TIME MARCHING SIMULATION MODELING IN ...

TIME MARCHING SIMULATION MODELING IN DESCENDING FLIGHT THROUGH THE VORTEX RING STATE Maria Ribera∗

Roberto Celi†‡

Alfred Gessow Rotorcraft Center Department of Aerospace Engineering University of Maryland, College Park

IN Wi,j Influence coefficient for near-wake circulation M Local Mach number p, q, r Roll, pitch, and yaw rates R Rotor radius r Spanwise distance from the axis of rotation uF , vF , wF Linear velocity components of the helicopter u Control vector V Velocity at the vortex filament VBM Velocity due to blade motion Vind Induced velocity Vman Velocity to to angular rates V∞ Linear velocity of the helicopter wtip Flap bending displacement at the blade tip y State vector β Equivalent rigid blade flapping angle βp Precone Γi Circulation ζ Wake age φ, θ, ψ Helicopter roll, pitch, and yaw angles ψ Reference azimuth angle

Abstract

This paper presents in detail the results of a simulation of a helicopter in a steep descent following a step reduction of collective pitch. The descent velocity increases through the vortex ring state and into the windmill brake state. The simulation is performed using a coupled rotor-fuselage dynamic model, including a nonlinear finite element model of the rotor blade flexibility, tightly coupled with a free vortex wake model capable of capturing the distortions of the wake geometry during maneuvers. The results indicate that the simulation model including the tight coupling with the free wake is numerically well behaved. The dynamic behavior of the helicopter is dominated by the effects of the vortex bundle approaching, and then crossing, the rotor plane. The consequences of the maneuver are relatively benign: velocities and rates are never dangerously high, and return to reasonable steady state values once the vortex ring is convected away from the rotor. The time varying geometry of the crossing regions has a powerful effect on the flapping response, and therefore Introduction on the flight dynamics of the helicopter. Because the asymmetries of the vortex ring and the details of the Sophisticated aerodynamic models are needed for interaction with the rotor plane are so important, moaccurate predictions in a variety of practical problems mentum or momentum-like theories cannot fully capof helicopter flight dynamics, such as the response ture the physics, and therefore cannot predict many to pilot inputs in moderate and large amplitude uneffects of vortex ring state from first principles. steady maneuvers, trim in steep, high-g turns, trim and response in descents, including near and through Notation the vortex ring state (VRS), and the off-axis response CL , CD , CM Local lift, drag, and moment coefficients to pilot inputs. Although in some cases momentum theory based models can be adequate with the apCT Rotor thrust coefficients propriate selection of tuning parameters, the most Ibi,j Influence coefficient for bound circulation accurate predictions from first principles require free ∗ Doctoral Candidate, [email protected] vortex wake models coupled with detailed models of † Professor, [email protected] rotor and fuselage dynamics. ‡ Presented at the 63rd AHS Annual Forum, VirThe paper describes some aspects of the formulaginia Beach, VA, May 1-3, 2007. tion and validation of such a simulation model, in 1

which a finite element based rotor model and large amplitude fuselage dynamic equations are coupled with a free wake model capable of capturing correctly the wake geometry distortions during maneuvering flight. This model, recently completed [1], can describe steady state flight conditions, both in straight, descending and maneuvering flight, and the free flight response to pilot inputs, with no restriction on the amplitude of the inputs or of the helicopter response. In the paper, the model is applied to the numerical study of the dynamics of a single main rotor helicopter in a steep descent, in which vortex ring state conditions are encountered. The literature on the vortex ring state is extensive, and only selected key references will be cited here. Newman et al. [2] and Johnson [3] provide comprehensive analyses of the literature, and comparisons of the experimental data currently available. Simple models, generally based on momentum theory, can be used when the descent angles are not very large. In some early work, Wolkovitch [4] and Peters and Chen [5], used momentum theory to map the boundaries of the vortex ring state. Heyson’s model [6] explains power settling and autorotation, and provides contour maps for the power required by the rotor as a function of glide speed, glide angle and rotor pitch angle. More recently, several studies have proposed simple inflow models [3, 7, 8, 9, 10] to describe steep or axial descent flight, including operation in the VRS. These parametric extensions of momentum theory can be adequate with the appropriate selection of tuning parameters, however they are not based on first principles, and are unable to capture the dynamics of the flow field of the rotor wake. On the other hand, they are computationally efficient, suitable for real-time simulations, and when properly tuned they show good agreement with experiment for the basic performance and motion quantities. The most accurate numerical models of VRS are those based on free wake models and computational fluid dynamics (CFD). Currently, they are too computationally intensive for real-time simulations, but are far more realistic and physically rigorous than momentum theory-based models. Leishman et al. [11, 12] studied the instabilities associated with the onset of the VRS using a time-accurate free wake model [13]. This model can capture the formation and development of the vortex ring state and can be used both to obtain steady state solutions as well as the time-integrated response to maneuvers in descending flight. Brown et al.[2, 14] developed a vor-

ticity transport model of the rotor wake, which they successfully applied to both axial and forward flight descents. Their results allowed for a numerical mapping of the VRS boundary that complemented that obtained through experiments and flight test. Moreover, the interference effects between the rotor wake and the fuselage and tail during the descent cases were also investigated. In all of these models, motion of the rotor and time histories of control inputs can be arbitrary, but prescribed. The focus is on aerodynamics research, and fuselage and rotor blade dynamics are essentially neglected. Although there has been considerable recent activity in coupling flight dynamics and comprehensive analyses with advanced free vortex wake or CFD models, very few studies of the dynamics of helicopters in steep descents using these advanced simulation capabilities exist in the literature. Houston and Brown [15] compare the results for finite-state induced inflow methods with the vorticity transport wake model. While for shallow descents the differences were small, for steeper descents, such as those required for autorotation, there were some discrepancies between the two models and the higher accuracy of the inflow predicted by the vorticity transport method was essential to correctly predict autorotation performance. A one-way coupling of a flight dynamics model with the Bhagwat–Leishman free wake [13] was used in Ref. [16] to perform aeroacoustics calculations for pop-up maneuvers starting from a shallow descent. In this study, the trajectory of the maneuver, including rigid blade motion, was computed without the free wake model. The resulting time histories were then applied to the wake in the form of prescribed velocities, controls, and blade motions, for the subsequent aeroacoustic analyses. The general objective of this paper is to contribute to the fundamental understanding of the dynamics of helicopters in steep descents, including flight in the vortex ring state. The specific objectives of the paper are: 1. To describe the solution methodology for the tight coupling of a free vortex wake model with a coupled rotor-fuselage models, for time marching simulations; 2. To present the results of the simulation of a maneuver that takes a single main rotor helicopter into an initially axial descent with increasing rate of descent, through vortex ring state conditions

and beyond;

