An Anechoic Wind Tunnel for the Investigation of the Main-Rotor/TailRotor Blade Vortex Interaction Con J. Doolan* and Damien Leclercq# #
*
[email protected] Lecturer School of Mechanical Engineering The University of Adelaide, SA, 5005
[email protected] Research Fellow School of Mechanical Engineering The University of Adelaide, SA, 5005
ABSTRACT The interaction of a helicopter’s main rotor tip vortex with the tail rotor is an important source of noise and vibration, yet it is still poorly understood. An important limiting case is the orthogonal blade vortex interaction (OBVI) where a three-dimensional vortex structure is cut by the tail rotor blade. It has been discovered that the blade unsteady surface pressures during OBVI are controlled by the tip vortex axial flow component. Recent aerodynamic testing of the OBVI confirming this result is reviewed along with a description of an anechoic wind tunnel for aero-acoustic OBVI research. The design of a new wind tunnel contraction and a single bladed rotor rig are also discussed.
Introduction
therefore an area of great interest to rotorcraft designers, manufacturers and operators.
The flow field about a helicopter airframe is dominated by vortices generated at the tips of the main rotor blades. The interaction of these vortices with other rotor blades and parts of the airframe is a source of unwanted noise and vibration and is
There are three broad classifications that can describe rotorcraft interactional aerodynamics. These are the main-rotor blade vortex interaction (MR BVI), the main-rotor/tail-rotor blade vortex interaction (MR/TR BVI) and the main-rotor/tailboom interaction.
Tail Rotor Blade Vortex Interaction Tail Rotor
Rotor Rotation
Main Rotor Flight Velocity
Main Rotor Tip Vortex 1 Main Rotor Blade Vortex Interaction Main Rotor Tip Vortex 2
Figure 1 Schematic illustrating two types of blade vortex interaction that occur on a rotorcraft in forward flight. Figure 1 illustrates MR BVI and MR/TR BVI on an idealised rotor arrangement in forward flight. Only two main rotor tip vortices are shown in this figure to enhance the clarity of the following description. The MR BVI occurs when tip vortex 1 interacts with a following main rotor blade. Tip vortex 2 however, is ingested by the tail rotor disc and consequently, this is called the MR/TR BVI.
Figure 2 shows a typical noise spectrum from a helicopter in forward flight [1] and demonstrates the significance of both main rotor and tail rotor vortex interactions to the overall noise level. Usually, main rotor vortex interactions are responsible for the familiar ‘thumping’ noise of helicopters and tail rotor vortex interactions are responsible for what is commonly called ‘tail rotor
Presented at the 6th Australian Vertiflite Conference on Helicopter Technology, Melbourne, VIC, March 19-22, 2007. Copyright © 2007 by the American Helicopter Society International, Inc. All rights reserved.
burble’. Much research and development effort has been expended in understanding and modelling the main rotor vortex interaction (see for example, Ref. [2]). Much less research however, has been aimed at understanding the interaction of the main rotor tip vortex and the tail rotor, despite this being a significant noise source in the rotorcraft acoustic field. In fact, tail rotor interaction noise has been studied sporadically for less than 30 years and only recently has the fluid dynamic aspects of the interaction been clarified through a series of experiments [3-8].
tail rotor. As the vortex moves over the blade, the vortex core is ‘cut’ or separates into two components. The dynamics of how the vortex achieves this cut determines the unsteady loading on the surface of the blade.
Figure 3 The Orthogonal Blade Vortex Interaction.
Figure 2 Typical noise spectrums from a helicopter in forward flight showing the relative significance of both main rotor and tail rotor interaction noise [1]. A recent, excellent review of helicopter tail rotor interactions has been performed by Ref. [9] and the reader is referred to this review to understand further aspects of MR/TR BVI such as the effects on control and handling. This paper will summarise existing research results concerning the OBVI and highlight future research needs, especially those relating to noise. The paper will then describe the anechoic wind tunnel at the University of Adelaide including the design of a new contraction section for OBVI research. The paper will conclude with a brief discussion of the design of a scale rotor-rig for OBVI aero-acoustic measurements.
