aiaa-1989-1319-451

State-Space Model for Unsteady Airfoil Behavior and Dynamic Stall

J. Gordon Leishman * Gilbert L. Crouse, J r . t

Center for Rotorcraft Education a n d Research, Depar tment of Aerospace Engineering,

University of Maryland, College Park , M D 20742

Abstract A t ime-domain m o d e l h a s been f o r m u l a t e d t o r ep re -

sent, a t an eng inee r ing level of approximat ion, t h e u n - s t e a d y l if t , d r a g and p i t ch ing m o m e n t c h a r a c t e r i s t i c ~ of a two-dimensional airfoil unde rgo ing d y n a m i c stall. The m o d e l is g iven as a set o f f i r s t o r d e r differential s t a t e equa t ions ; (1) an e igh t s t a t e l i nea r a t t a c h e d flow so lu t ion de r ived f r o m indic ia l r e sponse func t ions valid f o r compress ib le flow, (2) a t h r e e s t a t e so lu t ion f o r t h e progres- s ive non l inea r effects o f t r a i l i ng e d g e flow s e p a r a t i o n a n d , (3) a s ingle s t a t e so lu t ion f o r t h e ca t a s t roph ic l ead ing e d g e flow s e p a r a t i o n wh ich i s cha rac t e r i s t i c o f d y n a m i c stall. T h e d y n a m i c s o f e a c h p a r t o f t h e m o d e l are coup led i n s u c h a w a y to a l low progress ive t r ans i t i on b e t w e e n t h e airfoil s t a t i c s t a l l a n d t h e d y n a m i c s t a l l character is t ics . A n i m p o r t a n t f e a t u r e o f t h e m o d e l is t h a t t h e effects o f flow compress ib i l i ty a r e i nc luded a n d a s s u c h t h e m e t h o d is pa r t i cu l a r ly useful i n t h e pe r fo rmance , ae roe l a s t i c response , a n d r ea l - t ime s imu la t ion analys is o f he l i cop te r ro to r s . To va l ida t e t h e m o d e l , co r r e l a t i ons a r e p r e s e n t e d w i t h u n s t e a d y fo rce a n d m o m e n t d a t a f r o m osci l la tory p i t c h t e s t s o n NACA 0012, H H - 0 2 a n d SC-1095 airfoils.

Nomenclature Sonic velocity Coefficients of indicial functions Elements of the system state matrix Exponents of indicial functions Elements of the system output matrix Airfoil chord Chordwise force coefficient Pressure drag coefficient Pitching moment about the 1/4chord Zero lift pitching moment Normal force coefficient Maximum normal force coefficient Critical normal force coefficient Normal force curve slope Vortex lift center of pressure Separation point location Mach number Nondimensional pitch rate = c k / V Distance travelled in semi-chords = 2Vt l c Time Noncirculatory time constant = c/a Time constant for separation point movement

'Assistant Professor, Member AHS, AIAA. tGraduate Fellow, Member AHS, Student Member AIAA.

Presented as paper 89-1319 at the A I A A / A S M E / A S C E / A H S / A S C 80th Structures, Structural Dynamics and Malerials Conference, Mo- bile, Alabama, April 3-5, 1989. Copyright 01989 by J.G. Leishman and G.L. Crouse. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

Time constant for leading edge pressure response Time constant for vortex lift Time constant for vortex traverse over chord Free-stream velocity State variables Angle of attack Compressibility factor = 4- Indicia1 response function

S u b s c r i p t s

( . h c Refers to aerodynamic center (.)A4 Refers t o pitching moment about the 114-chord

(.)e Refers to pitch rate (.)a Refers to angle of attack

S u p e r s c r i p t s

(dC Refers t o circulatory loading (.)I Refers to noncirculatory (impulsive) loading

( . ) p Refers to potential (attached flow) loading Refers to loads with trailing edge separation

(.)" Refers to vortex loading

Introduction

T h e successful design of advanced rotorcraft requires the ability to confidently predict the large unsteady and vibratory loads generated and transmitted by the rotor system. The capability to accomplish this has improved significantly in recent years as a result of advances in the analytical modeling of blade structural dynamics, the rotor wake geometry and unsteady aerodynamics. While rotor structural dynamic modeling has now reached a good level of maturity, the development of accurate and computationally efficient aerodynamic models which represent the unsteady behavior of the blade sections, still poses a major challenge to the rotor analyst. Generally the analyst needs to determine an appropriate compromise between the accuracy of a given aerodynamic model and the need to keep computational requirements within practical limits. In most circumstances, these requirements are conflicting and the rotor analyst is forced to resort to a relatively simple representation for the aerodynamics. Unfortunately, this can restrict the range of flight conditions over which the analysis can be applied, and thereby severely limit its generality as a practical design tool.

Within a helicopter rotor flowfield, the blades encounter complex time varying changes in aerodynamic angle of attack due t o imposed control inputs, the dynamic motion of the blades and local variations in inflow velocity due the complex three-dimensional vortex wake system created by each blade. Understanding and modeling the effects of this complex time varying flowfield on the unsteady aerodynamic behavior of the blade sections is one of the major challenges still facing rotor analysts. In many rotor operating regimes, unsteady aerodynamic effects are of low magnitude and can be justifiably neglected in any arialysis. However, if the

angle of attack of the blade sections becomes large enough, dynamic stall may occur. Typically, this occurs on the retreating blade under conditions of high blade loading and in high speed forward flight. In general, the rotor operational limitations, i.e. vi- bration, aeroelastic stability, maximum control loads and fatigue limits are all determined by the onset of transient flow separation such as dynamic stall.

