A Fully—Implicit 2‐D Navier-Stokes Algorithm for Unstructured Grids

A fully-implicit algorithm is developed for the two-dimensional, compressible, Favre-averaged Navier-Stokes equations. It incorporates the standard k-? turbulence model of Launder and Spalding and the low Reynolds number correction of Chien. The equations are solved using an unstructured grid of triangles with the flow variables stored at the centroids of the cells. A generalization of wall functions including pressure gradient effects is implemented to solve the near-wall region for turbulent flows using a separate algorithm and a hybrid grid. The inviscid fluxes are obtained from Roe's flux difference split method. Linear reconstruction of the flow variables to the cell faces provides second-order spatial accuracy. Turbulent and viscous stresses as well as heat transfer are obtained from a discrete representation of Gauss's theorem. Interpolation of the flow variables to the nodes is achieved using a second-order accurate method. Temporal discretization employs Euler, Trapezoidal or 3-Point Backward differencing. An incomplete LU factorization of the Jacobian matrix is implemented as a preconditioning method. The accuracy of the code and the efficiency of the solution strategy are presented for three test cases: a supersonic turbulent mixing layer, a supersonic laminar compression corner and a supersonic turbulent compression corner.     

Large Eddy Simulations Using an Unstructured Grid Compressible Navier - Stokes Algorithm

Large Eddy Simulation (LES) of the decay of isotropic turbulence and of channel flow has been performed using an explicit second-order unstructured grid algorithm for tetrahedral cells. The algorithm solves for cell-averaged values using the finite volume form of the unsteady compressible Jittered Navier-Stokes equations. The inviscid fluxes are obtained from Godunov's exact Riemann solver. Reconstruction of the flow variables to the left and right sides of each face is performed using least squares or Frink's method. The viscous fluxes and heat transfer are obtained by application of Gauss' theorem. LES of the decay of nearly incompressible isotropic turbulence has been performed using two models for the SGS stresses: the Monotone Integrated Large Eddy Simulation (MILES) approach, wherein the inherent numerical dissipation models the sub-grid scale (SGS) dissipation, and the Smagorinsky SGS model. The results using the MILES approach with least squares reconstruction show good agreement with incompressible experimental data. The contribution of the Smagorinsky SGS model is negligible. LES of turbulent channel flow was performed at a Reynolds number (based on channel height and bulk velocity) of 5600 and Mach number of 0.5 (at which compressibility effects are minimal) using Smagorinsky's SGS model with van Driest damping. The results show good agreement with experimental data and direct numerical simulations for incompressible channel flow. The SGS eddy viscosity is less than 10% of the molecular viscosity, and therefore the LES is effectively MILES with molecular viscosity.     

SYMMETRIC AND EXTERIOR POWERS OF CATEGORIES

We define symmetric and exterior powers of categories, fitting into categorified Koszul complexes. We discuss examples and calculate the effect of these power operations on the categorical characters of matrix 2-representations.