Orthogonal BVI The OBVI is a particular, limiting case of the MR/TR BVI. As discussed above, tip vortices are shed from the main rotor and convect into the path of the tail rotor blades when the aircraft is in forward flight. The actual geometry of the interaction depends on the particular flight condition. The OBVI represents a particular subset of MR/TR BVI and is thought to occur in straight forward flight [9]. Figure 3 defines the OBVI as considered in our research. It consists of a concentrated tip vortex moving with respect to a blade that represents the
Unlike other forms of BVI, the OBVI is highly three-dimensional and is controlled by both the tangential and axial flow components of the concentrated tip vortex. It has been shown [10,11] that rotor blade tip vortices have significant axial flow within their cores due to the momentum deficit within the main rotor turbulent wake. This axial flow component persists within the core for significant wake ages (over 230 degrees of main rotor revolution [9]) and therefore plays an important role when the vortex interacts with tail rotor blades.
OBVI Aerodynamic Experiments One of the authors (Refs. [3-7]) has recently performed a series of wind tunnel tests investigating rotor tip vortex velocity distributions and how rotor tip vortices interact with a representative tail rotor blade. A summary of this work will now be presented as a review of previous work in OBVI measurements. Figure 4 is a schematic representation of the experimental set-up used to simulate a main-rotor tip vortex in a low speed wind tunnel. It consists of a single-bladed rotor rig that has a unique pitching arrangement so that the rotor blade is only at incidence when it passes the working section of the wind tunnel. Two experimental rigs were developed. The first was used in a 1.15 m x 0.85 m low speed wind tunnel and this arrangement is described in Figure 4. It had a rotor radius of 0.75 m with a NACA 0015 geometry using 100 mm chord. In these experiments, an instrumented NACA 0015 blade of 152.4 mm chord was placed 13.12 chord lengths downstream of the rotor centre-line to simulate a tail rotor blade. The
instrumented blade contained a chordal array of 30 miniature Kulite pressure sensors mounted to measure the surface pressure around the blade during a vortex cut or OBVI. The blade was translated across the tunnel working section so that transducers were positioned in the path of the vortex core. The second series of experiments were performed in the larger University of Glasgow 2.65 m x 2.04 m Argyle wind tunnel (Figure 5). A 1.6 m radius blade was used here again with a NACA 0015 geometry using 160 mm chord. An instrumented blade (275 mm chord, NACA 0015 profile) was placed 14.55 blade chord diameters downstream of the vortex generator axis as part of the research program to simulate OBVI. The aim of these tests was to investigate the effect of Reynolds number and scale on the OBVI event.
(a) Schematic
(b) Photograph viewed from wind tunnel settling chamber. Figure 5 OBVI vortex generator in University of Glasgow 2.65 m x 2.04 m Argyle wind-tunnel. (a) Side View
A summary of the vortex properties for the two experiments is given in Table 1. The values are quoted at the hot-wire measuring position which corresponds to the leading edge of the instrumented blade. It can be seen that the nondimensional circulation strength Γ is nearly ΩRc
(b) Plan Figure 4 Schematic illustrating OBVI experiment as built in the University of Glasgow 1.15 m x 0.85 m low speed wind tunnel. A hot-wire anemometer was used to measure the velocity components of the tip vortex generated by the rotor rigs. Figure 6 shows a typical velocity measurement as obtained from the smaller rotor rig with time non-dimensionalised with free-stream velocity and instrumented blade chord. Important features to note are the easily identified tip vortex structure in the upper plot, which shows the vertical velocity component which is dominated by the vortex tangential velocity. The lower plot of Figure 6 shows the transverse velocity component which aligns with the tip vortex axis and hence shows the axial flow within the vortex core.
identical between the two rigs. However, the core size is much bigger for the large rotor rig. This is due to the higher wake age allowing increased viscous diffusion of the vortex core. Despite this increased wake size, a strong vortex interaction was measured in the Argyle wind tunnel.
Figure 6 Rotor tip vortex velocity measurements [5]. Table 1 Tip Vortex Parameters Small Rotor Rig Large Rotor Rig (20 m/s) (40 m/s) Mtip Re(tip)
0.18 3.95 x 105
0.30 1.0 x 106
Re(blade)
3.1 x 105
1.0 x 106
rc/c (core size)
0.065±0.031
0.27±0.037
Γ ΩRc Wander (% blade c)
0.144±0.022
0.140±0.028
29.5%
1.8%
Figure 7 Schematic of the vortex cutting process. Typical surface pressure measurements obtained during an OBVI event are shown in Figure 8. In this figure, the axial flow component is arranged so that it is flowing away from the upper surface and towards the lower surface. Figure 8 presents the results in the form of a carpet plot, with nondimensionalised distance and time both represented. The surface pressure coefficient is arranged so that a positive number indicates suction and a negative number indicates pressure. In addition, the pressure coefficients in Figure 8 have been manipulated to show only the unsteady component. This was done by subtracting the steady state surface pressure from the unsteady pressure measurements during the post-experiment analysis.
