modeling dynamic stall of sc-1095 airfoil at high mach number

American Institute of Aeronautics and Astronautics

1

Modeling Dynamic Stall of the SC-1095 Airfoil at High Mach Number

Brian M. Clark*, Jeremy J. Bain†, Lakshmi N. Sankar‡, and J.V.R. Prasad§

School of Aerospace Engineering, Georgia Institute of Technology

The Leishman-Beddoes method of determining airloads for an airfoil undergoing dynamic stall is studied over a wide range of Mach numbers. To validate the method for higher Mach numbers where there is less available experimental data, a Computational Fluid Dynamics solver is utilized to provide airload predictions for comparison to the Leishman-Beddoes results. It is found that even for high Mach numbers the Leishman-Beddoes method provides reliable predictions for lift coefficient. However, at the higher Mach numbers pitching moment is sometimes overpredicted at high angle of attack. This is seemingly due to an inability to accurately determine the center of pressure in the high speed unsteady flow environment.

Nomenclature a = speed of sound, ft/s c = airfoil cord, ft CL = lift coefficient CM = pitching moment coefficient CN = normal force coefficient f = separation point, x/c k = reduced frequency k0, k1, k2 = center of pressure fit parameters M = Mach number t = time, s x = chordwise dimension, ft α = angle of attack, deg αm = mean angle of attack, deg αc = cyclical angle of attack, deg

* Graduate Research Assistant, School of Aerospace Engineering, Student Member AIAA. † Research Engineer, School of Aerospace Engineering, Student Member AIAA. ‡ Regents Professor and Associate Chair (Academic), School of Aerospace Engineering, Associate Fellow AIAA. § Professor, School of Aerospace Engineering, Associate Fellow AIAA.

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition4 - 7 January 2010, Orlando, Florida

AIAA 2010-877

Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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I. Introduction ynamic stall is a complicated, nonlinear problem in aerodynamics, and one that is important in helicopter performance, as it can greatly alter key design parameters such as lift and

pitching moment. Airfoil oscillation leading to dynamic stall tends to delay onset of stall and increase lift compared to static airfoil performance, but also leads to spikes in pitching moment and increased torsional loads. As such, a rotor designer must be able to model dynamic stall to predict performance and produce a reliable rotor.1

Dynamic stall is characterized by the unsteady airloads resulting from flow separation, vortex formation, subsequent shedding, and finally flow reattachment. These nonlinear aerodynamic effects do not lend themselves to a closed form solution. Computational Fluid Dynamics (CFD) with a full Navier-Stokes solver tends to be the best tool to accurately predict performance of an airfoil undergoing dynamic stall. However, since it is highly computationally intensive, it remains suboptimal for extensive design and optimization.

Many semi-empirical models have been developed to model dynamic stall, but as they are based in part on experimental data of a limited number of airfoils at a limited range of Mach numbers, their suitability for high Mach numbers is somewhat uncertain.

J.G. Leishman and T.S. Beddoes jointly developed a popular semi-empirical method of modeling dynamic stall that has proven to provide good estimates for lift, drag, and pitching moment of an airfoil undergoing dynamic stall with a very reasonable level of computational power. It is applicable to an airfoil in arbitrary motion, and has been extensively compared against data for oscillating and ramping airfoils, but mostly for lower Mach numbers.

Since their method was first introduced, it has been refined by Leishman and others to account for sweep effects2, trailing edge flaps3, and other aerodynamic problems more complicated than the relatively simple 2-D flow over an airfoil. It is one of the most widely used methods of determining airloads for an airfoil undergoing dynamic stall, and it has been utilized in several comprehensive aerodynamic codes such as Rotorcraft Comprehensive Analysis System (RCAS) and the wind turbine analysis program Aerodyn. As such, understanding how the Leishman-Beddoes method’s predictions compare to the flow physics is of great value.

II. Method In this paper, airloads on a 2-D SC-1095 airfoil are computed for flows from M=0.3 to M=0.7.

The movement of the airfoil is modeled by

€

α =αm +αc sin 2kMac

t

(1)

Testing was conducted for a variety of values for αm and αc, yielding results for attached

flow, light stall with flow separation, and deep dynamic stall. A program utilizing the Leishman-Beddoes method was written. Results from this program

are compared to results from the CFD solver to assess the ability of the Leishman-Beddoes method to accurately represent the loading on the airfoil.