following first-order, state-space, implicit form

˙ y, u; t) = 0 f (y, (1) 3. To highlight the most important physical phenomena, as they emerge from the simulation, and are integrated as a special case of differentialthat affect the flight dynamics and the rotor dy- algebraic equation (DAE) system, i.e., one with no alnamics of the helicopter. gebraic equations, using the DAE solver DASSL [19], which is a variable-step, variable-order method in which the derivatives are approximated by backward differentiation formulae. Mathematical Model In the present study, the simulation always starts from a trimmed condition, although this is not reRotor and fuselage dynamics The flight dynamics model used in the present study quired if suitable initial conditions for the rotor and (Refa. [17, 18]) is based on a system of coupled nonlin- fuselage states, and for the wake geometry and vorear rotor-fuselage differential equations in first-order, tex strength, are available through other means. The state-space form. It models the rigid body dynamics trim procedure is described in detail in Ref. [1], and of the helicopter with the non-linear Euler equations. it has also been reported in Refs. [18] and [20]. The time-accurate free wake model used in the The aerodynamic characteristics of the fuselage and empennage are included in the form of look-up ta- present study can be advanced in time with arbitrary bles. The dynamics of the rotor blades are modeled azimuth angle step size ∆ψ, and therefore it can be with coupled flap, lag and torsion, a finite element coupled with the rest of the helicopter model at any discretization and a modal coordinate transformation desired points of the integration. The inputs required by the free wake to advance to reduce the number of degrees of freedom. its solution by one azimuth step are the following: Wake model 1. The linear and angular velocities of the heliFree wake methods model the flow field using vortex copter, uF , vF , wF , and p, q, r. These velocities filaments that are released at the tip of the blade, are required to calculate the velocity field at the and which are discretized in time (ψ) and space (ζ). vortex filament: A schematic of the wake discretization is shown in Figure 1. The distortions of the wake geometry due V [r(ψ, ζ)] = V∞ + Vind [r(ψ, ζ)] + to maneuvers are taken into account without a priori +Vman [r(ψ, ζ)] (2) assumptions on the geometry. The free wake model used in the present study is where V∞ is the portion of the free stream vethe Bhagwat and Leishman free wake [13], which is a locity associated with the linear velocities of the time-accurate free wake model with a five-point cenhelicopter, Vind is the inflow velocity, and Vman tral difference scheme to describe the spatial derivais the portion of the free stream velocity associtive, Dζ and a predictor-corrector with second orated with the angular velocities of the helicopter. der backward (PC2B) scheme for the time derivative, both second order accurate. The bound circulation is 2. The velocity VBM at all the required points of obtained using a Weissinger-L lifting surface model. the blade due to the blade motion. These velociAlthough this method is potentially subject to nuties are required for the calculation of the bound merical instabilities, it is not restricted by the flight circulation with the Weissinger-L model: condition. Because time marching methods do not Ns need to enforce any type of boundary condition, they X   Ibi,j + IN Wi,j Γj = are suitable for transient conditions in which periodj=1 icity can not be enforced, and therefore relaxation methods can not be used rigorously. Such flight con{(V∞ + VBM + Vman + VF W ) · n}i (3) ditions include unsteady maneuvering flight and descending flight in the vortex-ring state. 3. The equivalent rigid blade flapping angles β(ψ), given by: Coupling of free wake and rotor-fuselage models Except for the wake equations, the equations of motion of the helicopter model are expressed in the

β(ψ) =

wtip (ψ) + βp R

(4)

because the current implementation of the wake assumes that the blade motion consists of rigid body flap only.

Vy (ψi , r), Vz (ψi , r), are available and are provided to the wake model, together with the circulation Γ(ψi , r).

The free wake model returns the inflow distribution λ(ψ, r). The methodology used in the present study is shown schematically in Figure 2. The DAE solver is called repeatedly inside an azimuth loop, which covers the entire duration of the integration. Each pass of the azimuth loop advances the simulation by an azimuth step ∆ψ, which is also the azimuth step at which the solution for the wake and that for the rest of the helicopter model are synchronized. It is important to note that the DAE solver uses a variable step size, dependent on the requested local error tolerances, which may be larger or smaller than ∆ψ. If the synchronization time does not coincide precisely with the end of one integration step, the solution is integrated just past the synchronization time, and the values of the states are reconstructed using the same BDF formulas that the solver is using at that time [19]. Therefore, the interpolated values have the same accuracy as the actual solution. In principle, it is also not necessary that the azimuth step used to solve the wake equation be the same as the value ∆ψ used for the synchronization. However, there is no obvious reason to choose different azimuth steps and, in fact, the same azimuth step has been used in the present research. More in detail, the overall integration proceeds as follows (note that, because the rotor speed is assumed to be fixed, time t and azimuth ψ are used interchangeably):

4. The circulation Γ(ψi , r) is held fixed, and is used to advance in time the wake geometry and inflow calculations to the time ti . At this point, the updated induced velocity λ(ψi , r) is available, and is provided to the rotor-fuselage model.

1. The simulation is started from a trimmed condition. Appropriate initial values of all the rotor and fuselage states at ψ = 0 are extracted from the vector of trim variables. 2. At t = 0, the following information is also available from the trim solution: (i) the induced velocities λ(ψ, r) at trim; (ii) the corresponding distribution of circulation over the rotor, Γ(ψ, r); and (iii) the geometry of the wake vortices. 3. With the inflow held fixed, the rotor and fuselage equations are integrated in time until the time ti corresponding to the azimuth angle increment ∆ψ. At this point, the state vector y(ti ), the equivalent rigid blade flapping β(ψi ), and the blade velocity components Vx (ψi , r),

5. The reference azimuth angle is incremented by ∆ψ, and steps 3 and 4 are repeated. Steps 3 through 5 are repeated until the desired end of the integration. In the current use of the term by the rotorcraft community, the procedure above is considered a “tight” coupling of the wake and the rotor model, in the sense that wake and rotor-fuselage equations exchange information at multiple time (or azimuth) points over a rotor revolution. This is in alternative to a “loose” coupling, where the exchange occurs every one or more rotor revolutions. It should be pointed out, however, that in the present study wake and rotor-fuselage models are each solved over the time step ∆ψ with the solution of the other model held constant. Although ∆ψ can be arbitrarily small, the two portions of the model are still integrated separately, and not simultaneously, and therefore a term like “semi-tight” would probably be a better characterization of the procedure. Results The results shown in this study refer to a helicopter similar to the UH-60, at 16,000 lbs and an altitude of 5250 ft. The descent is performed from an initial hovering condition, and is generated by a decrease of collective of 5 inches over 1 second. Each rotor blade is modeled with 4 finite elements, and 5 blade modes are retained in the modal coordinate transformation (2 flap, 2 lag and 1 torsion). The free wake has been modeled with a single tip vortex per blade, extending 4 free turns downstream of the rotor. A 10o discretization is used for both the time and space derivatives. The maneuver triggers an initially vertical descent that progresses toward the VRS and beyond. Because only the collective control is perturbed, and subsequently all controls are held fixed, the maneuver does not evolve as a perfectly axial descent. However, because of the high rates of descent that develop, this simulation provides an opportunity to analyze some

interesting aspects of the behavior of the helicopter duced by the vortex bundle is less visible in the rear in general, and of the rotor in particular, in flight view. At t = 4.89 sec the helicopter starts to roll regimes that include the vortex ring state. to the left. The vortex ring is now above the right half of the tip-path-plane , but still intersects the Wake geometry left half. At t = 5.82 sec the vortex bundle starts Figure 4 shows the side view of the free wake tip to be convected above the rotor, with a distorted vortex filaments at 6 time points during the descent. shape that is only very roughly parallel to the tipThe nose of the helicopter points to the left of the path-plane . Eventually, the upwash flow reorganizes figure. When the helicopter starts descending, the the wake above the rotor into a more regular and wake filaments are not convected as far downstream axisymmetric geometry and the helicopter roll and as they do in hover, and the increased proximity of pitch motions are consequently reduced. the vortex filaments causes the vortices to start to ”bundle”. At time t = 1.40 sec (corresponding to 6 Rigid body response full revolutions after the start of the maneuver), time The time history of the pitch attitude θ of the heliat which the collective has been decreased by the full copter for this maneuver is shown in Fig. 6(b). Two 5 inches from the hover trim value, two vortex bun- seconds into the maneuver, a large nose-up pitch andles are already visible, one in the far wake and one gle develops, which reaches a maximum value of 14 approaching the rotor. In addition, the vortex fila- deg at about 4 sec. After this peak, θ decreases to a ments start to move above the rotor tip-path-plane . nose-down -5 deg, and then oscillates about a mean At t = 2.56 sec (11 revolutions), the vortex filaments value of approximately 5 deg. The time history of are first convected above the tip-path-plane , and the the pitch rate q is shown in Fig. 6(c). The pitch rate vortex bundle previously part of the far wake is now shows fairly large oscillations with a maximum amapproaching the plane of the rotor. Six revolutions plitude of 15 deg, which last throughout the entire later, at t = 3.96 sec, the entire wake is bundled maneuver. Figure 7(b) shows the time history of the around the plane of the rotor, forming the actual vor- roll attitude φ. At the beginning of the maneuver, the tex ring, a toroidal vortex bundle that characterizes helicopter rolls slightly to the starboard. Then, after and defines the phenomenon. Notice that the vortex approximately 5 sec, a sharp roll to the left begins, ring is not evenly aligned with the tip-path-plane , to a maximum value φ = 26 deg after 5.77 seconds. and the uneven loads produced on the rotor cause the After that, the helicopter attitude returns slowly to helicopter to pitch and roll. At t = 4.89 sec (21 revo- a nearly wings-level steady state value. The correlutions), the vortex ring begins to separate from the sponding roll rate p, shown in Fig. 7(c), oscillates rotor, starting from the rear portion of the disk, but around p = 0 deg/sec until approximately t = 5 sec, still remains very close to the tip-path-plane . After at which time a large, doublet-like behavior occurs, 25 revolutions, corresponding to t = 5.82 sec, the up- followed, around t = 7 sec, by a return to a near zero ward flow generated by the increasing rate of descent steady value. pushes the vortex bundle above the rotor. The wake Figure 8(b) shows the time history of the thrust geometry starts to reorganize itself above the rotor, coefficient C . The thrust coefficient remains fairly T although the vortex bundle remains still close to the constant at the beginning of the maneuver. There is tip-path-plane . Eventually, the wake becomes again a sudden loss of thrust around 5 seconds of the maperiodic and forms a clean helical pattern above the neuver, typical of the lift losses associated with the rotor. At this point the helicopter is operating in the vortex ring going through the rotor, followed by a windmill-brake state, and is extracting power from spike. Eventually, C goes back to a constant value T the airflow. as the wake restructures into a helical pattern again. The rear view of the free wake geometry is shown for the same instants in the maneuver in Fig. 5. The rear view at t = 1.40 sec is very similar to the side view, with the vortex filaments first convecting over the rotor and the bundles forming downstream of the wake. Rear and side views are also very similar at next two instants shown in the figure, i.e., at t = 2.56 and t = 3.96 sec, except that the nose up pitch pro-

