared against experimental results and CAMRAD/JA Figure 5.3 Wake shape for hover Figure 5.4 Evolution of thrust coefficient until reaching periodicity for hover Figure 5.5 Blade loading distribution for hover Figure 5.6Blade loading for hover case as reported by Betoney, Celi and Leishman, original from ref [78] Figure 5.7 Wake shape for advance ratio of Figure 5.8 Evolution of thrust coefficient until reaching periodicity. For advance ratio of Figure 5.9 Wake shape for advance ratio of Figure 5.10 Evolution of thrust coefficient until reaching periodicity. For advance ratio of Figure 5.11 Wake shape for advance ratio of Figure 5.12 Evolution of thrust coefficient until reaching periodicity. For advance ratio of Figure 5.13 Trimmed values for collective, shaft tilt angle and longitudinal cyclic Figure 5.14 Initial wake geometry from an initial solution for hover case before collective input Figure 5.15 Collective transient Input for initial 200deg/sec rate Figure 5.16 Thrust coefficient evolution in time for 200 degrees per second rate collective pitch input Figure 5.17 Thrust coefficient evolution in time for 200 degrees per second rate collective pitch input, taken from reference [78] Figure 5.18 Blade flapping response for a 200 degrees per second rate collective pitch input Figure 5.19 Wake evolution time t=0.1856s ( up ) and t= s(bottom) Figure 5.20 Wake evolution time t=0.3371s ( up ) and t= s(bottom) Figure 5.21 Wake evolution time t=0.6402s ( up ) and t= s(bottom) IX

15 X Assessment of a State Space Free Wake Model

16 Nomenclature Latin symbols A ζ a b c C(k) C lα C l dl E τ ec v i, j, k K τ k L L M β M M β M I N b p, q, r q r r c Δr c r o Re Differentiation matrix Airfoil axis of rotation location in fractions of the chord Airfoil semi-chord Blade chord Theodorsen function Airfoil section lift slope Airfoil section lift coefficient Differential length element Legendre elliptic integral, second class Blade section index Plunging vertical position Perpendicular distance from a vortex element longitudinal, lateral and vertical axes Legendre elliptic integral, first class Reduced frequency Vortex filament length Change in vortex filament length Blade aerodynamic flapping moment Mach number Blade flapping moment Blade Inertial term Number of blades Rotational velocities Induced velocity by a vortex element Position vector Vortex filament core radius Change in vortex core radius Initial Vortex filament core radius Reynolds number XI

17 s U U t U n V V ex V ind V man V θ w g wup wlow Distance traveled in semi-chords 2Vt / c Resultant local velocity Tangential local velocity Normal local velocity Velocity vector External influencing velocity Induced velocity Maneuvering velocity Vortex core swirl velocity Gust velocity perturbation Wake upper surface Wake lower surface Greek symbols α α eff α L β β o δ ε φ s Γ γ b λ L 2 μ ϕ ν μ Ω ω ω ψ ψ s τ θ o θ i θ c Blade section angle of attack Blade section effective angle of attack Vortex filament core radius power coefficient Blade flapping angle Instantaneous blade flapping angle Dissipative term for vortex core deformation Vortex core deformation coefficient Velocity potential Wagner function Vortex strength Bounded vortex strength Normal inflow velocity ratio V ind / ΩR Laplace transform Laplace operator Doublet singularity potential Dynamic viscosity Advance ratio V / ΩR Rotational velocity Oscillation frequency Vorticity vector Azimuthal position Kussner function Source singularity potential Chord lengths traveled Vt/ c Blade collective pitch Blade twist Blade lateral cyclic XII

18 θ s θ ζ Blade longitudinal cyclic Blade local overall pitch Wake age Acronyms AoA BCVE CVC ODE PDE MOL IGE IBC A/RPC RT CFD BEM CHARM CAMRAD FFW NURBS BVI RHS LHS BC 2PCD2 4PCD4 2PU1 3PU2 5PBU4 Angle of Attack Basic Curve Vortex Element Constant vorticity contour Ordinary differential equation Partial differential equation Method of Lines In ground effect Individual Blade Control Aircraft / Rotorcraft Pilot Couplings Real time Computational Fluid Dynamics Boundary Element Method Comprehensive Hierarchical Aeromechanics Rotorcraft Model Comprehensive Analytical Model of Rotorcraft Aerodynamics and Dynamics Fast free wake Non rational B-Splines Blade vortex interactions Right hand side Left hand side Boundary Condition 2 nd Order, 2 point central difference 4 th Order 4 point central difference 1 st Order 2 point Upwind 2 nd Order 3 point Upwind 4 th Order 5 point Biased Upwind XIII

19 XIV Assessment of a State Space Free Wake Model

20 Chapter 1:Introduction 1.1 Helicopter Rotor Wake, a Context Interest on rotor wake analysis has always been a point of focus of rotorcraft analysis and design, its importance is undeniable and has been kept in current status within the research community as improvement of knowledge and tools have allowed the exploration of such phenomenon and its influence on engineering applications. While several approaches are available to consider the important effect of wake on rotorcraft performance, the need to explore in detail and emulate the nature of this complex fluid structure has increased to meet demanding operational requirements. High loading on blades, maneuver loads, unsteady structural dynamics, vibration management, major structural components fatigue, close to ground operations and on-board ship operations, all have shown their own demanding nature related with rotor wake, a reality that traditional models and methods may not meet completely when faced with convergent design philosophies and analysis based on early life cycle assessment to boost development. Some consideration to performance improvement trough time is observed in Figure 1.1 where the growth of the figure of merit is showed in history according to weight demands. Such a growth in efficiency and increased performance is the result of improvement of design and analysis methods during time, and their connection can be observed when considering reviews like that of Johnson [1] where the milestones of rotorcraft aeromechanics are exposed in relation to the technical background. Thus, the importance of improvement of wake behavior analysis, as 1

21 already exposed, is a main driver to explore new rotor concepts that could require alternative approaches to those of traditional concepts and shapes. Figure 1.1 Figure of merit for different types of helicopters ref [2][3] Considering specific aspects of the rotorcraft development cycle, aerodynamic-flight mechanics coupled applications, especially real time applications, are evidently seen as advantageous tools as, progressively, computational capabilities and methods make of them a feasible option for analysis and design providing an on the way analysis option pointing towards communion with simulation based operational evaluations. The idea of including a medium-high fidelity wake model into a flight mechanics simulation may allow a diversity of opportunities, for system integration accounting for driving physics that may be ignored or surrogated otherwise. To provide an example consider the Aircraft / Rotorcraft Pilot Couplings (A/RPC s) [4], these unfavorable oscillations could be better assessed during an early phase with the help of real time models that take into account localized phenomena such as blade behavior dependant on In-Ground-Effect (IGE) complex flow field. In another example, Individual Blade Control (IBC) [5] laws and models could be enforced and evaluated from a direct review, from simultaneous coupling, instead of looking at the behavior history, thus reducing the development loop time while increasing certification chance of success. Similarly, vibration reduction by implementation of active methods can be considered well into preliminary design stages model based design, giving the possibility, among others, of considering the overall operational performance output resultant from the contribution of such technologies. 2

22 In many ways, rotor flow field, i.e. the rotor wake, has an undeniable effect not only on system performance (Figure 1.2), but on the efforts behind the conception and development of the system, becoming worth of attention since it is a main driver influence in the already complex world of rotorcraft analysis and development. Following the aforementioned ideas, this work is oriented towards the inclusion of a rotor wake model within a basic mechanical model of a helicopter rotor aiming for future expansion, progressively including additional and more detailed physical influences into the model. This work is oriented to capture the knowledge related to a specific method of wake representation that is considered well suited for rotorcraft aeromechanics analysis, thus setting an initial point on the effort to provide improved rotorcraft capabilities. Figure 1.2 Rotorcraft aerodynamics environment ref [6] 3

23 1.2 Thesis Objectives General Objective To implement a rotor wake model to obtain mechanical aerodynamic coupled solutions in a state-space form Specific Objectives To define the elements of a state space representation of a free wake method. To explore the convenience of a state-space rotor free wake To implement a state-space rotor free wake model To validate the implemented method for specific flight conditions 1.3 Thesis Scope This work is limited to implementation and test of a free wake model in State-Space form coupled with blade flapping mechanics for an isolated rotor. Work has been supported with comprehensive literature reviews and the methods, techniques and theories required achieving implementation of the method intending to show its potential towards flight mechanics applications, in this case, specifically for rotorcraft. 1.4 Thesis Outline This thesis is arranged along six chapters. Chapter two presents the background supporting the development and implementation of rotor wake models, several models available in literature are presented and discussed around their applications to rotorcraft flight mechanics, limitations and advantages. Chapter three develops the state-pace model for rotor wake, describes the alternative Lagrangian definition of the 4

24 problem for wake elements, the coupling with blade mechanics and the method of solution by using the Method of Lines (MOL). Chapter four gathers diverse sources around implemented model for validation purposes, steady and unsteady cases are discussed for each model component. Chapter five includes specific cases of study for the integrated model. Chapter six presents the conclusions, recommendations for further work. 5

25 Chapter 2: Background 2.1 Rotor wake features In general, rotor wake structure is complex and its behavior could be far from that analytically defined, at some points its behavior could be observed as chaotic while in others it still shows some deterministic patterns. In any case, wake nature complexity is undeniable and work to cover the gap of this field is of high importance. As result of the research community efforts, substantial observations have been collected in regard to the major features of rotor wake, making possible a description around its main components and the way they dynamically interact. In accordance to studies such as that of Gray ([7] and [8]) and overviews like that of Conlisk [9], rotor wake following blade motion is a fluid structure with specific components, effects and properties. Summarizing its morphology, rotor wake can be depicted by three main components: the blade tip vortex, vortex inboard sheet and root vortex all of them acting on three regions of influence denoted as near, mid and far wakes. With this brief description as introduction, this chapter is committed to provide a description of such structures along with their formulations and the effects and issues related with wake influences and potential Tip Vortex Tip vortex is a high strength vortical fluid element generated by the sudden change in circulation near the blade tip. Since for a rotor, the largest velocity is achieved near the 6

26 tip zone, this area is consequently the most aerodynamically loaded and the difference of velocities on the upper and lower surfaces is increased, causing the tip vortex to have the strongest effect of all components [9]. Following its generation, vortex is convected usually from the point of largest loading, following the rotational flow field, along what is called the near and mid wake fields, beyond this influence zones tip vortices may evolve into an analog structure to those observed in fixed wing aircraft with two major contra rotating vortices, as is the case of forward flight condition. While not the only element, is usual to see models that only account for tip vortex due to its high impact as will be described in subsequent sections. Figure 2.1 Helicopter rotor wake scheme ref [10] Inboard vortex sheet When observed, this fluid structure could be described as a continuous thin sheet being shed along the trailing edge of the blade [8], it contains several vortex filaments shed from the point where the flow abandons the inboard portions of the blades trailing edge. The Vortex sheet keeps a high influence in the immediate zone behind the blade, near wake, causing significant effect in both inflow and blade loading. Its inclusion in models is yet important. For the case of fast or real time (RT) applications, sometimes inboard vortex sheet could be replaced by using an unsteady 2D airfoil model, accounting for the bounded vorticity, [11] and only the strong tip vortex is consider with the intention of preserving a low computational cost [12]. 7

27 2.1.3 Root Vortex Usually characterized by low strength when compared against the tip vortex, root vortex still represents, together with the inboard vortex sheet an important component when defining proper representation of rotor flow field and blade loading. Nevertheless, while it can be included, the gradual circulation reduction towards blade root causes the vortex to be weak or over-shaded by the other two wake components. From experiments it was observed that the elements can be found existing within three main regions as usually described [13]: Figure 2.2 Wake major components Near wake It is considered the nearest region to the blade, where the wake is shed from the rotor blade trailing edge, and completes the formation of the tip vortices. This region is of high importance due to the influence of its behavior on blade bound vorticity and thus on blade dynamics that define structural dynamic loads as well as acoustic behavior. Near wake can be consider from a practical point of view, not formally defined though, to cover a region within the first two revolutions after leaving the blade since is during those two revolutions that complete wake contraction takes place [9]. 8

28 2.1.5 Mid-wake It is the immediate region where an interaction with other components of the helicopter, such as fuselage and tail rotor, takes place. The relevance of this portion is the effect of unsteady flow field behavior on structural components as well as the performance effects due to the presence of foreign elements in the wake field Far wake It implies the region beyond the main craft body where influence of vorticity and flow perturbation state still generates a clear effect on velocity distribution over the rotor and consequently the flight mechanics. Far field influence can be usually considered for ground effect and interaction with platforms for shipboard and offshore operations where wake behavior derives from unsymmetrical fields generated by the un-uniform presence of ship major elements. A representation of the wake influence zones can be seen in Figure 2.3. Figure 2.3 Wake influence regions. Near field : red, Mid field: green, Far field: blue (original from ref [14]) 9