Many experimental t e ~ t s l - ~ have shown that the distinguish- ing feature of dynamic stall compared with static stall is the shedding of significant concentrated vorticity from the airfoil leading- edge region. This vortex disturbance is subsequently swept over the airfoil chord and induces a strong moving pressure wave on the airfoil surface. These pressure changes result in increases in airfoil lift and large nose-down pitching moments well in excess of the static values. For repeated excursions into stall, considerable hysteresis in the force and moment behavior can also arise. These conditions may lead to reduced or negative pitch damping, and if stall is sufficiently severe this may excite the blade torsion mode a t the natural frequency, which can lead to a dynamic instability known as stall flutter. Thus, to define the rotor operating envelope, i t is necessary to be able to predict this dynamic stall phenomenon and model its consequences on the dynamic response of the rotor.

The unsteady aerodynamic response of an airfoil to a specific time history of forcing can now be determined with considerable detail and accuracy using computational fluid dynamic (CFD) methods. For example, numerical solutions to the unsteady Navier-Stokes equations are now becoming increasingly feasible5*' and have shown some recent success in modeling dynamic stall. However, these aerodynamic solutions are complex and the required computational resources are, for most rotor analyses, prohibitive. Nevertheless, CFD methods are providing considerable insight t o the aerodynamic problems encountered by rotorcraft7. It appears, however, tha t for the foreseeable future more approximate aerodynamic solution methods must continue to be used in most rotor performance and aeroelasticity analyses. Clearly, this poses somewhat of a dilemma for the analyst as the consequences of complex viscous effects must be modeled within the practical constraints imposed by the computational enormity of overall rotor performance and/or aeroelasticity analysis.

To provide some representatibn of the unsteady ae;odynamic behavior of the blade sections. a number of fairlv so~histicated . . semi-empirically based models have been developed (e.g. Refs. 8-11). Many of these semi-empirical models have reasonable ap- proximations for the unsteady aerodynamics under attached flow conditions, however for the dynamic stall regime they often rely heavily on the synthesization of wind tunnel data from unsteady airfoil tests. In the interests of computational simplicity, many models sacrifice physical realism and so may have limited generality in application. Despite these limitations, developed methods have met with good success and have been shown to give significant improvements in performance prediction capability for helicopters. However, with the increasing overational demands - . t ha t are placed on helicopters and the increasing use of advanced blade technology and modern airfoil sections, there is still a fun- damental requirement for improved aerodynamic models that can be used with greater confidence in rotor design procedures.

I t is the purpose of this paper t o document the results of some recent research directed towards improving the representation of unsteady aerodynamics within a comprehensive rotor analyses, including the important effects of flow compressibility and dynamic stall. The method is presented in the time domain which is a necessary prerequisite to fully account for the flowfield encountered by a helicopter rotor. The approach used is to express the unsteady airloads as a series of elementary systems which are described by first order ordinary differential state equations (ODE's), i.e. state space form. Preliminary results with this formulation were first reported in Refs. 12 and 13. The state space

form has a number of advantages, particularly when applied to aeroelasticity analyses: (1) The form of the aerodynamic representation is particularly elegant. (2) The representation is compati- ble with the structural dynamic equations13 and can be integrated simultaneously in time. (3) The model is given as a continuous time representation and no particular form of discretization or numerical solution technique is imposed. Any standard ODE solver can be used to integrate the state equations. (4) Under attached flow regime the airloads are linear functions of the forcing, thus the state space representation lends itself to eigenanalyses. This however, cannot be done under stalled conditions unless some form of linearization is performed.

The approach of using differential equations to describe the unsteady aerodynamic behavior of a 2-D airfoil was first adopted by Tran ei aL9 who used experimental airfoil da ta from small amplitude pitch oscillations to estimate the equation parameters. The objective here was to obtain a linearized model for the unsteady aerodynamics through stall, and thereby provide a model whereby blade aeroelastic stability (with stall) could be computed using Floquet theory. Reasonable success has been demonstrated with this method, although the quantitative prediction capability for the airloads could probably be improved. Furthermore, this method requires a large data base of unsteady data for a particular airfoil section over a wide range of Mach numbers. Experimen- tally, these data are not easy to obtain. For general application to a variety of airfoil sections, it makes more sense to establish a model which (a) is based on well proven classical unsteady aerodynamic methods for attached flows and (b) can be used to e s i i m a i e the dynamic stall characteristics from the static stall characteristics. However this requires, a t a minimum, the modeling of key factors which affect the dynamic stall mechanism and must in- clude both leading edge separation and trailing edge separation effects together with leading edge vortex shedding where appropriate. These effects must be modeled within the practical computational constraints imposed by the aeroelastic or performance analysis of the rotor system.

In this paper, it is demonstrated how a relatively elegant and computationally efficient method can be created and used to cred- ibly reproduce the dynamic stall characteristics of an airfoil section given, primarily, the static (nonlinear) airfoil lift and pitching moment characteristics. The method draws on the work of both bed doe^'^^^^^^^ and Leishman and Beddoes", to establish a system of first order ODE's which represent specific aspects of the unsteady airfoil behavior. The method is useful in the performance and aeroelastic response analysis of helicopter rotors and propellers. To support the development of the method, illustra- tive comparisons with experimental da ta are presented for the unsteady lift, pitching moment and drag on three typical helicopter rotor airfoils undergoing oscillatory pitch forcing into dynamic stall.