The presence of these thrust fluctuations has been documented before in the literature, from experiment ([21, 22]) and simulation ([12, 24]). The main rotor power required as a function of time is shown in Fig. 8(c). There is a sharp initial decrease with the step input of collective, and a subsequent slower decrease as the increase in the rate of descent brings the power requirement to almost zero. At that point,

the onset of the vortex ring state occurs, accompanied by large fluctuations of the power required. As the helicopter gets deeper into the vortex ring state condition, the power required becomes negative, and the rotor extracts energy from the airflow. Eventually, the required power remains constant and slightly negative, indicating that the helicopter would be in autorotation for the remainder of the simulation. It should be pointed out that the rotor speed is assumed to be constant throughout the maneuver, and therefore the effects of rotor acceleration and deceleration on the power balance are not modeled. The time histories of the forward and heave translational velocities u and w are shown in Figs. 9(b) and 9(c). The initial pitch up of the helicopter, Fig. 6(b) produces a backward tilt of the thrust vector, which induces a backward motion of the helicopter, which reaches a rearward speed of about u = −15 ft/sec. As the rotor reacts with a nose-down tip-path-plane tilt and a nose-down pitching moment, the helicopter accelerates forward up to about u = 15 ft/sec, eventually reaching a fairly steady forward velocity u ≈ 10 ft/sec. The heave velocity w is shown in Fig. 9(c). The application of the collective step input is followed by a downward acceleration, with w ≈ 30 ft/sec at about one second after the perturbation collective reaches its final value. It should be noted that, at this point, the ratio of the climb speed Vc , which is the negative of the vertical velocitypin body axes, and the hover induced velocity vh = CT /2, which is 0.0549 in non-dimensional form, or 39.78 ft/sec, is about Vc /vh = −0.75, which in the literature is associated with a rotor operating in the vortex ring state [12]. The helicopter then continues its downward acceleration due to the thrust losses associated with the approach and crossing of the vortex ring through the tip-path-plane . A maximum value of w = 67 ft/sec is reached 5.33 sec into the maneuver, corresponding to a ratio Vc /vh = 1.68. As the maneuver progresses, the vertical speed reaches a steady value of approximately 63 ft/sec. This corresponds to a ratio Vc /vh = 1.58, which is not quite the value |Vc | > 2vh theoretically required for a windmill brake state based on momentum theory, although conditions typical of windmill break state can already observed in the behavior of the wake. It should also be noted that at this point the maneuver is no longer purely axial, and therefore the transition velocities for axial flight are not precisely applicable. Figure 10(b) shows the time history of the lateral translational velocity v. The roll attitudes that

develop for t > 5 sec produce a corresponding increase in the sideward velocity, which reaches almost v = −40 ft/sec, and eventually settles at a nearly steady value of approximately v = −25 ft/sec. This results in a total velocity component in the x − y body axis plane, i.e., not including the descent rate, of 27 ft/sec and a sideslip of -27 deg. Because the controls are assumed to be fixed during the entire maneuver, except for the initial collective input, the resultant flight condition is clearly not a purely axial descent, but a steep descent with a total velocity V = 68.5 ft/sec, a sideslip angle of βF = 21 deg and a flight path angle of γ = −81 deg. The simulation of a purely axial descent would require a coordination of all four pilot inputs to constrain all other motions of the helicopter. Also, because no pedal is applied to counteract the effects of the reduction of the required main rotor torque, the heading of the helicopter, shown in Fig. 10(c) changes considerably during the maneuver. At the end of the 15 seconds of the simulation, the helicopter has turned a full 90 degrees to the left from the initial orientation. The change in heading is slower than the decrease in torque and power required by the main rotor, but it follows a similar pattern. Main rotor response The time history of the rotor flapping response in shown in Fig. 11. Figure 11(b) shows the time history of the collective flap β0 . The collective input produces an immediate decrease of about 3 degrees. Then, β0 oscillates about a mean value of about 1.5 deg up to t = 6 sec approximately, at which point it increases very quickly to a peak value of β0 = 4 deg, before settling to a steady value of abot β0 = 2.5. The longitudinal and lateral flapping coefficients β1c and β1s (arising from a Multiblade Coordinate Transformation) are shown as a function of time in Figs. 11(c) and 11(d), respectively. Both β1c and β1s initially decrease, i.e., the rotor tilts back and to the right, by about 2 deg and slightly less than 1 deg, respectively. After the first second, β1c shows large oscillations throughout the descent, with maximum value of β1c = 8 deg around t = 6 sec. These oscillations are the primary reason for the large variations in the fuselage pitch angle and rate observed in Fig 6. The lateral flapping β1s also oscillates during the first 5 seconds of the maneuver, although both the mean value, equal to 0.5 deg, and the amplitude of the oscillations, of about 1 deg, are not as large. Around t = 6 sec, however, there is a sharp peak of β1s , to

a right tilt of almost 4 deg, following which it recov- Other response quantities ers and settles at a steady value around 1 deg to the The induced velocities at the rotor plane at the same right. 6 time points used in the previous discussion are Figure 12 shows the rotor lag response. The col- shown in Fig. 14 (note that larger scales will often be lective lag ζ0 , shown in Fig. 12(b), increases by 4 deg used for the quantities at t = 4.89 sec and t = 5.82 in one second as the collective is decreased, i.e., the sec, where the strong perturbations generated by the magnitude average lag displacement is reduced sub- vortex bundle are most visible: if the same scale had stantially (recall that the lag angle ζ is negative for a been used for all six cases, many interesting features backward displacement in the lead-lag direction). Af- at the other time points would have been very difter the rapid initial increase, ζ0 continues to increase ficult to identify). Compared to the trimmed hover more slowly, until it becomes positive (effectively a condition, reducing the collective first has the effect lead angle). At around t = 6 seconds, there is a of reducing the inflow ratio over the entire rotor disk rather sudden drop in ζ0 of about 1.5 degrees, and by an average of 0.03. As the maneuver progresses from that point it slowly evolves towards a steady and the wake filaments get closer to the wake, the invalue of around ζ0 = 1 deg. The mean values of the flow increases slightly towards the tips, but remains cosine and sine lag coefficients, ζ1c and ζ1s , shown in the same inboard of the blade. However, as the vortex Figs. 12(c) and 12(d), are much smaller in magnitude ring forms around t = 3.96 sec, the inflow distribution than the constant coefficient ζ0 , with values close to changes dramatically. The effects of the vortex ring zero. However, between t = 3 and t = 7 sec, both intersecting the tip-path-plane can be clearly seen in the cosine and sine coefficients show a high frequency the rear portion of the disk, where the vortex ring oscillation, with an amplitude that is fairly random, generates a strong upwash, with a strong downwash just inboard of it. The bias towards the rear is due with a maximum of about 1.5 deg. to the fact that at this particular instant, the vortex Finally, the time histories of the torsional response ring is starting to separate from the tip-path-plane in are shown in Fig. 13, again, in multiblade coordi- the front, while it remains still close in the rear. As nate format. The collective torsion φ0 , shown in time progresses to t = 4.89 sec and the vortex ring Fig. 13(b), starts at a value of −1 deg in hover, indi- starts to be convected above the rotor, the induced cating a slight average nose-down twist. The collec- velocities slowly return to a more regular distribution, tive input increases φ0 by 1 deg in the first 3 seconds although on the left, or retreating side of the disk the of the maneuver. For 3 < t < 5 sec , φ0 decreases to influence of the bundled vortices is still clearly visi−3 deg in a high-frequency oscillatory way, following ble. At this time point, the induced velocity reaches which it increases rapidly to a nearly zero twist an- both its largest and the lowest value. A region of upgle, that remains for the rest of the maneuver. The wash remains in the rear of the rotor towards the tip, longitudinal and lateral torsion components, φ1c and but the inflow distribution is dominated by the large φ1s , in Figs. 13(c) and 13(d), respectively, start with downwash on the advancing side. A second later, values of 0 deg in hover, and remain at about that t = 5.82, the vortex ring has completely separated value, except during the period corresponding to the from the rotor, and while moving upstream of the passage of the vorticity ring through the rotor disk, wake structure, still exerts some influence over the during which large, high-frequency oscillations occur. front of the rotor. As the wake reorganizes itself into Clearly, the most notable feature of the rotor re- a clean helical structure above the rotor, t = 9.31, sponse is the reaction to the wake vorticity as it pro- the inflow again becomes almost uniform. gressively bundles below the rotor disk, crosses it, Figure 15 shows the perpendicular blade sectional and is convected above it. The vortex ring has an velocity, UP , which includes the effects of inflow, deirregular, nonaxisymmetric geometry and this, cou- scent speed, and blade flapping. The UP component pled with the linear and angular motions that the he- decreases with the initial application of the colleclicopter progressively develops, generate lateral and tive step, both because of the decrease in induced longitudinal components of the rotor response. The velocities associated with the lower thrust level, and interaction between the rotor and the vortex ring because the descent of the helicopter produces an upmanifests itself with substantial unsteadiness in the wash through the rotor that decreases UP (positive rotor response. The magnitude of this transient re- for a downward flow). But as the vortex ring forms sponse is high, especially in flap and torsion. around the rotor tips, it is the influence of the in-