29 With the general rotor wake flow field description at hand a more detailed observation allows us to consider additional characteristics that belong to wake morphology. While every component is important, high attention has been given in literature to the vortex filament possibly because it is arguably the basic brick upon all the other elements can be built for mathematical representation. Chapter 3 will describe vortex filament in terms of its representative mathematics for its application to rotor wake modeling. 2.2 Rotor wake models Diverse methods and models have been developed to describe rotor wakes making use of a wide range of fundamental solutions. Some have been developed from basic definition of momentum theory while others are based on advanced CFD approaches with high fidelity/high cost results. In between, different formulations are found to have succeeded to demonstrate that they can be attractive and cost/effective options when it comes to rotorcraft analysis and design. Extending from the introductory section, the aim of this section is to explore those methods that could provide increased fidelity while keeping computational cost low enough to be included into flight mechanics or, hopefully, real time applications since the intention of adding dependable analysis capabilities, when designing rotorcraft systems, is a sensible goal when considering that design risk tend to propagate downstream along the development process. Research on rotor wake modeling has also explored different options originated from three main approaches: The methods that are based on the actuator disc concept, modeling the rotor as an actuator disc approaching to the problem by considering the change of momentum in the disc boundary, focusing on the inflow velocities induced by the downstream flow. From this first approach, steady and dynamic definitions have been developed ([15] [16][17][18] [19] [20]). Lagrangian methods, the second major approach, are based precisely on the Lagrangian definition of Navier-Stokes equations, formulating the transport of vorticity concentrated on flow elements that could be located in space by linking such elements to spatial point, analog to particles, called Lagrangian markers that move in space according to the so-called Lagrangian or material coordinate system. This approach has also been widely used and its capabilities have been successfully applied to fixed and 10

30 rotary wing problems, becoming into a popular and relatively low cost option while retaining a powerful physical definition that allows analysis of detailed problems. The remaining Eulerian approach is in opposition to Lagrangian method, a formulation that uses a fixed point reference frame not located on the particle itself. Eulerian methods contain a wide variety of solutions ranging from full Navier-Stokes solvers to the popular so-called panel methods among others (see Figure 2.4 and Figure 2.5). Fidelity of Eulerian solutions may vary according to the main assumptions as well (viscosity, compressibility, vorticity may or may not be considered). Such methods are popular and trough inclusion into commercial software packages their evolution has also reached high levels of sophistication. Despite their flexible formulation some Eulerian methods, may show a detriment in efficiency due to its relatively high computational demand, something that constrains their use when it comes to couple this solvers to inertial or elastic problems. Nevertheless, alternative panel methods formulations have been contemplated as possible candidates to be included in high fidelity real time simulations due to their alternative fast solution strategy. Further details on these state space methods include capabilities to cover formulations of vortex filaments and shed wakes among other topics upon computational capabilities, besides of the inflow models, in a more generic way looking for minimization of dependence on semi empiric tuning and computational time reduction. While this classification is general, a more dedicated classification is presented by Leishman [11], Conlisk [9] and van Ozenoort [21]. In this work the description of Van Hoydonck et al [22] will be followed providing general descriptions of the methods. 11

31 Figure 2.4 Wake inflow model families [21] Figure 2.5 Eulerian methods [9] 12

32 2.2.1 Dynamic Inflow Methods As stated by Peters et al. [23], the foundations of these methods rely on the basic Glauert s momentum theory [24] which basically relates the downwash generated by the rotor to its thrust by considering the change of momentum in the mass of air across an infinitely thin disc that represents the rotor. While relatively simple and widely used to define systems features it shows some main limitations, such as the inability to consider off axes responses, an indetermination in vortex ring state and lack of consideration of ground effect are the most relevant for helicopter analysis or the effects on the blade itself. Development of this initial conception has represented the commitment of diverse researchers aiming to solve the above mentioned limitations. Translation of the method to a dynamic definition was first developed by Carpenter and Fridovitch [25] who related, in a differential equation the inflow perturbations to thrust changes in a linear and nondimensional definition. Later on, the work of Pit and Peters ([23], [15], [16]) went further to make the method valid for forward flight and hover as well by defining three different inflow states, uniform inflow, lateral and longitudinal inflow states. These inflow states allowed accounting for the off-axis response limitations. Despite of the capabilities and practical definition of the dynamic inflow model, it has been observed along the years that it does require tuning and improvements. In response, Peters and his collaborators have worked to improve the method. First a problem related with the inversion of signs for the dynamic rates resultant from cyclic inputs that was improved [17]. Other improvement came later with a generalization of the Pitt-Peters model made by Peters-He ([18], [19]) who defined radial and azimuthal states. The subsequent work of Peters-Morillo [20] translated the model into a statespace model. Additional improvements include semi-empiric modifications to account for vortex ring state ([26], [27], [28]) and ground effect considerations [29]. During long time, dynamic inflow, and its diverse improvements has been a widely used tool for flight mechanics simulation, its versatility, though dependant on semi-empirical tuning, has served to its porpoises with success. Nevertheless, the model is limited to fixed states and fails to model highly distorted wake conditions like those observed in large maneuver accelerations or vortex ring state, besides, while it keeps a global description of the rotor aerodynamics it still lacks the fidelity to represent local phenomena derived from blade motion, blade shape or blade distribution. 13

33 Dynamic inflow model has been widely used and due to its low computational cost it has been implemented in flight simulations ([30], [31], [32]). An extended overview of dynamic inflow methods can be found in ref ([17], [33]) Boundary Element Methods (BEM) wake models Before going further in this topic, considerations around rotor wake experimental observations and main features are to be revisited. Representation of the presence of rotor wake was presented by Goldstein [34] as a helical surface downstream the rotor, emulating experimental observations. Although rotor wake is a complex dynamic system, assumptions were taken to facilitate modeling and practical application. This was done by including some limitations like a fixed wake pitch with a rigid definition allowing, at some extent, accounting for wake influence on rotor inflow velocities. Such influence can be considered, but not only, by formulating the flow behavior parting from a potential velocity. Those methods that use that velocity potential are usually known as panel methods, Boundary Element Methods (BEM) or potential methods in a more general classification. The basic formulation implies a model that neglects vorticity, viscosity and compressibility, thus resulting in a solution of the Navier-Stokes equations that shapes solutions from of the Laplace equation 2 = The overall solution of the equation can be obtained by considering elemental flows as basic solutions, thus the use of doublets, sources, vortex rings and vortex filaments became typical allowing for discretization of immerse bodies, such as wings and fuselages. The influence of the singular flow solutions for practical use was developed by Hess and Smith [35], providing the discretized integration form of the different elements. In many opportunities mixed formulations would be used to achieve better results depending on the case, besides there is also the chance of redefining the basic solutions by using nonlinear definition of the aforementioned elementary flows. 14

34 Figure 2.6 Aircraft configuration discretization with panel methods definition [36] Method s boundary conditions demand a normality condition that could be represented at convenience by the Newman or the Dirichlet conditions. The two boundaries are differentiated, in practical terms as follows: Newman boundary condition neglects any normal flow across a representative surface of a given body, represented by a constant potential equal to zero. For Dirichlet condition, a fixed potential is defined along the surface boundary and permeable or impermeable condition of normal flow could be proposed. Thus for Newman the boundary condition is given by n = While for Dirichlet, = f(b) 2.3 Defining a potential value for every point on the surface b. An additional condition is required when considering lifting bodies or surfaces. Kutta condition demands a smooth detachment from the body, meaning tangential velocities. 15

35 For the case of a rotor blade, detachment point may correspond to the blade trailing edge. Translated to potential formulation, the Kutta condition and its revised Morino s version [37] demand that the potential difference, also known as potential jump, at the detachment point corresponds to the difference between the potentials of the encountering flows, i.e. the flows running over the upper and lower surfaces of the blade. Thus the Morino-Kutta condition that defines the potential of the trailing flow is given by μ w = wup wlow = w 2.4 The graphical interpretation of this condition is shown in Figure 2.7. Figure 2.7 Wake potential strength and Kutta condition reinforcement for an airfoil and finite wing [36] With the values of the wake potential already defined, it is possible to generate a geometry representative of the wake. To do so, usually rectangular panels with doublets or vortex rings elements are used. Whether dealing with a rigid or prescribed wake or a so-called free wake method can be adapted to the particular application ranging from steady state to time marching problems. For the case of a body and wake (blade wake 16

36 system), solution of the resulting system of equations is composed by the integration of the elemental flow solutions, is then given by x, y, z = 1 4π μn b+w 1 r ds 1 4π b 1 ς r ds When solved, usually in a numeric fashion, this equation obtains the values of the potentials for body and wake and for body only. Latter extraction of the local velocities on each panel is possible accounting for the free stream potential. Wake models based on potential solutions are able to represent the inner portion of the wake, shedding a surface defined usually by doublets or vortex rings, it is unable though of modeling tip vortex effects having to use other strategies like wake rollup, which is of common use in fixed wing aerodynamic analysis. Figure 2.8 Wake shape in hover, panel method doublet based solution [38] 17

37 Extended discussion on these methods is available in reference [36] and implementation cases in time domain on a helicopter rotor are available in reference [39] and [40] where comparisons of results against of those of Caradonna-Tung [41] are made Vortex filaments for rotor wake model With vortex filament as the basic singularity, these wake models apply for a diversity of problems in fluid mechanics including fixed wing and rotary wing problems. Generally known as vortex filament models these approaches aim to represent line elements, or thin surfaces for the case of vortex lattices, that transport the vorticity embedded into them along its length. In essence simpler than other potential approaches this type of wake model is very flexible showing convenient properties that are translated into less computational expense, for example, in opposition to panel methods, vortex lines are more flexible to include wing/blade tip vortices reducing the need of a wake rollup auxiliary definition [36]. The WAKENET3-Europe [42] project for the fixed wing case, intended for operations analysis is an example of focused applications of vortex filament used in an extended scale. For the rotary wing case this method has become into the core model for rotor wake analysis and simulation showing a reduced computational cost while keeping proper fidelity when compared to other methods that could lack one or the other [21]. Its applications are found in different rotorcraft flight states, from OGE condition for brownout [43] and ship operations qualification to high performance maneuvers. Its formulation is well known and widely used, while still keeping research on in regard to basic characteristics like vortex core and stability issues [13]. In its basic or simplest form, a vortex filament is represented by, usually but not limited to, straight segment lines that when assembled resemble the entire vortical filament in space (see Figure 2.9). 18

38 Figure 2.9 Rotor tip vortex filament as observed by shadowgraphy [44] The straight vortex segment which form the main brick for mathematical formulation follows a basic potential formulation driven by the so-called Helmhortz laws [36] which state the following 1. The strength of a vortex filament is constant along its length. 2. A vortex filament cannot start or end in a fluid (it must form a closed path or extend to infinity). 3. The fluid that forms a vortex tube continues to form a vortex tube and the strength of the vortex tube remains constant as the tube moves (vortex lines will remain vortex elements with time) From these fundamentals, the vortex filament influences its surrounding flow field and is simultaneously subject to the influence of the neighboring fluid structures, as observed schematically in Figure 2.10 which shows a straight line vortex segment influencing a point p in space. 19

39 Figure 2.10 Velocity induced by a straight vortex segment [45] In its general form [12], vortex filament wakes are more related with a Lagrangian conception since the vortex filaments, represented by straight or curved segments, behave as material lines whose motion can be tracked by using points contained within the segments called Lagrangian markers. For the case of vortex filament wake applications, Navier-Stokes equation is reformulated to consider a velocity V and a vorticity ω ([12], [45]). ω t = V ω + ω V + ν ω 2.6 Thus, changes in vorticity of a given element are related to the instantaneous value of vorticity ω and the local velocity V. For this equation, in the right hand side, the first term represents vorticity convection, second is the strain term and the last is the diffusive term. Now, to consider the motion of the segments the Lagrangian definition of motion of a particle applies and the convective equation is defined as dr dt = V ; r t = 0 = r

40 Despite its simple formulation, implementation of equation (2.7) to wake models/solutions demands significant effort since motion of each vortex segment is influenced by the remaining segments, meaning a self and mutually induced velocity for the entire wake which to be properly modeled has to be defined by using a large number of segments, consequently increasing the computational cost. Most of applications of vortex filament to rotor wake problem make use of a definition that describes rotor wake with two main parameters, azimuthal position ψand wake age ζ. The former represents the angular position of the blade along its rotating path, it depends on rotor speed Ω and time t, while the latter, wake age, represents the angular position of a given Lagrangian marker in the wake vortex filament [11]. Schematics of rotor wake definition and terms meaning is shown in Figure Figure 2.11 rotor coordinate system and Lagrangian vortex filament discretization [11] Although it is the most common, straight vortex segment is not the only possibility for vortex filament representation. Alternative vortex element formulations like the curved vortex elements or the vortex particles and constant vorticity contour (CVC) among others may offer advantages according to the problem at hand. The Basic Curve Vortex Element (BCVE) defined by Bliss, Teske and Quackenbush [46] is an element based on the approximate Biot-Savart integration for a parabolic arc 21

