Spectral methods for CFD

One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched

Development of an integrated BEM approach for hot fluid structure interaction

The progress made toward the development of a boundary element formulation for the study of hot fluid-structure interaction in Earth-to-Orbit engine hot section components is reported. The convective viscous integral formulation was derived and implemented in the general purpose computer program GP-BEST. The new convective kernel functions, in turn, necessitated the development of refined integration techniques. As a result, however, since the physics of the problem is embedded in these kernels, boundary element solutions can now be obtained at very high Reynolds number. Flow around obstacles can be solved approximately with an efficient linearized boundary-only analysis or, more exactly, by including all of the nonlinearities present in the neighborhood of the obstacle. The other major accomplishment was the development of a comprehensive fluid-structure interaction capability within GP-BEST. This new facility is implemented in a completely general manner, so that quite arbitrary geometry, material properties and boundary conditions may be specified. Thus, a single analysis code (GP-BEST) can be used to run structures-only problems, fluids-only problems, or the combined fluid-structure problem. In all three cases, steady or transient conditions can be selected, with or without thermal effects. Nonlinear analyses can be solved via direct iteration or by employing a modified Newton-Raphson approach

Arbitrary Lagrangian-Eulerian form of flowfield dependent variation (ALE-FDV) method for moving boundary problems

Flowfield Dependent Variation (FDV) method is a mixed explicit-implicit numerical scheme that was originally developed to solve complex flow problems through the use of so-called implicitness parameters. These parameters determine the implicitness of FDV method by evaluating local gradients of physical flow parameters, hence vary across the computational domain. The method has been used successfully in solving wide range of flow problems. However it has only been applied to problems where the objects or obstacles are static relative to the flow. Since FDV method has been proved to be able to solve many complex flow problems, there is a need to extend FDV method into the application of moving boundary problems where an object experiences motion and deformation in the flow. With the main objective to develop a robust numerical scheme that is applicable for wide range of flow problems involving moving boundaries, in this study, FDV method was combined with a body interpolation technique called Arbitrary Lagrangian-Eulerian (ALE) method. The ALE method is a technique that combines Lagrangian and Eulerian descriptions of a continuum in one numerical scheme, which then enables a computational mesh to follow the moving structures in an arbitrary movement while the fluid is still seen in a Eulerian manner. The new scheme, which is named as ALE-FDV method, is formulated using finite volume method in order to give flexibility in dealing with complicated geometries and freedom of choice of either structured or unstructured mesh. The method is found to be conditionally stable because its stability is dependent on the FDV parameters. The formulation yields a sparse matrix that can be solved by using any iterative algorithm. Several benchmark stationary and moving body problems in one, two and three-dimensional inviscid and viscous flows have been selected to validate the method. Good agreement with available experimental and numerical results from the published literature has been obtained. This shows that the ALE-FDV has great potential for solving a wide range of complex flow problems involving moving bodies

The instanton method and its numerical implementation in fluid mechanics

A precise characterization of structures occurring in turbulent fluid flows at high Reynolds numbers is one of the last open problems of classical physics. In this review we discuss recent developments related to the application of instanton methods to turbulence. Instantons are saddle point configurations of the underlying path integrals. They are equivalent to minimizers of the related Freidlin-Wentzell action and known to be able to characterize rare events in such systems. While there is an impressive body of work concerning their analytical description, this review focuses on the question on how to compute these minimizers numerically. In a short introduction we present the relevant mathematical and physical background before we discuss the stochastic Burgers equation in detail. We present algorithms to compute instantons numerically by an efficient solution of the corresponding Euler-Lagrange equations. A second focus is the discussion of a recently developed numerical filtering technique that allows to extract instantons from direct numerical simulations. In the following we present modifications of the algorithms to make them efficient when applied to two- or three-dimensional fluid dynamical problems. We illustrate these ideas using the two-dimensional Burgers equation and the three-dimensional Navier-Stokes equations

Status of research at the Institute for Computer Applications in Science and Engineering (ICASE)

Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science is summarized

Interacting errors in large-eddy simulation: a review of recent developments

The accuracy of large-eddy simulations is limited, among others, by the quality of the subgrid parameterisation and the numerical contamination of the smaller retained flow structures. We review the effects of discretisation and modelling errors from two different perspectives. We first show that spatial discretisation induces its own filter and compare the dynamic importance of this numerical filter to the basic large-eddy filter. The spatial discretisation modifies the large-eddy closure problem as is expressed by the difference between the discrete 'numerical stress tensor' and the continuous 'turbulent stress tensor'. This difference consists of a high-pass contribution associated with the specific numerical filter. Several central differencing methods are analysed and the importance of the subgrid resolution is established. Second, we review a database approach to assess the total simulation error and its numerical and modelling contributions. The interaction between the different sources of error is shown to lead to their partial cancellation. From this analysis one may identify an 'optimal refinement strategy' for a given subgrid model, discretisation method and flow conditions, leading to minimal total simulation error at a given computational cost. We provide full detail for homogeneous decaying turbulence in a 'Smagorinsky fluid'. The optimal refinement strategy is compared with the error reduction that arises from grid refinement of the dynamic eddy-viscosity model. The main trends of the optimal refinement strategy as a function of resolution and Reynolds number are found to be adequately followed by the dynamic model. This yields significant error reduction upon grid refinement although at coarse resolutions significant error levels remain. To address this deficiency, a new successive inverse polynomial interpolation procedure is proposed with which the optimal Smagorinsky constant may be efficiently approximated at a given resolution. The computational overhead of this optimisation procedure is shown to be well justified in view of the achieved reduction of the error level relative to the 'no-model' and dynamic model predictions