duced velocities that dominates the distribution of UP , despite the continually increasing descent velocities. At t = 3.96 seconds, the wake induced upwash is particularly large in the rear of the rotor towards the tip. The area inboard of the formation of the vortex ring experiences positive values of UP due to the large inflow there, which counteracts the velocities due to the continuing descent. The same can be said about the next instant, t = 4.89, at which the velocity reaches very large positive values due to the large downwash in the retreating side. Eventually, the wake reorganizes and UP becomes again dominated by the upwash due to the descent. The evolution of the angle of attack α over the rotor for the same six instants during the axial maneuver is shown in Fig. 16. Recall that the angle of attack α at any given section is given by the difference between the geometric angle of attack, which is composed of the pitch of the blades due to the applied controls, the built-in twist and the elastic deformation of the blade in torsion; and the induced angle of attack φ, defined as φ = tan−1 UP /UT . The changes in α depend primarily on three factors: (i) the decrease in collective pitch, which affects the total geometric angle of attack; (ii) the upward flow through the rotor disk due to the descent velocity; and (iii) the change in induced velocities. Figure 16 indicates the following sequence of events. The initial step down in collective results in an overall decrease in α, despite the initially smaller UP , because of the lower geometric pitch angle. However, as the rate of descent continues to increase, the angle of attack increases thanks to the higher values of the upwash through the rotor, especially in the inboard blade region. As the vortex ring approaches and then crosses the rotor plane, the concentrations of vorticity generate load fluctuations, and also areas of very high and very low angles of attack, often in close proximity of each other. While some portions of the rotor disk are near or in stall conditions, others produce negative lift. As the descent progresses towards the windmill break state, the areas of negative angle of attack disappear, but the angles of attack in the descent remain much higher than during hover due to the decreased value of UP caused by the high descent speeds achieved. Figure 17 shows the lift coefficient CL at different instants during the axial descent. Recall that CL is a function of both angle of attack α and local Mach number M . Initially, the lift decreases (compared with the hover values, not included in this paper), because of the lower collective. At t = 2.56 sec the

upwash generated by the descent is already increasing CL over the entire rotor disk. As the vortex ring first approaches and then crosses the rotor plane, t = 3.96 and t = 4.89, areas of stall or near stall appear, while other parts of the rotor inboard of the vortex ring formation produce negative lift as the angle of attack becomes negative due to the high induced velocities found there. This behavior of the angle of attack and the lift coefficient is typical of the vortex ring state, and has been observed previously, both in simulation [24] and in experiment [21]. The large regions of negative lift that occur at t = 4.89 sec produce the thrust losses previously observed in Fig. 8(b). At t = 5.82 sec, however, the lift increases considerably all over the rotor, which produces the rapid increase of CT also visible in Fig. 8(b). Figure 18 shows distribution of CL M 2 , which is proportional to the elemental lift, at the same time points. The distribution of CL M 2 loosely resembles that of CL in Fig 17, but some areas of high CL , especially in the inboard portions of the disk do not translate into large CL M 2 because the local Mach number is low. Figure 19 shows the distribution of rCL M 2 , which is proportional to the local contribution to the flapping moment. The distribution is nearly axisymmetric, with the important exception of the period during which the vortex ring first approaches and then crosses the rotor plane. The evolution of rCL M 2 helps explain the flapping behavior observed in Fig. 11. For example, the reduction of flapping moment at the beginning of the maneuver, while keeping an axisymmetric distribution, justifies the corresponding reduction in coning. As the vortex bundle approaches the plane of the rotor, the unsteady airloads associated with it translate into some oscillations in the flapping response. At t = 3.96 seconds, the flap moment is very large on the rear of the rotor and, because of the nearly 90 deg lag in flapping response, produces a large lateral tilt of the tip path plane, and the peak of β1s . As the maneuver continues and the vortex ring goes through the rotor significantly large spikes and fluctuations occur [12]. At t = 4.89 sec, there is still a strong positive flap moment region in the rear portion of the disk, which sustains the strong lateral flapping. The picture is less clear on the retreating side, especially between 240 and 300 deg, where simultaneous regions of strong positive and negative contributions to flap moment are visible. The net result is a reduction of longitudinal flapping. At t = 5.82 sec, the effects of the vortex

ring begin to diminish, the distribution of rCL M 2 begins to take again a more regular shape, and the oscillations of longitudinal and lateral flapping begin to subside. Figure 20 shows the distribution of the elemental profile drag, CD M 2 . At the beginning of the descent, the drag decreases from the initial hovering condition (not shown) because of the decrease in collective, reaching very low values, as both the angle of attack α and the Mach number M are low. However, as the vortex ring crosses the rotor plane, the regions of very high and very low angles of attack previously observed cause large values of CD : at t = 3.96 sec, a concentrated band of very high drag begins to appear in the tip regions on the rear portions of the rotor. As the maneuver continues, t = 4.89 sec, this region is larger and extends over the outer portions of the fourth quadrant of the rotor, where the vortex ring still affects strongly the inflow distribution. At t = 5.82, the drag is returning to a more uniform distribution, with only an elevated area on the retreating side around ψ = 270o . As the maneuver continues and the wake returns to an organized helical structure, the values of the drag return to normal, although slightly higher than in hover. The decrease in profile drag following the reduction in collective clearly justifies the decrease in the lag angle seen in Fig. 12(b). As the maneuver continues, the vortex ring goes through the rotor producing large gradients of angle of attack, regions of high drag, and pronounced unsteadiness, which in turn cause the development of longitudinal and lateral lag components, and pronounced lag vibrations. The distribution of the moment coefficient CM during the maneuver is shown in Fig. 21. The moment coefficient is a function of both angle of attack α and Mach number M . The reduction of collective reduces CM slightly in the first instants of the maneuver. At t = 2.56 sec, α increases slightly and so does CM . At t = 3.96 sec, the area of high α near the tip in the rear of the rotor, where the Mach number is also high, produces some areas of nose-up CM . At t = 4.89 sec, the angle of attack reaches values lower than −10o in the fourth quadrant of the rotor, which increase CM into large nose-up values, along with some very low values further towards the tip, where α reaches values higher than 20 deg. As the vortex ring leaves the plane of the rotor, the lower α that occurs in the outer half of the rotor disk is associated with mild nose-down moments, although some high nose-down pitching moments are found near the root, where α