41 filament. When compared with traditional straight segments, BCVE allowed for significant improve in accuracy when predicting the vortex filament flow field. Additionally, since the curved definition allows accounting for longer segments, thus computational cost are reduced, while maintaining accuracy, despite of the increased analytical complexity of the parabolic arc, schematics of geometric definition for BCVE are shown in Figure Figure 2.12 Geometry for the Biot-Savart definition of curved vortex segments [46] Sections 3.2 and 3.3 will address filament vortex model with deeper description and formulation for rotor wake implementation Rigid/Prescribed wake models Prescribed geometries for wake modeling have been a feasible and practical approach for wake study and even today these methods can be properly used when conditions do not demand a more detailed analysis, e.g. time accurate highly unsteady conditions. Feasible cases may come usually in the form of steady flight conditions, where large accelerations or changes in attitude may not occur or can be neglected. 22

42 Definition of rigid wake morphology usually follows that observed during experimentation, i.e. where a helical flow structure is formed below the rotor, being convected downstream and affecting the inflow velocities at the disc plane. In its basic formulation these observations led to a basic representation of a rigid helical pattern keeping a fixed wake pitch without accounting for wake contraction or local geometry variations. Generally, work on rigid wake models led to improvements on the classic helical model, wake contraction, wake sweeping in forward flight conditions were included later along time. This kind of rigid wakes can be formulated to depend on the rotor advance ratio μ rotor trust coefficient C T and the angle of attack of the tip path plane α tpp as well as on semi-empirical tuning. Prescribed wakes result from the inclusion of experimental observations but opposite to rigid wake, prescribed wake accounts for some degree of predicted distortion. The work of Landgrebe and his collaborators, condensed in references [47], [48] and [49], presents a method where a prescribed wake model is developed from experimental observations that led to include parameters that are dependent on blade azimuth ψ and wake age ζ, formulating an envelope and a shape functions to provide a generalized wake model. The former describes the amplitude variation with the wake age while the latter describes the characteristic distribution of the tip vortex distortion locations with wake age phased by azimuth. This work was supported by the use of charts [47] that provided information to determine tip vortex location coordinates for a wide range of steady state conditions including blade/vortex interaction locations [47]. Figure 2.13 Prescribed distorted wake (left) and undistorted rigid wake (right) [47] 23

43 One additional method of prescribed wake representation was provided by Beddoes [50] who suggest an approach where the position of the tip vortices are estimated by assuming a given inflow distribution over the rotor disc. The process is split in two parts, first a prescribed distorted wake model is used (like the previously described) making use of a radial averaged downwash distribution, then with the tip vortices lines defined following a skewed cylinder shape wake induced velocities are calculated by locating a pair of equivalent constant strength straight vortex lines at critical points on the spiral defined by the tip vortex lines (see Figure 2.14). The method thus avoids the inboard shed wake and complex calculations. A detailed implementation of the method is described in reference [51]. To this day, rigid wake representation models are still a valid approach and use of rigid wake can be considered to reduce computational cost by implementing this concept in a hybrid approach like that seen in reference [52] where a finite volume approach is coupled with a rigid wake to solve a rotor blade problem. Diverse alternatives are available in the literature related with this relatively simple method and evolution of the method overview can be found in reference [53]. 24

44 Figure 2.14 Location of equivalent vortex lines on critical trailing vortex system (top) and perspective view of the distorted vortex system (bottom) [51] 25

45 2.2.5 Free wake models This concept is valid for a vortex filament representation or a boundary element method. Free wake models are basically divided into two different types, relaxation and time marching approaches. Relaxation method is dependent on wake periodicity assumption and is applicable only to steady-state conditions. On the other hand time marching methods provide the best representation of wake definition and evolution but it comes to a prize. Time marching methods are susceptible to numerical instabilities and also to the inherent instabilities of the wake and require careful formulation for the solution scheme [12]. Figure 2.15 Free wake development for a rotor in forward flight by using a relaxation method [39] As mentioned above, implementation of relaxation methods requires the assumption of periodicity in the wake at the rotational frequency, blade azimuth location is frozen in time and the Lagrangian markers or lattice/panel control points on each wake element 26

46 are updated until a defined convergence criterion is reached all following an initial condition which is usually given by a rigid or prescribed wake solution [12]. Relaxation methods can be defined using different approaches, Johnson [54] developed a solution based on influence coefficients for a general wake formulation for fixed and rotary wings. A semi implicit scheme was applied by Miller and Bliss [55] and later Bagai and Leishman [56] proposed a pseudo-implicit technique based on a predictor corrector method. The second type, the time marching approach, in opposition to relaxation method, allows exploring transient conditions. This method is suitable to study operations where transient flow field interactions are important, for example, near ground operations, transient maneuvers or autorotation flight. Diverse methods have been developed by several researchers. CHARM [57] implements a reduced order hierarchical fast vortex method which generates significant reductions in computational cost. CAMRAD [54] implements a general wake where distortions are calculated using a influence coefficient method. Widely spread, the methods developed by Leishman, Bagwath and Ananthan have gone through both types of free wakes ([58] [59],[60], [61],[62]) Vortex ring / Vortex tube wake models Basset et al. ([63], [64]) formulated a disc that contained the vorticity shed at different stations along the blade radius. Vorticity, at a given blade section, was defined from the blade load distribution at that same station, thus a disc composed by several concentric vortex rings was generated and its motion is the result of the sum of freestream velocity and the averaged downwash velocity, the process was repeated at a given step to account for the formation of a vortex disc during a complete revolution. Since several vortex rings are located in the disc, self and mutual influence of every vortex ring on the others is evaluated. The resulting series of vortex discs were arranged in rigid or flexible formations to form the rotor wake. 27

47 Figure 2.16 Free wake development for a rotor in forward flight by using a relaxation method [64] Zhao and Prasad ([65], [66]) proposed a vortex tube able to deal with transitional conditions and to represent uncoupled wake bending, skewing and spacing effects. Implementation of the method for flight mechanics applications is found in reference [21]. A more recent proposal is that of Palo et al ([67], [68]) presented a Fast Free Wake (FFW) model intended for wake-body interaction in real time. The model consists in a series of vortex rings being realized every certain time form the rotor disk being allowed to move freely interacting with all the other vortex elements without restrictions. Wake deformation is modeled by ring motion and change in radius with a vorticity strength that is defined by the instantaneous rotor thrust value. In his papers Palo claims low computational costs implementation in real time applications with good results. Some disadvantages are discussed though, since the rings are generated only at the tip radius the model presents limitations to predict the induced velocities over the inner portion of the blade, something that was proposed to be solved by including additional rings in the inner area on the downside it limits itself to interact not with blades but with an actuator disk, leaving out several interesting properties of the rotor system. An interesting model for a relaxation solved free wake was proposed by Van Hoydonck ([69], [70]) where, accounting for tip vortices only, a vortex tube is defined with NURBS primitives, taking advantage of the reduced number of points for wake geometry because of NURBS formulation. Intended for ship qualification applications, this method was tried for In Ground Effect (IEG) conditions showing interesting potential for its 28

48 development. Examples of wake development of this method are shown in Figure 2.17, Figure 2.18 and Figure Figure 2.17 NURBS defined rotor wake vortex tube [69] Figure 2.18 Wake trimming iteration process [70] 29

49 Figure 2.19 Source panels flight deck representation and interaction with rotor wake IGE [70] Fast vortex/panel methods Based on the work of Appel [71] Barnes and Hut [72] and Greengard and Rokhlin [73], these methods are specifically intended to reduce the computation time for N-body problems in galaxy simulation. Similar to those problems, free wake defined by Lagrangian segments can be treated with the same approach. As explained in reference [74] the main characteristic of fast methods is a combination of a spatial grouping scheme that associates the interacting vortices into localized structures and a formal approximation to the net inducing effect of the vortices in a group which is invoked for interactions between well-separated groups. Basic formulation of the method define a grid of cubic boxes containing the flow domain, first interactions between elements of a group are calculated and then a group to group influence is computed, based on influence coefficients, multipole expansion and Taylor expansion. Advanced methods like the multi-level hierarchical decomposition the grid is organized in a hierarchical level and the sub-groups are evaluated using efficient far field approximations. The hierarchy for this model is in its basic level contains individual elements. Reductions in computation time for this method are reduced for a N element problem from a O(N 2 ) to O(NLogN) number of operations, reducing the required memory in the same orders. 30

50 Figure 2.20 Octree structure and vortex interactions for fast vortex/panel methods [74] State-space free wake Several advantages come with this type of method. Because of its formulation it can be directly linked with other problems formulated in the same way, resulting in strong coupling for the different dynamic, aerodynamic and elasto-dynamic problems. A first formulation a free wake in state space was developed by Johnson ([75], [76]) who defined a linear model where the states were generalized coordinates describing the inflow and the inputs were generalized coordinates describing circulation. The model required an impulse response of the wake induced downwash λ to the rate of change of circulation with time for the shed wake and of an impulse response λ to circulation for the trailed wake. Celi [77]reformulated a state space free wake by using the method of lines (MOL) which is a technique for the numerical solution of partial differential equations (PDE) in which 31

51 only the dependency on space is discretized in finite difference form, the result is a PDE converted into an ordinary differential equation (ODE). In reference [[77]] as initial step Celi formulated the model using a rigid vortex wake and uniform inflow, using the exact existing solution for such particular condition [11]. Equations were first order reduced with MOL and then solved by using publicly available DE solvers DE/STEP and DASSL. With an ODE system defined a linearized system was extracted. The formulation proved to be convenient and flexible and it was extended by Letoney, Celi and Leishman [78]. In this case, instead of using a rigid wake with exact solution a free wake was used with non uniform inflow. Again using the MOL, a state-space model was obtained defining the states as the spatial positions of collocations points of Lagrangian markers on the wake representative filament (wake age). A linearized model was extracted by using finite difference approximations. The resulting state-space models are large and to solve for this problem the state vector is approximated by using shape functions. The model couples the free vortex wake with blade flapping and unsteady airfoil aerodynamic model also formulated in state space representation [79] to account for the shed vorticity in order to reduce computational requirements and keeping importance of tip vortex influence. Good agreement was found when compared with other formulations keeping fidelity with blade spanwise distributions of inflow and lift. 2.3 Rotor Wake models issues and considerations Blade Aerodynamics Blade aerodynamic representation is the source of load distribution definition and thus is linked to circulation distribution, bound vorticity and strength of the shedding wake, besides it is also used to account for interaction between the blade planform and aspects like noise generation [45]. Depending on the problem at hand, several methods are available to model blade aerodynamics. Panel method is a flexible and economic way to represent blade geometry with good fidelity, and is built usually by using a combination of singularities, sources for thickness and vortex rings or doublets for camber line, on flat or curved panels (Figure 2.21). 32

52 Figure 2.21 Geometry discretization for panel method for a blade [80] Lifting line models like that of Weissinger [81] or alternative adaptations, like that of Phillips-Snyder [82] overcome the initial limitations of the Prandtl method i.e. allowing to include arbitrary shapes like swept or skewed geometries or lifting surface systems (Figure 2.22) by using a segmented line composed of deformable horseshoe vortices. The principle stays for rotary wing where rotational velocities are included as function of blade radius (Figure 2.23). Vortex lattices are also available to represent a lifting surface, more computationally efficient than panel methods or highly defined finite volume grids. Figure 2.22 Systems of lifting surfaces using Phillips lifting line adaptation [82] 33

53 Figure 2.23 Lifting line for rotary wing [11] Unsteady aerodynamic loads come from different sources and for a rotor blade such loads can come in the form of a combination of forcing (input) due to collective and cyclic blade pitch, twist angle elastic torsion blade flapping velocity, elastic bending, induced velocity from trailed wake and blade-vortex interactions (BVI). Decomposition of the velocity field, as provided in [11] is shown in Figure 2.24 and Figure Such decomposition is discriminated for blade motion and flow field structure, the former representing the different effects that are sensitized into the alteration of the local angle of attack and the latter represents the effects due to flow field interactions according to source and behavior, whether periodic or aperiodic. Zooming into a blade section, the different effects are represented by variations of local velocities, i.e. angular and linear accelerations, which are represented by a pitch velocity, plunging due to flapping and lead-lag motion around the feather, flapping and rotor axes respectively (Figure 3.5). Unsteady airfoil models cover a wide variety and several authors have approached the problem in diverse ways, from Theodorsen [83], Wagner [84], Kussner [85] and von Karman and Sears [86] frequency domain models and Lomax et.al ([87],[88]) compressibility corrections to the Leishman-Beddoes [89] dynamic stall models, such models account for the different sources of aerodynamic loading in the transient state. While frequency domain models are well known and suited for most of the periodic motion behavior an equivalent definition in state-space form can be formulated and thus coupled with other dynamic models. The model followed in this work is that proposed by Leishman and Nguyen [79] for subsonic attached unsteady flow. 34

54 Figure 2.24 Blade Velocity build-up [11] Figure 2.25 Velocity field decomposition [11] Maneuvering Flight The capability of modeling maneuvers is a key aspect to properly design rotorcraft with satisfactory performance, flying and handling qualities. Maneuvers, for the case of rotor wake are integrated by the V ex term. In this term angular rates about the roll, pitch and yaw axes are included, thus linking this highly influential elements with wake generation. Definition of this velocity term considers the maneuver rate vector (p, q, r) and the position r of wake elements. Thus, external velocities are included by [12] V ex = p, q, r (x, y, z)

