Modelling local incidence

The local incidence at azimuth station ф and radial station r can be expanded as a linear combination of contributions from a number of sources, as indicated by eqn 3.300:

r; t) = apitch + atwist + aflap + aWh + ainflow (3.3°°)

The component apitch is the contribution from the physical pitch of the blade applied through the swash plate and pitch control system. The atwist component includes con­tributions from both static and dynamic twist; the latter will be discussed below in the Rotor dynamics subsection. The aflap component due to rigid blade motion has been fully modelled within the Level 1 framework; again we shall return to the elastic flap contribution below. The aWh component corresponds to the inclination of incident flow at the hub. Within the body of this chapter, the modelling of ainflow has been limited to momentum theory, which although very effective, is a gross simplification of the real helical vortex wake of a rotor. Downwash, in the form of vorticity, is shed from a rotor blade in two ways, one associated with the shedding of a (spanwise) vortex wake due to the time-varying lift on the blade, the other associated with the trailing vorticity due to the spanwise variation in blade lift. We have already discussed the inflow component associated with the near (shed) vortex wake due to unsteady motion; it was implicit in the indicial theory of Beddoes and Leishman. Modelling the trailing vortex system and its effect on the inflow at the rotor disc has been the subject of research since the early days of rotor development. Bramwell (Ref. 3.6) presents a comprehensive review of activities up to the early 1970s, when the emphasis was on what can be described as ‘prescribed’ wakes, i. e., the position of the vortex lines or sheets are prescribed in space and the induced velocity at the disc derived using the Biot-Savart law. The strength of the vorticity is a function of the lift when the vorticity was shed from the rotor, which is itself a function of the inflow.

Solving the prescribed wake problem thus requires an iterative procedure. Free wake analysis allows the wake vorticity to interact with itself and hence the posi­tion of the wake becomes a third unknown in the problem; a free wake will tend to roll-up with time and hence gives a more realistic picture of the flowfield downstream of the rotor. Whether prescribed or free, vortex wakes are computationally intensive to model and have not, to date, found application in flight simulation. As distributed flowfield singularities, they also represent only approximate solutions to the underlying equations of fluid dynamics. In recent years, comprehensive rotor analysis models are beginning to adopt more extensive solutions to the three-dimensional flowfield, using so-called computational fluid dynamics techniques (Ref. 3.3). The complexity of such tools and the potential of the achievable accuracy may be somewhat bewildering to the flight dynamicist, and a real need remains for simpler approximations that have more tractable forms with the facility for deriving linearized perturbations for stability analy­sis. Earlier, in Section 3.2.1, we referred to the recent development of wake models that exhibit these features (Refs 3.28, 3.29), the so-called finite-state wake structures. Here, the inflow at the rotor is modelled as a series of modal functions in space-time, each satisfying the rotor boundary conditions and the underlying continuity and momentum equations, through the relationship with the blade lift distribution. The theory results in a series of ordinary differential equations for the coupled inflow/lift which can be appended to the rotor dynamic model. Comparison with test results for rotor inflow in trimmed flight (Ref. 3.29) shows good agreement and encourage further development and application with this class of rotor aerodynamic model.