Jhs 000139 American Heli Soc

Prediction of Vortex Ring State Boundary of a Helicopterin Descending Flight by Simulation

Pierre-Marie Basset Chang Chen J. V. R. Prasad Sebastien KolbONERA Georgia Institute of Technology ONERA

Salon-de-Provence, France Atlanta, GA Salon-de-Provence, France

A helicopter rotor in descending flight may encounter its own wake resulting in a doughnut-shaped ring around the rotordisk, known as the vortex ring state (VRS). While VRS is a region of descending flight of unsteady flow through therotor, a determination of the precise boundary surrounding this region has posed a challenge to the researchers over theyears. Several criteria have been proposed in the literature for the determination of VRS boundaries. This paper considersprediction of VRS boundaries using three different criteria, viz., zero transport velocity of rotor tip vortices, bifurcation ofequilibria, and zero heave damping. Two different inflow models previously developed at ONERA and the Georgia Instituteof Technology are used in the VRS boundaries predictions. It is shown that, within the accuracy of the inflow models used,the VRS boundary predictions are significantly influenced by the criterion used.

Nomenclature

CT rotor mean thrust coefficientCTo rotor thrust coefficient at hoverDT0 collective pitch angleDTC lateral cyclic pitch controlDTS longitudinal cyclic pitch controlDTA tail rotor collective pitch anglek empirical factor used in the Newman criterionkx empirical factor used in the tip vortices criterionkz empirical factor used in the Wolkovitch criterionNring number of vortex ringsNb number of bladesS rotor disk areaT amplitude of rotor thrust fluctuationT mean rotor thrust(Uhel, Vhel, Whel) helicopter velocity vector in the helicopter

coordinate systemVH rotorcraft horizontal velocity in the Earth

coordinate frameVicMR,VisMR inflow gradients of the first harmonic inflow

model for the main rotorVih rotor-induced velocity at hoverVimMR,VimFAN rotor mean induced velocity, MR for the main

rotor, FAN for the tail rotor fan in finVTVDES nondimensional velocity of tip vortices and

adapted parameter for the tip vortices criterion(Eq. (8))

VtvX in-plane component of tip vortex velocityVtvZ vertical component of tip vortex velocityVZ rotorcraft vertical speed in the Earth coordinate

frame (positive in climb)Corresponding author; email: [email protected] received June 2006; accepted November 2007.

VRS vortex ring state empirical factor used in the tip vortices criterion blade angle of attackD descent angle, 2 arctan( VZVH ) rotorcraft horizontal speed nondimensionalized by

the blade tip speed (advance ratio) rotorcraft vertical speed normalized by

rotor-induced velocity at hover (positive in climb),VZ / Vih

rotor-induced velocity normalized by inducedvelocity at hover (positive in climb), Vi / Vih

air densityi induced velocity used in Newman criterion

normalized by rotor-induced velocity at hover rotorcraft horizontal speed normalized by

rotor-induced velocity at hover, VH / Vihx translational velocity used in the Newman

criterion normalized by rotor-induced velocity athover

z vertical velocity used in the Newman criterionnormalized by rotor-induced velocity at hover

WTV wake transport velocity used in the Newmancriterion normalized by rotor-induced velocityat hover

WTVE effective wake transport velocity used in theNewman criterion normalized by rotor-inducedvelocity at hover

rotor rotational speed, bank and pitch angles

Introduction

Over certain ranges of forward speed and descent rate, a helicopterrotor may encounter its own wake, resulting in a doughnut-shaped ring

139

140 P.-M. BASSET JOURNAL OF THE AMERICAN HELICOPTER SOCIETY

around the rotor disk, known as the vortex ring state (VRS). Flight in VRScondition can be dangerous as it may cause an uncommanded increasein descent rate, excessive thrust and torque fluctuations, vibration, andloss of control effectiveness. As simple momentum theory is no longervalid for a rotor in VRS, modeling of rotor inflow in the VRS contin-ues to daunt researchers. While routine operations of a helicopter in theVRS are restricted, a better understanding of the VRS problem and theability to accurately predict the VRS boundaries provide certain advan-tages. For example, a detailed understanding of the VRS problem maylead to the development of an automated system for VRS avoidance.Also, the ability to accurately predict the VRS boundaries may lead to abetter utilization of the safe operational envelope to facilitate significantnoise abatement, e.g., through segmented steep approaches for civilianhelicopters.

Typical aerodynamic phenomena associated with VRS include un-steadiness of flow, excessive thrust and torque fluctuations, and a signif-icant increase in vibration. Castles and Gray (Ref. 1) measured inducedvelocity for multiple rotor configurations in VRS and observed consider-able fluctuation in the distribution of induced velocity. Yaggy and Mort(Ref. 2) studied the steady and oscillating rotor thrust in descending flight.Their test results showed a loss in steady thrust and large oscillations ofrotor thrust in the VRS. Washizu et al. (Ref. 3) conducted experimentsto measure unsteady aerodynamic characteristics of a rotor operatingin the VRS. Empey and Ormiston (Ref. 4) tested a 1/8-scale model ofthe AH-1G helicopter in a wind tunnel. Their data revealed significantthrust oscillations in descent conditions. Wang (Ref. 5) applied the clas-sical vortex theory in axial descent with a linear decay of circulationof trailing vortices owing to the effect of fluid viscosity and interactionof induced flow with opposite free-stream flow. The resultant inducedvelocity curve from Wangs model matched well with the experimentaldata from Ref. 1. Xin and Gao (Refs. 6, 7) conducted whirling beamtests in both axial and inclined descent conditions. Remarkable fluctu-ations in both rotor thrust and torque were observed, especially in theregion of VZ/Vih = 0.6 to 0.8 (VZ/Vih is the rotor vertical speednormalized by the induced velocity at hover). The loss in the mean rotorthrust was also clearly indicated in the same region. Leishman et al. (Ref.8) applied a time-accurate free-vortex wake scheme in their study. Theonset and development of the VRS was viewed as the result of spatialand temporal wake instability. Brown et al. (Ref. 9) developed a vortic-ity transport model in their study of blade twist effects on a rotor in theVRS.

Dynamic behaviors of a single-rotor helicopter operating in the VRScondition include uncommanded drop in descent rate and loss of controleffectiveness. Basset and Prasad (Ref. 10) explored the use of nonlinearanalysis tools such as the bifurcation method in gaining an understand-ing of the flow dynamics associated with the VRS. In the investigationsreported in Refs. 11 and 12, a flight test campaign dedicated to the studyof VRS was conducted using a single main-rotor helicopter. The test pro-gram identified the main flight characteristics in the VRS condition as asudden increase in descent rate. Similar to the finding from Drees andHendal (Ref. 13), the VRS effects were not observed beyond forwardspeeds of VH /Vih = 1.0 (VH /Vih is the helicopter forward speed normal-ized by the rotor-induced velocity at hover). Brand et al. (Refs. 14, 15)conducted an extensive flight test program to evaluate VRS effects onthe V-22 tilt-rotor aircraft. While degradation of control effectiveness inthe vertical axis was often present for a single main-rotor configuration,uncommanded roll response was more obvious in VRS encounters for atilt-rotor configuration.

