Reduced Order Modeling Incompressible Flows

The details: a) Need stable numerical methods; b) Round off error can be considerable; c) Not convinced modes are correct for incompressible flow. Nonetheless, can derive compact and accurate reduced-order models. Can be used to generate actuator models or full flow-field model

Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

We investigate p-multigrid as a solution method for several different discontinuous Galerkin (DG) formulations of the Poisson equation. Different combinations of relaxation schemes and basis sets have been combined with the DG formulations to find the best performing combination. The damping factors of the schemes have been determined using Fourier analysis for both one and two-dimensional problems. One important finding is that when using DG formulations, the standard approach of forming the coarse p matrices separately for each level of multigrid is often unstable. To ensure stability the coarse p matrices must be constructed from the fine grid matrices using algebraic multigrid techniques. Of the relaxation schemes, we find that the combination of Jacobi relaxation with the spectral element basis is fairly effective. The results using this combination are p sensitive in both one and two dimensions, but reasonable convergence rates can still be achieved for moderate values of p and isotropic meshes. A competitive alternative is a block Gauss-Seidel relaxation. This actually out performs a more expensive line relaxation when the mesh is isotropic. When the mesh becomes highly anisotropic, the implicit line method and the Gauss-Seidel implicit line method are the only effective schemes. Adding the Gauss-Seidel terms to the implicit line method gives a significant improvement over the line relaxation method

A Comparative Study of Different Reconstruction Schemes for a Reconstructed Discontinuous Galerkin Method on Arbitrary Grids

A comparative study of different reconstruction schemes for a reconstruction-based discontinuous Galerkin, termed RDG(P1P2) method is performed for compressible flow problems on arbitrary grids. The RDG method is designed to enhance the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution via a reconstruction scheme commonly used in the finite volume method. Both Green-Gauss and least-squares reconstruction methods and a least-squares recovery method are implemented to obtain a quadratic polynomial representation of the underlying discontinuous Galerkin linear polynomial solution on each cell. These three reconstruction/recovery methods are compared for a variety of compressible flow problems on arbitrary meshes to access their accuracy and robustness. The numerical results demonstrate that all three reconstruction methods can significantly improve the accuracy of the underlying second-order DG method, although the least-squares reconstruction method provides the best performance in terms of both accuracy and robustness

External and Turbomachinery Flow Control Working Group

Broad Flow Control Issues: a) Understanding flow physics. b) Specific control objective(s). c) Actuation. d) Sensors. e) Integrated active flow control system. f) Development of design tools (CFD, reduced order models, controller design, understanding and utilizing instabilities and other mechanisms, e.g., streamwise vorticity)

A Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Unstructured Tetrahedral Grids

A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier-Stokes equations on unstructured tetrahedral grids. The RDG method, originally developed for the compressible Euler equations, is extended to discretize viscous and heat fluxes in the Navier-Stokes equations using a so-called inter-cell reconstruction, where a smooth solution is locally reconstructed using a least-squares method from the underlying discontinuous DG solution. Similar to the recovery-based DG (rDG) methods, this reconstructed DG method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional DG methods. Unlike rDG methods, this RDG method does not need to judiciously choose a proper form of a recovered polynomial, thus is simple, flexible, and robust, and can be used on unstructured grids. The preliminary results indicate that this RDG method is stable on unstructured tetrahedral grids, and provides a viable and attractive alternative for the discretization of the viscous and heat fluxes in the Navier-Stokes equations

Numerical Evaluation of P-Multigrid Method for the Solution of Discontinuous Galerkin Discretizations of Diffusive Equations

This paper describes numerical experiments with P-multigrid to corroborate analysis, validate the present implementation, and to examine issues that arise in the implementations of the various combinations of relaxation schemes, discretizations and P-multigrid methods. The two approaches to implement P-multigrid presented here are equivalent for most high-order discretization methods such as spectral element, SUPG, and discontinuous Galerkin applied to advection; however it is discovered that the approach that mimics the common geometric multigrid implementation is less robust, and frequently unstable when applied to discontinuous Galerkin discretizations of di usion. Gauss-Seidel relaxation converges 40% faster than block Jacobi, as predicted by analysis; however, the implementation of Gauss-Seidel is considerably more expensive that one would expect because gradients in most neighboring elements must be updated. A compromise quasi Gauss-Seidel relaxation method that evaluates the gradient in each element twice per iteration converges at rates similar to those predicted for true Gauss-Seidel

Ducted wind turbine optimization and sensitivity to rotor position

The design of a ducted wind turbine modeled using an actuator disc was studied using Reynolds-averaged Navier–Stokes (RANS) computational fluid dynamics (CFD) simulations. The design variables included the rotor thrust coefficient, the angle of attack of the duct cross section, the radial gap between the rotor and the duct, and the axial location of the rotor in the duct. Two different power coefficients, the rotor power coefficient (based on the rotor swept area) and the total power coefficient (based on the exit area of the duct), were used as optimization objectives. The optimal value of thrust coefficients for all designs was nearly constant, having a value between 0.9 and 1. The rotor power coefficient was sensitive to rotor gap but was insensitive to the rotor's axial location for positions ranging from upstream of the throat to nearly half the distance down the duct. Compared to the design that maximized rotor power coefficient, the design for maximal total power coefficient was characterized by a smaller angle of attack, a smaller rotor gap, and a downstream placement of the rotor. The insensitivity of power output to the rotor position implies that a rotor placed further downstream in the duct could produce the same power with a considerably smaller duct exit area and thus a greater total power coefficient. The design for that maximized total power coefficient exceeded Betz's limit with a total power coefficient of 0.67