Practical application of adjoint variables to real-time computation of off-rotor induced velocities

The finite-state inflow model is widely used in real-time simulations. Coupling such a model with rotor blade flapping requires that the induced velocity be computed at the rotor disk, but the model also allows the flow to be computed at any location in the flow field above the rotor plane. However, to compute the flow field below the rotor disk (such as is required for co-axial rotors) requires computation of the adjoint variables (along with the normal state variables) including a time delay. Since the adjoint variables must be marched backwards in time, this can pose a problem in real time analysis. In this paper, computation of the adjoint variables (and flow below the disk) is addressed in the time domain. For illustrative purposes, the parameters for the two-blade Harrington co-axial rotor are used. A step input is given to collective pitch in hover. The blade sectional lift is then calculated based on combined blade-element theory and on dynamic wake modeling (including blade flapping). These equations are first time-marched forward to give the conventional state variables in the time domain. The co-state theorem is then introduced to calculate the co-states and the induced velocity below the rotor. Two alternatives methods are explored in order to compute the adjoint variables with time-delay. The first is the convolution method (in which at every time steps the adjoint variables are computed by a closed-form convolution). The second method is to march backwards in time for the co-states (i.e., adjoint variables). Two methods are considered for this second method: 1.) time marching bacwards at every three time steps, and 2.) time-marching backwards once at the end of the domain of interest. The various methods are compared for computational efficiency and numerical accuracy.