55 V ex = qz ry i + pz rx j + (qx py)k 2.9 Where the first term in the right hand side represents the effect of the pitch and yaw rates on wake, generating a longitudinal skewing distortion. The second term is the lateral velocity component that results from the roll and yaw rates generating a lateral skewing distortion. The third term effect is observed as an asymmetric axial stretching wake distortion. These maneuver generated velocities have significant effect on induced velocities and hence in rotor loads and blade flapping, being as well one of the main reasons behind off-axis helicopter response [12]. Figure 2.26 Evolution of effects of maneuver flight on wake [90] 36

56 2.3.3 Ground effect Rotorcraft operations In Ground Effect (IGE) have grown in importance since problems like brownout and ship operations have gain importance. Inclusion of wake influence is also important to develop protocols for deck operations ([91], [92]). Two main methods are used to account for ground effect. Method of images uses a second identical rotor having the same aerodynamic load and wake geometry but inverted and located at ground level in a flat configuration. This configuration causes a self influence of equivalent strength elements, imitating the presence of a flat ground. The remaining method is the surface singularity method. This method is, in essence, defined as those explained in section The method allows for increased flexibility since does not require a flat ground to be applied. Geography can be defined by locating singularities like sources with a boundary condition enforcing tangent flow (e.g. Newman B.C.), the resultant wake geometry will be affected by the influence of those singularities. Figure 2.27 Ground effect nodel by method of images [12] 37

57 Methods describing interaction between rotor, wake and ground can be found in reference [93] Compressibility Considerations are made for compressibility effects on blade sections (airfoils), or surfaces by compressibility corrections, wake can also be perturbed by compressibility conditions. Its formulation is defined by neglecting the initial assumption of incompressibility for Navier-Stokes equation and linearizes the continuity and Euler equations by using pressure and density perturbations which leads to consider the compressible effects by using the compressible velocity potential given by the Prandtl- Glauert equation. This method was applied by Szymendera [45] testing the method up to M=0.88, showing the method to be able to keep validity in incompressible regimes while being able to predict the increase in lift nearby the blade tip compressible region. For his study Szymendra used a vortex lattice method to represent the blade but poor agreement with results was seen near the leading edge, it is recommended to use panel methods to overcome the singularity Wake/Body interaction One of the main goals of implementing a free wake is the possibility to explore interactions between the flow field and the bodies or structures immersed in that field. For the case of a helicopter interactions are relevant to define the influence of fuselage or near structures like in ground effect or bodies in the flow field like external loads subject to perturbations in flight conditions. In similar fashion to ground effect modeled by panel methods, immerse bodies can be included following a similar process; an example of unsteady interactions for helicopters is given by Lorber and Egolf [94], where forward flight is explored. A more recent application is that of CHARM model that allows for a global interaction by using advanced fast panel methods, Wachspress, Quackenbush and Boschitsch [74]. 38

58 Figure 2.28 Fully interacting rotor- wake-surface system for complex configurations [74] Problems generated by the use of this approach appear when a vortex element passes near the surface element. Because of the reduced distance the vortex element will tend to induced extremely high velocities at the surface panel. To solve this problem, a tessellation strategy can be used where the control point on the surface panels where projected outwards, generating new points for velocity calculation far from the control points [95]. The importance of interaction body or blade with vortex elements is usually related with noise generation, considerations and examples for blade vortex interactions (BVI) can be found in references ([96], [97], [98], [99]) The real time problem Computational times for real time applications result in a demanding requirement for model efficiency and/or computational capabilities. The desire to include increased fidelity wake models results from the possibilities to include other elements in the already complex human-machine interface loop. Also, predictive analysis allows for design of control systems with a virtual validation. For the specific case of rotorcraft wake, increased fidelity has come to a price of high computational cost, even more noticeable with the already mentioned demands. With wake models like those already 39

59 presented, possibilities allow to include free wake models in the form of state-space formulations or fast multi-pole panel methods or vortex-ring models, showing that there is an open possibility to include the effects of complex phenomena like wake-body interaction. The work of Horn and Bridges [91] presents what could be called a guideline for implementation of free wake models into a real-time flight simulation environment, setting a parametric study to explore a proper balance between fidelity and computational cost. The main parameters used were taken from the free wake model definition including 1. The number of vortex filaments emanating along the span of each rotor blade. 2. The number of vortex along each filament modeled as the rotor wake propagates downstream (for how long a vortex segment s influence is active) 3. The length of each vortex element, which for a time running model is linked to the temporal resolution used for updating the structure of the wake and for calculating the wake induced velocities. Also, parameters that included the global simulation (flight mechanics/wake model) were identified. Seven parameters were used 1. Time step of flight dynamics model 2. Number of blade elements in flight dynamics model 3. Time step of free wake model 4. Maximum number of vortex filaments in the full span 5. Amount of full-span wake modeled as opposed to far-field (how many revolutions are required with full-span wake generation). 6. Use of prescribed wake models mixed with free wake models. 7. Wake model update frequency. From the combination of parameters, 19 virtual experiments resulted, all tested for a hover flight condition and two forward flight situations at 45 knots and 90 knots. A baseline test was executed for comparison of results with a weighted error cost function. Additional comparisons were done against dynamic inflow model. Some of the conclusions addressed included 1. When observing small amplitude on-axis response, the difference between a high fidelity wake model and the simple finite state inflow 40

60 were negligible in many cases, and thus real-time free wake model show no real advantage 2. The use of coarse time increments for wake update (45 degrees sweep) and a 4 per revolution induced velocity update seemed to produce a reasonably accurate response with dramatic increase in execution speed. The use of curved vortex elements appears to be helpful to achieve realtime performance due to its longer length. 3. Increase resolution appeared to have significant impact on the response of the vehicle for the free wake model. 41

61 Chapter 3:State-space rotor - wake formulation Along the two previous chapters, the context surrounding the rotor wake problem was exposed showing different options for solution and related issues that were discussed according to several researchers. Following that initial insight, in this section, a particular solution, the State-Space formulation, will be adopted in order to explore in an initial form the ways to include such a wake model into a rotor mechanics model, aiming for future extension. More precisely, the approach adopted by Celi [77] will be addressed with more detail trying to take advantage of its formulation to face basic models for rotor mechanics and maneuvering inside a state space formulation. Supporting this purpose the current chapter will follow the work of Leishman, Bagai and Baghwat [12] and Betoney, Celi and Leishman [78], with more emphasis on the latter, which is an extension of Celi s seminal work for a free wake and includes the main elements for a flight mechanics expansive model. 3.1 Assumptions and constraints Since implementation of the method into a flight mechanics application may be a complex matter, the problem definition and solution method limitations are to be considered. The work of Rivera [100] provides a detailed and well considered set of initial assumptions to handle rotorcraft flight mechanics problems including a high 42

62 fidelity free wake model. In analog fashion, the initial assumptions and considerations for this work are: - A rigid blade representation. - Only flapping motion is considered. - Dynamic stall and compressibility on blade or vortex are not included. - Constant rotor speed. - Rotor is fixed and isolated. - Free stream velocity is defined as input. While limited, the above assumptions allow space for exploration in regard to the basic mechanics/aerodynamics interactions, providing insight into blade motion and reactions to wake velocity field influence in a specific and identified way. Additionally such an approach provides the need to generate couplings in model formulation and computational structure, driving to strategies to do so. It is important to mention that more mature models, e.g. including dynamic stall and compressibility for blade analysis and vortex filament formulation are extensible from the models presented in the available literature that is followed in this work. 3.2 A Lagrangian approach for wake In this section, a dissection of the state-space model of Betoney Celi and Leishman [78] will be performed focusing on main formulation and model integration. As introduced in section State Space wake formulation considers rotor wake as a set of filaments being convected from the trailing edge of the moving blade, such filaments can be represented in the form of finite, straight vortical elements bounded by points in space which serve as position markers. It is on those so-called markers where velocity influence from the surrounding flow field (wake filament motion) can be accounted for. In that order, a Lagrangian, particle oriented formulation is taken, thus considering the Navier-Stokes equations in velocity-vorticity form (introduced in section 2.2.3), the change of flow vorticity in a fluid element is defined by [12] ω t = V ω + ω V + ν Δ ω

63 Where the first term in the RHS of the equation describes vorticity convection, the second represents the strain of a given element and the remaining term represents the flow diffusive behavior. Adding to the above formulation, a Lagrangian definition can also be related to vorticity transport in the fluid, thus, consideration of vortical line elements as particles is allowed, represented by material points inside the element, being transported along a velocity field V. The specific case of a rotor blade shedding a trailing wake (Figure 3.1) shows vorticity being concentrated and transported along fine rotating filaments with viscous cores that, when compared to the length scales of the potential flow problem, are small enough to justify the assumption of a limited viscous effect within the core region [11], this leads to a practical discretization in the form of vortex lines segments, straight or curved, with a given vorticity distribution [77]. Such vortex filaments are trailed from the blade trailing edge ([56], [54]) and convected downstream according to local velocity field conditions. In a more practical approach, usually only blade tip vortex filament is modeled, keeping good fidelity though, due to the dominant influence of tip vortex behavior (section 2.1.1), saving thusly an important amount of computational cost. Inboard wake components then are modeled to account for overall vorticity intensity, defined by local circulation on blade sections, along the blade radius and condensed in the tip vortex line ([56] [78]). With a filament discretization defined by several straight lines (see Figure 2.11), an analogy with particle dynamics can be made by considering (not strictly though) the start and end points of every segment, known as markers, as if they were particles moving according to the velocity field conditions. Thus, if such an assumption is made, behavior of markers in time, within a given velocity field, can be defined by dr dt = V t, r t = 0 = r Where r is the spatial radius and V(t) is the local velocity at point r. Now, reviewing the wake helicoidal general behavior, presented in Figure 3.1, it is apparent that equation (2.7) can be redefined to track filament markers in terms of the helicoidal wake structure, then, position of a given marker can be given in terms of azimuthal position ψ, serving as an implicit form of time [11] defined from rotor speed 44

64 Ω and a spatial variable ζ which represents the angle of the marker relative to its point of origin, that is, generally the blade tip or near the tip and it is known as the wake age. Figure 3.1 Rotor wake filament scheme (original from [12]) Thus for time variable, above considerations lead to ψ = Ω t 3.1 Then it follows that equation (2.7) equivalent form is dr(ψ, ζ) dt = V[r ψ, ζ ] 3.2 Now, wake age term can be defined considering wake filament formation sequence, given a starting position at blade tip, related to an starting instant t ζ0 when the marker is first trailed from the blade tip which is angularly positioned at (ψ ζ). Thus, relating blade position and wake age yields ψ ζ = Ωt ζ0, consequently ζ = Ω(t t ζ0 ) 3.3 Which conveniently leads to modification of equation (3.2) on RHS, resulting into 45

65 dr(ψ, ζ) dt = dr ψ, ζ dψ dψ dt + dr ψ, ζ dζ dζ dt dr(ψ, ζ) dt = Ω dr ψ, ζ dψ + dr ψ, ζ dζ Finally, equation (3.2) becomes dr ψ, ζ dψ + dr ψ, ζ dζ = 1 V[r ψ, ζ ] 3.4 Ω To solve equation (3.4), a numerical approach is required due to the complex behavior of its components, especially for RHS velocity term which will be developed in more detail in next section. Note that LHS of equation (3.4) can be discretized using different available schemes whose selection depends upon stability, computational cost, accuracy and influence of numerical behavior on the physics involved ([11][77]). For this work, section 3.5 presents the discretization method for LHS of equation (3.4) based on the Method of Lines. 3.3 Induced velocity field Velocity field is included into free wake model by developing the RHS of equation (3.4). The term V[r ψ, ζ ] is composed by three major velocity components, freestream velocity V, induced velocity V ind and the external velocity component V ex which comprises alterations on the velocity field due to a diversity of sources like general maneuvers[12]. Thus, the velocity term is given by V r ψ, ζ = V + V ind + V ex 3.5 Of especial complexity, the induced velocity term V ind can be a computationally expensive term due to the integration process required to calculate the mutual induced velocity generated by every vortex segment that compose the wake filament. 46

66 Considering a straight vortex segment representation and complementing the initial description provided in section this method formulation will be described in this section and its linkage to the specific application to rotorcraft will be developed. Thus, formally presented, and retaking the fundamental vortical straight segment depicted in section 2.2.3, influence of a three dimensional straight vortex segment on a given point is dictated by the Biot-Savart law, which in its general form is given as [36] V ind = Γ 4π dl r r 3 = Γ 4π dl (l AB r 1 ) l AB r Where Γ is the circulation strength or intensity of a vortex element, and dl is the differential length of the vortex element and the remaining notation is geometrically shown in Figure 2.10 and Figure 3.2. After convenient vector treatment, induced velocity v ind at a point p by one straight line vortex depending on geometrical vectors segment given by [36] v ind = Γ 4π r 1 r 2 r 1 r 2 2 l AB r 1 r 1 l AB r 2 r For this case, l AB is the vector that lays along the points A and B shown in Figure 3.2, while r 1 and r 2 are the vectors defined from the points A and B to the influenced point P. This formulation, results in a convenient expression for numerical computations. Note though, that Biot-Savart model becomes singular when the distance from the filament element and the evaluation point becomes zero and thus the model requires further considerations to be used in practical applications, which are related with the vortex core. The vortex core is defined to account for dissipation (viscosity [12]) effects and it can be included into the Biot-Savart induced velocity equation by inserting a compensating term that will allow avoiding the singularity generated when the magnitude of the perpendicular distance r approaches to zero. 47