Methodology

The objective is t o derive a concise but comprehensive description of the unsteady aerodynamic behavior of a 2-D airfoil undergoing dynamic stall as a finite number of first order differential state equations. The background to this form of representation can be found in many excellent texts on modern control system theory, such as Refs. 17 or 18. In general, an nth order differential system with m inputs and p outputs may be represented by n first order differential equations

k = A x + B u with the output equations

where z = d+/d t ; The vector x = z;,i = 1,2 , ..., n is a n x 1 column vector called the state-vector; u is a m x 1 input column

vector; y is an output column vector; A , B, C and D are coefficient matrices of appropriate dimensions. The inputs u to the system are the angle of attack and pitch rate, and the outputs y are the required lift, drag and pitching moment. In general, the states describe the internal behavior of the system and are sim- ply the information required a t a given instant in time to allow the determination of the future outputs from the system given future inputs. In other words, the state of the system determines its present condition. It should be noted that it is desirable to obtain a system with a minimum number of states (since each state imposes extra computational overhead) while a t the same time maximizing the performance of the system in terms of aerodynamic prediction capability.

Attached Flow Behavior

A prerequisite in any unsteady aerodynamic theory is the ability to accurately represent the unsteady aerodynamic response under attached flow conditions. Classical theories in this category for incompressible flows have been formulated by c he odor sen'^, Greenbergzo, sears2', and von Karman and searsz2. In addition, Wagnerz3 has obtained a solution for the indicial response. The indicial response is particularly useful since the state equations representing the unsteady airfoil behavior can be obtained by the direct application of Laplace transforms to the indicial response to get the aerodynamic transfer function.

While the well known Wagner indicial function is used in some form in many aeroelastic analyses, strictly speaking it is re- stricted to incompressible flows. However, most practical aerodynamic problems involve compressibility to some degree and there is no exact equivalent of Wagner's function for a compressible flow. Nevertheless, as shown by various researchers, including M a z e l ~ k i ~ ' ~ ~ ~ , ~eddoes" , ~ o w e 1 1 ~ ~ and Leishmanm, a practical representation of the indicial responses can be made using an ex- ponential series approximation. By suitably generalizing the indicial response functions in terms of Mach number, as shown by Leishman30, the corresponding state equations may be obtained in a relatively general form.

In Ref. 30, the indicial functions are assumed to be idealized into two parts. One part of the response is for the initial noncirculatory loading which comes from piston theoryz4. This result is valid for any Mach number for time equal t o zero. For subsequent time, pressure waves from the airfoil propagate at the local speed of sound and the loading decays rapidly with time from its initial value. In fact, these time dependent noncirculatory loads may be considered the compressible analog of the apparent mass terms used in many incompressible analyses. The second part of the indicial response is due to the circulatory loading which builds up quickly to the steady state value. The behavior of the indicial lift response is shown in Fig. 1 for a step change in angle of attack a t Mach numbers of 0.2 and 0.8 in comparison with the classical Wagner function. By convention, the functions are plotted versus nondimensional time S = 2Vt l c which corresponds to the relative distance travelled by the airfoil in terms of semi-chords.

In general, the indicial normal force and quarter-chord pitching moment response to a step change in angle of attack a and a step change in pitch rate q can be written as

M=O (Wagner)

10 - l -- M = 0.2 ( - . - M = 0.8

/.- ./.-.

o ! . , . , . , . , . , . , . , . , 0 2 4 6 8 10 12 14 16

Distance travelled In semi-chords, S

Figure 1: Indicia1 lift response to a step change in angle of attack

where the indicial response functions 4:, q5:, d:,, 4:, 4:, 4fM and 4:, are expressed in terms of both aerodynamic time S and the Mach number M. These functions are fully defined in Ref. 30. The superscripts C and I refer t o the components of circulatory and noncirculatory (impulsive) loading respectively and the sub- script M refers to the pitching moment contribution. The second term in Eq. 4 represents the contribution to the pitching moment due t o a Mach number dependent offset of the aerodynamic center, xac , from the airfoil quarter-chord axis and must be obtained from either experiments or CFD codes. At subsonic speeds, the aerodynamic center lies close t o the 114-chord although for tran- sonic speeds the effective aerodynamic center moves quickly t o the vicinity of the mid-chord as the free-stream Mach number approaches unity.

To illustrate the form of the state equations, consider the normal force response to changes in angle of attack. The circulatory indicial response function is written as

and the noncirculatory function as

R o m ~ e d d o e s ' ~ , the constants of the circulatory lift function are A1 = 0.3, A2 = 0.7, bl = 0.14 and b2 = 0.53 and the function is generalized t o different Mach numbers by scaling the exponents by P2. From Leishmanm, the noncirculatory time constant, T, = Z(,TI is given based on an approximation to the exact linear theory results of ~ o r n a x ~ ' where TI = c l a and

The circulatory normal force response to a variation in angle of attack can be written in terms of the differential state equations

where x = dxldt . For the circulatory component, the pitch term q can be coupled into the above equations by using the angle of attack a t the 3/4-chord, i.e.

The output equation for the normal force coefficient is given by

where 2 x / P is t h e force-curve-slope for linearized compressible flow. Similarly, the noncirculatory normal force due t o angle of at tack can be written a s t h e differential s ta te equation

1 13 = -- 1 3 + a ( t ) = a3323 + a ( t )

I(, Tz (11)

with the ou tput equation for the normal force coefficient given by

T h e remaining (five) s ta te equations for t h e pitching moment and pitch ra te terms can be derived in a similar way, and a re given in Ref. 25 and 26.

