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A pseudo-implicit predictor-corrector method with five-point central differencing in space has been developed for the solution ofthe governing differential equations ofthe rotor free-wake problem. It is shown that a relaxation implementation of this method exhibits good convergence characteristics for the rotor tip vortex locations. The method is also generalized to allow unequal azimuth and trailed vortex segment angles, which for some applications, help retain the fidelity of the results. while keening eomontational reauirements to a minimum. The results are comnared with exuerimental mea-. " . surements of the rotor tip vortex geometries and flow field velocities in hover and forward flight with good agreement. Nomenclature c Blade chord. m Rotor thrust coefficient, Tlpn R~(QR)= Unit vectors in the x, y and z directions, respectively Number of blades Number of tip vortex filaments Position vector of a point on a vortex filament, m Tip vortex viscous core radius, m Radial position from which vortex filaments are trailed, mRotor radius, m Time, s Rotor thrust, N Tangential velocity, m/s Free-stream velocity, m/s Induced velocity vector, m/s 5 Distance along trailed tip vortex, rad. P Rotor advance ratio, V-IQR v Kinematic viscosity of air, m2/s * Azimuth angle, rad q b Blade azimuth angle, rad. qW Tip vortex age, rod. Q Rotational frequency of rotor, rod/s

A free-wake model has been developed to predict the wake geometries of tandem, tilt-rotor and coaxial twin-rotor systems. The basic formulation is based on a pseudo-implicit, predictor-corrector, relaxation scheme with five-point central differencing and velocity averaging in space. Results are presented for three rotor configurations for several advance ratios ranging from relatively low speed to moderately high speed forward flight conditions. Comparisons have been made between computed and measured tip vortex geometries for the coaxial configuration, with gwd overall agreement. Nomenclature 52 Rotational frequency of rotor, radls '4 Azimuth angle, rad c Blade chord, m x... Position vectors of rotor hub 2 relative to hub 1, m C, Rotor thrust coefficient, T I ,~R~(~~R )~ 5 Distance along trailed wake filament, rad CQ Rotor torque coefficient, T I~~R~( Q R )~R A* Discretized azimuthal step size, rad h Perpendicular distance from vortex element to evaluation ~i~~~~~i~~d angular step along vortex filament, point, m A O A I , J , k Unit vectors in the x , y and z directions, respectively Number of blade azimuthal steps in a rotor revolution Introduction J m x k , , , Number of collocation points on a vortex filament The ability to accurately predict the rotor aerodynamic environment Nb Number of blades is essential for the development of helicopters with enhanced perfor-N, Number of vortex filaments mance, reduced vibration, and improved acousticcharacteristics. While Q Rotor torque, Nm blade-element/momentum considerations provide areasonable predic-7 Position vector of an evaluation (collocation) point, m tion of rotor thrust and power, vortex methods with their ability to r~ Vortex core radius, m resolve the flow to a higher resolution, are required to delineate localrv Radial position along blade from which tip vortex filaments i x d phenomena such as blade vortex interactions (BVD, which conare trailed, m tribute to vibrations and acoustics. R Rotor radius, m Experimentally, the blade tip vortices have been found to be the T Rotor thrust, N most dominant smctures in the flow field. Unlike the case for fixed ? Local velocity at a point, mls wing aircraft, these vortices remain close to the plane of the rotor and f, Free-stream velocity, d s have a powerful influence on the spatial and temporal variations in the kd Induced velocity, d s aerodynamic loading on each blade. For multi-rotor systems, the prohx, y, Cartesianc,,,rdinate system, origin at hub center of lem becomes even more acute since the mutual interference between Rotor shaft longitudinal tilt angle, rad rotors results in a much greater and more complicated wake distortion. 9 Pa Blade coning angle, md In addition, there is an increased likelihwd of BVI for multi-rotor sysr Circulation, m2/s terns, since the wake from one rotor can he ingested directly by the 4 Uniform induced inflow ratio other, e.g. Ref. 1. P Rotor advance ratio, V,/Q R Vortex methods determine the rotor induced velocity field by comv Kinematic viscosity of air, m2/s bining, at a poi...

Mathematical results are derived for the schlieren and shadowgraph contrast variation due to the refraction of light rays passing through two-dimensional compressible vortices with viscous cores. Both standard and small-disturbance solutions are obtained. It is shown that schlieren and shadowgraph produce substantially different contrast profiles. Further, the shadowgraph contrast variation is shown to be very sensitive to the vortex velocity profile and is also dependent on the location of the peak peripheral velocity (viscous core radius). The computed results are compared to actual contrast measurements made for rotor tip vortices using the shadowgraph flow visualization technique. The work helps to clarify the relationships between the observed contrast and the structure of vortical structures in density gradient based flow visualization experiments. Nomenclature c f Cr I 1 nb p p~ r, 0, Z rc R o0 Y F~ 7o /r p p~o 17 f2 Unobstructed height of schlieren light source in cutoff plane, m Blade chord, m Focal length of schlieren focusing mirror, m Rotor thrust coefficient, T/(pTz~2R 4) Image screen illumination, Zm/m 2 Distance from vortex to shadowgraph screen, m Number of blades Pressure, N/m 2 Ambient pressure, N/m 2 Cylindrical coordinate system Vortex core radius, m Non-dimensional radial coordinate, (r/rc) Rotor radius, m Tangential velocity, m/s Specific heat ratio of air Circulation (strength of vortex), m2/s Non-dimensional quantity, ( F 2 p~/8n2~poor 2) Refractive index of fluid medium Refractive index of fluid medium at reference conditions Gladstone-Dale constant, ma/kg Density, kg/m a Density at ambient conditions, kg/m a Non-dimensional density,

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