Momentum theory has been successfully used for rotor inflow mod-eling in hover, climb, and even forward flight conditions. However, mo-mentum theory breaks down in descending flight due to the collapse ofa smooth slipstream. As pointed out in Ref. 16, the region may begin

at a rate of descent equal to about 23% of the hover-induced velocityand persist until the rate of descent exceeds 125% of the hover-inducedvelocity. Nevertheless, researchers have developed various methods inextending simple momentum theory for descending flight. One of theearliest efforts can be traced back to Glauert (Ref. 17). Recent attemptshave come from He et al. (Ref. 18) and Jimenez (Ref. 19). They indi-vidually formulated parametric extensions of momentum theory in theflow model to remove the modeling singularity in VRS and renderedsimulation models covering the full range of flight conditions. Perhapsthe most comprehensive parametric extension of momentum theory hasbeen Johnson (Ref. 20). On the basis of a broad review of availablewind-tunnel and flight test data for rotors in the VRS, a model was de-veloped suitable for simple calculations and for real-time simulations.Using the developed VRS model, Ref. 20 showed negative (unstable)heave damping for a certain range of descent rates, resulting in the VRSboundary being defined in terms of the stability boundary of the vehicleflight dynamics. It is to be noted that models based on parametric ex-tension of momentum theory need to further establish direct dependenceon critical rotor parameters such as rotor solidity and blade twist, andso on.

VRS Boundaries

A number of criteria for arriving at VRS boundaries have been pro-posed over the years. These criteria include a region of roughness (Ref.13), thrust or torque fluctuation (Refs. 3, 6, 7), mean thrust reduction(Ref. 21), zero velocity of tip vortices (Ref. 22), blade-flapping fluctua-tion (Ref. 8), wake breakdown (Ref. 23), heave stability (Ref. 12), androll stability (Refs. 14, 15).

The most common way to display a VRS boundary is to use free-stream velocity components, VH and VZ , normalized by hover-inducedvelocity Vih. The area within the VRS boundary indicates the regionwhere the effects of VRS are significant in terms of a chosen criterion.Another commonly used variable in the depiction of VRS boundaries isdescent angle (D) with D = 90 representing axial descent. A summaryof several VRS boundaries from previous studies is included in Fig. 1.What follows is a brief description of the various criteria that have beenused by the researchers in arriving at the VRS boundaries shown assubplots in Fig. 1.

Region of roughness

From the investigation of a rotor in the VRS (Ref. 13), Drees andHendal identified a region of roughness. In this region, the rotor behaviorwas rough with respect to attitude and control. Unexpected loss of altitudeand a large nose-down pitching motion were also observed. The regionof roughness ranged from VZ/Vih = 0.62 to VZ/Vih = 1.53 in axialdescent, extending in inclined descent to VH /Vih = 1.0.

Thrust fluctuation

Washizu et al. conducted an experiment for a rotor in descent con-ditions (Ref. 3). Unlike other wind-tunnel tests, Washizus experimentswere performed using a track system to minimize facility effects that weretypically associated with wind-tunnel experiments. During the tests, theVRS boundary based on the magnitude of T/T was derived, whereT and T are the amplitude of fluctuation and mean value of thrust,respectively. Two reference values of T/T were constructed: 0.15 and0.30. When T/T = 0.15, the corresponding boundary extended fromaxial descent to inclined descent with the forward velocity componentVH /Vih < 1. When T/T = 0.30, the corresponding boundary mainly

APRIL 2008 PREDICTION OF VORTEX RING STATE BOUNDARY OF A HELICOPTER IN DESCENDING FLIGHT 141

0 0.5 1 1.52.5

2

1.5

1

0.5

0

V Z/V

ih

VH /Vih

Drees

0 0.5 1 1.52.5

2

1.5

1

0.5

0

V Z/V

ih

VH /Vih

Washizu (T/T = 0.15)Washizu (T/T = 0.30)

0 0.5 1 1.52.5

2

1.5

1

0.5

0

V Z/V

ih

VH /Vih

Wolkovitch upper limitWolkovitch lower limit

0 0.5 1 1.52.5

2

1.5

1

0.5

0

V Z/V

ih

VH /Vih

Peters

0 0.5 1 1.52.5

2

1.5

1

0.5

0

V Z/V

ih

VH /Vih

Xin and Gao

0 0.5 1 1.52.5

2

1.5

1

0.5

0

V Z/V

ih

VH /Vih

Betzina (CT /CTo < 1)Betzina (0.6 < CT/CTo < 0.7)

0 0.5 1 1.52.5

2

1.5

1

0.5

0

VV

Z/

ih

VH /Vih

Leishman, 10%Leishman, 12%Leishman, 15%Leishman, 20%

0 0.5 1 1.52.5

2

1.5

1

0.5

0

V Z/V

ih

VH /Vih

French Flight tests: Vz dropFluctuations increaseVz stabilizationONERA VRS model

0 0.5 1 1.52.5

2

1.5

1

0.5

0

V Z/V

ihVH /Vih

Thrust fluctuations < 2.5%Thrust fluctuations > 2.5%AsymmetriesV-22 roll-offNewman

Fig. 1. A summary of VRS boundaries from the literature.

covered the inclined descent region with the descent angle ranging from45 to 80.

Wolkovitch criterion

To predict the VRS boundary in descent condition analytically,Wolkovitch (Ref. 22) considered a flow model consisting of a slipstreamwith uniform flow at any section of the rotor surrounded by a protectivetube of vorticity. The tube was made up of tip vortices leaving the ro-tor. It was assumed that unsteady vortex ring flow was associated witha breakdown of this protective tube of vorticity. Consequently, VRS on-set occurred when the net velocity of tip vortices became zero. With thevertical convection speed of the tip vortices as the mean value betweenthe airspeed outside the wake (VZ ) and inside the wake (VZ + Vim), thecondition for VRS entry corresponded to

VZ = Vim2 (1)

As for VRS departure, Wolkovitch utilized a coefficient kz to take intoaccount the distance above the rotor where the vorticity would accumu-late, corresponding to

VZ = kz Vim2 , 1 kz 2 (2)

A value of 1.4 was used for kz by Wolkovitch. According to Ref. 12,Wolkovitchs boundaries were close to those of experimental data at lowforward speed. Nevertheless, the boundaries were not consistent withother experiments as VRS could be encountered at any forward speed.However, Wolkovitchs ideas on critical vortex transport speed were laterextended by other researchers (e.g., Ref. 23).

Peters and Chen modified Wolkovitchs method by removing severalinconsistencies in the flow model and taking into account the wake skewangle (Ref. 24). After modification, Peters boundary showed no VRSfor VH /Vih > 0.62 and predicted VRS over a wider range of VZ/Vih.

Torque fluctuation

Xin and Gao observed an irregular variation of the rotor torque at aboutVZ/Vih = 0.28 (Refs. 6, 7). Torque fluctuations were more severe forD = 60 and D = 75 than in axial descent. They further found that asthe descent angle decreased, the torque fluctuations also decreased andfinally disappeared below D = 40.

On the basis of observations from the tests, Xin and Gao pointed outthat there were three problems associated with Peters VRS boundary.First, Peters boundary showed that the rotor entered the VRS even forsmall descent rates. Second, no occurrence of VRS existed for VH /Vih >0.62. Third, VRS occurred at every descent angle. Xin and Gao thusproposed an improved VRS boundary as shown in the subplot of Fig. 1.This new boundary was more consistent with data from flight and modeltests.