To obtain t h e total airloads under at tached flow conditions, the individual components of loading are linearly combined to obtain the overall aerodynamic response. For example, t h e total normal force coefficient CL is given by

and a similar equation holds for the pitching moment. T h e net unsteady aerodynamic response in at tached flow can be described in terms of a two input , two output system where the inputs are the airfoil angle of at tack and pitch rate and the outputs are t h e unsteady normal force (lift) and pitching moment. I t can be readily shown t h a t by rearranging all (eight) s ta te equations for at tached flow conditions, they can be represented as a diagonal canonical set of s t a t e equations

where the matrices are of the form

T h e total aerodynamic lift and pitching moment response t o a n arbi trary t ime history of n and q can be obtained from the above s ta te equations by integrating using a n ODE solver (see Ref. 26).

Referring t o Fig. 2, the unsteady chord force (in-plane force) and pressure drag on t h e airfoil may also be obtained in terms of the s t a t e variables already defined. For a fixed-wing problem the wing has a high in-plane stiffness and so the chord force component rarely participates in the aeroelastic problem. However, for a helicopter rotor the relatively low lead/lag stiffness of the blades makes t h e chord force component much more significant for the aeroelasticity problem.

F tom the ou tput equation (Eq. 10) the effective angle of attack of t h e airfoil, a ~ , due t o t h e shed wake (circulatory) terms can be written in te rms of the states rl and 1 2 as

Figure 2: Force resolution on a thin airfoil in unsteady flow

T h e corresponding chord force in potential flow Cg is given in terms of a~ a s

which involves t h e appropriate combination of the states 2.1 and 12. Thus , as a by-product of the above system representation for t h e unsteady lift, the necessary information may be extracted from t h e system t o obtain the unsteady chord force component. Finally, the instantaneous pressure drag can be obtained by re- solving the components of the normal force and chordwise forces through the geometric angle of attack a using

C L ( t ) = C&(t) sin a ( t ) - C g ( t ) c o s a ( t ) (I61

Further details and validation of the unsteady chord force and pressure drag representation are given by ~ e i s l l m a n ~ ~ .

Nonlinear Aerodynamics and Dynamic Stall

Having established the differential s ta te equations which govern the at tached flow (linearized) behavior of the airfoil, it is required to extend the analysis to encompass the nonlinear airfoil behavior and dynamic stall. T o d o this, it is necessary to identify and model the key features of the stall process, rernembering t h a t this must be done using a minimum number of state equations. I t is also required to at tr ibute a physical significance to each of the s ta te equations t h a t are defined.

Stall Onset

T h e most crucial aspect of the modeling of dynamic stall is the identification and representation of the conditions for the onset of leading edge separation. T h e implementation of any criterion must also be sufficiently general t o allow for the prior history of t h e airfoil motion, including angle of attack and Mach number variations.

Evans and Mort33 present a useful criterion equivalent to a critical leading edge pressure and associated pressure gradient which may be used t o denote the onset of static leading-edge stall. This criterion was subsequently evaluated by 13eddoes14 within the context of rotor airfoil performance, under both steady and unsteady conditions. For practical purposes, Beddoes determined t h a t although under time-dependent forcing conditions the

Trailing Edge Separation

NACA 001 2, woods*

0 .4 ...8.- NACA 001 2. ~ c ~ r o s k r y " C

s L O]], 0 0 .1 .2 .3 .4 .5 .6 .7 .8 .9

Mach number, M

Figure 3: Stall boundary for the NACA 0012 airfoil section

pressure gradient on the airfoil a t a given angle of attack was significantly modified, it was possible to predict the onset of leading edge separation (and hence, dynamic stall) using a criterion in which the attainment of a critical local leading-edge velocity (pressure) was the primary factor. The analysis was subsequently extended by ~ e d d o e s ' ~ to encompass higher Mach number flows, where the attainment of a critical leading edge pressure was again used to denote the onset of shock induced stall.

In application the leading edge pressures P are related to the normal force CN, so it is possible to obviate the need to com- pute airfoil pressures by transforming the calculation to the CN domain. From an analysis of airfoil static test data, a critical value of CN(static)= CN, may be obtained which corresponds to the critical pressure for separation onset a t the appropriate Mach number. In practice CNl is close to the maximum static CN. Thus a Mach number dependent separation onset (stall) boundary may be defined. A typical boundary for the NACA 0012 airfoil is shown in Fig. 3.

For unsteady conditions, there is a lag in CN(t) with respect to the forcing (above), however there is also a lag in the leading edge pressure response P ( t ) with respect to C N ( t ) Thus for an increasing angle of attack, the lag in the leading edge pressure response results in the critical pressure being achieved a t a value of CN and hence a t a higher angle of attack than the quasi-steady case. Thus, this mechanism significantly contributes to the overall delay in the onset of dynamic stall. To implement the critical pressure criterion under unsteady conditions, a first order lag may be applied to CN(t) to produce a substitute value C&(t) with the presumption tha t whatever properties apply to P ( t ) must also apply to CEy(t). This representation may be written as the differential state equations

where the input to the above equation is the total unsteady lift under attached flow conditions, C;. The time constant T p is a function of Mach number and can be determined empirically from unsteady airfoil data. The value of T p is largely independent of airfoil shape but is more dependent on Mach number1'. Thus by monitoring the value of Ch( t ) the onset of leading edge/shock induced separation under dynamic conditions will be initiated when Ch( t ) exceeds the critical CNl (M) boundary. Furthermore, if the value of Ch( t ) is monitored throughout the calculation into stall, then it may be used as an indicator for the conditions which per- mit flow reattachment, i.e. if Ch( t ) < Cjv1

A phenomenon tha t is also involved in most types of airfoil stall is progressive trailing edge separation. The associated loss of circulation due to trailing edge separation introduces a nonlinear force and moment behavior, especially with the cambered airfoils more typically used on modern helicopters. ~ i l b ~ ~ ~ suggests tha t trailing edge separation may play a significant role in the onset of dynamic stall. However, as also discussed by Wilby, experimental tests have indicated tha t the occurrence of trailing edge separation is suppressed by increasing pitch rate. The dynamic stall process may then be initiated by leading-edge separation or shock induced separation if supercritical flow is allowed t o develop. Even so, when the primary source of separation is at the leading edge or the foot of a shock wave it appears tha t this is generally sufficient to promote some separation a t the trailing edge and hence initiate some nonlinear behavior in the force and moment response.