Much effort was spent on accurately controlling the speed of the large rotor rig. This effort was repaid by the fact that much reduced vortex wander was measured in the Argyle wind tunnel. The axial flow within the vortex core complicates the unsteady loading of the tail rotor blade. As the vortex is cut by the blade, each side of the vortex responds in a different manner, controlled by the direction of the axial flow with respect to the blade surface. Figure 7 illustrates how the vortex responds just after it is cut. For the side where the axial core velocity is away from the blade surface, the core thins. The reverse is true for the side where the axial core velocity is towards the surface of the blade. Here, the vortex core enlarges or bulges in response to the cut. In many respects, the cutting of a main rotor tip vortex has many similarities to the impulsive blocking of an incompressible jet.
Figure 8 Surface pressure measurements obtained during an OBVI. (a) Upper surface, (b) Lower surface.
The blade surface pressure can be related to the velocity components of the vortex. For the upper surface where the axial flow is away from the blade, the interaction consists of an intense suction peak about the leading edge of the blade. This is due to the initial cutting of the vortex and the inertia of the axial core flow reducing pressure about the leading edge. After the vortex has travelled approximately 30% of the chord, the suction peak has reduced to an almost constant level across the chord. The maintenance of the suction peak is possibly due to a tornado-like effect where the vortex is transferring mass from the blade boundary layer into the vortex core. On the lower surface (Figure 8 (b)), the axial flow component is directed towards the blade, resulting in a very different response. An impulsive pressure pulse is measured about the leading edge, thought to be due to the impulsive blocking of the core jet. At approximately 20% of the chord, most of the pressure pulse has diminished and a small suction ridge begins to form which is maintained over the entire chord. This suction ridge is thought to be due to a region of localised separation resulting from an interaction of the blade boundary layer and vortex tangential components. As shown, the unsteady surface pressure varies greatly on either side of the blade during an OBVI event. If the surface pressures are integrated about the chord for each sampling period, impulsive force and quarter chord pitching moment coefficients can be obtained.
the vortex strikes the leading edge ( tV∞ c ~ 1 − 2 ) the normal force increases rapidly, followed by a gradual decay. Quarter chord pitching moment (Figure 9(b)) shows a sudden sign reversal, indicating the effect of the three-dimensional vortex as it passes the quarter chord point.
Future Research Needs While a good understanding has now been obtained of the fluid dynamic processes controlling OBVI, much further work is now required. Our aim, as engineers, should be to use our knowledge to design quiet rotorcraft. Before we can do this, a greater understanding of the relationship between the OBVI aerodynamics and noise emission must be obtained. Once this relationship can be understood new rotorcraft noise prediction tools can be developed that contain tip vortex axial flow components. These models can be placed with larger rotorcraft wake models to give designers a more accurate representation of the flow field and hence, better assessment of future designs. This is the underlying principle behind the University of Adelaide OBVI aero-acoustic program.
University of Adelaide Anechoic Wind Tunnel As stated above, the scope of the work has now shifted to relating the aerodynamics at the blade surface to the far-field noise. Hence a new research program at the University of Adelaide has commenced which will investigate the aeroacoustics of the OBVI and methods for its control. Of special importance for the new work is to develop good quality data for the validation of computer models. To carry out this work, an existing anechoic wind tunnel is being upgraded to perform OBVI experiments, with a focus on relating the unsteady blade loading to noise generation. This part of the paper will describe the existing facility, the design of a new contraction section for OBVI research and conclude with a description of the OBVI experimental rig. Existing Anechoic Facility
Figure 9 Unsteady normal force (a) and quarter chord pitching moment (b) coefficients during OBVI. Figure 9(a) shows the variation in normal force coefficient during OBVI. It can be seen that after
The University of Adelaide anechoic wind tunnel is shown in schematic form in Figure 10. Air is supplied from large compressed air tanks which are filled using a compressor. The air discharges though a combined silencer and settling chamber before passing through a contraction to exhaust over in an anechoic chamber. The current outlet dimensions of the contraction section are 50 mm x 50 mm. The anechoic chamber has a size of 2 m x 2 m x 2 m and an acoustic cut-off frequency of 200
Hz. The maximum speed of the wind tunnel is 70 m/s, however the usual working speed is 30 m/s.