Mean thrust reduction

Betzina observed a significant mean thrust reduction in VRS (Ref. 21).As a result, the thrust ratio CT /CTo was used as an indicator, where CTand CTo represented mean thrust coefficient and the thrust coefficient athover, respectively. It was shown that the lowest thrust ratio centered ona descent angle of about D = 75 and VH /Vih = 0.3, and extended fromD = 60 to D = 90.


Blade-flapping fluctuation

Recently, a free-vortex wake method has been developed by Leishmanet al. (Ref. 8), which shows that in addition to thrust and torque fluctuation,blade-flapping fluctuation may also be a concern as a result of unsteadyairloads found near or in the VRS. An excessive blade-flapping angle(greater than 10% of the mean) may lead to piloting difficulties. As such,contours of excessive blade-flapping fluctuation were used as indicationof VRS onset.

Newman criterion

Newman developed a wake transport criterion for VRS assessment(Ref. 23) and defined an effective wake transport velocity, WTVE, asfollows:

WTVE =

k22x + (z + i )2 (3)where k represents the relative effectiveness of the in-plane velocity com-ponent. A critical value of WTVE existed indicating an onset of flowbreakdown in the wake stream tube, denoted as WTVECRIT. The true waketransport velocity at this critical condition was expressed with respect tothe critical effective velocity:

W T V =

2WTVECRIT + (1 k2)2x (4)

On the other hand, the mean induced velocity i was represented as

i = 1WTV

(5)

Thus, the boundary for the onset of flow breakdown was given by

z =

2WTVECRIT k22x i (6)On the basis of experimental observations from Drees and Hendal (Ref.13), Newman selected the values of the empirical factors as k = 0.65and WTVECRIT = 0.74.

Heave stability

During a recent ONERA flight test, several behaviors were observed,especially in the VRS region (Ref. 12). During the phase leading to VRS,the crew would first feel an increased level of vibration, followed by asudden increase in the rate of descent. Increasing the collective would notprevent the helicopter from a further increase in its descent rate. Duringthe descent, the helicopter was very unstable and was difficult to control.The flight tests also established that the VRS effects disappeared beyondcertain forward speed. The final VRS boundary was determined basedon the following criteria: (1) an increased level of vibration, (2) a suddenincrease in the descent rate, and (3) exiting from VRS due to stabilizationof the descent rate.

Roll stability

The goal of the testing of the V-22 tilt-rotor aircraft was to determinethe criteria for a quasi-steady state VRS boundary (Refs. 14, 15). With anincrease in descent rate from hover, the test team first observed an increaseof thrust fluctuations was first observed. As the situation degraded, thepilots experienced an uncommanded roll response, which defined theVRS boundary. Figure 1 shows Newmans VRS boundary superimposedon the V-22 test data, indicating that the VRS boundaries for tilt-rotoraircraft and for conventional helicopters were remarkably similar.

Rotor Inflow Modeling

As simple momentum theory is no longer valid for a rotor in VRS,modeling of rotor inflow in VRS becomes very challenging. Followingis a discussion of two different rotor inflow models: the ONERA inflowmodel and the Georgia Tech ring vortex model.

ONERA inflow model

ONERA developed an inflow model (Ref. 19) based on experimentaldata, mainly the Dauphin 365N flight test data (e.g., Ref. 11), as thishelicopter has been the one used for research at the French flight testcenter in Istres. The purpose of this model was to provide an extensionof the momentum theory that could be applied in steep approaches, evenin case of VRS. The approach for building the model was first arrivedat for axial descent using the wind-tunnel data of Ref. 1 as sketched inFig. 2. The model was then further developed by using the Dauphin flighttest data. A first harmonic inflow model was built such that the power orcollective curves obtained by trim computations in descending flights atdifferent forward speeds matched the ones measured in flight tests. Thefirst harmonic inflow gradients were based on the Meijer-Drees modelwith a modification of the longitudinal inflow gradient (VicMR) in order toavoid any discontinuity when the mean inflow became zero (during thechange between the helicopter and windmill modes).

The working steps of the ONERA inflow model are as follows: first,at each forward speed below a critical advance ratio, an interpolationis performed between the helicopter and windmill branches (for higherforward speeds, the momentum theory can still be applied). From thesetwo branches, two points are identified (e.g., points 1 and 2 as shown inFig. 2). At each of the two points, two conditions are imposed to ensurecontinuity and differentiability of the function Vim = f (VZ ). These fourconditions enable the calculation of the four coefficients of a third-orderinterpolating polynomial. Second, the mean inflow given by the momen-tum theory is augmented by an extra term of induced flow. Indeed, mostof the wind-tunnel data (e.g., Ref. 1) show that the mean inflow is strongerin descending flight than the inflow predicted from momentum theory.During descent, tip vortices merge in a vortex ring at the periphery of the

Fig. 2. Rotor-induced velocity variation with descent rate from theONERA inflow model (example of comparison in vertical descentwith experimental data from Ref. 1).


rotor, create air recirculation, and thus increase the induced flow. Thisextra induced flow is represented in the model by a transition functionwith three main parameters: a maximum magnitude term (A0), a verticalspeed (Vz0) at which the magnitude reaches its maximum (A0), and anexponent on which depends the curvature of the added nonlinear func-tion (Refs. 19, 25). Both A0 and Vz0 depend on forward speed. Theseparameters are determined to match the flight test data. Moreover, themodel is formulated using nondimensional variables. Thus, the modelcorresponds, in principle, to a generic two- to four-ton class helicopter.

One advantage of using flight test data is to avoid the uncertainties ofwind-tunnel measurements regarding the proximity of the tunnel wallswhich may interact with the large airflow recirculating around the rotor inthe VRS (as is questionable in Ref. 1). However, any kind of experimentaltest has its drawbacks. In flight tests, for example, one main difficulty isto assess the induced flow through the rotor. One method is to deducemean inflow from the measured main rotor power after subtracting fromit an estimate of profile power. Another method is to make use of thein-flight airflow measurements below the rotor as described in Ref. 12.

Ring vortex model

One drawback of momentum theory is its inability to account forthe interaction between the rotor wake and the surrounding airflow in adescending condition (Ref. 26). The effects of this interaction may be lesssignificant at hover or in climb. Nevertheless, as a helicopter increasesits descent rate, the interaction becomes more and more intense due tolarger velocity gradients between the upflow outside the wake and thedownflow inside the wake. To take these flow interactions into account,several of the current authors proposed a new inflow model at the GeorgiaInstitute of Technology (Refs. 2729). The so-called ring vortex modelsupposes that, due to the flow interaction, there exists a series of vortexrings located at the rotor periphery. These vortex rings move downwardalong the wake when the helicopter descends at a low rate. As the rateof descent increases, the vortex rings tend to accumulate near the rotortip. As the rate of descent increases, the vortex rings move upward alongthe wake. A new vortex ring is formed with every blade rotation, i.e.,2 /Nb second. Therefore, the locations of these discrete vortex ringscan be determined by the product of convection velocity of the vortexrings and 2m/Nb (where m is an integer representing the numberingof vortex rings).

Each vortex ring induces additional normal velocity at the rotor disk.The flow field of a vortex ring can be computed based on elliptic integrals(Ref. 30). One advantage of utilizing vortex rings is that the effect ofvortex rings is nonuniform with respect to relative distance between therings and the rotor disk. The closer a vortex ring is to the rotor disk,the larger the magnitude of normal velocity at the disk. The resultingnonuniform effect conforms to what has been observed in test data. Also,using a random number of vortex rings, it is possible to capture the scatterin induced velocity measurements observed in experiment.

