Philip S. Beran

Analytical Sensitivity Analysis of an Unsteady Vortex Lattice Method for Flapping Wing Optimization

This work considers the design optimization of a flapping wing in forward flight with active shape morphing, aimed at maximizing propulsive efficiency under lift and thrust constraints. This is done with an inviscid three-dimensional unsteady vortex lattice method, whose lack of fidelity is offset by a relatively inexpensive computational cost. The design is performed with a gradient-based optimization, where gradients are computed with an analytical sensitivity analysis. Wake terms provide the only connection between the forces generated at disparate time steps, and must be included to compute the derivative of the aerodynamic state at a time step with respect to the wing shape at all previous steps. The cyclic wing morphing, superimposed upon the flapping motions, is defined by a series of spatial and temporal approximations. The generalized coordinates of a finite number of twisting and bending modes are approximated by cubic splines. The amplitudes at the control points provide design variables; increasing the number of variables (providing the wing morphing with a greater degree of spatial and temporal freedom) is seen to provide increasingly superior designs, with little increase in computational cost. I. Introduction HE design and optimization of artificial flapping wing flyers presents considerable difficulties in terms of computational cost: the complex physical phenomena associated with the flight (unsteady low Reynolds number vortical flows in conjunction with a nonlinear elastic wing surface undergoing large prescribed rotations and translations) may require a high-fidelity computational tool. Furthermore, the search optimization process typically requires many function evaluations to converge to a relevant optimum. Lower fidelity numerical tools may help alleviate the burden, either used during the search process in conjunction with a higher-fidelity model 1

Uncertainty quantification of limit-cycle oscillations

Different computational methodologies have been developed to quantify the uncertain response of a relatively simple aeroelastic system in limit-cycle oscillation, subject to parametric variability. The aeroelastic system is that of a rigid airfoil, supported by pitch and plunge structural coupling, with nonlinearities in the component in pitch. The nonlinearities are adjusted to permit the formation of a either a subcritical or supercritical branch of limit-cycle oscillations. Uncertainties are specified in the cubic coefficient of the torsional spring and in the initial pitch angle of the airfoil. Stochastic projections of the time-domain and cyclic equations governing system response are carried out, leading to both intrusive and non-intrusive computational formulations. Non-intrusive formulations are examined using stochastic projections derived from Wiener expansions involving Haar wavelet and B-spline bases, while Wiener-Hermite expansions of the cyclic equations are employed intrusively and non-intrusively. Application of the B-spline stochastic projection is extended to the treatment of aerodynamic nonlinearities, as modeled through the discrete Euler equations. The methodologies are compared in terms of computational cost, convergence properties, ease of implementation, and potential for application to complex aeroelastic systems.

Numerical Analysis of Store-Induced Limit-Cycle Oscillation

Store-induced limit-cycle oscillation of a rectangular wing with tip store in transonic flow is simulated using a variety of mathematical models for the flowfield: transonic small-disturbance theory (with and without inclusion of store aerodynamics) and transonic small-disturbance theory with interactive boundary layer (without inclusion of store aerodynamics). For the conditions investigated, assuming inviscid flow, limit-cycle oscillations are observed to occur as a result of a weakly subcritical Hopf bifurcation and are obtained at speeds lower than those predicted 1) nonlinearly for clean-wing flutter and 2) linearly for wing/store flutter. The ability of transonic small-disturbance theory to predict the occurrence and strength of this type of limit-cycle oscillation is compared for the different models. Differences in unmatched and matched aeroelastic analysis are described. Solutions computed for the clean rectangular wing are compared to those computed with the Euler equations for a case of static aeroelastic behavior and for a case of forced, rigid-wing oscillation at Mach 0.92.

K-ε and Reynolds stress turbulence model comparisons for two-dimensional injection flows

Two-dimensional steady e owe elds generated by transverse injection into a supersonic e ow are numerically sim- ulated by integrating the Favre-averaged Navier -Stokes equations. Fine-scale turbulence effects are modeled with compressible K-≤ and second-order Reynolds-stress turbulence models. These numerical results are compared to numerical results of the Jones -Launder K-≤ model and experimental data. The credibility of the Reynolds-stress turbulence model relative to experimental data and other turbulence models is demonstrated by comparison of surface pressure proe les, boundary-layer separation location, jet plume height, and descriptions of recirculation zones and e ow structureupstream and downstream of the jet. Results indicate that theReynolds-stress turbulence model correctly predicts mean e ow conditions for low static pressure ratios. However, it is also observed that, as the static pressure ratio increases, the boundary-layer separation point moves farther upstream of the jet and pre- dictions become lessconsistent with experimental results. The K-≤ results are lessconsistentwith theexperimental resultsthanthoseassociatedwiththeReynolds-stressturbulencemodel.Finally,unlikethe K-≤results,nonphysical vorticity phenomena upstream of the jet plume are not observed in the Reynolds-stress turbulence model results. This phenomenon is shown to coincide with strong gradients in the wall functions used to compute πt.

