SUMMARYA numerical continuation method for the compressible Reynolds-Averaged Navier-Stokes equation with the Spalart-Allmaras turbulence model is presented and applied to the flow around a 2D airfoil. Using continuation methods it is possible to study the steady flow states of a system as a parameter such as angle of attack is varied. This approach allows unstable solutions to be calculated, which are important for understanding the nonlinear dynamics of the system. Furthermore, this method can be used to find any multivalued solutions that exist at a single parameter value. The eigenvalues of the system are calculated using the Cayley transform to precondition the eigenvalue solver ARPACK. The eigenvalues are important as they show the stability of the solutions as well as accurately detect parameter values at which bifurcations take place.

A “dual-time” method for solving the three-dimensional Euler equations describing the compressible flow about wings undergoing arbitrary motions and deformations is presented. A finite-volume formulation is chosen where the volumes distort as the wing moves or deforms. Independent motion of the inner and outer boundaries of the grid is permitted with a sequence of grids generated using transfinite interpolation. An implicit real-time discretisation is used, and the equations are integrated in a fictitious pseudo time. This approach allows the real-time step to be chosen on the basis of accuracy rather than stability. It also permits the acceleration techniques commonly used to speed up steady flow calculations to be used when marching in pseudo time, without compromising real-time accuracy. A two-dimensional version of the method has also been developed and results for both two and three-dimensional transonic flows are presented and compared with experimental data where available.

A method for computing unsteady incompressible viscous ows on moving or deforming meshes is described. It uses a well-established time-marching ÿnite-volume ow solver, developed for steady compressible ows past rigid bodies. Time-marching methods cannot be applied directly to incompressible ows because the governing equations are not hyperbolic. Such methods can be extended to steady incompressible ows using an artiÿcial compressibility scheme. A time-accurate scheme for unsteady incompressible ows is achieved by using an implicit real-time discretization and a dual-time approach, which uses a technique similar to the artiÿcial compressibility scheme. Results are presented for test cases on both ÿxed and deforming meshes. Experimental, numerical and theoretical data have been included for comparison where available and reasonable agreement has been achieved.

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SUMMARYReduced-Order Models (ROMs) have been the focus of research in various engineering situations, but it is only relatively recently that such techniques have begun to be introduced into the CFD ÿeld. The purpose of generating such models is to capture the dominant dynamics of the full set of CFD equations, but at much lower cost. One method that has been successfully implemented in the ÿeld of uid ows is based on the calculation of the linear pulse responses of the CFD scheme coupled with an Eigensystem Realization algorithm (ERA), resulting in a compact aerodynamic model. The key to the models is the identiÿcation of the linear responses of the non-linear CFD code. Two di erent methods have been developed and reported in literature for linear response identiÿcation; the ÿrst method linearizes the CFD code and the second method uses Volterra theory and the non-linear code. As these methods were developed independently they have not previously been brought together and compared. This paper ÿrst explains the subtle, but fundamental di erences between the two methods. In addition, a series of test cases are shown to demonstrate the performance and drawbacks of the ROMs derived from the di erent linear responses. The conclusions of this study provide useful guidance for the implementation of either of the two approaches to obtain the linear responses of an existing CFD code.