To obtain an appreciation of the flow quality available from the existing facility, Pitot surveys at various downstream distances from the contraction outlet are displayed in Figure 11. As shown, the wind tunnel can provide an extremely uniform test flow with a boundary layer on the order of 1.5 mm. In addition, turbulence measurements at the wind tunnel centre-line indicate a turbulence intensity of 0.37%.
Silencer Inlet air Contraction Anechoic Chamber
Model
Figure 10 Schematic of University of Adelaide anechoic wind tunnel. 35.00 Dynamic Pressure (mbar)
30.00
25.00 X=0 mm (Centre) 0mm X=0 mm (Centre) 20mm X=0 mm (Centre) 40mm X=0 mm (Centre) 60mm X=0 mm (Centre) 80mm X=0 mm (Centre) 100mm X=0 mm (Centre) 150mm X=0 mm (Centre) 195mm
20.00
15.00
10.00
5.00
0.00 -35.0
-30.0
-25.0
-20.0
-15.0
-10.0
-5.0
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
Y position (mm) -5.00
Figure 11 Dynamic pressure measurements across existing outlet at 30 m/s flow velocity. Wind Tunnel Upgrade Program While the existing wind tunnel is of excellent quality, the outlet size is too small for our planned OBVI program. The contraction has therefore been re-designed to allow a greater flow area in the anechoic chamber. The OBVI tests require a two-dimensional test flow through a 275 mm x 75 mm outlet. To fit to the existing settling chamber, this requires a contraction ratio of 3.67, which is outside the normal recommendations for contraction design [12]. Therefore, in order to obtain a test flow of high quality, we must ensure that the flow does not separate from the walls of the contraction. Also, the uniformity of the flow at the outlet must be within an acceptable tolerance (a standard deviation of 1% was the specification used for our design study). Many techniques exist to compute the flow field within the contraction. One of most popular techniques is to use a steady Reynolds
Averaged Navier-Stokes (RANS) method to compute the flowfield. While this technique will compute the separation point and flow uniformity, it is computationally expensive, requiring many hours on a PC to solve a particular test case. For a design exercise, it was though that this method was too time-consuming. Instead, a potential flow solver [13] was employed to obtain an inviscid solution of the contraction flow field. A thin, laminar boundary layer was assumed to exist on the surface of the contraction and the potential flow solution was used as the boundary condition for the outer surface of the boundary layer. Boundary layer solutions were obtained using Thwaite’s method [14]. Figure 12 shows a solution of the flow within the new contraction calculated using the potential flow solver. A mesh refinement study determined that a 150 x 20 x 20 mesh
was sufficiently accurate for this study. Table 2 summarises the technical details of the new contraction. The most important information to note is that a 5th order polynomial was selected for the contraction shape and that a 275 mm constant area extension was required after the contraction to prevent flow separation. Also, the available flow area is reasonable (as defined by the Reynolds number based on momentum thickness) as is the flow uniformity at the exit.
are identical and present a more challenging environment for an attached boundary layer. For this reason, the flow in the corner region was selected for further analysis using a viscous flow solver.
The potential flow solution gives the designer velocity and hence pressure along the surface of the contraction. The surface pressure coefficients along the surface are shown in Figure 13. Here, pressure coefficient is defined as:
U C p = 1 − U exit
2
(1)
Pressure coefficient was calculated along three surface lines. These were positioned axially along the centre of the top surface, along the corner joint between the top and side surfaces and along the centre of the side surface. As shown, the corner and top pressure coefficients
Figure 12 Potential flow solution for new contraction section for the University of Adelaide anechoic wind tunnel. Note only right half of contraction is displayed and U* is velocity normalised by the inlet velocity. Flow is from left to right.
