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this paper an unstructured multigrid algorithm is used as an iterative solution procedure for the discrete equations arising from an implicit time discretisation of the unsteady Euler equations on tetrahedral grids. To calculate unsteady flows due to oscillating boundaries, a novel grid movement algorithm is introduced, in which an elliptic equation with a nonlinear diffusion coefficient is used to define the displacement of interior grid nodes. This allows large grid displacements to be calculated in a single step. The multigrid technique uses a edge--collapsing algorithm to generate a sequence of grids, and a pseudo--timestepping smoother. On the coarser grids, no grid motion is used. Instead, surface normals are rotated consistently and transfer/interpolation weights are based on the time-averaged grid coordinates. A 2D NACA0012 test case is used to validate the program

This paper presents a timestep stability analysis for a class of discretisations applied to the linearised form of the Navier-Stokes equations on a 3D domain with periodic boundary conditions. Using a suitable definition of the `perturbation energy' it is shown that the energy is monotonically decreasing for both the original p.d.e. and the semi-discrete system of o.d.e.'s arising from a Galerkin discretisation on a tetrahedral grid. Using recent theoretical results concerning algebraic and generalised stability, sufficient stability limits are obtained for both global and local timesteps for fully discrete algorithms using Runge-Kutta time integration. Subject classifications: AMS(MOS): 65M10, 65M20, 65M60, 76-08, 76N10 Key words and phrases: Navier-Stokes, method of lines, stability analysis This work was supported by Rolls-Royce plc, DTI and EPSRC. Oxford University Computing Laboratory Numerical Analysis Group Wolfson Building Parks Road Oxford, England OX1 3QD E-mail: giles@comlab.oxford.ac.uk April, 1997 1

The flutter characteristics of the first AGARD standard aeroelastic configuration for dynamic response, Wing 445.6, are studied using an unsteady Navier-Stokes algorithm in order to investigate a previously noted discrepancy between Euler flutter characteristics and the experimental data. The algorithm, which is a three-dimensional, implicit, upwind Euler/Navier-Stokes code (CFL3D Version 2.1), was previously modified for the time-marching, aeroelastic analysis of wings using the unsteady Euler equations. These modifications include the incorporation of a deforming mesh algorithm and the addition of the structural equations of motion for their simultaneous time integration with the governing flow equations. In this paper, the aeroelastic method is extended and evaluated for applications that use the NavierStokes aerodynamics. The paper presents a brief description of the aeroelastic method and presents unsteady calculations which verify this method for Navier-Stokes calculations. A lin...

Fellowship sponsored by Dryden Flight Research Center. Specifically, I would like to thank Dr. Kajal K. Gupta and the rest of the STARS group at Dryden Flight Research Center for their generous support of this research. I would like to express my sincere appreciation to my major advisor, Dr. Andrew S. Arena, for his enthusiastic support and guidance in this research. In addition to being a source for inspiration in academics and research, he has also been an excellent role model during my time at OSU. Similarly, I would like to thank the other members of my committee, Dr. P. M. Moretti and Dr. G. E. Young, for their efforts in furthering my education. I would also like to thank my parents, Timothy M. and Marsha L. Cowan, for their early efforts at molding me into who I am today. Their ongoing support and encouragement is much appreciated. Finally, I would especially like to thank my wife, Leslie, for her understanding and support during the past two years. I will be eternally grateful for her love and devotion.

This paper analyses the accuracy and numerical stability of coupling procedures in aeroelastic modelling. A two-dimensional model problem assuming unsteady inviscid flow past an oscillating wall leads to an even simpler one-dimensional model problem. Analysis of different numerical algorithms shows that in general the coupling procedures are numerically stable, but care is required to achieve accuracy when using very few time steps per period of natural oscillation of the structure. The relevance of the analysis to fully three-dimensional applications is discussed.

With advanced subsonic transports and military aircraft operating in the transonic regime, it is becoming important to determine the effects of the coupling between aerodynamic loads and elastic forces. Since aeroelastic effects can contribute significantly to the design of these aircraft, there is a strong need in the aerospace industry to predict these aero-structure interactions computationally. To perform static aeroelastic analysis in the transonic regime, high fidelity computational fluid dynamics (CFD) analysis tools must be used in conjunction with high fidelity computational structural dynamics (CSD) analysis tools due to the nonlinear behavior of the aerodynamics in the transonic regime. There is also a need to be able to use a wide variety of CFD and CSD tools to predict these aeroelastic effects in the transonic regime. Because source codes are not always available, it is necessary to couple the CFD and CSD codes without alteration of the source codes. In this study, an aeroelastic coupling procedure is developed which will perform static aeroelastic analysis using any CFD and CSD code with little code integration. The aeroelastic coupling procedure is demonstrated on an F/A-18 Stabilator using NASTD (an inhouse

Studies on Numerical Simulations of Fluid-Structure Interaction Problems Wei Hao In this thesis, numerical simulations of Fluid-Structure Interaction (FSI) are studied. Computer codes for solving the Navier-Stokes equations using meshfree methods are developed with the capability of handling different boundary conditions depending on the flow type. Good agreement with published and analytical results has been achieved for representative numerical examples. Preliminary results are presented for a panel flutter problem as well as an eigenvalue analysis of unsteady flows. ii ACKNOWLEDGMENTS I would like to take this opportunity to thank my advisor, Professor Wing Kam Liu, for offering me the opportunity to pursue this work. His dedication and enthusiasm for research have always been an inspiration for me over the past few years. I am also grateful to Professor Ted Belytschko and Professor Brian Moran for serving on my thesis committee and for providing many helpful suggestions. I am esp...

Abstract. In this report, we address aerodynamic shape optimization problems including uncertain operating conditions. After a review of robust control theory and the possible approaches to take into uncertainty, we propose to use Taguchi robust design methods in order to overcome single point design problems in Aerodynamics. The latter techniques produce solutions that perform well for the selected design point but have poor off-design performance. Under the conduct of Taguchi concept, a design with uncertainties is converted into an optimization problem with two objectives which are mean performance and its variance, so that the solutions are as less insensitive to the uncertainty of the input parameters as possible. Furthermore, the Multi-Criterion Evolutionary Algorithms (MCEAs) are used to capture a set of compromised solutions (Pareto front) between these two objectives. The flow field is analyzed by Navier-Stokes computation. In order to reduce the number of expensive evaluations of fitness function, Response Surface Modelling (RSM) is employed to estimate fitness value using the polynomial approximate model. The proposed approach is applied to the robust optimization of the 2D high lift devices of a business aircraft, by maximizing the mean and minimizing the variance of the lift coefficients with uncertain free-stream angle of attack at landing and takeoff flight conditions respectively