In this paper, the potential flow equations are converted to ordinary differential equations through a Galerkin approach in which velocity and pressure potential functions are expanded in terms of closedform solutions to Laplace's equation. Because the method gives differential equations for the flow in terms of a relatively few generalized coordinates (that represent modes of the flow field) the resultant equations can be used effectively in preliminary design, real-time simulations, and dynamic eigenvalue analysis for aeroelasticity. This new theory is more general than the Peters-He dynamic wake model since it has a more rigorous derivation and includes inflow modes previously neglected in the Peters-He model. Results are presented in the frequency domain for simple harmonic motion. The axial velocity components are obtained by this new methodology on and off the disk, for axial and skewed flows, and for different pressure distributions and are compared with the Peters-He model and with an exact solution obtained by a convolution integral. INTRODUCTION Dynamic inflow models go back over 50 years in terms of their development and application. Sissingh first formulated a mathematical model that described how gradients in inflow could result from cyclic pitch changes or from fuselage pitch and roll rates. He further showed how these inflow gradients could create cyclic changes in blade angle of attack that significantly changed the actual rotor response. Curtiss and Shupe refined this idea into a lift deficiency function for rotor cyclic variations. Ormiston and Peters showed how the formulation could be generalized into an inflow model * AIAA Fellow Copyright © 2002 The American Institute of Aeronautics and Astronautics Inc. All rights reserved. for both uniform and side-to-side variations in inflow. Peters generalized these concepts to the unsteady case by including apparent mass terms to account for the time delay involved in the development of the rotor flow field. Comparisons with static data in hover and forward flight showed that the previous methodologies were inadequate in forward flight. Pitt and Peters used principles of potential flow theory to show that these concepts could be developed from first principles without ad hoc assumptions on time delay or on the effects of forward flight. They obtained an unsteady flow model from potential functions that gave spectacular correlation with wind tunnel response data throughout the frequency and advance ratio range. However, despite the unquestioned success of this inflow model, it still remained a theory with only three inflow degrees of freedom: 1) uniform, 2) fore-to-aft, and 3) side-to-side. Later, Peters and He showed how the Pitt-Peters ideas could be truly generalized to a theory with an arbitrary number of inflow harmonics and an arbitrary number of radial shape functions per harmonic. They showed how the general theory could reduce in special cases to the old Pitt-Peters model as well as to Loewy theory and Prandtl tip-loss theory. This new theory was used to correlate unsteady wind tunnel data from the NASA Langley wind tunnel for various planforms, thrust coefficients, and advance ratios. The match in both steady and unsteady distributions was excellent The Peters-He model represents a mature dynamic inflow model that is now used in many production codes, including 2GCHAS (Government code), FLIGHTLAB (Advanced Rotorcraft Technology), COPTER (Bell helicopter), ONERA-DFVLR (European Community) and many university applications. More recent extensions of the theorv include applications to the off-axis coupling problem, and to ground effect in flight simulation. 1 American Institute of Aeronautics and Astronautics (c)2002 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. The inflow models described in the above paragraphs are formulated such that the states of the model represent induced flow distributions on the rotor disk (expanded radially as polynomials and azimuthally as Fourier coefficients). The first-order differential equations for these states depend on the free-stream impingement angle, flight speed, and thrust level. However, they are all in closed form in terms of a mass matrix and a damping matrix, such that they are easily assembled either into a comprehensive code, an eigenvalue analysis, or a flight simulator. This makes the model extremely efficient. However, in many of the applications, such as the ground effect work, it is necessary to find not just the normal flow at the disk, but all three components of flow off of the disk as well. Peters and Morillo presented a consistent methodology for computing all three components of the flow in axial flow, both on and off of the rotor disc within the context of a finite-state model. The model is formulated in a manner fairly similar to previous work (in that the potential functions in ellipsoidal coordinates are used). However, in contrast with the previous work, the states represent velocity potentials rather than individual flow components. In addition, all potential functions are considered (not just the ones that have a pressure discontinuity across the disk); and the derivation of the equation coefficients is done in a more consistent and rigorous manner than in the earlier derivations. The result is a simpler derivation and a more complete inflow theory for the velocity both off and on the disk. Previous dynamic inflow models in axial flow are shown to be special cases of the new model when offdisk coupling is neglected. This new methodology based on a Galerkin approach provides the exact solution on as well as off the disk for the axial velocity component in axial flow. In this paper, the concepts introduced by Peters and Morillo are extended to analyze skewed flow cases. FLUID DYNAMICS EQUATIONS The three-dimensional potential flow equations (momentum and continuity equations) for the pressure and velocity fields P and v , with a free-stream velocity V^are f;-Jp-^ (1) V • v = 0 (2) These equations have been non-dimensionalized by defining P as pressure divided by pVj, v as induced velocity divided by V^ and time as a reduced time T, (i.e., time multiplied by VJR.) The variable £ is the non-dimensional coordinate along the free-stream line, positive upstream. All lengths are divided by the rotor radius R. Figure 1 shows the coordinate system. Fig.l Coordinate system. From continuity, Eq. (2), it is observed that v can be expressed by a velocity potential that satisfies Laplace's equation. It can also be shown that P satisfies Laplace's equation. Therefore, P can be expressed as a summation of pressure potentials, 0; and v can be expressed as a summation of the gradient of velocity potentials, V. NEW FORMULATION Pressure Potentials* and Velocity Potentials To transform Eqs. (1) and (2) by a Galerkin method, it is required to expand the pressure potential, 4>, and the velocity potentials, *P, in terms of complete set of functions that satisfy Laplace's equation. In addition, they have to fulfill the boundary conditions for pressure in the case of <&, and for velocity in the case of V. The boundary conditions for pressure are given by a discontinuity across the rotor disk. The use of an ellipsoidal coordinate system, has the advantage that any odd function in v, will allow a representation of a discontinuity across the rotor disk. An additional advantage of using an ellipsoidal coordinate system is that an analytical solution of Laplace's equation is known and can be expressed as,
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