In the ring vortex model, downward velocity due to vortex rings isintegrated into the induced velocity calculated by momentum theory.The concept works well in the VRS and windmill phases. Nevertheless,in axial descent and inclined descent at low forward speed (VH /Vih upto 0.6204), momentum theory fails to predict a transition phase betweenthe helicopter and the windmill branches. To address this problem, thesimple momentum theory equation for induced velocity is modified to

(

2.72(1 + 2))2

+ 2 + ( + )2 = 1 (7)

where , , and are the normalized values (normalized by Vih) ofdescent rate, induced velocity, and forward speed, respectively. The ad-

2.5 2 1.5 1 0.5 0 0.50

0.5

1

1.5

2

2.5

3

3.5

VZ /Vih

V i/V

ih

Test, VH /Vih = 0.0Test, VH /Vih = 0.23Test, VH /Vih = 0.69Test, VH /Vih = 0.92Test, VH /Vih = 1.16Test, VH /Vih = 1.85RVM, VH /Vih = 0.0RVM, VH /Vih = 0.23RVM, VH /Vih = 0.69RVM, VH /Vih = 0.92RVM, VH /Vih = 1.16RVM, VH /Vih = 1.85

Fig. 3. Normalized inflow curves calculated with the ring vortexmodel for a generic helicopter (test data source obtained from Ref.20).

ditional term ( 2.72(1+2) )2, which is analogous to the parachute drag term,modifies equilibrium curves for inflow dynamics, creating a steady-statetransition between helicopter and windmill branches in axial and steepdescents. Its effect diminishes at other flight conditions.

To demonstrate the effectiveness of the ring vortex model, normal-ized inflow curves are calculated for a generic helicopter in Fig. 3 (testdata source from Ref. 20). Note that the forward speeds are kept con-stant during the test with Vx/Vh at 0.0, 0.23, 0.69, 0.92, 1.16, and 1.85.Generally, results from the ring vortex model agree with the experi-mental data. At high forward speed (Vx/Vh = 0.92, 1.16, 1.85), pre-dicted inflow curves from the ring vortex model tend to get closer withsimple momentum theory. In these conditions, vortex rings are furtherswept away from the rotor disk and their influence on rotor inflow isdiminished.

Methodology

To identify by simulation the sensitive parameters for the VRSphenomenon and their associated critical values (which may lead toan entry into this regime), two complementary approaches can beenvisaged:

1) Equilibrium computations can be conducted to reproduce the con-ditions of steady descending flight by sweeping a large set of velocitycouples (VH , VZ ) and by changing the value of one parameter, such asgross weight or other flight conditions.

2) Time simulations of different types of descending flight (constantslope, segmented, or decelerated approaches) can be performed with arotorcraft flight dynamics model coupled with an automatic pilot module.

The present paper focuses on the first approach (equilibrium computa-tion). The second one (time simulations) is addressed in Ref. 31. Indeed,the objective of this paper is to compare three methods and associated cri-teria for the prediction of VRS boundaries in steady flight condition withan unaugmented helicopter, i.e., without an automatic pilot or a stabilityaugmentation system.


The three methods discussed in detail hereafter are as follows:1) Point-by-point method: a NewtonRaphson algorithm associated

with tip vortices criterion;2) Bifurcation method: a continuation algorithm associated with bi-

furcation criterion;3) Heave stability method: a NewtonRaphson algorithm associated

with heave stability criterion.

Point-by-point method

The helicopter overall simulation tool (HOST code created by Euro-copter, Ref. 32) makes use of the NewtonRaphson algorithm for trimcomputation similar to most rotorcraft flight dynamics codes. The six-degree-of-freedom rigid body dynamics are driven to steady state in aniterative process searching for the values of the four controls (collective,longitudinal cyclic, and lateral cyclic controls of the main rotor and col-lective control of the tail rotor), as well as two attitude angles (pitch androll angles). With this method, the trim is calculated point by point foreach flight condition.

With the HOST code, a series of trim sweeps are conducted for eachvalue of the studied parameters:

1) sweep of forward speed VH from 0 to 50 km/h with a step of 5 km/h,and

2) for each forward speed, a sweep of descent rate VZ from 0 to25 m/s with a step of 1 m/s.

Each series of trim sweeps contains 286 trim calculation points. Eachof these points corresponds to a steady descending flight condition witha slope given by the (VH , VZ ) couple. At each equilibrium point, the tipvortices criterion is used to determine whether this flight condition mayrisk a VRS onset.

The tip vortices criterion is an extension of the Wolkovitch criterion(Ref. 22). As mentioned earlier, the basic principle is that the rotor mayencounter a VRS situation when the convection speed of the main rotorblade tip vortices is too small, resulting in vorticity accumulation aroundthe rotor and air recirculation. The vertical convection speed of the tipvortices is assumed to be the mean value between the airspeed outsidethe wake (VZ ) and the airspeed inside the wake (VZ + Vim).

As pointed out earlier, the Wolkovitch criterion does not take intoaccount the effect of forward speed very well. As a result, this criterionpredicts that a VRS may occur at any forward speed (see Fig. 1). However,such an occurrence is not physically feasible because above a certainforward speed, the vortex wake is swept backward preventing the rotorfrom flying in its own wake.

The extension of the Wolkovitch criterion has already been presentedwith the consideration of wake skew angle (Refs. 12, 19). In terms of traveldistance with respect to the rotor plane, the in-plane

(

VtvX)

and normal(

VtvZ)

components of the tip vortices velocity do not have similar ef-fects. Therefore, the modified criterion has the following nondimensionalform:

VTVDES =

(

Vtvxkx Vih

)2

+(

VtvzVih

)2

(8)

where VtvX =(Vx,ROTOR)2 + (Vy,ROTOR)2 and VtvZ = VzROTOR + Vim2 .

Comparisons with experimental data (wind-tunnel and flight test data)have led to the following empirically tuned values. The weight coefficienton the in-plane component kx is set at 4, while is determined to be 0.25by giving an overall agreement with the experimental data. One note ofcaution is that the ONERA VRS model in Fig. 1 adopted a value of0.2. This parameter provides a way of bringing a certain margin withrespect to the VRS domain. Higher values of give rise to predictionsof wider VRS boundaries. The core VRS area (in which the aircraft

will encounter the VRS) can be assessed by using = 0.2. A wider riskzone can be obtained by using = 0.3, for which all the test points areinside the predicted domain. In the rest of the paper, is set to 0.25 as itis the value for which the limits given by this criterion pass through theflight test points and may be considered as the VRS onset boundaries

Bifurcation method

The principle of this method is presented in Ref. 25. Here, only themain concepts are restated. In the study of nonlinear system dynamics,the well-suited continuation algorithm (described, e.g., in Ref. 33) en-ables the determination of an equilibrium curve even in the presence offolds (vertical tangents), which represent multiple equilibrium points forthe same set of parameters. Such complex patterns may occur dependingon the nonlinear terms within the system dynamics. Rotorcraft flight dy-namics is well known for involving many nonlinear effects, for example,the complex aerodynamics of rotor blades, its interactions with the othercomponents (airframe, tail rotor, etc.), inertial coupling, and so on. How-ever, the application of nonlinear analysis to rotorcraft flight dynamicsremains rare (Refs. 10, 25, 3437).

