Wake analysis methods
By analysis of his carefully conducted series of smoke-injection tests, Landgrebe5 reduced the results to formulae giving the radial and axial coordinates of a tip vortex in terms of azimuth angle, with corresponding formulae for the inner sheet. From these established vortex positions the induced velocities at the rotor plane may be calculated. The method belongs in a general category of prescribed-wake analysis, as do earlier analyses by Prandtl, Goldstein and Theodorsen, descriptions of which are given by Bramwell. These earlier forms treated either a uniform vortex sheet as pictured in Fig. 2.11 or the tip vortex in isolation, and so for practical application are effectively superseded by Landgrebe’s method. More recently, considerable emphasis has been placed on free-wake analysis, in which modern numerical methods are used to perform iterative calculations between the induced velocity distribution and the wake geometry, both being allowed to vary until mutual consistency is achieved. This form of analysis has been described for example by Clark and Leiper6. Generally the computing requirements are very heavy, so considerable research effort also goes into devising simplified free-wake models which will reduce the computing load. Calculations for a rotor involve adding together calculations for the separate blades. Generally this is satisfactory up to a depth of wake corresponding to at least two rotor revolutions. A factor which helps this situation is the effect on the tip vortex of the upwash ahead of the succeeding blade – analogous to the upwash ahead of a fixed wing. The closer the spacing between blades, the stronger is this effect from a succeeding blade on the tip vortex of the blade ahead of it; thus it is observed that when the number of blades is large, the tip vortex remains approximately in the plane of the rotor until the succeeding blade arrives, when it is convected downwards. In the ‘far’ wake, that is beyond a depth corresponding to two rotor revolutions, it is sufficient to represent the vorticity in simplified fashion; for example free-wake calculations can be simplified by using a succession of vortex rings, the spacing of which is determined by the number of blades and the mean local induced velocity. Eventually in practice both the tip vortices and the inner sheets from different blades interact and the ultimate wake moves downward in a confused manner.
By analysis of his carefully conducted series of smoke-injection tests, Landgrebe5 reduced the results to formulae giving the radial and axial coordinates of a tip vortex in terms of azimuth angle, with corresponding formulae for the inner sheet. From these established vortex positions the induced velocities at the rotor plane may be calculated. The method belongs in a general category of prescribed-wake analysis, as do earlier analyses by Prandtl, Goldstein and Theodorsen, descriptions of which are given by Bramwell. These earlier forms treated either a uniform vortex sheet as pictured in Fig. 2.11 or the tip vortex in isolation, and so for practical application are effectively superseded by Landgrebe’s method. More recently, considerable emphasis has been placed on free-wake analysis, in which modern numerical methods are used to perform iterative calculations between the induced velocity distribution and the wake geometry, both being allowed to vary until mutual consistency is achieved. This form of analysis has been described for example by Clark and Leiper6. Generally the computing requirements are very heavy, so considerable research effort also goes into devising simplified free-wake models which will reduce the computing load. Calculations for a rotor involve adding together calculations for the separate blades. Generally this is satisfactory up to a depth of wake corresponding to at least two rotor revolutions. A factor which helps this situation is the effect on the tip vortex of the upwash ahead of the succeeding blade – analogous to the upwash ahead of a fixed wing. The closer the spacing between blades, the stronger is this effect from a succeeding blade on the tip vortex of the blade ahead of it; thus it is observed that when the number of blades is large, the tip vortex remains approximately in the plane of the rotor until the succeeding blade arrives, when it is convected downwards. In the ‘far’ wake, that is beyond a depth corresponding to two rotor revolutions, it is sufficient to represent the vorticity in simplified fashion; for example free-wake calculations can be simplified by using a succession of vortex rings, the spacing of which is determined by the number of blades and the mean local induced velocity. Eventually in practice both the tip vortices and the inner sheets from different blades interact and the ultimate wake moves downward in a confused manner.
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