Variationally derived closure models for large eddy simulation of incompressible turbulent flows

Abstract

We present a variationally consistent method for deriving residual-based closure models for incompressible Navier-Stokes equations. The method is based on the fine-scale variational structure facilitated by the variational multiscale framework where fine scales are driven by the residuals of the Euler-Lagrange equations of the resolved scales in the balance of momentum and conservation of mass equations. A bubble-functions based approach is applied directly to the fine-scale variational equation to derive analytical expressions for the closure model. Variational consistency of the model lends itself to rigorous linearization that results in quadratic rate of convergence of the method in the iterative solution strategy for the nonlinear equations. The method is shown to work for a family of linear and quadratic hexahedral and tetrahedral elements as well as composite discretizations that are comprised of hexahedral and tetrahedral elements in the same computational domain. Numerical tests with the proposed model are presented for various classes of turbulent flow problems to show its generality and range of applicability. The test cases investigated include Taylor-Green vortex stretching, statistically stationary wall-bounded channel flows, and modeling the effects of the geometry of the leading edge of the plate on the instability of the boundary layer that leads to flow separation and flow reversal over flat plates of finite thickness. K E Y W O R D Shexahedral and tetrahedral elements, hierarchical variational multiscale method, large eddy simulation, residual-based turbulence models, residual-free bubbles, stabilized finite elements

A Multiscale Vision-Illustrative Applications From Biology to Engineering

“…A unique feature of this class of methods is that the solution of the fine-scale variational equation does not require a priori assumptions on the structure of the subgrid scale. Subsequently, the hierarchical VMS framework was proposed in Masud and Franca (2008) and Masud and Scovazzi (2011), which resulted in variationally derived closure models for incompressible turbulent flows (Calderer and Masud, 2013;Masud and Calderer, 2011;Masud and Zhu, 2021), as shown in Figs. 12 and 13.…”