Reduced order Euler equations for unsteady aerodynamic flows: Numerical techniques
Fluid eigenmodes can be used to construct extremely efficient reduced order unsteady aerodynamic models of the Euler equations, which can also be used to perform highly efficient aeroelastic analyses. This is done by transforming the large set of primitive variable (density, velocity, etc.) equations into a much smaller set of decoupled (modal) equations. The technique of constructing a reduced order aerodynamic model is quite general and applies to most any CFD solver, regardless of the size of the primitive variable computational model. However, in practice the construction of reduced order representations of the CFD model is limited by the ability to accurately calculate the fluid eigenmodes. The difficulty of this calculation increases dramatically with the number of degrees of freedom in the primitive variable model. In this paper, two techniques are presented for calculating the eigenvalues and the corresponding eigenvectors of the large nonsymmetric matrices typically encountered with the discretized Euler equations: a Lanczos reduction and a Modified WYD reduction. The Lanczos reduction procedure is limited to small computational meshes (O(103) DOF). The Modified WYD reduction is developed here to extend the reduced order modelling technique to primitive variable systems with O(104) DOF. Results presented in this paper demonstrate that a reduced order aeroelastic model created from a primitive variable system of this size predicts the onset of flutter within 0.5% of the conventional, refined mesh CFD based analysis, but at 10-4 the cpu cost per analysis point.