The numerical implementation of the continuation algorithm inASDOBI (an ONERA program developed to study fixed-wing flight dy-namics, Refs. 38, 39) is based on the repetition of four steps (see Fig. 4):Step 1: finding a point on the equilibrium curve,Step 2: predicting the direction of the tangent,Step 3: predicting the next point, andStep 4: correcting the predicted point.

Step 3 is based on an AdamsBashforth-like integration method,whereas steps 1 and 4 are based on the NewtonRaphson scheme adaptedto the implicit problem of n equations with (n + 1) variables. Here,n is the number of states, and the (n + 1)th variable is a parameteror control, here designated by u. As can be seen, the continuation al-gorithm is more advanced compared with the basic NewtonRaphsonalgorithm.

By denoting X as the state vector,.

X as its time derivative, U as thecontrol vector,

.

X = F(X, U ) as the set of differential equations, and(X0, U0) as an equilibrium solution, the stability property of each equi-librium point (X0, U0) can be characterized by calculating the Jacobianmatrix of the system dynamics:

DX F(X0, U0) [

Fi (X0, U0)x j

]

(9)

If one eigenvalue of the Jacobian matrix has a positive real part, thesystem is unstable at this equilibrium point. In both Figs. 4 and 5, theupper and lower parts of the equilibrium curve are stable, whereas themiddle part within the fold is unstable.

Fig. 4. Continuation algorithm for calculating an equilibrium curve.


Fig. 5. Calculation of bifurcation locus.

Bifurcation points are special equilibrium points that indicate a changeof system stability. On the equilibrium curve of both Figs. 4 and 5, theblack points between the stable and unstable parts are called turningpoints. These points, also known as saddle nodes, belong to the family ofthe real bifurcations associated with the nullification of a real eigenvalue.The mathematical translation of this property involves adding one extraequation to the n equations of the equilibrium problem. The equilibriumpoint (X0, U0) is a real bifurcation (i.e., turning point), if the Jacobianmatrix has at least one zero eigenvalue. Thus, the extra equation, forcharacterizing the real bifurcation points, is set by writing that the deter-minant of the Jacobian matrix is null. Hence, the bifurcation locus can befound if the continuation algorithm is applied to this augmented systemof (n + 1) equations by considering two control parameters (u1, u2) toset implicit problem ((n + 1) equations for (n + 2) variables); see Fig. 5.

As described in Ref. 25, the rotorcraft flight dynamics simulationcode HOST was coupled with the ONERA ASDOBI nonlinear analysistool. The inflow gradients VicMR and VisMR of the first harmonic inflowmodel are always included (in Ref. 25 and the present work). In the firstapplication of this method to the study of flight dynamics in descendingflight (Ref. 25), the first level of the rotor models available in HOST wasused, i.e., an analytical rotor disk model with quasi-steady blade-flappingdynamics. Later, second-order blade flap and lead-lag dynamics wereconsidered by using the blade element model. As usual, the flapping ()and lead-lag ( ) motions were described up to the first harmonics leadingto 12 more states:

X = {Uhel, Vhel, Whel, Phel, Qhel, Rhel, , ,VimMR, VicMR, VisMR, Vim F AN , (10)0, 1c, 1s, 0, 1c, 1s, 0, 1c, 1s, 0, 1c, 1s}

The effect of the rotor model on the bifurcation loci is shown in Fig.6. This is for a four-bladed generic helicopter with a gross weight of3500 kg. The curves show the flight conditions for which there wouldbe a change of stability in descending flight. The bifurcation loci withcontinuous and dotted lines were calculated with the analytical rotor diskmodel (a continuous line for quasi-steady and a dotted line for first-orderflapping dynamics). These two results are nearly the same as using ablade element model (shown as a dashed line in Fig. 6), except that it isseen that the type of rotor model (either a rotor disk model or a bladeelement model) plays a role in the prediction of the values of the trimvariables, i.e., trim control and attitude angles. However, the bifurcation

loci are nearly identical in the classical (VH , VZ ) diagram shown as thefirst subplot in Fig. 6.

Our first assumption seems, therefore, to be valid: the flapping andlead-lag dynamics do not play a significant role in the VRS boundaries inthe (VH , VZ ) diagram, at least in the case of the helicopter studied. More-over, the computational time required to calculate the bifurcation locuswith the blade element model is excessive compared with the analyticalrotor model. Hence, for the subsequent HOST and ASDOBI bifurcationcomputations, the analytical rotor model with quasi-steady flapping wasselected as in Ref. 25.

Heave stability method

From the inflow velocity results in Figs. 2 and 3, it is noted thatthere is not only an increase in the magnitude of induced velocity overthe prediction from momentum theory but also a steeper and varyinggradient of inflow curve.

According to Ref. 27, the derivative of the inflow curve from momen-tum theory can be obtained as follows:

dd

= 12

+ 2

2 + 4 (11)

In the range from = 0 to = 1.5, the absolute value of dd frommomentum theory is always less than 1. This indicates that with momen-tum theory, a change in descent rate is associated with || < ||,whereas with either the ONERA inflow model or the ring vortex model,the result is || > || (for values of roughly in the range of 0.5to 1.5).

The overall effect of a decrease in descent rate ( > 0) on blade an-gle of attack () and hence on heave damping (T/) with momentumtheory is

[

(

1 + dd

)]

< 0 T/ < 0 (12)

Here, the symbol stands for proportional to. With either the ONERAinflow model or the ring vortex model, the overall effect is

[

(

1 + dd

)]

> 0 T/ > 0 (13)

Thus for values of roughly in the range of0.5 to1.5, with momentumtheory, a decrease in descent rate results in a decrease in rotor thrust.This is the case where the vehicle vertical dynamics has stable heavedamping, i.e., T/ < 0. However, with the ONERA inflow model orthe ring vortex model, an increase in descent rate from a value of in theabove-mentioned range gives rise to a decrease in rotor thrust, resultingin unstable heave damping, i.e., T/ > 0.

Results and Discussion

ONERA (Ref. 12) conducted flight tests of a helicopter in steep de-scents. The 11.93-m-diameter rotor has a solidity of 0.083 and a 10.2blade twist.

VRS boundaries from tip vortices and bifurcation criteriaobtained by ONERA

For the prediction of VRS boundaries based on the tip vortices crite-rion and the bifurcation criterion, the examples of calculations presentedhere by ONERA are for the cases of the helicopter in steady descendingflight at sea level (in International Standard Atmosphere condition) for


Quasi-steady blade flappingFlapping velocityFlap and lead-lag dynamics

0 10 20 3016

12

8

4

VH km/ h

VZ m/ s

0 10 20 303

4

5

6

7 DT0 DEG Collective pitch

0 10 20 30

0

1

2

DTC DEG Lateral cyclic pitch

0 10 20 300.81.21.6

22.4

DTS DEG Longitudinal cyclic pitch

0 10 20 302224262830 DTA DEG Tail rotor collective pitc h

0 10 20 301

2

3

4

5 DEG Bank angle

0 10 20 30

1.6

2

2.4 DEG Pitch angle

0 10 20 30

0

4

8UHEL M/S Longitudinal helicopter speed

0 10 20 30

0.4

0.8

1.2

VHEL M/S Lateral helicopter speed

0 10 20 304

8

12

16 WHEL M/S Vertical helicopter speed

0 10 20 3016

20

24

28VIM-MR M/S Main rotor mean induced flo w

0 10 20 3036

40

44

48VIM-FAN M/S Tail rotor mean induced flow

VRS domain Blade element modelAnalytical rotor disk model

VH km/h

VH km/h

VH km/h

VH km/h

VH km/h

VH km/h

VH km/h

VH km/h

VH km/h

VH km/h

VH km/h

Fig. 6. Comparison of the bifurcation loci calculated with different rotor models.