Projection methods for reduced order models of compressible flows

Two different projection methods, Galerkin projection and direct projection, are developed for reduced-order modeling applications. The projection methods are used to identify low-dimensional systems of ordinary differential equations to represent the dynamics of a compressible, two-dimensional, inviscid flow-field under oscillatory forcing. Proper orthogonal decomposition is used to identify a small number of fluid modes to serve as the basis functions for the projections. Performance is evaluated relative to a high-order numerical model in terms of accuracy, order reduction, and computational efficiency. The treatment of boundary conditions, and stability of the reduced-order model are addressed in detail. The methods developed in this paper are suitable for general application to the Euler equations. With the addition of dissipation parameters, both the Galerkin projection and direct projection methods are tractable, stable, and properly treat the boundary conditions.

Reduced order modeling of a two-dimensional flow with moving shocks

Abstract The objective of this paper is to demonstrate the ability of proper orthogonal decomposition, in combination with domain decomposition, to produce accurate reduced order models (ROMs) for two-dimensional high-speed flows with moving shock waves. To demonstrate this ability, a blunt body flow with quasi-steady shock motion is considered. The blunt body flow contains a strong bow shock that is moved via a change in inlet Mach number and angle of attack. Accuracy is quantified by comparing surface pressures obtained from the ROMs with those from the full order simulation under the same free stream conditions. The order reduction, and computational performance of the ROM is also quantified relative to the full order simulation. The robustness of the ROM to varying flow parameters is explored. A non-Galerkin quasi-implicit steady state implementation is considered.

Applications of multi-POD to a pitching and plunging airfoil

Multi-POD is a new proper orthogonal decomposition (POD) based reduced order modeling (ROM) technique for modeling flows on deforming grids. Presented is the application of multi-POD to flow about a pitching and plunging airfoil. The multi-POD technique expands the parameter space in which POD is applied through selection of the best available ROM for a given set of grid deformations. For application to unconstrained pitching and plunging motion of an airfoil, multi-POD is shown to be effective when trained for forced grid motion, reducing the training requirements significantly. A three-orders of magnitude reduction in the number of degrees of freedom is also shown in the use of POD/ROM for the aeroelastic problem.

Airfoil Pitch-and-Plunge Bifurcation Behavior with Fourier Chaos Expansions

A stochastic projection method is employed to obtain the probability distribution of pitch angle of an airfoil in pitch and plunge subject to probabilistic uncertainty in both the initial pitch angle and the cubic spring coefficient of the restoring pitch force. Historically, the selected basis for the stochastic projection method has been orthogonal polynomials, referred to as the polynomial chaos. Such polynomials, however, result in unacceptable computational expense for applications involving oscillatory motion, and a new basis, the Fourier chaos, is introduced for computing limit-cycle oscillations. Unlike the polynomial chaos expansions, which cannot predict limit-cycle oscillations, the Fourier chaos expansions predict both subcritical and supercritical responses even with low-order expansions and high-order nonlinearities. Bifurcation diagrams generated with this new approximate method compare well to Monte Carlo simulations.

POD-Based reduced-order models with deforming grids

Proper orthogonal decomposition based reduced order modeling (POD/ROM) is examined with deforming grids. POD/ROM is a technique that operates in an index-space for computations, not typically accounting for grid dynamics. Two model problems are presented to demonstrate the method of accounting for the effects of grid deformation on POD/ROM. The analytical solution of flow about an oscillating cylinder and potential flow over an oscillating panel. The accuracy and robustness of POD/ROM on deforming grids are compared to that of rigid grid POD/ROM. Deforming grid POD/ROMs are found to require more modes than rigid grid POD/ROMs for similar accuracy levels. In addition, for deforming grids, POD/ROMs are less accurate when the grid deformation is significantly altered from the deformations seen in the POD/ROM development. To address these issues, a new technique is developed that compares the relative grid motion between the POD/ROM creation and execution. The technique determines the current relative grid deformation and selects the best POD/ROM from those available.

Reduced-Order Model Development Using Proper Orthogonal Decomposition and Volterra Theory

A new approach for generating reduced-order models of fluid systems was developed using proper orthogonal decomposition in combination with Volterra theory. The method involves identifying fluid basis functions with proper orthogonal decomposition and applying systems realization theory to generate a low-dimensional model for the scalar coefficients. The method was tested on a two-dimensional inviscid flow over a bump with forcing. Eight fluid basis functions were identified, and the eigensystem realization algorithm was used to identify an eight-state, reduced-order model. Time histories of both the reduced-order coefficients and the expanded flowfield data accurately tracked the full-order results in both amplitude and phase (average error less than 5%). The reduced-order model demonstrated four-orders-of-magnitude reduction in compute time relative to the full system, which represents a computational improvement on the same order as the reduction in degrees of freedom.