is much higher. The large decrease in the constant torsion component φ0 shown in Fig. 13(b), is associated with a lower overall nose down pitching moment while in the vortex ring state, as the regions of very high nose-up CM are cancelled by broad regions of very low CM . Finally, the distribution of required induced and profile torque for this maneuver are shown, respectively, in Figs. 22 and 23. Figure 22 shows the distribution of rCL M 2 sin φ, which is proportional to the induced torque. As expected, the required induced torque decreases during the maneuver, and it becomes slightly negative (i.e., torque is provided to the rotor, by the airflow) as the descent becomes established. The profile torque is generally much lower than the induced torque, although as the drag increases (Fig. 20), some regions of localized high profile torque occur in the early stages of the vortex ring state, and at t = 5.82 sec there is one such large region on the retreating side, corresponding to the vortex bundle crossing the rotor disk. The main rotor power required, previously shown in Fig. 8(c), initially decreases because of the lower induced torque, but then increases because of the higher profile torque that occurs around t = 5.82 sec. Conclusions This paper presented the results of a simulation of a helicopter in a steep descent following a step reduction of collective pitch. The descent velocity increased through the vortex ring state and into the windmill brake state. The simulation was performed using a coupled rotor-fuselage dynamic model, including a nonlinear finite element model of the rotor blade flexibility, tightly coupled with a free vortex wake model capable of capturing the distortions of the wake geometry during maneuvers. The full model could not be validated for these specific flight regimes because of the lack of sufficiently detailed experimental data, especially for wake geometries, distribution of aerodynamic parameters over the rotor disk, and rotor blade dynamics. However, the model was extensively correlated with flight test data in other flight conditions, and components of it with a broad spectrum of wind tunnel and flight test data. The results of the present study suggest the following key conclusions: 1. The simulation model including the tight coupling with the free wake is numerically well behaved. Even in the potentially critical phase

of the tip vortices bundling into a vortex ring, tightly interacting with each other, approaching the rotor plane and crossing it, no numerical instabilities or other computational difficulties were observed.

it emerges from the present study. They might be improved by allowing the rings to move at an angle with the rotor, as opposed to parallel to it, by allowing them to take at least an elliptical shape not centered with the rotor shaft, and by allowing them at least some simple distortion rather than being flat. State-space, physicsbased models of the phenomenon, suitable for stability analyses and flight control system design, remain unavailable.

2. The dynamic behavior of the helicopter is dominated by the effects of the vortex bundle approaching, and then crossing, the rotor plane. For the specific example of the paper, the effects were strongest in the retreating side and the rear portion of the disk, resulting in strong Acknowledgments lateral flapping and noticeable, but not as strong, This research was supported by the National Rolongitudinal flapping. The rotor-vortex ring intorcraft Technology Center under the Rotorcraft Centeraction also triggered significant unsteadiness ter of Excellence Program. The authors would like to in all aerodynamic and dynamic response quanthank Dr. J. G. Leishman and Dr. S. Ananthan for tities. providing a copy of the maneuver free wake code, and 3. Overall, the consequences of the maneuver de- for many useful discussions. scribed in this paper were relatively benign for the single main rotor helicopter considered. Be- References cause of the desire to isolate the effects of aircraft-wake interactions, the maneuver is not [1] Ribera, M., “Helicopter Flight Dynamics Simufully representative of typical steep descents, belation With a Time-Accurate Free Wake Model,” cause the rotor speed was assumed to be conPh.D. Dissertation, Department of Aerospace stant, and no corrective action by the pilot was Engineering, University of Maryland, College assumed. However, even in these “fixed stick” Park, MD, 2007. conditions, the roll and pitch attitudes were [2] Newman, S. J., Brown, R., Perry, J., Lewis, S., never dangerously high, and returned to reasonOrchard, M., and Modha, A., “Comparative Nuable steady state values once the vortex ring was merical and Experimental Investigations of the convected away from the rotor, well above it. At Vortex Ring Phenomenon in Rotorcraft,” Prothat point, all velocity components also began to ceedings of the 57th Annual Forum of the Amerstabilize, to values that would not affect safety ican Helicopter Society International, 2001. of flight. 4. In the critical phases of the interaction with the rotor disk, the vortex ring did maintain a clearly identifiable geometry, which however was neither axisymmetric nor flat or parallel to the rotor plane. The time varying geometry of the crossing regions had a powerful effect on the flight dynamics of the helicopter, because it determined magnitude and direction of the flapping response, and therefore of the tip-path-plane tilt. 5. Because the asymmetries of the vortex ring and the details of the interaction with the rotor plane are so important, momentum or momentum-like theories cannot fully capture the basic physics of the vortex ring state, and therefore cannot predict it from first principles. Simplified vortex ring state models based on discrete rings also miss some of the fundamental physics, as

[3] Johnson, W., “Model for Vortex Ring State Influence on Rotorcraft Flight Dynamics,” Proceedings of the AHS Fourth Decennial Specialists’ Conference on Aeromechanics, 2004. [4] Wolkovitch, J., “Analytical Prediction of Vortex-Ring Boundaries for Helicopters in Steep Descents,” Journal of the American Helicopter Society, Vol. 17, No. 3, 1972, pp. 13–19. [5] Peters, D. A., and Chen, S.-Y., “Momentum Theory, Dynamic Inflow, and the Vortex Ring State,” Journal of the American Helicopter Society, Vol. 27, No. 3, July 1982, pp. 18–24. [6] Heyson, H. H., “A Momentum Analysis of Helicopters and Autogyros in Inclined Descent, with Comments on Operational Restrictions,” NASA TN D-7917, October 1975.

[7] He, C. J., Lee, C., and Chen, C. W., “Fi- [16] Hennes, C. C., Chen, H., Brentner, K. S., S, nite State Induced Flow Model in Vortex Ring A., and Leishman, G. J., “Influence of TranState,” Journal of the American Helicopter Sosient Flight Maneuvers on Rotor Wake Dynamciety, Vol. 45, No. 4, October 2000, pp. 318–320. ics and Noise Radiation,” AHS 4th Decennial Specialist’s Conference on Aeromechanics, San [8] Chen, C., Prasad, J. V. R., and Basset, P. M., Francisco, CA, January 21–23 2004. “A Simplified Inflow Model of a Helicopter Rotor in Vertical Descent,” Proceedings of the 60th [17] Theodore, C., and Celi, R., “Helicopter Flight Annual Forum of the American Helicopter SociDynamic Simulation with Refined Aerodynamic ety International, 2004. and Flexible Blade Modeling,” Journal of Aircraft, Vol. 39, No. 4, July-August 2002, pp. 577– [9] Chen, C., and Prasad, J. V. R., “Theoretical In586. vestigations of a Helicopter Rotor in Steep Descent,” Collection of Technical Papers - AIAA Modeling and Simulation Technologies Conference, 2005.

[18] Ribera, M., and Celi, R., “Simulation Modeling in Climbing and Descending Flight with Refined Aerodynamics,” Proceedings of the 62nd Annual Forum of the American Helicopter Society, [10] Peters, D. A., and He, C., “Modification of Phoenix, AZ, May 9-11 2006. Mass-Flow Parameter to Allow Smooth Transition Between Helicopter and Windmill States,” American Helicopter Society, Vol. 51, No. 3, July [19] Brenan, K. E., Campbell, S. L., and Petzold, L. R., The Numerical Solution of Initial Value 2006, pp. 275–278. Problems in Differential-Algebraic Equations El[11] Leishman, J. G., Bhagwat, M. J., and Anansevier Science Publishing Co., New York, 1989. than, S., “Free-Vortex Wake Predictions of the Vortex Ring State for Single Rotor and Multi- [20] Ribera, M., and Celi, R., “Simulation ModelRotor Configurations,” Proceedings of the 58th ing in Steady Turning Flight with Refined AeroAnnual Forum of the American Helicopter Socidynamics,” Proceedings of the 31th European ety International, Montr´eal Canada, 2002. Rotorcraft Forum, 2005. [12] Leishman, J. G., Bhagwat, J. G., and Anan- [21] Betzina, M. D., “Tiltrotor Descent Aerodynamthan, S., “The Vortex Ring State as a Spatially ics: A Small-Scale Experimental Investigation and Temporally Developing Wake Instability,” of Vortex Ring State,” Proceedings of the 57th American Helicopter Society International AeroAnnual Forum of the American Helicopter Socidynamics Meeting, San Francisco, CA, January ety International, Washington, D.C., May 9–11 2002. 2001. [13] Bhagwat, M. J., and Leishman, J. G., “Sta[22] Washizu, K., Azuma, A., K¯oo, J., and Oka, T., bility, Consistency and Convergence of Time “Experiments on a Model Helicopter Rotor OpMarching Free-Vortex Rotor Wake Algorithms erating in the Vortex Ring State,” Journal of of Time-Marching Free-Vortex Rotor Wake AlAircraft, Vol. 3, No. 3, May-June 1966, pp. 225– gorithms,” Journal of the American Helicopter 230. Society, Vol. 46, No. 1, January 2001, pp. 59–71. [14] Brown, R. E., Line, A. J., and Ahlin, G. A., [23] Ananthan, S., Leishman, J. G., and Ramasamy, M., “The Role of Filament Stretching in the “Fuselage and Tail-Rotor Interference Effects Free-Vortex Modeling of Rotor Wakes,” Ameron Helicopter Wake Development in Descendth ican Helicopter Society 58th Annual Forum, ing Flight,” Proceedings of the 60 Annual FoMontr´eal, Canada, June 11–13, 2002. rum of the American Helicopter Society International, 2004. [24] Brown, R. E., Newman, S. J., Leishman, J. G., [15] Houson, S. S., and Brown, R. E., “Rotor-Wake and Perry, F. J., “Blade Twist Effects on RoModeling for Simulation of Helicopter Flight Metor Behaviour in the Vortex Ring State,” Prochanics in Autorotation,” Journal of Aircraft, ceedings of the 28th European Rotorcraft Forum, Vol. 40, No. 5, September-October 2003. 2002.