One theory which models separated flow regions on 2-D bodies is attributed to Kirchhoff and is reviewed in Refs. 35 and 36. A specific case of Kirchhoff flow is a simple model for the trailing edge separation phenomenon (Fig. 4) in which the airfoil normal force coefficient, CN, may be approximated as

where 2 a is the force-curve-slope for incompressible flow, f is the trailing edge separation point and a is the angle of attack. Thus, if the separation point can be determined it is a trivial calculation to determine the normal force. In practical cases, this expression may be extended t o encompass compressible flows where 27r is replaced by the force-curve-slope at the appropriate Mach number

As first shown by ~ e d d o e s ' ~ , in order to implement this procedure the relationship between the effective separation point, f , and the angle of attack, a , can be deduced from the airfoil static lift behavior by rearranging Eq. 20 to solve directly for f . The relationship between a and f can be generalized empirically in a fairly simple manner using the relations

1 - 0.3exp {(a - al)/S1) if a < ffl

= { 0.04 - 0.66exp {(a1 - a)/S2) if ff > a1 (21)

The coefficients Sl and Sz define the stall characteristic, while a1 defines the break ~ o i n t corresponding to f = 0.7. I t should be noted tha t f x 0.7 closely co~responds to the static stall angle for most airfoil sections. S l , S2 and a1 are determined empirically for each airfoil and vary with Mach number. Using the above equations, the reconstructed lift versus angle of attack relation- ships are shown for the NACA 0012, HH-02 and SC-1095 airfoils a t a Mach number of 0.3 in Fig. 5. This procedure has also been thoroughly validated for higher Mach numbers and other

Figure 4: Kirchhoff model for separated flow past a flat plate

A NACA 001 2 O HH-02 0 SC-1095 - Reconstruction

I I I 1

-10 0 10 20 30 Angle of attack, a (deg.)

Figure 5: Reconstruction of the stat ic lift behavior using the Kirchhoff model

airfoils, and may be applied to almost any airfoil if the stat ic stall characteristic is known apriori.

A general expression for the pitching moment behavior cannot be obtained from Kirchhoff theory and an alternative empirical relation must be formulated. From the airfoil s tat ic d a t a , t h e center of pressure a t any angle of at tack may be determined from the rat io C M / C ~ (allowing for the zero lift moment CM,,). T h e variation can be plotted versus the corresponding value of the separation point and fitted in a least squares sense to the form

where ICo = (0.25-X,,) is the aerodynamic center offset from the 114-chord. T h e constant K1 gives the direct effect on the center of pressure due t o the growth of the separated flow region and the constant helps describe the shape of the moment break a t stall. T h e values of 1 6 , K l , and m can be adjusted for different airfoils, a s necessary, t o give the best moment reconstruction. Typically, m = 2. Stat ic moment reconstructions for the NACA 0012, HH-02 and SC-1095 airfoils are shown in Fig. 6.

An expression for the chord force Cc may also be deduced from the Kirchhoff solution for the trailing edge stall problem

Due t o viscous effects on the pressure distribution, the airfoil does not realize 100% of t h e chord force which would be attained in potential flow. Allowance for this nonrealization is made through the factor q which can obtained empirically from static airfoil test da ta . Typically q = 0.95. Note t h a t for inviscid flow, q = 1 and f = 1 so for s teady conditions CL = 0. However for unsteady conditions CL # 0 as the angle of at tack in Eqs. 20 and 23 must be replaced by the effective angle of attack a~ (i.e. Eq. 14). It should be noted t h a t under unsteady conditions the instantaneous pressure drag can even become negative. Further details on modeling the unsteady chord force and pressure drag are given by Leishman in Refs. 11 and 30.

For unsteady flow there will exist a modified separation point location d u e t o t h e temporal effects on the airfoil pressure dis-

A NACA 001 2 O HH-02 0 SC-1095 - Reconstruction

I 1 1 i 1 0 10 20 30

Angle of attack, a (deg.)

Figure 6: Reconstruction of the stat ic r i~oment behavior

tribution and t h e boundary layer response. A relatively simple open loop procedure was first developed by ~ e d d o e s " and later by Leishman and Beddoes" t o represent the variation of the time- dependent location of the trailing edge separation point. This procedure can be used a s a means of extending the evaluation of nonlinear forces and moments into the dynamic regime via the application of the Kirchhoff theory, as above.

T h e procedure is performed by firstly incorporating the airfoil unsteady pressure response via Eqs. 17 and 18 which may then be used t o define an effective angle of attack af which gives the same unsteady leading edge pressure a s for the quasi-steady case

This value of af may be used t o determine a value for the effective separation point f' a t this af from the stat ic f versus a relationship in Eq. 21.