0 0.5 1 1.51.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

VH /Vih

V z/V

ih

Max. slope,(M = 2850 kg): 45.07Max. slope,(M = 3850 kg): 43.58

Max. slope,(M = 2850 kg): 27.22Max. slope,(M = 3850 kg): 27.22

Tip vortices criterion ( = 0.25)

Bifurcation criterion

+: Flight tests

Fig. 7. Comparison of the VRS boundaries computed with tip vorticesand bifurcation criteria.

two different gross weights (GW): 2850 and 3850 kg. The results areshown in Fig. 7, together with the flight test data from Refs. 12 and 19.

According to the model, increasing gross weight has the same globaleffect on the VRS domain as increasing altitude or temperature. Indeed

all these variations lead to an increase of the mean induced flow and thusalso to the increase of the critical descent rate for a VRS onset. However,working with nondimensional parameters, i.e., dividing the speeds (VH ,VZ ) by the theoretical mean inflow in hover (Vih) (which is related to thethrust coefficient CT ), tends to make the VRS domain less dependent ongross weight, altitude, and temperature. This comes from the fact that Vihand CT vary with gross weight, altitude, and temperature:

Vih =

GW2S

(14)

The discrepancy at the lower part of the VRS domain (high descentrates and low forward speeds) in Fig. 7 is explained in Ref. 25. Duringthe flight tests, the pilot identified the exit of the VRS in terms of sta-bilization of descent rate in the windmill mode following the drop fromthe helicopter mode. As sketched in Fig. 8, the folds of the equilibriumcurves at the bifurcations from the helicopter mode (circle in Fig. 8)correspond naturally to higher descent rates than those at the actual bi-furcations from the windmill mode (square in Fig. 8). HOSTASDOBIresults highlight that from the windmill branch the bifurcation towardthe helicopter branch occurs for lower descent rates due to the hysteresiseffect. On the other hand, the tip vortices criterion is adjusted by tuningcoefficients of the Wolkovitch criterion based on the flight test data. As a


Fig. 8. Why the bifurcation criterion predicts lower descent rates for the lower limit of the VRS boundary.

result, the boundary from the tip vortices criterion envelops these pointsat the lower boundary.

Another discrepancy in Fig. 7 concerns the upper limit. For forwardspeeds (VH /Vih) approximately between 0.5 and 1, the VRS boundaryfrom the tip vortices criterion occupies higher portion of the figure overthe one from the bifurcation criterion. This difference plays a prominentrole in the determination of the predicted maximum descent slope. Itshould be noted that the maximum slopes in Fig. 7 are large, but couldbe lower in practice in a tail-wind condition.

With the tip vortices criterion, the descent ratio (VZ /Vih) for a potentialVRS onset becomes lower in absolute value when the forward speedincreases. This tendency comes from the Wolkovitch criterion, the basisfor the tip vortices criterion (see the graph at the right upper corner ofFig. 1).

In the applied tip vortices criterion, a condition is added onto thein-plane velocity component (VtvX ) such that a VRS onset is no longerpredicted above a certain forward speed. But the vertical component(VtvZ ) of the tip vortices velocity still depends on the half value of themean inflow (Vim/2), as mentioned in the discussion of the Wolkovitchcriterion. In the model, the mean inflow decreases with forward speed.Hence, the critical value of the descent rate for a VRS onset becomeslower in absolute value when forward speed increases. More precisely,the minimum of the tip vortices criterion (VTVDES in Eq. (8)) isreached at a lower descent rate when forward speed is increased (Fig. 9).The derivative of this criterion with respect to the descent rate is nullwhen

d (VTVDES )dVz

= 0 if dVimdVz

= 2 or Vim = 2Vz (15)

It can be seen in Fig. 9 that the minima of the tip vortices criterion arereached when Vim = 2Vz . Besides, when forward speed increases, themean rotor inflow (Vim) is lower and increases more slowly with descentrate. Therefore, the inflow condition (Vim = 2Vz) is obtained by themodel at a lower descent rate when forward speed increases. However,this effect of forward speed on the variation of the minimum of thecriterion is reduced or truncated by using an parameter: the VRS ispredicted not at the minimum of VTVDES, but when its value becomeslower than (see Fig. 9, VRS-MR = 1 if VTVDES < ). This is why theupper limit of the VRS domain given by the tip vortices criterion remainsmore or less horizontal. The corresponding critical descent rate for the

VRS entry may be slightly less (Fig. 7), mainly because of the decreaseof the mean inflow with forward speed as mentioned before.

The bifurcation criterion predicts the contrary: the higher the forwardspeed, the higher the critical descent rate (from which a VRS may occur).This prediction is not as easy to explain because it relies on a numericalcriterion rather than on an explicit analytical criterion such as the tipvortices criterion. The bifurcation criterion is based on the change instability of the trim solutions, determined by computing the eigenvaluesof the Jacobian matrix of the helicopter flight dynamics. The discrepancyon the upper limit may be explained by looking at the turning points ofthe function Vz = f (DT 0) (as done for the lower limit differences inFig. 8). The bifurcation from the helicopter branch also corresponds tothe minimum of the required collective for trimming the helicopter whenthe collective is decreased from a hover or level flight condition, (see DT 0= f (Vz) in Fig. 9). When forward speed increases, this turning point ofthe Vz = f (DT 0) function or local minimum of the DT 0 = f (Vz) functionoccurs at a higher descent rate (see Fig. 9, the differences between thetrim curves for VH : 5, 20, and 30 km/h). The trim collective follows thesimilar variation trend as the main rotor power (power coefficient CP-MRin Fig. 9) and the average axial rotor inflow (Vim + Vz). As indicated inRef. 25, because this function reaches the minimum in question when

d (Vim + Vz)dVz

= 0 if dVimdVz

= 1 (16)

the bifurcation criterion provides similar results in descending flightsas the heave stability criterion, the dominant dynamics of such flightconditions.

In both cases, tip vortices and bifurcation criteria, the ONERA inflowmodel is utilized. Yet the coefficients of the tip vortices criterion (kx ,) are tuned to cover the VRS domain identified in the flight test data,whereas the bifurcation criterion reflects the stability of the helicopterflight dynamics model.

To illustrate this fundamental difference between the two criteria, timesimulations were performed from three different descending flight initialconditions, as shown in Figs. 10 and 11.