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III. Leishman-Beddoes Leishman-Beddoes consists of an attached flow model for linear airloads, and a separated flow model and a dynamic stall model for nonlinear airloads. The outputs for the attached flow subsystem are used as the inputs for the separated flow model, and the outputs from that subsystem are in turn used as inputs for the dynamic stall vortex shedding model. The attached flow model can be solved independently of the other subsystems, while the separated flow and dynamic stall subsystems interact through adjustments to shared time constants at each time step. In the attached flow subsystem, indicial response functions for a step increase in the angle of attack are used to determine loads using Duhamel’s integral. These indicial response functions are composed of a circulatory and non-circulatory (impulsive) component.

A trailing edge separation model based on the Kirchoff-Helmholtz model is used in Leishman-Beddoes. The actual separation point is not known, so an effective separation point, f, is derived from static airfoil data. This effective separation point is then used to provide a correction to normal force, pitching moment, and chord force calculations.

To model vortex effects, the dynamic stall subsystem is first triggered by a Mach number dependent critical normal force condition related to a leading edge pressure. This critical pressure represents either leading edge or shock induced separation. Once triggered, an incremental increase to normal force from vortex effects occurs while the strength of the vortex also exponentially decays and moves downstream based on empirically derived time constants. Modeling the position of the vortex over the airfoil in this way then allows a determination of a contribution of the vortex to pitching moment.

Refs. 4 and 5 outline the initial development of the Leishman-Beddoes method. The implementation of the method here required a modification to the circulatory component of the normal force to account for an asymmetric airfoil where the zero lift angle of attack occurred at an angle other than 0°. This was accomplished with a simple empirically derived angle offset.

IV. CFD Solver The CFD solver used in this study is the OVERFLOW version 2.0y code developed by

NASA. OVERFLOW is a 3-D time marching implicit Navier-Stokes code, but it is readily applicable to 2-D problems such as that considered here. The Kinetic-Eddy Simulation turbulence model is used.6,7 KES is a Large-Eddy Simulation (LES) turbulence model, where the largest turbulent length scales are directly resolved and smaller length scales are then modeled. In regions where grid resolution is very coarse, it transitions to a Very Large-Eddy Simulation (VLES) model while for very fine grids KES approaches a Direct Numerical Simulation (DNS) model. A C-grid is utilized, as shown in Fig. 1. The OVERFLOW code with KES turbulence model has been validated against experimental data for an oscillating 2-D airfoil undergoing deep dynamic stall, as shown in Fig. 2. The experimental data is from Ref. 8, which has been shown to have significant wind tunnel effects, so this validation run was conducted with the wind tunnel walls included in the CFD solution. Prediction of lift coefficient is very good. However, even with wind tunnel walls included in the CFD solution there is a small overshoot by the OVERFLOW solution for both pitching moment and drag. Otherwise, correlation for these two parameters is also very good.

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Figure 1. OVERFLOW Grid

Figure 2. Dynamic Stall Airloads, SC-1095 Airfoil M=0.3, αm=10° , αc=10°

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V. Results and Discussion Results from the OVERFLOW and Leishman-Beddoes solutions yield very good correlation

for CL across the range of Mach numbers studied, although in general the OVERFLOW results predict a lower lift on the downstroke of the airfoil, resulting in a larger hysteresis loop.

Correlation between pitching moment predictions was not quite as good. As in Fig. 2, peak moment due to vortex convection predicted by OVERFLOW exceeded that predicted by Leishman-Beddoes. Additionally, in some cases the Leishman-Beddoes pitching moment prediction exceeded that from OVERFLOW at high angle of attack; this was typically the case at higher Mach numbers.

Also similar to Fig. 2, the predictions from Leishman-Beddoes for peak drag lag below those from OVERFLOW. This was found to be the case across the range of Mach numbers studied.

Figure 3. Airloads, M=0.6, k=0.1, αm=7° , αc=4°

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Figure 4. Airloads, M=0.7, k=0.1, αm=5° , αc=2°

In the Leishman-Beddoes method, pitching moment is calculated by adding a component due to a convecting vortex to a component that includes circulatory, impulsive, and separated flow effects. This latter component, CMf has been found to be the component that drives the growth in pitching moment in Fig. 4.

€

CMf = k0 + k1 1− f( ) + k2sin π f 2( )( )CNf (2) CNf is the normal force component that models all normal force effects except for vortex lift, and the quantity in parenthesis models the center of pressure. Since CMf is the dominant pitching moment component at high angle of attack and lift force is fairly well represented, indicating that there is not significant error in normal force, this indicates that the Leishman-Beddoes method is

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calculating a larger center of pressure offset from the quarter chord than that of the OVERFLOW solution.