Secondly, the additional effects of the unsteady boundary layer response may be represented by applying a first order lag to the value of f' t o produce the final value for the unsteady trailing edge separation point f N . In s ta te variable form this additional lag in the response may be represented as

As in t h e case of Tp, the time constant Tf is Mach number dependent. Substantiation for the modeling of the above two components which contribute t o the overall unsteady trailing edge separation response have been given previously in Refs. I 1 and 15. Finally, the (nonlinear) normal force CL with the the modified (unsteady) trailing edge separation point f" is given by

and the pitching moment by Hence, the corresponding pitching moment C b produced by the vortex lift component will be given by CL = [KO + K l ( l - f") + Kz ~ i n ( ? r ( f " ) ~ ) ] C$ + C M ~ (28)

where C$ is the circulatory normal force coefficient and CMo is the zer+lift moment. T h e contributions of the other unsteady circulatory and noncirculatory moment terms are additive t o Eq . 28. Similarly, the chord force is written as

Modeling of Dynamic Stall

T h e general case of dynamic stall involves the formation of a vortex near t h e leading edge of the airfoil which subsequently sep- arates from the surface and is transported downstream. After the vortex detaches, the induced airloads appear to be qualitatively similar for different airfoils, for different Mach numbers and for different modes of forcing such a s oscillatory pitch and ramp motions34.

An approximate, but physically acceptable, model for the dynamic stall process has been formulated by viewing the vortex lift contribution C& as an excess circulation which is not shed into the airfoil wake until some critical condition is reached. T h e critical condition used here is of course when C h ( t ) exceeds CN,. At this point, catastrophic flow separation occurs and the accumulated circulation passes over the airfoil and into the wake. T h e vortex lift process can be represented by the differential s ta te equations

where

C$ [ l - ( 1 + f l ) 2 / 4 ] for TU 5 2Tui c u = { O for T,, > 2Tu/ (32)

as derived from the Kirchhoff approximation. T h e vortex lift is essentially generated by the time rate of change of circulation d I ' / d t K C, which also dissipates with a characteristic time con- s tan t T,. Consequently, when the rate of change of lift is low the vortex lift is being dissipated a s fast as it accumulates and in the limit a s the flowfield becomes steady, the airfoil characteristic will revert smoothly back to the stat ic nonlinear behavior.

Abrupt airloading changes occur when the critical conditions for leading edge or shock induced separation effects are met , i.e. C h ( t ) exceeds CN,. At this point there is a catastrophic loss of leading edge suction and the accumulated vortex lift is assumed t o s t a r t t o convect over the airfoil chord. When this occurs, the nondimensional t ime r, = 0 is defined t o track the vortex pas- sage. T h e rate a t which this vortex convection process occurs has been shown from a variety of experimental tests t o be somewhat less than half of the free-stream velocity, the actual rate which is also somewhat dependent on Mach number. During the vortex convection process, the vortex lift is assumed to continue t o accumulate via Eqs. 30-32 but the accumulation is terminated when the vortex reaches the airfoil trailing edge, i.e. after a suitable nondimensional t ime interval Tur. Assuming the airfoil main- tains a high angle of at tack, the airloads quickly approach their quasi-steady values, even though the angle of attack may still be changing.

T h e center of pressure on the airfoil also varies with the chordwise position of the vortex and will obtain a maximum value when the vortex reaches the trailing edge (r, = TUl). A fairly general representation of the center of pressure behavior (aft of 114-chord) can be approximated a s

CP, = 0.25 1 - cos [ (8

Both t h e vortex decay time constant T,, and the nondimensional t ime Tur have been determined statistically from a variety of dynamic stall test data" and appear to be relatively independent of both Mach number and airfoil shape.

Modifications to the Model

Although t h e above system equations describe, in a n open loop sense, t h e basic physical flow phenomena likely t o be encountered, the elements of the model are physically coupled. For example, trailing edge separation development will interact to some extent with the onset of leading edge separation and the subsequent vortex lift generation. In addition, under general forcing conditions, separation effects may be compounded by a the airfoil kinematics.

As shown in Ref. 1 1 , in most cases interactional effects can be readily represented by modifying the appropriate t ime constant associated with the behavior, i.e. by reducing or increasing the t ime constant associated with the process. T h e various strategies a re documented in Ref. 11 and are not repeated here. All the modifications a re incorporated into the algorithm using simple logic and the values of the time constants are updated during each pass through the algorithm.

Other modifications have been made t o the model based on experience with the discrete t ime version presented previously11. These included modifications to the chord force calculation under deep stall conditions and the modeling of secondary vortex shedding phenomena. In addition, the mean center of pressure during flow reat tachment from the deep stall regime was represented by using Eq. 28 with a different effective separation point t h a n for t h e lift. This parameter , fM, is computed by using the quasi-steady separation point location f,, (for the same angle of at tack, a j ) as an input t o the system

This requires the addition of a final s ta te , namely xlz, however the overall improvement in the correlation obtained with test d a t a justifies the inclusion of this ex t ra s ta te .

Total Aerodynamic Response

Thus , four additional s tates z9,zlo, x11,x12 are required t o represent the nonlinear aerodynamics. Essentially, this part of the model is a series of simple first order systems t h a t are connected by nonlinear gains. (This can be more easily seen if the the systems a re written ou t a s a simulation block diagram). T h e input t o the first part of the nonlinear model is the unsteady potential lift Ch, with t h e other system inputs being derived from the ou tput of the previous system. By suitable manipulation of the ou tputs from the various subsystems, the required total loadings can be obtained. For example, the total normal force coefficient CN under dynamic stall conditions is given by

with similar equations for the itching moment and drag force components.

Results and Discussion

T o evaluate the model, a computer program was developed using FORTRAN and implemented on a MicroVAX I1 computer. T h e

- NACA 0012 SIKORSKY SC-1095

HUGHES HH-02

Figure 7: Geometry of the airfoils considered in this study

program was used to study a variety of examples of the unsteady airloads on airfoils subject to prescribed forcing below stall, and the results were correlated with experimental da ta . For these particular calculations, the integration of the state equations was performed using the ODE solver DE/STEP given in Ref. 37, which is a general purpose Adams-Bashforth ODE solver with variable step size and variable order. Further discussion of the performance of this ODE solver is given in Ref. 38.