67 Figure 3.2 Biot-Savart influenced velocity by a vortex straight segment on a point in space This term can represent dissipation in several ways, depending on the core model to be used for which several options are available [11], all focused on representing the presence of viscosity through its effect on core velocity profile along the radius of vortex core. A vortex segment moving in space is described by three main features, the swirl velocity, its viscous growth in time, and its strain deformation [11]. The first is observed to be a function of the radius of the vortex from its origin decaying as moving away from the core, the second is a consequence of viscous effects and implies the finite dimension of the core in the real wake dynamics, something that has been assumed to be an infinitesimally thin tube that containing the vorticity [101] to allow analytical modeling. The third is the result of vortex motion in a three dimensional space, thus deformations will imply that an increase or decrease in length will result in a change in the cross sectional area of the vortex, analog to the behavior of a continuum solid material behavior when strained [102]. In a basic approach for vortex core swirl velocity, Rankine proposed a simple model for swirl velocity assuming a linear velocity distribution along the core radius while the flow outside the core behaves as a potential vortex, thus the conditional expression for Rankine s model is given by if 0 r 1 V θ r = Γ v 2πr c if r 1 V θ r = Γ v 1 2πr c r 3.8 Where rc is the core radius. Refinements of Rankine s model led to the Lamb-Ossen model which assumes laminar behavior for small vortex Reynolds numbers, developing the momentum equation in polar coordinates resulting 48

68 V θ r = Γ v 2πr c r 1 e αlr2, α L = In this model α L represents a power coefficient accounted to match experimental results. Latter on Vatistas [103] developed a flexible and more general method covering previous models according to the value of the power coefficient n. This formulations is convenient and useful since, for example, Scully model [104] and Rankine model can be obtained for n=1 and n = infinity respectively. V θ r = Γ 2πr c r (1 + r 2n ) 1/n 3.10 More recent revisions of the topic led Ramasamy and Leishman ([105], [106]) to propose a model in function of Reynolds number inside the vortex filament depending on vortex intensity Γ v and dynamic viscosity v which is defined as Re v = Γ v υ 3.11 This Reynolds definition for vortex filament allows accounting for turbulence effects on core vortex growth, which leads to define an additional dissipative term δ δ = 1 + a 1 Re v 3.12 With a 1 being an empirical constant dependent on distribution of eddy-viscosity across the vortex that usually corresponds to a 1 = 0.1 as reported by Bagai and Leishman [56] and later defined as a 1 = by Ananthan [107] which appears to be in better agreement with experimental observations, the later is used in this work. This dissipative term allows to consider turbulent behavior inside the vortex core and it is to this point where vortex swirl velocity and core growth come together since this dissipative term directly affects the core radius rc. 49

69 Figure 3.3 Core swirl velocity models To account for diffusion due to viscous effects on core growth Lamb-Oseen proposed a semi empirical laminar vortex model considering vortex core deformation in time r c t = 4αυt 3.13 Where α = [108]. Since vortical flow can initially show laminar flow like behavior as time increases, the vortex progressively turns into a turbulent flow problem, to account for this the Lamb-Ossen core growth model can modified by inserting the vortex Reynolds dependant dissipative term [12] r c t = 4αυδt 3.14 Additionally, a convenient minimum radius is defined for practical purposes to avoid numerical problems or singularities considering that at initial generation time to=0 50

70 rc=0, then, vortex decay is considered in comparison to an initial state (virtual time) and the core radius affected by the aforementioned set of considerations yields a model that considers laminar to turbulent viscous transition, which as posed by Ramasamy and Leishman [106] is expressed as r c = r 0 2 4αυ 1 + a 1 Re v (t t o ) 3.15 According to the work of Bagai and Leishman [108] the initial core size corresponds to r o = 0.25c while van Hoydonck used a value of r o = 0.08R taken from experimental measurements[109]. For this work the later value for core size is used. Presented in terms of wake age such that ζ = Ωt equation (3.15) yields into a convenient form given as r c ζ = 4αδν ζ ζ o Ω r o 2 + 4αδνζ Ω 3.16 With the vortex growth considered, a correction term K can be defined to avoid the aforementioned Biot-Savart singularity, then from the finite core model of Vatistas [110] this term is given by[11] K = 2 r c 2n + v 2n 1 n 3.17 Which accounts for Skully core model[104] if n=1.wich will be used here. Besides this model, an improved regularization is consider parting from the results of van Hoydonck [70], a modified K factor which provides correction for velocities at the beginning and end of a given vortex segment, conditioning the use of the term h only when the projection of the evaluation point P falls within the effective correction region around the vortex filament segment and between its endpoints, if the projection falls before point A, or after point B then the h term is uses the radial distances r 1 and r 2. Thus the regularization factor K v is given as 51

71 K v = d 2 r c 2n + d v 2n 1 n 3.18 Such that d = r 1 if cosθ 1 < 0 r 2 if cosθ 2 > 0 else Then the corrected Biot-Savart results in V ind = Γ 4π K v r 1 r 2 r 1 r 2 2 l AB r 1 r 1 l AB r 2 r Or in an alternative form V ind = Γ 4π d 2 r c 2n + d v 2n 1 n cosθ 1 cosθ 2 e 3.20 Where e is an unitary vector given by e = l AB r 1 l 12 r The cosine values and h, as referenced in Figure 3.2 and Figure 3.4, result in cosθ 1 = l AB r 1 l 12 r 1 cosθ 2 = l AB r 2 l AB r 2 with 52

72 l AB = r B r A r 1 = r P r A r 2 = r P r B h = r 1 sinθ 1 = r 2 sinθ 2 This representation results in a convenient programming definition. Figure 3.4 Schematics of interactions of induced velocities and blade loads 3.4 Blade model While several options are available to model blade aerodynamic loading (see section 2.3.1), for this case an unsteady state space airfoil model will be adopted. This type of model provides commonality of formulation with the wake state-space form and allows consideration of the unsteady character of the aerodynamic environment around the blade and as shown, it also provides the convenience of a time domain formulation. Such formulation is integrated to blade dynamics trough strip- theory, assuming two dimensional flow for high aspect ratio wings, and coupled to wake influence by 53

73 considering the wake induced velocity at every point over the blade where a strip airfoil is located, this section will depict the integration of theses aspects with the main blade flapping dynamics and rotor motion. By adopting the aforementioned approach computational cost reduction is expected, since only the blade tip vortex filament is considered, as a perturbation velocity at section level, leaving the inner shed vorticity to be treated as a near field wake integrated into the unsteady airfoil model, represented in practical terms by a lift deficiency function Considerations and assumptions Blade aerodynamic environment is developed following four main assumptions and considerations [111]. 1. Two-dimensionality: This assumption neglects the effects of three dimensional effects due to the high aspect ratio typical of rotary wings, (around 10). Except for blade tip where rotating flow formed at the tip breaks this assumption has been valid enough for most rotary wing analyses. Considering the unsteady nature of blade motion, this assumption relevancy rises from the relatively diminished spanwise propagation of disturbances as unsteadiness affecting the wing is increased for high aspect ratio wings. 2. Reduced frequency: The consideration of unsteadiness of the blade aerodynamics is described by the reduced frequency, a concept that accounts for the change of driving aerodynamic conditions by relating the excitation frequency with the velocity of the flow and the section chord, that is: k = ωc 2V 3.22 Reduced frequency allows defining a threshold for the dominance of unsteady effects, being considered that for values of 0 k 0.05~0.06 unsteady effects can be ignored. For the rotorcraft case typical reduced frequencies are around k=0.07 at 0.75R, beyond the point of quasi-steady assumption, further consideration of torsion modes and other aeroelastic effects can rise the unsteady influence to a value of k=0.2 [11], thus clearly, unsteady consideration is in fair consistency with the real operating conditions. 3. Variable flow and airfoil motion: Particular for rotary wings, this consideration covers the characteristic large fluctuations of motion observed in both longitudinal and vertical motions. This consideration is covered by an 54

74 assumption of infinitesimal motion but can be of limited pertinence if large variations, like large amplitude maneuvers are to be studied. 4. Near and far field effects: Opposite to fixed wing, where the wing leaves the perturbation, i.e. wake, behind, rotary wing is strongly influenced by its own perturbation as well as the perturbations of vicinity blades, (see section 2.1) Airfoil velocity sources Contribution of airfoil motion to generation of aerodynamic loads can be approached initially by considering three main types of motion that finally result into the approximate general behavior of the section. Such motions are plunging, pitching and lead-lagging. Pure plunging represents a vertical motion generating a velocity component normal to the airfoil chord opposed to the plunging direction. Pure pitching implies an isolated variation of angle of attack by changing the incidence angle around a given axis located at a point along the airfoil chord, being the resultant angle of attack independent of the stream velocity angle. Finally lead-lagging is a type of motion of particular interest for rotorcraft which implies a forward-backwards motion along the motion path and is present in a helicopter rotor in forward flight where the azimuthal variation of velocities generates motion of this type on the blade. Additionally, besides the motion components there is one type of velocity source that considers the presence of gusts affecting the airfoil, this kind of component is the link between the airfoil loading and the velocity field generated from the rotor wake. If considered. Figure 3.5 shows the different velocity sources schematics. For this work, the lead-lagging component will not be considered following the initial assumptions that include only flapping blade motion; the model is extendable though to include such kind of motion. 55

75 Figure 3.5 Unsteady blade section velocity components [11] By considering this set of motion components, a resultant upwash velocity results along the airfoil chord. Assuming linear behavior, such upwash results in [112] w a = ( + Uα) + α x ab 3.23 The first term represents the plunging velocity, the second term represents the effect of angle of attack variation and the third term is the pitch variation for a given pitch rate α around an axis x=ab, which if located at ¼ chord then a=-1/2 and with b =c/2. Additionally, an induced local downwash velocity λ lw generated by the trailing wake is to be accounted besides the velocity w b resulting from the airfoil bound vorticity. Thus, these velocities, along with the freestream velocity V, form the basic setting for the unsteady airfoil formulation The Theodorsen model As a starting point for the state space model the Theodorsen model [83] is revisited since as stated by Leishman, Theodorsen and Wagner functions and the state space 56

76 model are simply different representation of the same dynamical system [79], including same components and the same dynamic basis. Thus, considering harmonic motion, the unsteady lift load is given, preserving linearity, by the result presented by Theodorsen, in this case, for lift only L Total = 2πρV 2 b bα + α + V V 1 a C k 2 + πρv 2 b b V 2 + bα V b2 V 2 aα 3.24 With the correspondent lift coefficient resulting in C ltotal = 2π bα + α + V V 1 a C k 2 + πb V 2 + α V bαα V Theodorsen s model accounts for the aforementioned velocity field plus the acceleration of the mass of air surrounding the airfoil, thus, in a deeper exploration, the components of aerodynamic load in the RHS of equations (3.24 and 3.25) represent: 1. Circulatory loads include two sub-components. For the first term, inside the brackets, the quasi-steady formulation for angle of attack is found including the angle components that can be traced to the basic velocity decomposition of section 3.4.2, it accounts for the load originated from the bound circulation and upwash. Secondly, the term C(k) is found, which comes to be the so-called Theodorsen Function and essentially works as a dissipative term or lift deficiency function that includes a sort of damping effect originated from the wake induced velocity λ lw. Theodorsen function C(k) is defined as a complex function C(k)=F(k)+iG(k), defined by the Hankel functions of the second kind and Bessel first and second kind functions (see[113] [111] and [112]). When compared against the quasi-steady result, the deficiency effect caused by the Theodorsen function on the oscillatory behavior comes in the form of amplitude reduction 57

77 and phase lag effect on the circulatory load portion following the variation of the reduced frequency k, obtaining the steady state for k=0 [11]. 2. Non-circulatory loads, found in the second term in equations 3.24 and 3.25, is characterized by featuring acceleration terms, such terms represent the so-called virtual mass, or apparent mass effect, which in practical essence acts as an attenuation factor which emulates an inertia like effect originated by the mass and inertia of the air surrounding the airfoil [111]. While acceleration terms may be typically small for rotary wings in normal steady conditions, relative to blade mass properties, these non-circulatory loads are usually found relevant when the flow-airfoil interaction is suddenly altered, a sudden pitch change for instance, resulting into transient loads that dissipate along with the damping of the perturbation. Maneuvering flight is one of the particular circumstances where this apparent mass effect applies ([102] [114] [115]). In a simpler more basic analogy, the previous representation of an airfoil aerodynamic system could make more physical sense when compared to a second order mechanical system of the form mx + bx + cx = F, where m represents the moving mass, b represents a damping variable and c is a elastic constant. In the same way, if compared, the virtual mass definition becomes clear considering that is attached to the acceleration term, or terms, of the system, with a similar comparison for the damping effect which is related to the lift deficiency function. The relevance of Theodorsen model could be condensed in the legacy of his concept of lift deficiency function, the presence of such term allowed for the extension or generalization of the model by proper definition of the C(k) function according to the problem at hand, defining specific solutions for pure angle of attack oscillations, pure plunging and pitching oscillations according to a given input, known as forcing, according to the case. From this point, several derivatives, or alternatives, of the Theodorsen problem include the work of researchers like: 1. The returning wake Loewy 2. Sinusoidal gusts Sears 3. The indicial response- Wagner 4. Sharp-edged gust Kussner 5. Traveling sharp-edged gust Miles 6. Varying incident velocity Greenberg 58

