TS - Dean interactions in curved channel flow

A weakly nonlinear theory is developed to study the interaction of TS waves and Dean vortices in curved channel flow. The prediction obtained from the theory agree well with results obtained from direct numerical simulations of curved channel flow, especially for low amplitude disturbances. At low Reynolds numbers the wave interaction is generally stabilizing to both disturbances, though as the Reynolds number increases, many linearly unstable TS waves are further destabilized by the presence of Dean vortices

Special opportunities in helicopter aerodynamics

Aerodynamic research relating to modern helicopters includes the study of three dimensional, unsteady, nonlinear flow fields. A selective review is made of some of the phenomenon that hamper the development of satisfactory engineering prediction techniques, but which provides a rich source of research opportunities: flow separations, compressibility effects, complex vortical wakes, and aerodynamic interference between components. Several examples of work in progress are given, including dynamic stall alleviation, the development of computational methods for transonic flow, rotor-wake predictions, and blade-vortex interactions

Nonparametric Uncertainty Quantification for Stochastic Gradient Flows

This paper presents a nonparametric statistical modeling method for quantifying uncertainty in stochastic gradient systems with isotropic diffusion. The central idea is to apply the diffusion maps algorithm to a training data set to produce a stochastic matrix whose generator is a discrete approximation to the backward Kolmogorov operator of the underlying dynamics. The eigenvectors of this stochastic matrix, which we will refer to as the diffusion coordinates, are discrete approximations to the eigenfunctions of the Kolmogorov operator and form an orthonormal basis for functions defined on the data set. Using this basis, we consider the projection of three uncertainty quantification (UQ) problems (prediction, filtering, and response) into the diffusion coordinates. In these coordinates, the nonlinear prediction and response problems reduce to solving systems of infinite-dimensional linear ordinary differential equations. Similarly, the continuous-time nonlinear filtering problem reduces to solving a system of infinite-dimensional linear stochastic differential equations. Solving the UQ problems then reduces to solving the corresponding truncated linear systems in finitely many diffusion coordinates. By solving these systems we give a model-free algorithm for UQ on gradient flow systems with isotropic diffusion. We numerically verify these algorithms on a 1-dimensional linear gradient flow system where the analytic solutions of the UQ problems are known. We also apply the algorithm to a chaotically forced nonlinear gradient flow system which is known to be well approximated as a stochastically forced gradient flow.Comment: Find the associated videos at: http://personal.psu.edu/thb11

Computational aspects of the prediction of multidimensional transonic flows in turbomachinery

The analytical prediction and description of transonic flow in turbomachinery is complicated by three fundamental effects: (1) the fluid equations describing the transonic regime are inherently nonlinear, (2) shock waves may be present in the flow, and (3) turbomachine blading is geometrically complex, possessing large amounts of curvature, stagger, and twist. A three-dimensional computation procedure for the study of transonic turbomachine fluid mechanics is described. The fluid differential equations and corresponding difference operators are presented, the boundary conditions for complex blade shapes are described, and the computational implementation and mapping procedures are developed. Illustrative results of a typical unthrottled transonic rotor are also presented

Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems

Drawing upon the bursting mechanism in slow-fast systems, we propose indicators for the prediction of such rare extreme events which do not require a priori known slow and fast coordinates. The indicators are associated with functionals defined in terms of Optimally Time Dependent (OTD) modes. One such functional has the form of the largest eigenvalue of the symmetric part of the linearized dynamics reduced to these modes. In contrast to other choices of subspaces, the proposed modes are flow invariant and therefore a projection onto them is dynamically meaningful. We illustrate the application of these indicators on three examples: a prototype low-dimensional model, a body forced turbulent fluid flow, and a unidirectional model of nonlinear water waves. We use Bayesian statistics to quantify the predictive power of the proposed indicators