There are many good examples of unsteady airfoil behavior available in the published literature which can be used to illustrate the performance of the theory. However in the interests of brevity, representative examples of oscillatory pitch airfoil motion under light and deep dynamic stall conditions will be considered. Validation of the modeling under attached conditions has been previously considered in Refs. 25 and 26.

To avoid difficulties in comparing test data from different wind tunnels, all the experimental da ta are taken from the tests performed by McCroskey e t The selected set of data are for harmonic pitch oscillations a t various mean angles of attack with a constant oscillation amplitude of 10' a t a (nominal) reduced

frequency of 0.1. This reduced frequency is representative for the once-per-rev cyclic pitch change on a helicopter rotor. The Mach number in all the tests was 0.3. All the experimental data presented are the ensemble average of some 50 pitch cycles. The input was supplied to the model was obtained by performing an F F T on the the time history of the angle of attack forcing as measured in the experiment. The first three harmonics were then used to reconstruct the forcing.

Three airfoils were selected from Ref. 41 for this study; the NACA 0012 (as a baseline section), the HH-02 and the SC-1095. The latter two airfoils are typical modern helicopter rotor sections. The NACA 0012 is a 12% thick symmetric airfoil, whereas the HH-02 and SC-1095 are cambered airfoils with approximately 9.5% thickness to chord ratios. In addition, the HH-02 is considerably more cambered than the SC-1095 although has the distinc- tion of a large trailing edge tab which cancels most of the pitching moment associated with the camber. The geometries of the three airfoils are shown in Fig. 7.

It should be mentioned, tha t for the purposes of the comparisons with test da ta the lift curve slope 2 x l P was replaced by the appropriate quasi-static value obtained from the experimental data, as appropriate. Similarly, the aerodynamic center z,, was obtained from the quasi-static test da ta and was implemented via Eq. 28. The numerical values of the parameters used in the model for each airfoil section are given in Table 1.

Stall Onset, a( t ) = 5' + 10' sin wt

NACA 001 2

Fig.8 showns typical normal force, pitching moment and drag predictions compared with test da ta for stall onset conditions, where the angle of attack is just enough to initiate leading edge separation. Under fully attached flow conditions, nominally elliptical lift

SC- 1095

-5 0 5 10 15 -5 0 5 10 15


Figure 8: Comparison of theory with test data for the unsteady normal force, pitching moment and pressure drag a t stall onset

1379

NACA 001 2 L

Model ;Ma- * * * * ** .**

, --- Stat~c data

NACA 00 1 2

- Model '.

SC- 1 095

SC- 1095


Figure 9: Comparison of theory with test data for the unsteady normal force, pitching moment and pressure drag during moderately strong dynamic stall

and moment curves are obtained and this is consistent with pre- difficulties with stall onset conditions so difficulties of this nature dictions from linear theory. With the initiation of some Row sep- were not entirely unexpected. It should also be noted tha t consid- aration however, the loops become distorted near the maximum erable cycle-tecycle variability of the experimental airloads are angle of attack. In all cases, all three airfoils exhibit an increase obtained under these conditions. Overall, the correlation with in C N ~ , , over the static values due to the lag in the development the test da ta was quite acceptable bearing in mind that the stall of flow separation under unsteady conditions. I t is clear however, onset condition is where most aerodynamic models are likely to tha t the HH-02 and SC-1095 maintain attached flow to higher have difficulties. angles of attack with higher values of CN and thereby exhibit a superior performance to the the NACA 0012.

Out of all the likely cases of dynamic stall, the stall onset condition was found to be the most difficult to model. For all three airfoils, the critical CN, value was just exceeded and so these stall onset examples were a good test for the algorithm. It can be seen from Fig. 8 tha t the model does quite well in predicting the onset of stall for all three airfoils. The point of stall onset is perhaps seen most clearly in the drag characteristics. The NACA 0012 exhibits a somewhat greater maximum drag, indi- cating tha t it has lost more leading edge suction and penetrated slightly deeper into stall. Although stall onset was quite well predicted, the subsequent behavior during the downstroke of the motion was considerably more difficult t o predict, especially for the pitching moment. Previous studies" have also shown some

Moderate Dynamic Stall, a( t ) = 10' + 10°s inwt

Fig. 9 shows force and moment predictions in comparison with test da ta for a case of moderately strong dynamic stall. Under these conditions leading edge vortex shedding is initiated and the characteristic lift overshoot and strong nose down pitching mc- ment behavior are exhibited. Considerable hysteresis in the force and moment behavior is resent. All three airfoils exhibit a aual- itatively similar type of behavior, although there are certainly some quantitative differences. The NACA 0012 exhibits moment stall (points M) a t a lower angle of attack to either the HH-02 or the SC-1095, although the lift stall (points L) occurs a t approximately the same angle of attack for all three airfoils. The moment stall occurs just after onset of leading edge separation.

2.5 : NACA 001 2 HH-02

1.5 : \

CN - 4 /'- Model

0 Experlmenl --- Static data

NACA 001 2 HH-02 SC- 1095 V

- Model Experlrnent


Figure 10: Comparison of theory with test da ta for the unsteady normal force, pitching moment and pressure drag during deep dynamic stall

Both the NACA 0012 and the SC-1095 airfoils exhibit a well rounded moment break at the onset of dynamic stall, in comparison with,the HH-02 which has a very abrupt moment break. This suggests t ha t some trailing edge separation is present on the NACA 0012 and SC-1095 prior t o the onset of leading edge separation. This is consistent with the static airfoil chacteristics, since both the SC-1095 and NACA 0012 exhibit fairly gradual static stall characteristics which is symptomatic of trailing edge separation. This suggests tha t to some extent the static stall behavior is carried over into the dynamic stall regime.