78 References [113],[112] and [11] include elaborated descriptions of each of these problems, for this work, only Wagner and Kussner problems will be considered in the following sections towards the buildup of a state space unsteady airfoil model. While including the relevant physics aspects and being an ingenious solution, as stated by Leishman and Nguyen, Theodorsen s theory has a significant deficiency since the assumption of simple harmonic motion, upon which it is based, is strictly valid only at the flutter boundary. In addition, Theodorsen s method restricts the solution method of a rotor analysisi to the frequency domain, i.e., the harmonic balance method. For rotor analysis using time-marching techniques, the unsteady aerodynamic behavior of the blade sections must be properly modeled in the time domain, [79]. Considering the aforementioned, the state-space model used in this work, time domain defined, makes use of two developments around the Theodorsen model that leads to its formulation The indicial function approach, Wagner problem While the Theodorsen model is defined for harmonic motions, the aim for generality led to define a response of the system as result of arbitrary excitations or inputs. Wagner [84] considered a time domain approach to the unsteady airfoil problem proposing a solution based on an indicial function. An Indicial function is, simply put, the response of a dynamic system to an instantaneously applied disturbance at the instant t=0, i.e. a step function that is kept constant for t>0 [11]. Wagner s approach to the problem leads to an analog formulation to that of Theodorsen but for a more general problem, resulting for the unsteady lift coefficient C l t into C l t = πc δ t + 2παφ s 2V I c C l t = C l t + C l t 3.26 Note that equation (3.26) has as similar structure as that of Theodorsen s. Wagner model has a non-circulatory component, apparent mass, given by the first term and defined by a Dirac-delta as response to a step input while the second component corresponds to the quasi-steady state lift coefficient affected by the function φ(s) which serves as the lift deficiency function for this case and is called the Wagner function, defined in terms of a distance traveled in semi-chords s=ut/b also known as reduced time. 59

79 If only the circulatory lift component in equation (3.26) is considered, that is C l c t = 2παφ(s) 3.27 And knowing the function φ(s), the unsteady circulatory load for arbitrary inputs of angle of attack can be obtained by superposing the indicial responses with the Duhamel integral. This is possible in similar fashion as the case for a generic system perturbed by a step input, which is then, following the Duhamel superposition principle [113], able to consider general loads. Considering a step input for the case of t 0 a given system output y(t) results in the Duhamel integral y t = f 0 φ t + t df(t) φ t ς dς 0 dt 3.28 From this general Duhamel definition the f(t) term is the admittance function or forcing function which for this case is defined as f t = α(t), considering the reduced time s the circulatory load yields C l c t = 2πα α 0 φ s + 0 t dα ς dt φ s ς dς or 3.29 C l c t = 2πα e (t) For this result, the effective angle of attack α e (t), following the Duhamel principle, accounts for the time history that emerges from the wake induced effect. While the Wagner function has indeed an analytical solution. (see Gulcat [113]) R.T. Jones [116] proposed an approximated exponential alternative within 1% of the analytical solution, thus, such approximation is given by φ s e s e 0.3 s

80 This approximation has a noticeable advantage for simulation purposes since analytical solution may be complex to extract, even more if rotary wing applications are considered. This result is compared against the analytical solution in Figure 3.6 Figure 3.6 Wagner function comparison for analytical and approximated function Note that there exists an initial infinite peak followed by a drop up to a 0.5 offset, denoting in practical terms that the response of lift load to a sudden change of AoA is reduced by half of its steady value [111] The Sharp-edged gust, Kussner problem So far, blade sections have been covered by including the Wagner function lift analysis through the use of R.T. Jones approximation. Thus, blade sections, assuming negligible 3D effects due to large aspect ratio, account for the circulatory loads due to bound vorticity and the effect of its own shed wake. Now, as covered in sections 3.2 and 3.3, the influence of a high strength blade tip vortex filament has a direct influence on the 61

81 local airfoil section velocity. To account for this effect the sharp-edged gust problem, from Kussner [85] is introduced. Similarly to the Wagner problem, Kussner problem also considers arbitrary variations of angle of attack on the airfoil, but it considers that the angle of attack does not change immediately over the entire airfoil but rather gradually as the airfoil penetrates a gust field [117], in this way, the rather variable velocity field found in rotary wing velocity environments can be considered in similar fashion as presented in Figure 3.7 Figure 3.7 Effect of airfoil penetration in a gust field (top), equivalent effect for blade section under influence of a nearby vortex filament (bottom). Original from [11]. 62

82 Thus, since the blade is passing through the previously left wake, the gust field can be represented by the presence of one or several vortex filaments still in the way or nearby the blade (see Figure 3.1). The Kussner problem (also approached by Karman and Sears [118]) consist of a vertical gust having an effect on the airfoil such that w g = 0 if x > Vt c w o if x Vt c 3.31 In this way the Kussner problem represents, specifically, a progressive change in the quasi-steady angle of attack just as the leading edge of the airfoil starts to penetrate the gust field. Similarly to Wagner the resulting circulatory lift coefficient for this variation is given as C l cgust t = 2παψ s 3.32 Where ψ s is the so-called Kussner function, which also, like Wagner s, has an analytical solution but was conveniently approximated by Sears and Sparks [119] to be ψ s e 0.13 s 0.5 e 1.0 s 3.33 Trough Duhamel principle, with Kussner function defined, the evolution of the lift load in time results in C l cgust t = 2π w g 0 ψ s + 0 t dw g ς dt ψ s ς dς 3.34 Which again generates an effective angle of attack that is summed to that obtained by Wagner function application Note that, for Kussner case, the admittance function is defined by the gust distribution, which for the case of rotary wing is distributed along 63

83 the azimuth Comparison of the values of exact and approximate values for the Kussner functions are shown in Figure 3.8 Figure 3.8 Kussner function, analytical and approximate behavior State space unsteady airfoil model Now retaking the indicial responses functions from Wagner and Kussner, and applying directly the Laplace transform to them, then a set of aerodynamic state equations emerge. Thus, considering a generic indicial response function of the form[79][11] φ gen s = 1 A 1 e b 1s A 2 e b 2 s ; s = 2Vt c This time with s being the distance traveled measured in semichords, if variables T 1 and T 2 are introduced such that 64

84 T 1 = Assessment of a State Space Free Wake Model c and T 2Vb 2 = c 1 2Vb 2 Then φ s results φ gen t = 1 A 1 e t T 1 A 2 e t T 2 Thus, the indicial function is now conveniently defined in the time domain, If t=0 then φ gen 0 = 1 A 1 A 2 = 0 From this result the impulse response can be extracted to be t = A 1 T 1 e t T 1 A 2 e T 2 t T 2 Rearranging, the impulse response becomes t = A 1 b 1 2V c e b 1 2V c t A 2 b 2 2V c e b 2 2V c t At this point a convenient Laplace from can be used for the impulse response such that L t = A 1b 1 2V c p + b 1 2V c A 2b 2 2V c p + b 2 2V c With p being the Laplace variable (usually noted by s), rearranging L t yields 65

85 L t = A 1b 1 + A 2 b 2 2V c p + A 1 + A 2 b 1 b 2 2V c p 2 + b 1 + b 2V 2 c p + b 1 b 2V 2 c 2 2 This result finally leads to a state space form for a given aerodynamic load (lift for this case), thus considering the state space form for a given system x = Ax + Bu With the output given as y = Cx + Du The state system for this case is given as x 1 x 2 = 2V c b b 2 x 1 x α(t) 3.35 With the resulting normal force output C l t = C lα 2V c A 1 b 1 A 2 b 2 x 1 x From this state-space template the Wagner and Kussner functions can be included through the coefficients A and b that correspond, for the Wagner case, to those from R.T. Jones approximation in equation (3.30), thus A 1w = A 2w = b 1w = b 2w = 0.3 Replacing the specific values and rearranging, the resultant state system for arbitrary change in angle of attack for unsteady conditions, i.e., Wagner s indicial function yields 66

86 x 1 x 2 = b 1 b 2V 2 (b c 1 + b 2 ) 2V c x 1 x α(t) 3.37 For this case, as observed in Figure 3.6, Wagner s initial value comes to be one half, then for this specific case the additional initial state has to be included such that φ 0 = 1 A 1w A 2w = 0.5 C l c t = 2π b 1 b 2 2V c 2 (A 1w b 1w + A 2w b 2w ) 2V c x 1 x α t 3.38 Now, considering the Kussner case for a velocity perturbation caused by the presence of a nearby vorticity element or gust the initial velocity field, described in section 3.4.2, is altered by an additional transient velocity Δw g (t) due to the gust, as a result the local angle of attack is also altered such that Δα t = tan 1 Δw g(t) V Δw g(t) V To account for the resultant load perturbation ΔC l c t a similar process as that for Wagner is followed. Then, Kussner indicial function coefficients that correspond to those from Sears and Sparks approximation in equation 3.33, lead to G 1 = 0.5 G 2 = 0.5 g 1 = 0.13 g 2 = 1.0 Finally the resulting perturbation state is given by 67

87 z 1 z 2 = g 1 g 2V 2 (g c 1 + g 2 ) 2V c z 1 z Δw g (t) V 3.39 With its related output ΔC l c t = 2π g 1 g 2 2V c 2 (G 1 g 1 + G 2 g 2 ) 2V c z 1 z Finally for a given blade section r i the complete state system output is then defined as C l t = C l c t + C l nc t + ΔC l c t 3.41 With C l nc representing the lift coefficient due to the apparent mass effect C l nc = πb α V + V baα V Figure 3.9 Schematics for treatment of Blade 2D sections for unsteady conditions Blade flapping model For this work, blade dynamics is reduced to a second order rigid blade flapping model. Despite of its simplicity, this model displays the main particularities of the integration 68

88 between the aerodynamics and inertial dynamics of the blade trough its formulation of the non-conservative aerodynamic forces, i.e., the damping moment. I bf β + I bf Ω 2 β = M Lflap 3.43 For this blade model, flapping is defined in terms of flapping acceleration β, flapping angle β, blade flapping inertia I bf and the aerodynamic moment M Lflap. Aerodynamic moment term, around the flapping hinge, provides the aforementioned link between aerodynamics and inertial dynamics. Such linkage can be extracted by considering that the local lift for every section along the blade span can be given by l r i = 1 2 V r i 2 cc l 3.44 Thus the flapping moment results M β = R l r r dr r 0 ; r = r i R 3.45 This can alternatively be represented in a quadrature form intended for numerical solution. M β = i=n i=1 l i r i Δr ; i = No. of blade sections 3.46 Now, a special consideration is to be taken for the strength of the blade tip vortex filament. When methods like lifting line theory (see section 2.3.1) the vortex filament strength is solved either simultaneously or iteratively with the local strengths for every section s bounded vorticity. For this case, since local lift coefficient results from the unsteady state space airfoil model, then the local circulation can be defined, thus, alternatively local lift yields 69

89 l r i = ρv r i Γ b r i 3.47 Keeping the high aspect ratio assumption that leads to a 2D analysis the bounded vorticity at every section can be found. Γ b r i = 1 2 V r i cc l = l r i ρv r i 3.48 Thus finally the relation between blade flapping motionβ, bounded circulation Γ b and vortex filament strength Γ v can be considered by defining the maximum bounded vorticity value, typically located around 80% of the blade radius [78]. Γ v = max Γ b r i 3.49 Thus at every time step, depending on blade - vortex self influence, a vortex filament, curved or straight, will be formed with a given vorticity strength, along the blade or located at tip only upon preference Formulation of velocities for rotating blade With the unsteady formulation defined, an exportation of definitions from the two dimensional unsteady airfoil model to the environment of the rotating blade is required looking for equivalence of terms. Considering the motion of a given point on the blade, its position will be a function of the flapping angle and the pitch setting at that specific azimuthal position, as is the case for cyclic variation. The velocities involved, derived from such position parameters, result in the flapping velocity β and angular acceleration β, the pitch rate θ and pitch acceleration θ. For this case the formulation of unsteady section loads follows that presented by Johnson [120]. The lift load requires expressions for and α in terms of the different rotor degrees of freedom, thus considering equation (3.23) and grouping terms for equation (3.24) the resulting unsteady lift load can be arranged to 70