TRIMMED SOLUTION

Start Simulation: t=0 t = t + ∆t Integrate Equations of Motion at t

Save flap and blade velocities

FREE WAKE Calculate circulation

z

q

y Time-marching wake to t

Blade, N

ψ

Straight line segment approximation

r

Ω

p

Γv

h

x

l ζ

Blade, N-1 Lagrangian markers

Γv

Save inflow

NO

l +1

Is simulation complete?

l +2

YES

Curved vortex filament

End Simulation: t=T

Induced velocity from element of vortex trailed by blade N-1

Figure 1: Free wake discretization in the azimuth (ψ) Figure 2: Integration scheme procedure with a timeand filament (ζ) directions. marching free wake model coupled to the rotorfuselage model.

Collective perturbation, Δθ0 (in)

1 0 -1 -2 -3 -4 -5 -6 0

5

10

15

Time (sec)

Figure 3: Time history of the control input for an axial descent; V = 1 kt.

1.5

1.5

1.0

1.0

T= 1.40 sec (6 revs)

0.5

z/R

z/R

0.5

0.0

−0.5

−1.0

−1.0

−1.0

−0.5

0.0 x/R

0.5

1.0

−1.5 −1.5

1.5

1.5

z/R

z/R

0.0 x/R

0.5

1.0

1.5

1.0

1.5

1.0

1.5

0.5

0.0

0.0

−0.5

−0.5

−1.0

−1.0

−1.0

−0.5

0.0 x/R

0.5

1.0

−1.5 −1.5

1.5

1.5

T= 5.82 sec (25 revs) −1.0

−0.5

0.0 x/R

0.5

1.5

1.0

1.0

T= 3.96 sec (17 revs)

0.5

z/R

0.5

0.0

0.0

−0.5

−0.5

−1.0

−1.0

−1.5 −1.5

−0.5

1.0

T= 2.56 sec (11 revs)

0.5

−1.5 −1.5

−1.0

1.5

1.0

z/R

0.0

−0.5

−1.5 −1.5

T= 4.89 sec (21 revs)

−1.0

−0.5

0.0 x/R

0.5

1.0

1.5

−1.5 −1.5

T= 9.31 sec (40 revs) −1.0

−0.5

0.0 x/R

0.5

Figure 4: Side view of the free wake tip vortex geometry at different times for an axial descent; V = 1 kt.

1.5

1.5

1.0

1.0

T= 1.40 sec (6 revs)

0.5

z/R

z/R

0.5

0.0

−0.5

−1.0

−1.0

−1.0

−0.5

0.0 y/R

0.5

1.0

−1.5 −1.5

1.5

1.5

z/R

z/R

0.0 y/R

0.5

1.0

1.5

1.0

1.5

1.0

1.5

0.5

0.0

0.0

−0.5

−0.5

−1.0

−1.0

−1.0

−0.5

0.0 y/R

0.5

1.0

−1.5 −1.5

1.5

1.5

T= 5.82 sec (25 revs) −1.0

−0.5

0.0 y/R

0.5

1.5

1.0

1.0

T= 3.96 sec (17 revs)

0.5

z/R

0.5

0.0

0.0

−0.5

−0.5

−1.0

−1.0

−1.5 −1.5

−0.5

1.0

T= 2.56 sec (11 revs)

0.5

−1.5 −1.5

−1.0

1.5

1.0

z/R

0.0

−0.5

−1.5 −1.5

T= 4.89 sec (21 revs)

−1.0

−0.5

0.0 y/R

0.5

1.0

1.5

−1.5 −1.5

T= 9.31 sec (40 revs) −1.0

−0.5

0.0 y/R

0.5

Figure 5: Rear view of the free wake tip vortex geometry at different times for an axial descent; V = 1 kt.

Collective perturbation, Δθ0 (in)

1 0

-2 -3 -4 -5 -6

Pitch attitude, θ (deg)

15

0

5

10

15

(b) Pitch angle

10 5 0 -5

10 Pitch rate, q (deg/sec)

(a) Input

-1

0

5

10

5

15

(c) Pitch rate

0

-5

0

5

Time (sec)

10

15

Figure 6: Time history of the control input, pitch angle θ and pitch rate q for an axial descent; V = 1 kt.

Collective perturbation, Δθ0 (in)

1 0

-2 -3 -4 -5 -6

Roll attitude, φ (deg)

5

0

5

10

15

0 -5 -10 -15 -20 -25 -30

30 Roll rate, p (deg/sec)

(a) Input

-1

(b) Roll angle

0

5

10

15

20 10 0 -10 -20 -30 -40

(c) Roll rate

0

5

Time (sec)

10

15

Figure 7: Time history of the control input, roll angle φ and roll rate p for an axial descent; V = 1 kt.

Collective perturbation, Δθ0 (in) Main Rotor Power Required (HP) Main Rotor Thrust Coefficient, CT

1 0 (a) Input

-1 -2 -3 -4 -5 -6 -0.009

0

5

10

15

-0.008 (b) Thrust coefficient

-0.007 -0.006 -0.005 -0.004 -0.003 -0.002

2000

0

5

10

15

1500 1000

(c) Power required

500 0 -500 -1000 -1500

0

5

Time (sec)

10

15

Figure 8: Time history of the control input, thrust coefficient CT and main rotor power required QM R for an axial descent; V = 1 kt.

Collective perturbation, Δθ0 (in)

1 0

-2 -3 -4 -5 -6

Forward velocity, u (ft/sec)

20

0

5

10

15

15 10 5 0 -5 -10 -15

70 Vertical velocity, w (ft/sec)

(a) Input

-1

(b) Forward velocity

0

5

10

15

60 50 40 30 20 10 0

(d) Vertical velocity

0

5

Time (sec)

10

15

Figure 9: Time history of the control input and forward and vertical velocities, u and w respectively, for an axial descent; V = 1 kt.

Collective perturbation, Δθ0 (in)

1 0

-2 -3 -4 -5 -6

Lateral velocity, v (ft/sec)

10

0

5

10

0

15

(b) Lateral velocity

-10 -20 -30 -40

20 Yaw attitude, ψ (deg)

(a) Input

-1

0

5

10

15

0 -20

(c) Yaw attitude

-40 -60 -80 -100 -120

0

5

Time (sec)

10

15

Figure 10: Time history of the control input and the lateral velocity v yaw angle ψ for an axial descent; V = 1 kt.

Collective perturbation, Δθ0 (in)

1 0

(a) Input

-1 -2 -3 -4 -5 -6 4.0

0

5

10

15

10

15

10

15

10

15

3.5

β0 (deg)

3.0 2.5 2.0 1.5 1.0 0.5 0.0 8

(b) β0 0

5

β1c (deg)

6 4 2 0

(c) β1c -2

2

0

5

β1s (deg)

1 0 -1 -2 -3 -4

(d) β1s 0

5

Time (sec)

Figure 11: Time history of the flap coefficients, β0 , β1c and β1s for an axial descent; V = 1 kt.