Figure 5. Cmf (left) and Center of Pressure Offset (right) M=0.7, k=0.1, αm=5° , αc=2°

Figure 6. Calculated and Modeled Center of Pressure Offset, M=0.7

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The Leishman-Beddoes method uses a least squares curve fit to static airfoil data to determine the center of pressure to be used in Eq. 2. Fig. 6 shows the calculated center of pressure, (CM-CM0)/CN, as well as the curve fit center of pressure. As shown in Fig. 6, the curve fit appears to be fairly good for the M=0.7 case, particularly in the α=3-7° range considered in Fig. 4. This indicates that the center of pressure discrepancy may not be due to a poor modeling of the static center of pressure. Rather, the assumption that the static center of pressure matches the center of pressure for dynamic loads may not be valid at high angle of attack and high Mach number.

VI. Conclusions and Recommendations The Leishman-Beddoes method is a robust method that can provide good results even at higher

Mach numbers. Lift coefficient has been shown to be accurately modeled across the range of Mach numbers considered here, although under certain conditions there is a small to modest discrepancy in the peak lift and/or angle of attack at which peak lift occurs.

Pitching moment predictions have not been quite as good as lift, but are still of value. However, at high Mach number and high angle of attack the Leishman-Beddoes method begins to overpredict pitching moment contributions from the normal force due to circulatory and separated flow contributions. Since for these conditions the lift predictions appear to be fairly accurate, it is hypothesized that this is due to an inability of the Leishman-Beddoes method to model the center of pressure for high velocity unsteady flows.

The exact cause of the inability of the method to model the center of pressure at high angle of attack and high Mach number is important to understand to improve moment predictions in this region; then, perhaps a physically representative model can be developed to improve predictions. Accurate predictions of pitching moment are important for structural design of helicopter rotors, as well as accurate predictions from CFD-CSD codes where pitching moment can have a significant effect on angle of attack. As such, the center of pressure discrepancy would be a valuable area of future research.

References 1Leishman J.G. 2006. Principles of Helicopter Aerodynamics 2nd Edition. New York: Cambridge University Press. 2Leishman J.G. 1989. “Modeling Sweep Effects on Dynamic Stall,” Journal of the American Helicopter Society 34 (18), pp.

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8McCroskey W.J., McAlister K.W., Carr L.W., and Pucci S.L. 1982. “An Experimental Study of Dynamic Stall on Advanced Airfoil Sections,” Vols. 1, 2, &3 NASA TM-84245.

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10Carr L.W., McAlister K.W., and McCroskey W.J. 1977. “Analysis of the Development of Dynamic Stall Based on Oscillating Airfoil Measurements,” NASA TN D-8382.

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12McCroskey W.J. 1981. “The Phenomenon of Dynamic Stall,” NASA TM-81264. 13Leishman J.G. 1987. “Validation of Approximate Indicial Aerodynamic Functions for Two-Dimensional Subsonic Flow,”

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Helicopter Society, Washington D.C., May 20-22. 19Minnema J.E. 1998. “Pitching Moment Predicitions on Wind Turbine Blades Using the Beddoes-Leishman Model for

Unsteady Aerodynamics and Dynamic Stall,” Masters of Science Thesis, Department of Mechanical Engineering, University of Utah, Salt Lake City, UT.

20RCAS Theory Manual, Version 2.0, United States Army Aviation and Missile COMmand/AeroFlightDynamics Directorate (USAAMCOM/AFDD) Technical Report 02-A-005, US Army Aviation and Missile Command, Moffett Field, CA, June 2002.

21Bousman W.G. 2003. “Aerodynamic Characteristics of SC1095 and SC1094 R8 Airfoils,” NASA/TP-2003-212265, AFDD/TR-04-003.

22Hansen M.H., Gaunaa M., and Madsen H.A. 2004. “A Beddoes-Leishman Type Dynamic Stall Model in State-space and Indicial Formulations,” Riso National Laboratory, Roskilde, Denmark Riso-R-1354(EN).

23Bain J.J., Mishra S.S., Sankar L.N., and Menon S. 2008. “Assessment of a Kinetic-Eddy Simulation Turbulence Model for 3D Unsteady Transonic Flows,” 26th AIAA Applied Aerodynamics Conference, Honolulu, Hawaii, August 18-21.

24Nichols R.H. and Buning P.G. User’s Manual for OVERFLOW 2.1 version 2.1t, August 2008.

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modeling dynamic stall of sc-1095 airfoil at high mach number

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