At the beginning of the time simulations with the HOST code, aminor decrease of the collective is applied from its trim value. From theinitial approach conditions (a) and (c) (Fig. 10), the minor perturbationon the collective eventually leads to a large increase of the descent rate,symptomatic of a VRS onset (Fig. 11). In case (a), a longer time is


Vh=5 km/h sweep on VZ Vh=20 km/h sweep on VZ Vh=30 km/h sweep on VZ

16 12 8 4 03

4

5

6

30 km/h

20 km/h5 km/h

DT0 DEG Collective pitch

16 12 8 4 00

0.2

0.4

0.6 VTVDES-MR S.U. Tip vortices nondimensional speed (Eq. (8))

16 12 8 4 06

8

10

12

14 Vim+Vz m/s Mean axial airflow through the rotor

16 12 8 4 00

0.4

0.81 in VRS

0 out of VRS

VRS-MR S.U. Tip vortices VRS criterion

16 12 8 4 0

2.4E4

2.8E4

3.2E4

3.6E4

4E4

VZ m/s

CP-MR S.U. Main rotor power coefficient

16 12 8 4 010

20

30 VIM-MR M/S Main rotor mean induced flow

VZ m/s

VZ m/s

VZ m/sVZ m/s

VZ m/s

Fig. 9. Why the bifurcation criterion predicts higher descent rates for the upper limit of the VRS boundary in forward flight.

0 5 10 15 20 25 30 35-16

-14

-12

-10

-8

-6

-4

VH Km/h

VZ m/s

(a) (b)

(c)

(a) : VZ = 6 m/s, VH = 20 km/h (b) : VZ = 6 m/s, VH = 30 km/h (c) : VZ = 10 m/s, VH = 30 km/h

Fig. 10. Initial conditions for the subsequent tests in time simulation.

needed for the VRS onset compared to case (c). This is well predictedby the bifurcation criterion considering that both points are close to thebifurcation locus surrounding the VRS domain: case (a) is just outsidethe domain, whereas case (c) is slightly inside (see Fig. 10). Thus, it islogical that more time is needed for a VRS onset for case (a) than for case(c). Variations of descent rate are also well predicted by the bifurcationlocus compared with the time simulations (larger Vz variation in case (a)than in case (c)), although as explained in Fig. 8, the helicopter stabilizes

in the windmill regime at higher descent rates than the lower part of thebifurcation locus (after a bifurcation from the helicopter mode).

For case (b), the initial approach condition is outside the bifurcationlocus, but inside the VRS domain given by the tip vortices criterion (seeFig. 7). Even with a reduction of the collective twice the value appliedfor cases (a) and (c), the helicopter remains stable and no VRS entry isobserved in time simulations.

Therefore, the bifurcation method gives a way to determine the condi-tions at which the rotorcraft would be naturally unstable (without stabilityaugmentation system or automatic pilot). More precisely, the bifurcationlocus surrounds the conditions for which the helicopter model is po-tentially unstable (high descent rate after a small collective pitch drop).The bifurcation criterion strictly reflects the change in the stability of themodel. In contrast, the tip vortices criterion may predict VRS onset evenwhen the flight dynamics predicted by the model is stable. Thus, witha perfect model, the bifurcation criterion would be preferable. But thetuning coefficients of the tip vortices criterion may allow to compensatethe weakness of the model.

VRS boundary from the flight dynamics stability criterionobtained by Georgia Tech

The VRS boundary obtained from the flight dynamics stability crite-rion was calculated separately at Georgia Tech. While the Georgia Techbasic flight dynamics model is similar to that of ONERAs helicoptermodel, the biggest difference is the adoption of the ring vortex model.


(a) Descent VZ =6 m/s - H = 0M ISA M = 3.5T VH = 20 km/h

(b) Descent VZ =6 m/s - H = 0M ISA M = 3.5T VH = 30 km/h

(c) Descent VZ =10 m/s - H = 0M ISA M = 3.5T VH = 30 km/h

0 40 804.28

4.3

4.32

4.34

t (s)

DT0 DEG

0 40 80

16

12

8

4

t (s)0 40 80

20

20.4

20.8

21.2

21.6

t (s)

0 40 80

-0.4

0

0.4

0.8

t (s)

DT0 DEG

0 40 80

12

8

4

0

t (s)0 40 8029

30

31

t (s)

0 40 803.52

3.54

3.56

3.58

t (s)

DT0 DEG

0 40 80

16

12

8

4

t (s) 0 40 8029

30

31

t (s)

VH km/hVRS onset

VRS Onset

VH km/h

VH km/hVZ m/s

VZ m/s

VZ m/s

Fig. 11. Tests in time simulation from the three initial conditions presented in Fig. 10.

Comparisons between the ONERA inflow model and the ring vortexmodel can be summarized as follows.

The ONERA inflow model is a three-state first harmonic inflow model(with the VRS effect globally taken into account for the mean inflowterm). The finite-state inflow model (Ref. 40), on the other hand, canbe selected as a baseline inflow module over momentum theory in thering vortex model. Effects from blade taper, blade twist, and forwardspeed can be investigated with the finite state inflow model. The finitestate inflow model accounts for a three-dimensional unsteady wake fora lifting rotor based on the unsteady potential flow theory (Ref. 40).Similar to the adjustment associated with momentum theory, the massflow parameter (VT ) in the finite state inflow model can be modifiedto accommodate a steady-state transition. It will be essential to includea reasonable number of inflow states for future studies. In this study,seven inflow states are chosen for both radial and azimuthal variationup to first harmonic. Radial variation accounts for the spanwise distri-bution of inflow due to blade taper and blade twist, whereas azimuthalvariation takes care of the effect of the forward velocity. In the ring vor-tex model, the actual number of vortex rings is allowed to vary within agiven range in the simulation. By doing so, it is theorized that vortex ringsmay either survive beyond a nominal age with a slightly benign aerody-namic environment at a particular moment or prematurely burst due to aslightly adverse environment at another moment. With a random numberof vortex rings, magnitudes of induced velocities are expected to scatteraround their nominal values. This provides a pattern of data fluctuation,which can be utilized to compare rotor thrust and torque distribution fromexperiments. A different number of vortex rings also leads to differentVRS boundaries. In this study, the nominal number of vortex rings ischosen as two, while three vortex rings are also considered as a worstcase.

The helicopter model used by Georgia Tech is an in-house rigid bodyflight dynamics model coded in MATLABTM. It includes three trans-lational motions as well as roll and pitch motions. The blade-flappingmotion and yaw motion of the vehicle are assumed to be quasi-steady.The aircrafts gross weight was selected at 3500 kg. Notice this grossweight is slightly different from ONERAs selection in Fig. 7 (the topgross weight being 3850 kg). The NewtonRaphson algorithm is utilizedfor trim purposes. The eigenvalue for heave dynamics is subsequentlyidentified from the linearized model.

The predicted heave mode values (positive values indicating unstableheave mode) are shown in Figs. 12 and 13. Two different numbers ofvortex rings are given in these two figures. From the damping results, VRSboundaries can be described based on stability characteristics of heavemode, as given in Fig. 14. In Fig. 14, the smaller boundary correspondsto Nring = 2, whereas the larger boundary is Nring = 3. Two other VRSboundaries corresponding to the tip vortices and the bifurcation criteriaare also provided in Fig. 14 for a gross weight of 3850 kg.

When Nring = 2, the predicted VRS boundary based on the heavemode criterion appears circling around inner points of the test data. WhenNring = 3, the corresponding VRS boundary embraces almost all the testpoints. A stability strip is created between these two boundaries. Withinthe strip, even if the heave mode is stable at one moment, it may becomeunstable at another moment due to random variation in the number ofrings associated with the unsteady nature of flow in the VRS.

In a broader view, the boundary with Nring = 3 provides an advancewarning for a potential VRS occurrence. The boundary with Nring = 2gives the most confined condition. Consequently, these two VRS bound-aries may provide certain clues to the design of a VRS detection andavoidance system, although such a system may involve more compli-cated safety issues.