Collective perturbation, Δθ0 (in)

1 0

(a) Input

-1 -2 -3 -4 -5 -6 2

0

5

10

15

10

15

10

15

10

15

1

ζ0 (deg)

0 -1 -2 -3 -4 -5 -6 1.0

(b) ζ0 0

5

0.8

ζ1c (deg)

0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6

1.5

(c) ζ1c 0

5

ζ1s (deg)

1.0 0.5 0.0

(d) ζ1s -0.5

0

5

Time (sec)

Figure 12: Time history of the control input and lag coefficients, ζ0 , ζ1c and ζ1s for an axial descent; V = 1 kt.

Collective perturbation, Δθ0 (in)

1 0

(a) Input

-1 -2 -3 -4 -5 -6 1

0

5

10

15

10

15

10

15

10

15

φ0 (deg)

0 -1 -2 -3

(b) φ0 -4 4

0

5

3

φ1c (deg)

2 1 0 -1 -2 -3 -4

8

(c) φ1c 0

5

φ1s (deg)

6 4 2 0

(d) φ1s -2

0

5

Time (sec)

Figure 13: Time history of the control input and torsion coefficients, φ0 , φ1c and φ1s for an axial descent; V = 1 kt.

90

90

1

0.2

60

120

60

0.4

0.8

0.8

0.6

0.15

0.6

0.3

150

30

150

1

120

30 0.4

0.4 0.1

0.2

0.2

0.2 0.1

180

0

0.05 180

0 0

0 210

−0.1 210

330

330 −0.2

−0.05 240

300

240

−0.1

270

270

(a) T= 1.40 sec (6 revs) 90

(b) T= 4.89 sec (21 revs)

1

90

0.2

60

120

−0.3

300

1

0.2

60

120

0.8

0.8 0.15

0.6 30

150

0.15

0.6 30

150

0.4

0.4 0.1

0.2 180

0

0.1

0.2

0.05 180

0

0 210

0 210

330

330

−0.05 240

300

−0.05 240

−0.1

270

90

300

−0.1

270

(c) T= 2.56 sec (11 revs)

(d) T= 5.82 sec (25 revs)

1

90

0.2

60

120

1

0.2

60

120

0.8

0.8 0.15

0.6 30

150

0.15

0.6 30

150

0.4

0.4 0.1

0.2 180

0

0.1

0.2

0.05 180

0

0 210

330

−0.05 300 270

(e) T= 3.96 sec (17 revs)

−0.1

0.05

0 210

330

240

0.05

−0.05 240

300 270

(f) T= 9.31 sec (40 revs)

Figure 14: Distribution of induced velocities at different times for an axial descent; V = 1 kt.

−0.1

90

90

1

120

0.25

60

1

120

0.25

60

0.8

0.8 0.2

0.6 150

0.2 0.6

0.15

30 0.4

150

0.1

0.2 0

0

0.1

0.2

0.05

180

0.15

30 0.4

0.05

180

0

−0.05

−0.05

−0.1 210

330

−0.1

−0.15

210

330

−0.15

−0.2 240

300

−0.2 240

−0.25

270

300

−0.25

270

(a) T= 1.40 sec (6 revs) 90

(b) T= 4.89 sec (21 revolutions) 90

1

120

0.25

60

1

120

0.25

60

0.8

0.8 0.2

0.6 150

0.2 0.6

0.15

30 0.4

150

0

0

0.1

0.2

0.05

180

0.15

30 0.4

0.1

0.2

0.05

180

0

−0.05

330

−0.1

−0.15

210

330

−0.15

−0.2 240

300

−0.2 240

−0.25

270

300

−0.25

270

(c) T= 2.56 sec (11 revs)

(d) T= 5.82 sec (25 revs) 90

1

120

0.25

60

1

120

0.25

60

0.8

0.8 0.2

0.6 150

0.2 0.6

0.15

30 0.4

150

0

0

0.1

0.2

0.05

180

0.05

180

0

−0.05

330

−0.15

−0.1 210

330

−0.2 300 270

(e) T= 3.96 sec (17 revs)

−0.25

0 −0.05

−0.1 210

0.15

30 0.4

0.1

0.2

240

0 −0.05

−0.1 210

90

0

−0.15 −0.2

240

300

−0.25

270

(f) T= 9.31 sec (40 revs)

Figure 15: Distribution of blade sectional perpendicular velocity at different times for an axial descent; V = 1 kt.

90

90

1 60

120 0.8

60

30 0.2

30

0

0

10

0.4 0.2

5

180

15

0.6 150

10

0.4

20

0.8

15

0.6 150

1

120

20

5

180

0

−5

210

−5

−10

330

210

−10

330

−15 240

−15

−20

300

240

270

−20

300 270

(a) T= 1.40 sec (6 revs) 90

(b) T= 4.89 sec (21 revs) 90

1 60

120

60

30 0.2

30

0

0

10

0.4 0.2

5

180

15

0.6 150

10

0.4

20

0.8

15

0.6 150

1

120

20

0.8

5

180

0

−5

210

210

−10

330

−15 240

−15

−20

300

240

270

−20

300 270

(c) T= 2.56 sec (11 revs)

(d) T= 5.82 sec (25 revs) 90

1 60

120

60

30 0.2

30

0

0

10

0.4 0.2

5

180

15

0.6 150

10

0.4

20

0.8

15

0.6 150

1

120

20

0.8

5

180

0

−5

210

330

−10

300 270

(e) T= 3.96 sec (17 revs)

−20

0 −5

210

330

−15 240

0 −5

−10

330

90

0

−10 −15

240

300 270

(f) T= 9.31 sec (40 revs)

Figure 16: Distribution of sectional angle of attack at different times for an axial descent; V = 1 kt.

−20

90

90

1

120

60

1

120

1

60

0.8

1

0.8 0.8

0.6 150

30

0.6

0.4

30

0.6

0.4

0.4

0.2

0.8

0.6 150

0.4

0.2

0.2 180

210

0.2 180

0 0

0

−0.2

−0.2

−0.4

330

0

210

−0.4

330

−0.6 240

−0.6

−0.8

300 270

240

−1

(a) T= 1.40 sec (6 revs) 90

−1

(b) T= 4.89 sec (21 revs) 90

1

120

−0.8

300 270

60

1

120

1

60

0.8

1

0.8 0.8

0.6 150

30

150

0.6

0.4

30

0.6

0.4

0.4

0.2

0.8

0.6

0.4

0.2

0.2 180

210

0.2 180

0 0

0

−0.2

−0.2

−0.4

330

0

210

−0.4

330

−0.6 240

−0.6

−0.8

300 270

240

−1

(c) T= 2.56 sec (11 revs) 90

−1

(d) T= 5.82 sec (25 revs) 90

1

120

−0.8

300 270

60

1

120

1

60

0.8

1

0.8 0.8

0.6 150

30

150

0.6

0.4

30

0.6

0.4

0.4

0.2

0.8

0.6

0.4

0.2

0.2 180

210

330

0.2 180

0

0

0

0

−0.2

−0.2

−0.4

210

330

−0.6 240

300 270

(e) T= 3.96 sec (17 revs)

−0.8 −1

−0.4 −0.6

240

300 270

(f) T= 9.31 sec (40 revs)

Figure 17: Distribution of lift coefficient CL at different times for an axial descent; V = 1 kt.

−0.8 −1

90

90

1

0.3

60

120

1

0.3

60

120

0.8

0.8 0.2

0.6

0.2

0.6

30

150

30

150

0.4

0.4

0.1

0.2

0.1

0.2

180

0

0

180

0

−0.1

210

−0.2

330

−0.1

210

−0.2

330

−0.3 240

−0.3 240

300 270

300 270

(a) T= 1.40 sec (6 revs) 90

(b) T= 4.89 sec (21 revs) 90

1

0.3

60

120

1

0.3

60

120

0.8

0.8 0.2

0.6

0.2

0.6

30

150

30

150

0.4

0.4

0.1

0.2

0.1

0.2

180

0

0

180

0

−0.1

210

−0.2

330

210

−0.2

330

−0.3 240

300 270

300 270

(c) T= 2.56 sec (11 revs)

(d) T= 5.82 sec (25 revs) 90

1

0.3

60

120

1

0.3

60

120

0.8

0.8 0.2

0.6

0.2

0.6

30

150

30

150

0.4

0.4

0.1

0.2

0.1

0.2

180

0

0

180

0

−0.1

210

330

−0.2

300 270

(e) T= 3.96 sec (17 revs)

0 −0.1

210

330

−0.3 240

0 −0.1

−0.3 240

90

0

−0.2 −0.3

240

300 270

(f) T= 9.31 sec (40 revs)

Figure 18: Distribution of non-dimensional lift CL M 2 at different times for an axial descent; V = 1 kt.