It was interesting tha t both the HH-02 and SC-1095 exhibited a slightly greater CNMA, than the NACA 0012. Again, the static CNMA, gains are carried over somewhat into the dynamic regime. The gains in CNMA, for the HH-02 and SC-1095 are also reflected in the drag curves where higher peak values of CD are obtained as a consequence. On the other hand, the NACA 0012 clearly exhibits a lower peak value of CM than the other two airfoils. This suggests tha t the strength of the shed leading edge vortex

may be somewhat less for the NACA 0012. Overall, the model performed particularly well in predicting

the stall onset and the subsequent magnitude and phasing of the induced airloads. The peak values of the lift and minimum pitching moment coefficients were well predicted for all three airfoil sections. I t was clear that the slight differences in the onset of dynamic stall between the NACA 0012 and the other two airfoils could be predicted with the model.

Deep Dynamic Stall, cr(t) = 15" + 10' sinwt

Fig. 10 shows force and moment predictions in comparison with test da ta for a cases of deep dynamic stall. Under these conditions the airfoil reaches a high maximum angle of attack. Strong leading edge vortex shedding occurs giving significant increments in normal force, pitching moment and drag. As in the previous case, all three airfoils exhibit a qualitatively similar type of behavior. Both the HH-02 and SC-1095 exhibit increased values of CNMAX

NACA 0012 0.113

Table 1: Parameters used for each airfoil

over the NACA 0012. It is interesting, however, that while under static conditions the SC-1095 exhibits a gain in CNMAX of about 0.1 over the HH-02, under dynamic conditions there is almost no difference in CNMAx between these two airfoils. Similarly, the minimum values of CM are almost identical. This indicates tha t ..- while CNMAx may be a useful measure of the airfoils performance under static conditions, this is not necessarily an indication of the performance under unsteady conditions.

The aerodynamic model did reasonably well in predicting the magnitude and phasing of the airloads for this deep stall condition. I t should be noted tha t for all three airfoils there is evi- dence of secondary vortex shedding near the maximum angles of attack. This manifests itself as smaller secondary peaks in the normal force, pitching moment and drag behavior. This aspect of the behavior is also captured reasonably well with the model, being based on an effective Strouhal number for bluff body vortex shedding".

As in the previous cases, most of the differences between the model and the test da ta are apparent during the reattachment phase. In this regime, there are generally significant variations in the airloads from cycle to cycle due to the inherent random- ness of the flow. In addition, it appears tha t there may be some influence of the tunnel and/or test configuration on the airloads in this flow regime. The unsteady lift response of the NACA 0012 has also been compared with results from three different test f a ~ i l i t i e s ~ ~ , ~ ~ , ~ ~ . It appears during the onset of separation and during dynamic stall the airloads are comparable, however there appears significant variations between the measured airloads during reattachment. It should also be noted here that recent calculations of dynamic stall using the Navier-Stokes equations6 also show significant deviations from the test da ta in the flow reattachment regime.

Summary and Conclusions

Rotor aeroelasticity and performance analyses require versatile and relatively simple methods for evaluating the unsteady aerodynamic behavior of the blade sections. The main objective behind the work outlined in this paper has been to provide an improved model for the unsteady force and moment characteristics of an airfoil undergoing dynamic stall, but in a sufficiently simple manner and in a computational form that can be included within a comprehensive analysis of the rotor system. To this end, a state space formulation has been selected as particularly appropriate.

The main emphasis has been on the development of a fairly general model for the effects of dynamic stall tha t can be applied to a variety of conventional and advanced airfoils used for contem-

porary rotor designs. Based on the novel work of Beddoesl4>l5, individual flow features are represented in a sufficiently simple manner that permits inclusion within the overall sectional aer+ dynamics calculation.

The nonlinear effects on the airloads due to trailing edge separation have been implemented using the Kirchhoff flow theory as a means of relating the force and moment characteristics to the location and progression of the trailing edge separation point. Features of leading edge or shock induced separation have been reviewed and are implemented in terms of a representation of the unsteady leading edge pressure response in which the attainment of a critical value is used to denote the initiation of dynamic stall. Finally, the induced force and pitching moment behavior during dynamic stall have been represented in a physically realistic manner. All the above phenomena are modeled as a series of differential equations tha t can be implemented as a subroutine for the blade sectional aerodynamics.

Validation of the model has been conducted with 2-D test da ta for three rotor airfoils, namely the NACA 0012, HH-02 and SC-1095. Correlation with the test da ta was good, particularly in terms of predicting the onset of dynamic stall. The phasing of the dynamic stall induced airloads was also modeled quite well. It is considered tha t the model is sufficiently general to allow its application to other airfoil sections, a t least when engineering levels of prediction capability are required. The level of correlation obtained with the aerodynamic model provides considerable confidence when applied in the design of new rotors.

Acknowledgements

This work was supported by the U.S. Army Research Office under the Center of Excellence for Rotary Wing Technology Pro- gram a t the University of Maryland. Drs. Robert Singleton and Tom Doligalski were the technical monitors. The authors wish to thank Prof. Roberto Celi and Rotorcraft Fellow Khanh Nguyen of the University of Maryland for their stimulating discussions on the state-space formulation. The authors also wish to thank Prof. Inderjit Chopra of the University of Maryland for his continued encouragement during the course of this research.

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