90 L Total = 2πρVb + Uα bα + πρb 2 + Uα 1 a C k 2 baα 3.50 Notice that the term + Uα and its derivative presents the velocity components not in terms of pitch and heave separately but identifies the mean components of the normal velocity distribution along the chord. Thus for the case of rigid blade flapping in forward flight the velocity components correspond to + Uα U t θ U n 3.51 And α = θ + Ωβ 3.52 The local velocity V disserves additional consideration, since it is dependent upon the other velocity components. Keeping the large aspect ratio assumption and neglecting radial velocities, the tangent velocity (see Figure 3.10) to the blade leading edge for a forward flight condition is given by ([120], [112]) U t = ΩRμ cos τ sin ψ + e c RΩ 3.53 And for the normal velocity U n = ΩRμ sin τ + e c Rβ + ΩRμ cos τ cos ψ sin (β) 3.54 Where for both cases e c = r i R μ = V ΩR ; V = U t 2 + U n 2 and τ is the shaft tilt angle (around y axis only for this case). 71

91 Thus preserving the rigid blade assumption, control input pitch rates, whether from collective or cyclic, account for the changes in time of angle of attack along with the local blade flapping angle. The variation in plunging velocity is defined by the flapping velocity and acceleration which are provided by solving equation (3.43). Notice that for this case, wake induced velocity is not included into the normal velocity U n as is typical for blade element approaches, considering the previous development for a sharp edge gust from Kussner, induced velocity w g is included as a perturbation velocity that results into a perturbation of the local angle of attack in equation (3.39). Local blade twist θ i is included along with pitch control, that is, lateral cyclic θ c, longitudinal cyclic θ s and collective θ o inputs. Then the local total pitch results θ loc = θ o + θ i e i R + θ c cos ψ + θ s sin (ψ) 3.55 Thus the local angle of attack (with no wake perturbation) at a given section i=1 n along the blade span, to be used in equation (3.38), results in α loc i = U tθ loc U n V θ loc U n U t θ loc φ b

92 Figure 3.10 Local velocity components for forward flight One additional consideration for blade velocities is the reverse flow condition. This condition may generate a problem if equation 3.56 is considered when the blade finds a zero local velocity or velocity is reversed due to the sum of velocity components when in forward flight. To address this problem, the procedure adopted by Taamallah [121] is followed, which for the system of reference used in this work leads to define the angle φ b as φ b = arctan U n U t if 0 < U t φ b = sign U n π 2 arctan U n U t if U t

93 3.4.9 Induced velocity transformation Considering the induced velocity already presented in section 3.3 it can be noticed that such velocities are defined within the inertial reference frame, while for induced velocity applications to the blade element model the necessary velocity transformations are to be defined so such velocities can act on the shaft plane where the blade element model velocities are defined. Usual formulation include an inertial reference frame [X Y Z] a body frame [x b y b z b ] located at the center of gravity, a hub [x H y H z H ] and shaft [x s y s z s ] frames and a hinge frame [x h y h z h ] around which blade flapping and lagging rotations take place, additionally, a velocity frame [x V y V z V ] is included depending on the case; a basic scheme is shown in Figure For this particular case, only a basic transformation from the air velocity reference frame which is coincident with the inertial reference frame to the shaft plane reference frame is performed, thus for a given instant in time, the induced velocity acting on the blade will be defined as normal to the shaft plane along an evaluation line, being this one the projection of the flapping blade on such plane, in that order the velocity will be suitable since it is on this plane (hub plane for general case) where blade element acting velocities are usually defined [11]. Figure 3.11 Basic typical arrangement of reference frames for blade motion 74

94 Figure 3.12 Shaft plane rotations for induced velocity evaluation Thus considering the shaft axis longitudinal τ and lateral τ 2 inclinations the resulting transformation matrices respectively are R i = cosτ 2 sinτ 2 0 sinτ 2 cosτ 2 R j = cosτ 0 sinτ sinτ 0 cosτ Thus the transformation matrix results in RM = R i R j

95 With the induced velocity defined by equation (3.19) the transformed velocities result from V ind = u v w = RM u v w The Method of Lines (MOL) for rotor wake solution The Method of Lines (MOL) could be considered more a numerical approach or strategy for solution of Partial Differential Equations (PDE) than formal method since its implementation is linked to the particularities of the problem at hand. This approach, as presented by Schiesser [122], allows the solution of PDE by reducing the number of dependant variables to only one, usually replacing the spatial variable with algebraic approximations, thus a complex PDE problem can be represented by a system of Ordinary Differential Equations (ODE) with the resulting advantages that come with these type of equations. While useful and practical it is recommended that MOL be applied in a specific manner, depending on the problem at hand, with the arrangement of such system representing the specific challenge when facing particular problems. Several examples of related problems can be found in [122] and [123], in this work the specific problem of rotor wake dynamics will be described following the guidelines of Celi [77] Problem statement Revisiting equation (3.4) it can be seen that this equation is a PDE dependant on azimuthal position ψ (which is an implicit form of time) and wake age ζ on the left hand side (LHS) of the equation. To consider a solution, discretization can be defined for both, RHS and LHS sides of the equation. Since the RHS of the equation can be represented as depicted in section 3.3, this problem focuses on how to describe the discretization of LHS trough MOL to represent a PDE in terms of spatial (wake age) and temporal (azimuthal) variables as an ODE system dependant only on the temporal variable. 76

96 3.5.2 Auxiliary conditions Considering the background discussed in chapter 2, rigid wake solutions provide a the next step in the solution of free wake problems, this approach allows reduction of complexity and computational cost, as well as initial solutions in the numerical solutions of PDE s reducing transients in a marching time process or accelerating convergence in relaxed wake solution. Following the work of Celi, initial and boundary conditions can be proposed by considering the analytical solution for a constant, uniform inflow velocity λ i crossing the disc. Since equation (3.4) is first order in ψ and ζ, one initial condition and one boundary condition are required to obtain the solution of the intended ODE system. As proposed by Leishman [11] the velocity across the disk for such constant inflow condition is given by V = ΩRμi + ΩRλ i k 3.60 Thus equation (3.4) becomes dr(ψ, ζ) dψ + dr(ψ, ζ) = Rμi + Rλ dζ i k 3.61 For such case from the analytical solution, representing a so-called rigid wake in space results as r ψ, ζ = Rμζ + r v cos β 0 cos ψ ζ cosα s + sin β 0 sin α s i + r v cos β 0 sin ψ ζ j + Rλ i ζ + r v sin β 0 cos α s cos β 0 cos ψ ζ sin α s k 3.62 Where R is the rotor radius, μ and λ i are the advance ratio and inflow ratio respectively, β 0 is the blade flapping angle α s is the rotor shaft tilt angle (longitudinal) and r v is the location of the vortex filament along the blade radius (r v =R for tip vortex). From this convenient analytical solution initial and boundary conditions are obtained by setting ψ = 0 and ζ = 0 respectively, resulting for initial condition in 77

97 r x 0, ζ = Rμζ + r v cos β 0 cos ζ cosα s + sin β 0 sin α s r y 0, ζ = r v cos β 0 sin ζ 3.63 r z 0, ζ = Rλ i ζ + r v sin β 0 cos α s cos β 0 cos ζ sin α s and for boundary conditions it yields r x ψ, 0 = r v cos β 0 cos ψ cosα s + sin β 0 sin α s r y ψ, 0 = r v cos β 0 sin ψ 3.64 r z ψ, 0 = r v sin β 0 cos α s cos β 0 cos ψ sin α s Spatial differentiation of LHS (5PBU4) Celi described the implementation of several spatial discretization schemes based on finite differences approach. The different tested schemes covered those presented by Schiesser [122] while describing the effects on numerical stability and error behavior of each system. Schemes used included: - 2 nd Order, 2 point central difference (2PCD2) - 4 th Order 4 point central difference (4PCD4) - 1 st Order 2 point Upwind (2PU1) - 2 nd Order 3 point Upwind (3PU2) - 4 th Order 5 point Biased Upwind (5PBU4) While several discretization schemes can be used for MOL, according to need and convenience, Celi used a fourth-order five-point biased upwind (5PBU4) originally proposed by Carver and Hinds [124] for advective systems. Such approximation was reported to provide the best stability (see Figure 3.13 from ref. [77]) and was later on replicated by Betoney-Celi-Lesihman [78] for a more general application of free wake, following that result, only such latter scheme will be presented here which in terms of Schiesser first order hyperbolic PDE s numerical diffusion and oscillation is considerably reduced. For this reason we recommend the used of this method for the NUMOL solution of first order hyperbolic PDE s [122 p. 135]. 78

98 Figure 3.13 Linearized system poles for different discretization systems [77] Thus, for the spatial term in the RHS of equation (3.4) with k coordinate directions (x,y,z) and i (i=1: N ζ ) number of collocation points in the vortex filament (Lagrangian markers) and with a wake age step Δζ, with the specific 5PBU4 discretization scheme for middle points in the domain it results r k (ψ, ζ i ) ψ = 1 12Δζ r k ψ, ζ i r k ψ, ζ i 2 18r k ψ, ζ i r k ψ, ζ i + 3r k ψ, ζ i Ω V k ψ, ζ i While for domain border initial elements (i=1:3) are represented as r k (ψ, ζ 1 ) ψ 1 12Δζ 25 r k ψ, ζ r k ψ, ζ 2 36r k ψ, ζ r k ψ, ζ 4 3r k ψ, ζ Ω V k ψ, ζ

99 r k (ψ, ζ 2 ) ψ 1 12Δζ 3 r k ψ, ζ 1 10 r k ψ, ζ r k ψ, ζ 3 6r k ψ, ζ 4 + r k ψ, ζ Ω V k ψ, ζ r k (ψ, ζ 3 ) ψ 1 12Δζ r k ψ, ζ 1 8 r k ψ, ζ 2 + 0r k ψ, ζ 3 + 8r k ψ, ζ 4 r k ψ, ζ Ω V k ψ, ζ And last border element (i= N ζ ) is given by r k (ψ, ζ Nζ ) ψ 1 12Δζ 3 r k ψ, ζ Nζ 3 16 r k ψ, ζ Nζ r k ψ, ζ Nζ 1 48r k ψ, ζ Nζ r k ψ, ζ Nζ Ω V k ψ, ζ Nζ Now extracting the weighing coefficients of the algebraic approximations for every section of the domain (borders and inner points) a matrix arrangement can be built representing the spatial discretization of a single wake filament, thus the matrix A ζ is defined as A ζ = 1 12Δζ This particular matrix, called differentiation matrix, has to be used for the x, y and z directions and should be repeated as many times as the number of filaments required along the blade radius. Thus, Remembering that = 2 ψ 2, = ψ dr ψ, ζ = r k ψ, ζ dψ 80

100 dr ψ, ζ dζ = A ζ r k ψ, ζ Finally the rotor wake dynamic system of representative ODE s obtained by MOL can be represented as r k ψ, ζ = A ζ r k ψ, ζ + 1 V r(ψ, ζ) 3.71 Ω Which is the basic form of a state space system. 81

101 Chapter 4:Components Validation and vorticity coupling With model described, validation process is structured for each of its main components, that is, Biot-Savart model for induced velocity, MOL representation of wake geometry and unsteady blade section representation. 4.1 Biot-Savart law To test the Biot-Savart law model for induced velocity, a circular vortex ring will be used taking advantage of its analytic solution, which is derived from the Legendre s elliptic integrals of the first and second class. The approach was considered by Castles and de Leeuw [125] and has been used for validation by Baghwat and Leishman [62] and van Hoydonck [70] among others. A detailed description of the vortex ring analysis can be found in [126]. Consider a circular vortex ring on the xy plane, with center at the origin and with constant circulation strength, if a cylindrical coordinate system is considered the vortex ring will induce a velocity in radial and axial direction (because of symmetry the remaining component is neglected). Considering only the normal velocity for this case then vi z = 1 dψ = Γ v r p dr p 2πxR (AB + CDF)

102 Where A = K τ E τ B = x 1 + x + 1 d 1 d 2 C = d 1 + d 2 τe τ D = 1 τ 2 τ = d 2 d 1 d 2 + d 1 x = r p R Of particular interest are the K τ and E τ terms, representing the Legendre elliptic integrals of first and second class respectively. These integrals are defined in terms of the sweeping of the angular coordinate θ through a function φ and are given as E τ = K τ = 2 0 π τ 2 sin 2 φdφ dφ 1 τ 2 sin 2 φ An schematic representation of the circular vortex ring is show in Figure 4.1 Figure 4.1 Circular vortex ring influencing a point P in space 83

103 vz / R Assessment of a State Space Free Wake Model By evaluating the induced velocity on a series of point along the radius r on the plane xz the velocity in the z direction can be evaluated. Following the analytic solution, the numerical solution of the Biot-Savart law comes next, following the model described in section 3.3. Results of both analytic and numeric approaches are presented for comparison in Figure Induced velocity by a circular vortex ring analytic Biot-Savart x/r Figure 4.2 Comparison of induced velocity solutions for a circular vortex ring It can be observed that results from Biot-Savart model show a good agreement with those of the analytic solution for the same number of evaluation points. It is generally known that discretization, i.e. the number of elements that represent the actual geometry, is a driving factor when for solution. Thus a study of the effect of the number of straight segments, representative of the ring geometry, was made by using different number of elements while keeping the same evaluation points nev constant. The results for the variation of ring elements n are shown in Figure 4.3 and a more detailed study of finer gridding relative error is shown in Figure 4.4 for different angular spacing. 84

