Filtering in Turbulent Flows: Insights from Prof. Rob Stoll
Explore the concepts of filtering in turbulent flows from a lecture by Prof. Rob Stoll at the University of Utah. Learn about calculating 3D energy spectra, filtering equations for incompressible flow, and more. Discover the implications of applying filters to compressible Navier-Stokes equations and the conservation of mass and energy. Gain insights into Favre filtering and the resolution of viscous stress tensors in turbulent flows.
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Presentation Transcript
1 LES of Turbulent Flows: Lecture 6 (ME EN 7960-003) Prof. Rob Stoll Department of Mechanical Engineering University of Utah Fall 2014
2 Homework #2 Goal of homework assignment is to calculate 3D energy spectrum from isotropic data and to perform 3D filtering on a 3D isotropic turbulence dataset plot: E(|k|) vs. |k|=[k12+k12+k12] we estimate E(|k|)by binning all the energy within a d|k| shell in wavespace Calculating the 3D energy spectrum k2 Energy levels shells ln(E(|k|)) |k| d|k| k1 k3 ln(|k|) d|k|
3 LES filtered Equations for incompressible flow Mass: Momentum: Scalar: Resolved Energy: SFS stress: SFS flux: SFS Dissipation:
4 Filtering the compressible N-S equations What happens when we apply a filter to the compressible N-S equations? -Conservation of Mass for compressible flow: filtering each term => results in an SFS term! How can we avoid having a SFS conservation of mass? -Density weighted filtering: Formalized for compressible flow by Favre (Phys. Fluids, 1983) for ensemble statistics, a Favre (or density weighted) filter is defined by: where we note that as compressibility becomes less important and we can show that the conservation of mass becomes:
5 Filtering the compressible N-S equations We can use this to write the Favre filtered equations of motion (see Geurts pg 32-35 or Vreman et al. Applied Sci. Res. 1995 for details) -Conservation of Mass: -Conservation of Momentum: where the SFS terms are collected on the RHS of the equation and we now have both a resolved ( ) and SFS ( ) viscous contribution because is a function of the Favre filtered temperature and => nonlinear viscous stress tensor Recall: strain rate tensor => smooth viscous stress tensor The SFS stress tensor for the Favre filtered equations is given by which is obtained from the nonlinear term =>
6 Filtering the compressible N-S equations -Conservation of total energy: where the LHS contains the SFS terms created using the procedure used in Lecture 5 for kinetic energy transferred from resolved to SFSs pressure velocity SFS term (effect of SFS turbulence on the conduction of heat at resolved scales compressibility effects (vanishes for incompressible) conversion of SFS kinetic energy to internal energy by viscous dissipation SFS viscous stress term SFS heat flux term (Note qj is the heat flux vector) Typically assumptions that and are made eliminating a5 and a6 .