1.5 1 0.5 00.2

0.1

0

0.1

VZ /Vih

Dam

ping

of h

eave

mod

e

VH /Vih = 0.0VH /V ih = 0.10VH /Vih = 0.23VH /Vih = 0.30VH /Vih = 0.40VH /Vih = 0.50VH /Vih = 0.60

Fig. 12. Heave mode damping with Nring = 2.

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 00.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

VZ /Vih

Dam

ping

of h

eave

mod

e

VH /Vih = 0.0VH /Vih = 0.23VH /Vih = 0.50VH /Vih = 0.60VH /Vih = 0.69VH /Vih = 0.80VH /Vih = 0.92

Fig. 13. Heave mode damping with Nring = 3.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

VH /Vih

V Z/V

ih

Flight test: Vz drop

Flight test: Vz stabilization

Tip vortices criterionBifurcation criterionHeave stability criterion, N

ring = 2Heave stability criterion, N

ring = 3

Fig. 14. Comparisons of VRS boundaries with tip vortices, bifurca-tion,and heave stability criteria.

Among the three VRS boundaries in Fig. 14, the boundaries withthe bifurcation criterion and heave stability criterion show similar trendswith forward speed (except at low forward speed for the lower part ofthe boundary with the bifurcation criterion (see the previous explanationin Fig. 8). Both boundaries differ only in magnitude. This is no surpriseas both boundaries are obtained from flight dynamic stability analyses,whereas the boundary with the tip vortices criterion is obtained fromaerodynamics.

Concluding Remarks

This paper provides a brief review of various criteria used in theliterature for arriving at VRS boundaries of rotors in descending flight.Three different criteria are considered to study the impact of the choiceof a specific criterion on VRS boundary predictions. The first criterioninvolves zero transport velocity of rotor tip vortices. It is rooted in therotor flow behavior, and as such requires a detailed modeling of the flowenvironment of a rotor in descending flight. The second criterion involvesbifurcation of descending flight equilibria. The third criterion involvesthe loss of heave damping. The second and third criteria are rooted inthe loss of descent rate damping, and as such they require detailed flightdynamics modeling in descending flight.

Two different rotor inflow models for descending flight previouslydeveloped at ONERA and the Georgia Institute of Technology are usedin this study. The ONERA model, arrived at from test data, providesan empirical extension of the momentum theory to descending flight.The model is incorporated into the HOST flight dynamic simulationof a generic helicopter for its use in VRS boundary predictions. TheGeorgia Tech ring vortex model provides a heuristic extension of thefinite state inflow model to descending flight by considering additionalinduced velocity due to a random number of vortex rings. The model isincorporated into a MATLABTM based in-house flight dynamic model ofa generic helicopter for its use in VRS boundary predictions.

Results obtained in this study show that the VRS boundary predic-tions are consistent with flight test data. More importantly, in additionto this consistency, it is concluded that different criteria lead to differ-ent VRS boundary predictions. The similarities and discrepancies havebeen analyzed in this paper. It is suggested that future VRS studies takeinto account the respective effect of the inflow model and of the specificcriterion used.

Acknowledgments

The second author would like to acknowledge the financial supportfrom the DSO National Laboratories, Singapore, for his graduate studyprogram at Georgia Tech.

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24Peters, D. A., and Chen, S. Y., Momentum Theory, Dynamic Inflow,and the Vortex Ring State, Journal of the American Helicopter Society,Vol. 27, (3), July 1982, pp. 1824.

25Prasad, J. V. R., Chen, C., Basset, P.-M., and Kolb, S., SimplifiedModel and Nonlinear Analysis of a Helicopter Rotor in Vortex Ring Statefor Flight Simulation, 30th European Rotorcraft Forum Proceedings,Marseilles, France, September 1416, 2004.

26Johnson, W., Helicopter Theory, Princeton University Press,Princeton, NJ, 1980, pp. 98102.

27Chen, C., Prasad, J. V. R., and Basset, P.-M., A Simplified InflowModel of a Helicopter Rotor in Vertical Descent, American HelicopterSociety 60th Annual Forum Proceedings, Baltimore, MD, June 710,2004.

28Chen, C., and Prasad, J. V. R., A Simplified Inflow Model of aHelicopter Rotor in Forward Descent, AIAA-2005-622, 43rd AIAAAerospace Sciences Meeting and Exhibit, Reno, NV, January 1013,2005.

29Chen, C., and Prasad, J. V. R., Ring Vortex Model for Descendingflights, American Helicopter Society 61st Annual Forum Proceedings,Grapevine, TX, June 13, 2005.

30Basset, P.-M., and Omari, A., A Rotor Vortex Wake Model forHelicopter Flight Mechanics and Its Application to the Prediction of thePitch-Up Phenomenon, Paper no. H08, 25th European Rotorcraft ForumProceedings, Rome, Italy, September 1416, 1999.

31Taghizad, A., and L. Binet, L., Assessment of VRS Risk duringSteep Approaches, American Helicopter Society 62nd Annual ForumProceedings, Phoenix, AZ, May 911, 2006.

32Benoit, B., Dequin, A.-M., Basset, P.-M., Gimonet, B., vonGrunhagen, W., and Kampa, K., HOST, a General Helicopter Simu-lation Tool for Germany and France, American Helicopter Society 56thAnnual Forum Proceedings, Virginia Beach, VA, May 24, 2000.

33Kubicek, M., and Marek M., Computational Methods in BifurcationTheory and Dissipative Structures, Springer-Verlag, Berlin, 1983, pp.3942.

34Sibilski, K., Bifurcation Analysis of a Helicopter Nonlinear Dynam-ics, 24th European Rotorcraft Forum Proceedings, Marseilles, France,September 1517, 1998.

35Buler, W., Sibilski, K., Winczura, Z., and Zyluk, A., NonlinearDynamics of Helicopter with Slung Load Analysis by Continuation,AIAA-2001-110, 39th AIAA Aerospace Sciences Meeting and Exhibit,Reno, NV, January 811, 2001.

36Bedford, R. G., and Lowenberg, M. H., Use of Bifurcation Analysisin the Design and Analysis of Helicopter Flight Control Systems, 29thEuropean Rotorcraft Forum Proceedings, Friedrichshafen, Germany,September 1618, 2003.

37Bedford, R. G., and Lowenberg, M. H., Bifurcation Analysis ofRotorcraft Dynamics with an Underslung Load, AIAA-2004-4947,AIAA Atmospheric Flight Mechanics Conference and Exhibit, Provi-dence, RI, August 1619, 2004.

38Guicheteau, P., Bifurcation Theory: A Tool for Nonlinear Flight Dy-namics, Philosophical Transactions: Mathematical, Physical and Engi-neering Sciences, Vol. 356, No. 1745, Nonlinear Flight Dynamics ofHigh-Performance Aircraft, Oct. 15, 1998, pp. 21812201.

39Guicheteau, P., Stability Analysis through Bifurcation Theoryand Non-Linear Flight Dynamics, Nonlinear Dynamics and Chaos,AGARD Lecture Series 191, 1993, pp. 3.13.10, 4.14.11, 5.15.13.

40He, C. J., Development and Application of a Generalized DynamicWake Theory for Lifting Rotors, Ph.D. Dissertation, Georgia Instituteof Technology, School of Aerospace Engineering, Atlanta, GA, August1989.

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