Table 2 Contraction Technical Details Contraction Ratio 3.67 Contraction Length (mm)
550
Extension Length (mm)
275
Inlet Height (mm)
275
Outlet height (mm)
75
Width (mm)
275
Design Test Speed (m/s)
30
Contraction Shape
5th Order Polynomial
Reθ
600.72
Exit Plane Uniformity
0.41%
As mentioned above, a laminar viscous flow solver was developed for this design study. This was based on Thwaite’s method [14]. A laminar boundary layer was assumed, based on the research of Ref. [12]. Ref. [12] argue that a re-laminarisation process takes place downstream of a wind tunnel’s settling chamber screens and along the walls of its
contraction. This assumption has been shown to be reasonable and accurate for small contractions such as the one under consideration in this paper. The viscous flow solver computes the boundary layer height and skin friction coefficient along the contraction surface. Separation is deemed to occur if the skin
friction drops to zero. Figure 14 shows the results of the viscous flow solution for the new wind tunnel contraction. It shows that the skin friction coefficient remains positive along the contraction length indicating that separation will not occur. OBVI Aero-Acoustic Experiment The final section of this paper will present a brief outline of the planned OBVI aeroacoustic experiment. Figure 15 shows a schematic of the intended experimental layout. Figure 15(a) is a plan view and illustrates the how the experiment is intended to function. In this view, the method of generating a vortex transversely across the test cross-section is explained. The vortex generator is a single bladed rotor where the rotor blade pitch is varied in four equivalent (90°) phases of azimuth.
Figure 13 Pressure coefficient along contraction as calculated by the potential flow solver. When the rotor blade is pointing away from the test section (phase I), the rotor blade has zero angle of attack and generates no tip vortex. As the rotor blade turns towards the test section (phase II), the angle of attack is continuously varied until it reaches 10°, where it remains constant as the blade passes in front of the test section (phase III). During this phase of operation, the vortex generator creates a tip vortex of constant strength that is then convected downstream by the wind tunnel flow and subsequently interacts with the instrumented blade. After this phase of operation, the vortex generator blade is continuously pitched down (phase IV) until it is at zero angle of attack and the cycle repeats. Hence the vortex generator creates a continuous stream of three-dimensional vortices that are available to interact with an instrumented blade placed immediately downstream as shown in Figure 15(a).
Effective vortex cutting can only occur if the ratio of interacting blade thickness to vortex core diameter (known as the thickness ratio, T = t/rc) is sufficiently small. Previous experiments [3-7] have shown that successful vortex cutting can occur if 0.5 < T < 3.5. Based on rotor wake vortex measurements [15], it can be estimated that the minimum vortex radius (rc) generated will be 0.1-0.15 of the rotor chord. Hence the interacting blade may have a thickness t of up to 10.5 mm and a chord of up to 70 mm (if a NACA 0015 blade is used) for a successful experiment. Figure 15 is scaled for an interacting blade chord of 40 mm. This is an intermediate value designed to ensure clean vortex cutting and acceptable room for four (4) custom made internal miniature pressure transducers.
Figure 14 Skin friction coefficient and momentum boundary layer height along contraction length. The internal pressure transducers will be constructed using a microphone connected to a capillary tube, as designed previously by Leclercq and Bohineust [16]. Such an arrangement allows an accurate surface pressure measurement over a very small surface area, to prevent spatial averaging issues from affecting the measurements in the frequency range of interest. Provision will be made to vary the angle of incidence in order to measure the acoustic emission at various blade loadings so as to compare with previous studies of the fluid mechanics. The leading edge of the instrumented blade can be positioned very close to the vortex generator (1-2 vortex diameters) in order to obtain a strong acoustic signature. In reality, helicopter main rotor vortices travel approximately 3-4 main rotor chord lengths before striking the tail rotor. Previous studies ([3-8]) have not been able to achieve such close spacing of the instrumented blade to the vortex generator due to the constructional
layout of the wind tunnels used. A large spacing results in significant viscous dissipation in the core of the vortex so that it is considerably weakened when it finally interacts with the instrumented blade. In this project, there is an opportunity to investigate the OBVI at close range representing vortex travel more characteristic of actual rotorcraft (0.3 to 5 main rotor chord lengths). Close range testing will mean that a much higher strength vortex will be available resulting in strong surface pressure and acoustic signals.
impulsive blade loading during OBVI. Based on these results, the University of Adelaide have commenced a research program that will investigate the aero-acoustics of OBVI events. The aim of the program will be to obtain a better fundamental understanding of noise generation during OBVI and to obtain quality data for later validation of computer models. To support the research effort, the University of Adelaide anechoic wind tunnel will be upgraded to support a larger test flow. The design of a new contraction has been described in this paper along with the proposed experimental rig for OBVI research.