90

90

1

120

0.3

60

1

120

0.3

60

0.8

0.8

0.6

0.6

0.2

150

30

0.2

150

30

0.4

0.4 0.1

0.2 180

0

0

0.1

0.2 180

0

−0.1 210

330

−0.1 210

330

−0.2 240

300

−0.2 240

−0.3

270

300

−0.3

270

(a) T= 1.40 sec (6 revs) 90

(b) T= 4.89 sec (21 revs) 90

1

120

0.3

60

1

120

0.3

60

0.8

0.8

0.6

0.6

0.2

150

30

0.2

150

30

0.4

0.4 0.1

0.2 180

0

0

0.1

0.2 180

0

−0.1 210

330

240

210

300

330 −0.2 240

−0.3

270

300

−0.3

270

(c) T= 2.56 sec (11 revs)

(d) T= 5.82 sec (25 revs) 90

1

120

0.3

60

1

120

0.3

60

0.8

0.8

0.6

0.6

0.2

150

30

0.2

150

30

0.4

0.4 0.1

0.2 180

0

0

0.1

0.2 180

0

−0.1 210

330

300 270

210

(e) T= 3.96 sec (17 revs)

−0.3

0

−0.1 330

−0.2 240

0

−0.1

−0.2

90

0

−0.2 240

300 270

(f) T= 9.31 sec (40 revs)

Figure 19: Distribution of flap moment rCL M 2 at different times for an axial descent; V = 1 kt.

−0.3

90

0.01

90

1

120

60

0.009

0.8

60

30

30

0.007

0.4

0

0.035

0.4 0.2

0.006

180

0.04

0.6 150

0.2

0.045

0.8

0.008

0.6 150

0.05 1

120

0.03

0.005180

0

0.004

0.02

0.003 210

330

0.015 210

330

0.002 240

0.01

0.001

300 270

240

90

0

0.01 60

90

60

30

0.04

0.6 150

30

0.007

0.4 0.2

0.035

0.4 0.2

0.006 0

0.045

0.8

0.008

0.6

0.05 1

120

0.009

0.8

180

0

(b) T= 4.89 sec (21 revs)

1

150

0.005

300 270

(a) T= 1.40 sec (6 revs) 120

0.03

0.005180

0

0.004

330

0.015 210

330

0.002 240

0.01

0.001

300 270

240

0.05 60

90

60

30

30

0.035

0.2 0

0.007

0.4 0.2

0.03

180

0.008

0.6 150

0.4

0.009

0.8

0.04

0.6

0.01 1

120

0.045

0.8 150

0

(d) T= 5.82 sec (25 revs)

1

120

0.005

300 270

0

(c) T= 2.56 sec (11 revs) 90

0.006

0.025180

0

0.02

330

0.003 210

330

0.01 240

300 270

(e) T= 3.96 sec (17 revs)

0.005 0

0.005 0.004

0.015 210

0.025 0.02

0.003 210

0.025

0.002 240

300 270

0.001 0

(f) T= 9.31 sec (40 revs)

Figure 20: Distribution of elemental drag CD M 2 at different times for an axial descent; V = 1 kt.

90

0.02

120

1

120

60

60

0.015

0.8 0.6

0.08

0.8

0.06

0.6

0.01

150

0.1

90

1

150

30

30 0.04

0.4

0.4

0.005

0.2

0.2

0.02

0 180

180

0

0

0

−0.005

−0.02 −0.01 210

−0.04 210

330

330

−0.015 240

−0.06

−0.02

300 270

240

−0.025

(a) T= 1.40 sec (6 revs) 90

0.02

0.1

90

1

120

60

60

0.015

0.8 0.6

0.08

0.8

0.06

0.6

0.01

150

−0.1

(b) T= 4.89 sec (21 revs)

1

120

−0.08

300 270

150

30

30 0.04

0.4

0.4

0.005

0.2

0.2

0.02

0 180

180

0

0

0

−0.005

−0.02 −0.01 210

−0.04 210

330

330

−0.015 240

−0.06

−0.02

300 270

240

−0.025

(c) T= 2.56 sec (11 revs) 90

0.02 60

90

0.6

60

30

0.6

0.01

150

0.4

0.015

0.8

0.01

150

0.02 1

120

0.015

0.8

−0.1

(d) T= 5.82 sec (25 revs)

1

120

−0.08

300 270

30 0.4

0.005

0.2

0.005

0.2 0

180

0

0

180 −0.005

0 −0.005

−0.01 210

330

−0.01 210

330

−0.015 240

300 270

(e) T= 3.96 sec (17 revs)

−0.02 −0.025

−0.015 240

300 270

−0.02 −0.025

(f) T= 9.31 sec (40 revs)

Figure 21: Distribution of moment coefficient CM at different times for an axial descent; V = 1 kt.

0.02

90 120

0.02

90

1

1

120

60

60

0.01

0.8

0.015

0.8

0

0.6

0.6 150

150

0.01

30

30

0.005

0.2 180

−0.01

0.4

0.4

0

0

0.2

−0.02

180

0

−0.03 −0.04

−0.005

−0.05 210

330

210

−0.01

330 −0.06

−0.015 240

240

300 270

−0.02

(a) T= 1.40 sec (6 revs)

0.02

90

1

120

−0.08

(b) T= 4.89 sec (21 revs) 0.02

90

−0.07

300 270

1

120

60

60

0.01

0.8

0.015

0.8

0

0.6

0.6 150

150

0.01

30

30

0.005

0.2 180

−0.01

0.4

0.4

0

0

0.2

−0.02

180

0

−0.03 −0.04

−0.005

−0.05 210

330

−0.01

210

330 −0.06

−0.015 240

240

300 270

−0.02

(c) T= 2.56 sec (11 revs) 90

0.02 60

90

0.02 1

120

0.01

0.8

−0.08

(d) T= 5.82 sec (25 revs)

1

120

−0.07

300 270

60 0.015

0.8

0

0.6 150

30 −0.01

0.4 0.2

0.6 150

0

0.005

0.2

−0.02

180

0.01

30 0.4

−0.03180

0

−0.04

0 −0.005

−0.05 210

330

210

330

−0.01

−0.06 240

300 270

(e) T= 3.96 sec (17 revs)

−0.07 −0.08

−0.015 240

300 270

−0.02

(f) T= 9.31 sec (40 revs)

Figure 22: Distribution of induced torque rCL M 2 sin φ at different times for an axial descent; V = 1 kt.

90

0.012

90

1

120

1

120

60

60

0.16

0.8

0.8 0.01 0.6

0.14

0.6

150

30

150

0.4

30 0.4

0.008

0.2

0.12

0.2 0.1

180

0

0.006180

0 0.08

0.004 210

330

0.06 210

330

0.04

0.002 240

300

240

270

270

0

(a) T= 1.40 sec (6 revs) 90

(b) T= 4.89 sec (21 revs) 0.012

90

1

120

0.02

300

60

0.012 1

120

60

0.8

0.8 0.01

0.01

0.6

0.6

150

30

150

0.4

30 0.4

0.008

0.2

0.008

0.2

180

0

0.006180

0

0.004 210

330

0.004 210

330

0.002 240

0.002

300

240

270

0

0.045

90

0.012 1

120

60

60 0.8

0.04

0.8 0.6 150

0

(d) T= 5.82 sec (25 revs)

1

120

300 270

(c) T= 2.56 sec (11 revs) 90

0.035

30

0.01 0.6 150

30 0.4

0.4 0.03

0

0.008

0.2

0.2 180

0.006

0.025 180

0

0.006

0.02 0.004

0.015 210

210

330

330

0.01 0.002 240

300 270

(e) T= 3.96 sec (17 revs)

0.005

240

300 270

0

(f) T= 9.31 sec (40 revs)

Figure 23: Distribution of profile torque rCD M 2 cos φ at different times for an axial descent; V = 1 kt.