Validity of the Inflow Equation

The inflow equation as given by Eq. 2.126 or Eq. 2.137 is widely employed for practical calculations involving rotors in climbing and descending flight in both axial and forward flight. However, a nonphysical solution will always be obtained if there is a descent (upward) component of velocity normal to the rotor disk that is between 0 and 2t>, (i. e., if —2Vi < Voo since < 0 in level flight). Under these conditions there can always be two possible directions for the flow and there can be no well-defined slipstream boundary as was assumed in the physical model. Therefore, the momentum theory cannot be applied under these conditions. For example, this can occur when the rotor disk is at steep angles of attack.

With the numerical solution several things may occur if these restrictions are ignored. First, the simple fixed-point iteration method may not converge. This will always be the case for axial flight (/z. = 0) when —2 < Vc/vi < 0. Second, the fixed-point iteration method may converge, but to a nonphysical solution. In such cases caution should be exercised and the results should probably not be used unless validated by other means. The Newton – Raphson method will generally converge under all conditions, but again the solutions will be nonphysical in the range —2ty < Voo sin a < 0. The Newton-Raphson method is also sensitive to the initial conditions and may converge to different nonphysical solutions if inappropriate initial conditions are used. Generally it is assumed that ко = – JCt /2 and this will be satisfactory when the rotor is in the normal working state. However, this initial condition will cause the method to fail for descents (i. e., Vc < 0). Here, convergence of the Newton-Raphson method to the proper physical result can generally be ensured if ко = —kc. Overall, results from the numerical solution to the inflow equation should be used with sensible cross checks to the analytic results wherever possible.