Comprehensive velocity measurements will be obtained using a hot-wire anemometer system. Surface pressure measurements will also be recorded simultaneously with far-field noise measurements.
Acknowledgements The University of Adelaide research work is supported by the Sir Ross and Sir Keith Smith Fund. The University of Glasgow research was jointly supported by the UK EPSRC, Ministry of Defence and Westland Helicopters plc.
Summary Recent research results concerning the orthogonal blade vortex interaction have been summarised. It was shown the axial flow component within the core of the main rotor tip vortex is responsible for much of the
Phase IV Rotor Flow Direction
Convecting Tip Vortex Phase III
Phase I
Phase II
Tunnel Wall
Instrumented Blade
(a) Plan Vortex Generator
Tip Vortex
Pressure Transducers
Flow Direction Potential Core (with boundary layer correction) Shroud
Provision for Axial Movement of Blade
Rotor Drive
(b) Side Elevation
Figure 15 Schematic of orthogonal blade vortex interaction experiment.
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Orthogonal Vortex Interaction,” Journal of the American Helicopter Society, July, pp 1-7, 2001. [4] Doolan, C.J., Coton, F.N. and Galbraith, R.A.McD. “Surface Pressure Measurements of the Orthogonal Vortex Interaction,” AIAA Journal Vol. 39, No. 1, 2001. [5] Doolan, C.J., Coton, F.N. and Galbraith, R.A.McD. “Three-Dimensional Vortex Interactions with a Stationary Blade,” The Aeronautical Journal, December 1999.
[6] Green, R.B., Doolan, C.J. and Cannon, R.M. “Measurements of the Orthogonal Blade-Vortex Interaction using a PIV technique,” Experiments in Fluids, Vol 29, 2000. [7] Wang, T., Doolan, C.J., Coton, F.N. and Galbraith, R.A.McD. “An Experimental Study of the Three-Dimensionality of Orthogonal BladeVortex Interaction,” AIAA Journal, Vol. 40, No. 10, 2002. [8] Marshall, J.S. and Khrishnamoorthy, S. “On the Instantaneous Cutting of a Columnar Vortex with Non-Zero Axial Flow,” Journal of Fluid Mechanics, Vol. 351, 1997. [9] Coton, F.N., Marshal, J.S., Galbraith, R.A.McD. and Green, R.B. “Helicopter Tail Rotor Orthogonal Blade Vortex Interaction,” Progress in Aerospace Sciences, Vol. 40, 2004. [10] Thompson, T.L., Komerath, N.M. and Gray, R.B. “Visualization and Measurement of the Tip Vortex Core of a Rotor Blade in Hover” J. Aircraft, Vol. 25, (12), December, 1988. [11] Bagwhat. M.J. and Leishman, J.G. “Correlation of Helicopter Rotor Tip Vortex Measurements”, AIAA Journal, Vol. 38, (2), February, 2000. [12] Bell, J.H. and Mehta, R.D. “Contraction Design for Small Low-Speed Wind Tunnels,” NASA Contractor Report, NASA-CR-177488, August 1988. [13] Jasak, H., Weller, H.G. and Nordin, N. ” Incylinder CFD simulation using a C++ objectoriented toolkit,” SAE Technical Paper 2004-010110, 2004. See also www.openfoam.org [14] Cebeci, T. and Bradshaw, P. Momentum Transfer in Boundary Layers, McGraw Hill, New York, USA, 1977. [15] Martin, P.B. and Leishman, J.G. “Trailing Vortex Measurements in the Wake of a Hovering Rotor Blade with Various Tip Shapes,” Presented at the 58th Annual Forum of the AHS International, Montréal Canada, June 11-13, 2002. [16] Leclercq, D.J.J. and Bohineust, X. “Investigation and modelling of the wall pressure field beneath a turbulent boundary layer at low and medium frequencies,” Journal of Sound and Vibration, Vol. 257, No. 3, 2002.