104 relative error Assessment of a State Space Free Wake Model Figure 4.3 Influence of discretization on induced velocity for a circular vortex ring =1 o =5 o =10 o =30 o x/r Figure 4.4 Relative error for different number of elements for Biot-Savart numeric and analytic solutions 85

105 Now, considering the regularization presented in section 3.3, the induced velocity distribution result of the inclusion of the damping described by equation (3.8) is compared against the vortex ring analytic solution in Figure 4.5. For this case the core radius r c was set such that it would be 8% of the ring radius. Figure 4.5 Induced velocity for a vortex ring, analytic solution vs regularized Biot_Savaart 4.2 MOL wake geometry representation Validation of MOL method for wake representation was performed by imitation of the analytic solution defined by equation (3.62) and with the same parameters presented by Celi [77], μ = 0.1, λ = 0.05 for the first case and μ = 0.3, λ = 0.05 for the second. A total wake length of two revolutions with a length of 720 degrees and 80 wake elements was used. Solutions can be observed in Figure 4.7 and Figure 4.8 Wake density was studied by checking the ability of the code to imitate the analytic solution through variations of the wake meshing density. Several evaluations were performed using different wake marker spacing, from 50 degrees to 5, checking for 86

106 both, RMS reduction and the number of evaluations executed by the ODE solver. Results obtained are shown in Figure 4.7, where the RMS is given by RMS ka :k Mol = 1 3N w N w 3 i=1 j =1 n n 1 r i,j r 2 i,j 4.2 It is observed that the number of evaluations tend to increase significantly if a tight spacing is used, below 10 degrees, this value is considered for this work since a matching value for the Biot-Savaart evaluation of previous section may imply a strong increment in computational time, considering that the presented number of evaluations correspond to the complete state space system. Figure 4.6 RMS and Number of evaluations(ncall) evolution for different wake angular spacing. 87

107 Figure 4.7 Comparison of wake solution, analytic solution of eq (blue) and MOL solution (red) for advance ratio of Figure 4.8 Comparison of wake solution, analytic solution of eq (blue) and MOL solution (red) for advance ratio of

108 4.3 Unsteady blade section Following the velocity components described in section resulting two relevant types of motion, i.e., pure plunging, pure variation of angle of attack. This is done looking for compliance with the required flow field conditions to which the blade is submitted Pure Angle of attack oscillations Pure angle of attack oscillation case is explored considering the case of thin airfoil under harmonic oscillation from the battery of experiments performed by Pizialli [127]. Specifically case No. 18 was considered for which the conditions are provided as α = 4.06 ± 2.30 Freq. = cps ν = ; reduced frequency Vel. = fps M = R e = x10 6 For this case results were also compared against the Theodorsen model which uses the Theodorsen function C(k) as solved through the Bessel functions [11] and also against the same model using an approximation for C(k) given as[111] C k k i k i 4.3 Note from results in shown in Figure 4.9 that the lift coefficient is in good agreement with the experimental results from Piaziali for C L vs α. The difference with both, approximation and analytic solution, for Theodorsen function present a deficiency to imitate the experimental results, showing similar lift slope but it does not match the magnitude of the net lift coefficient. Results showing a decomposition of the lift components is shown in Figure

109 Figure 4.9Lift coefficient for unsteady harmonic AoA oscillation 90

110 0.7 Unsteady Incompressible Lift Response C l total C l circulatory C l non-circulatory C l t Figure 4.10 Load discrimination for harmonic AoA oscillation Harmonic plunge oscillations For plunging motion the results obtained by Grey and Liiva [128] are used, the motion forcing for this case is represented by α = 9.75 V k = ; reduced frequency M = 0.2 The motion for this case includes the variation in plunging velocity which for this case is given by α = 9.75 ωδ cos ωt V 91

111 Clc Assessment of a State Space Free Wake Model With Δ = 0.3 for this case. Results are shown in Figure 4.11 and the resulting angle of attack (forcing) is show in Figure The resulting lift coefficient shows good agreement with the experimental despite some lag can be observed. 1.5 Unsteady Incompressible Lift Response Plunging Grey-Liiva State space t Figure 4.11 Lift coefficient output for harmonic plunging motion 92

112 15 Angle of Attack (forcing) [deg] t Figure 4.12 Angle of attack due to plunging Blade load integration Considering the numerical procedure to integrate the loads along the blade radius, a quadrature, a given number of blade elements is to be defined suitable to character of the numerical solution. For this purpose, equation (3.46) is integrated using quadrature in order to explore the influence of blade load formulation to the number of blade sections located along the blade radius such that ec i = r i. For this case a generic one R blade rotor is defined first with the analytical integration for quasi-steady airfoil section and integrated, then, the discretized blade will be defined using the quadrature and then both models were compared to check for consistency and the required meshing density to emulate the analytical load at lowest computational cost and convenience of formulation. Results for blade sections ranging from 5 to 20 are comparatively shown in Figure The results indicate that a distribution of 20 elements provide a better approximation to the analytical response for blade moment and match the number of segments used by Rivera [100] but differs from those of Betoney et al [78] where 10 sections are used. Parting from the results of the quadrature test, to keep the lowest number of related dynamic states, a good compromise may result with 15 blade sections. 93

113 Integrated Aerodynamic Moment [Nm] Assessment of a State Space Free Wake Model Comparison of Flapping Moment Integration Analytic Integration ec=5 ec=10 ec=15 ec= Revs Figure 4.13 Flapping moment integration for hover 4.4 Uniform inflow distribution - hover As already presented in section 3.2 and 3.5.2, equation (3.4) has an exact analytic solution for the uniform inflow case (constant velocity trough the wake)[77], represented by equation (3.62) and repeated here, by components r ψ, ζ x = Rμζ + r v cos β 0 cos ψ ζ cosα s + sin β 0 sin α s r ψ, ζ y = r v cos β 0 sin ψ ζ r ψ, ζ z = Rλ i ζ + r v sin β 0 cos α s cos β 0 cos ψ ζ sin α s This set of equations, represent the wake in terms that can be derived considering the basic condition of uniform inflow that can be obtained from momentum theory and thus can be related to a specific thrust coefficient C T and vortex strength Γ v. This analytic solution thus results in a convenient checking tool for inflow velocity 94

114 resulting from a given vortex filament distribution, in this case that described by equation (3.62). Consider a hover case for a given C T = , corresponding for design weight of the UH-60 helicopter as reported by Howllet [129], for this specific condition the uniform inflow ratio is given by λ = C T Now, considering that the total thrust for a N b bladed rotor is given as T L b N b Such that the lift per blade L b corresponds to L b = C TρπR 2 ΩR 2 N b Then if a lift distribution is assumed such that the peak l r i max is located at 80% radius [97] the maximum lift per unit span l r i max can be expressed as l r i max = 2L b R And retaking equations (3.48) and (3.49), the bound vorticity distribution is obtained Γ b r i = l r i ρv r i Followed by the maximum bound vorticity Γ v = max Γ b r i Considering the regularization and viscous diffusion of section 3.3 the eddy dissipative term (equation (3.12) )for this case δ = 1 + a 1 Γ v ν 95

115 In this way, the properties of the wake filament are related to the thrust generated by a rotor with an assumed uniform inflow distribution. For UH-60 configuration the wake setting is shown in Figure 4.14 Figure Blade-wake configuration for hover case under uniform inflow Figure 4.15 shows the influencing elements (red) and the evaluation points along the radius, and located on the shaft plane (blue). The results of the inflow ratio distribution, generated by the wake filaments, compared against the result obtained from the uniform flow is shown in Figure 4.16, where a fair matching can be observed along the radius and along the azimuth since for all four evaluation points sets the induced velocity is the same.. The variation observed towards the tip is the result of the regularization of section 3.3 which keeps the velocity value at the tips finite avoiding singularities. If not used, high values for induced velocity are obtained as shown in Figure For this case, four wake revolutions were considered, which provided the best results without accounting for wake truncation, this setting is not far from the suggested three wake revolutions by Celi [77] and Betoney, Celi and Leishman.[78]. 96

116 Figure 4.15 Evaluation points along the blade radius (blue) and wake influencing elements (red) Figure 4.16 Induced velocity along the rotor radius. Comparison between uniform inflow and vortex filament analytic wake. 97

117 Figure 4.17 Induced velocity along the rotor radius. Regularization OFF 4.5 Linear inflow distribution - forward flight In this section a different assumption is made for forward flight, considering not a uniform inflow but a linear inflow distribution along the longitudinal axis, for the azimuth positions corresponding to 0 and π following the model presented by Drees [130] for forward flight condition. For the case of μ > 0.15 the time averaged longitudinal linear inflow variation is given according to Glauert [131] by λ i = λ o 1 + k x r cosψ 4.5 Where 98

118 C T λ i = λ o = 2 μ λ i 4.6 And the k x term is a weight factor that represents the deviation of the inflow from the uniform value obtained from momentum theory[11] and has several definitions result of different authors, in this work as mentioned above, that of Drees which is defined as k x = cosχ 1.8μ 2 sinχ 4.7 The term χ is the wake skew angle due to the longitudinal advance velocity component expressed as And χ = tan 1 μ x μ z + λ i 4.8 μ x = μsinα s μ z = μcosα s The experiments of Elliot et al. [132] [133] are used to test the induced velocity calculation method in the case of forward flight. A set of inflow distributions at different azimuth stations along the rotor radius is available from such experiments and for different advance ratios. For this work those results for μ = 0.15 are considered. 99

119 Figure 4.18 Rotor wake system for forward flight Results for μ = 0.15 case are presented in Figure 4.19 compared to the instantaneous inflow from the present method and the linear inflow distribution of Drees. While the results for the aft section of the rotor ψ = 0 shows good agreement with the simplification of Drees and at some extent, along the outer portion, with the data from the experiments the forward portion ψ = π does not, showing irregular peaks along the radius. This is due to the close proximity of the vortex filament, which induces the 100

120 irregular velocity behavior, at the aft section this is not the case, since the wake filaments are convected away from the evaluation points, causing the velocity behavior to be smoother. This can be observed in Figure 4.18 This is a similar case to that observed by Tauzing [97] where the instantaneous induced velocity presented the similar behavior. Induced velocities at the relevant azimuth stations are shown in Figure 4.20, where it can be observed that for azimuth stations ψ = π/2 and ψ = 3π/4 the induced velocity behavior is still similar to that of uniform inflow. Figure 4.19 Instantaneous induced velocity, longitudinal distribution 101

121 Figure 4.20 Instantaneous induced velocity distribution at different azimuth stations The time averaged values for induced velocities were obtained for one revolution for which Δψ = 5 o, for a total of 72 steps, velocities, as expected the results show better agreement when compared against linear inflow distribution and experimental results. In Figure 4.21 the longitudinal distribution can be observed, this time, the forward portion of the rotor, ψ = π, does not show the irregular behavior and goes along with both, data and linear distribution. Figure 4.22 shows the overall behavior for all four azimuth positions on which evaluation points were located, these time averaged results still show agreement with the uniform inflow for the azimuth stations coincident with ψ = π/2 and ψ = 3π/4, while the longitudinal stations show a more or less symmetrical behavior. For this case, the relevant properties of the each element of the model have been dissected looking for further integration into a free wake / rotor integration. Such is the aim of the next chapter. 102

122 Figure 4.21 Time averaged induced velocity, longitudinal distribution Figure 4.22 Time averaged induced velocity distribution at different azimuth stations 103

123 Figure 4.23 Instantaneous induced velocity, azimuth distribution 4.6 Free wake periodicity Despite the nature of a free wake may not necessarily imply a strict periodic behavior, for some conditions, depending on the case, rotor free wake does show a periodic behavior within a given threshold. The importance of this kind of behavior is its influence on transients dissipation and convergence analysis, for transient and relaxation approaches respectively. For the former case, evaluation of periodicity will allow, among other things, defining initial conditions for specific cases when a steady solution is set before a sudden control input for instance. In that order, the same RMS error of section 4.2, but this time the overall difference for wake location points is considered at each rotor revolution, thus evolution of transients can be kept under check when possible. In such order of ideas, the rotor wake should behave periodically within a tolerance if properly modeled for a causal set of conditions. The RMS given in this case is 104

124 RMS n :n 1 = 1 3N b N w N b N w 3 i=1 j =1 k =1 n n 1 r i,j,k r 2 i,j,k 4.9 Where the new summation term accounts for the 1/Rev behavior. A general case of μ > 0.15, C T = and τ = 5 is examined for a rotor with N b = 4 blades. Results for RMS are shown in Figure 4.24 where the change of RMS is shown in time given by RMS dev = RMS n RMS n 1. Figure 4.24 RMS behavior for wake periodicity study case For this case, the RMS deviation shows that transients dissipate after the sixth revolution, keeping a periodic behavior after that, Figure 4.25 shows how wake geometry superposes for revolutions after RMS is fixed for revolutions 7 (solid blue line), 9 (red dots) and 11 (yellow stars). 105

125 Figure 4.25 Superposition of wake markers for different revolutions after periodic behavior is achieved (7 th rev.- solid blue line, 9 th rev.- red dots and 11 th rev. yellow stars). 106