Modeling the unsteady aerodynamic forces on small-scale wings
S. L. Brunton and C. W. Rowley
AIAA Paper 2009-1127, 47th AIAA Aerospace Sciences Meeting , January 2009.
The goal of this work is to develop low order dynamical systems models for the unsteady aerodynamic forces on small wings and to better understand the physical characteristics of unsteady laminar separation. Reduced order models for a fixed, high angle of attack flat plate are obtained through Galerkin projection of the governing Navier-Stokes equations onto POD modes. Projected models are compared with direct numerical simulation (DNS) to show that they preserve qualitative behavior such as coherent structures. It is shown that in flows with Reynolds number 100, even a two degree of freedom model is sufficient to capture high angle of attack laminar vortex shedding. Next, the classical theories of Theodorsen and Wagner are compared with DNS for a number of pitch and plunge maneuvers of varying Strouhal number, reduced frequency, pitch amplitude and center. In addition to determining when these theories break down, the flow field structures are investigated to determine how the theories break down. This is an important first step toward combining and extending classical unsteady aerodynamic models to include high angle of attack effects. Theodorsenís model for the lift of a sinusoidally pitching or plunging plate is shown to agree moderately well with DNS for reduced frequencies k < 2.0. One major observation is that the classical aerodynamic models all begin to disagree when the effective angle of attack, either determined by Strouhal number in the plunging case or angle of attack excursion in the pitching case, exceeds the critical stall angle where vortex shedding and laminar separation become prominent. Velocity field and body force data for a flat plate are generated by 2D direct numerical simulation using an immersed boundary method for Reynolds number 100-300, and regions of separated flow and wake structures are visualized using Finite Time Lyapunov Exponents (FTLE) fields, the ridges of which are Lagrangian coherent structures (LCS).
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