Explicitly uncoupled VMS stabilization of fluid flow

Abstract Variational multiscale methods have proven to be an accurate and systematic approach to the simulation of turbulent flows. Many turbulent flows are solved by legacy codes or by ones written by a team of programmers and of great complexity so implementing a new approach to turbulence in such cases can be daunting. We propose a new approach to inducing a VMS treatment of turbulence in such cases. The method adds a separate, uncoupled and modular postprocessing step to each time step. Adding this step requires the ability to solve a Stokes problem either on the same mesh or to solve and interpolate between the postprocessing step’s mesh and the code’s mesh. We prove stability and convergence for the combination and quantify the VMS dissipation induced. Numerical experiments confirming the theory are given. In particular, the performance of the two step, modular VMS method is comparable to a monolithic (fully coupled) VMS method for the benchmark problem of decaying homogeneous turbulence.

[1]  Unconditional Stability for Numerical Scheme Combining Implicit Timestepping for Local Effects and Explicit Timestepping for Nonlocal Effects , 2003, math/0311363.

[2]  Luigi C. Berselli,et al.  On the Large Eddy Simulation of the Taylor–Green vortex , 2005 .

[3]  Volker John,et al.  Large Eddy Simulation of Turbulent Incompressible Flows - Analytical and Numerical Results for a Class of LES Models , 2003, Lecture Notes in Computational Science and Engineering.

[4]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[5]  Volker John,et al.  Analysis of Numerical Errors in Large Eddy Simulation , 2002, SIAM J. Numer. Anal..

[6]  W. Bangerth,et al.  deal.II—A general-purpose object-oriented finite element library , 2007, TOMS.

[7]  M. Kronbichler,et al.  An algebraic variational multiscale-multigrid method for large eddy simulation of turbulent flow , 2010 .

[8]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[9]  Monika Neda,et al.  Numerical Analysis of Filter-Based Stabilization for Evolution Equations , 2012, SIAM J. Numer. Anal..

[10]  Danesh K. Tafti,et al.  Comparison of some upwind-biased high-order formulations with a second-order central-difference scheme for time integration of the incompressible Navier-Stokes equations , 1996 .

[11]  Gunar Matthies,et al.  Nonconforming, Anisotropic, Rectangular Finite Elements of Arbitrary Order for the Stokes Problem , 2008, SIAM J. Numer. Anal..

[12]  G. Lube,et al.  Local projection stabilization for incompressible flows: equal-order vs. inf-sup stable interpolation. , 2008 .

[13]  Gert Lube,et al.  Analysis of a variational multiscale method for Large-Eddy simulation and its application to homogeneous isotropic turbulence , 2010 .

[14]  T. Hughes.  Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods , 1995 .

[15]  Max Gunzburger,et al.  Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms , 1989 .

[16]  T. Hughes,et al.  Large Eddy Simulation and the variational multiscale method , 2000 .

[17]  Rickard Bensow,et al.  Residual based VMS subgrid modeling for vortex flows , 2010 .

[18]  R. Rannacher,et al.  Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization , 1990 .

[19]  Leo G. Rebholz,et al.  Modular Nonlinear Filter Stabilization of Methods for Higher Reynolds Numbers Flow , 2012 .

[20]  G. Taylor,et al.  Mechanism of the production of small eddies from large ones , 1937 .

[21]  Petr Knobloch.  On a Variant of the Local Projection Method Stable in the SUPG Norm , 2009, Kybernetika.

[22]  L. Berselli,et al.  Mathematics of Large Eddy Simulation of Turbulent Flows , 2005 .

[23]  Volker John,et al.  A variational multiscale method for turbulent flow simulation with adaptive large scale space , 2010, J. Comput. Phys..

[24]  L. R. Scott,et al.  A quasi-local interpolation operator¶preserving the discrete divergence , 2003 .

[25]  Mihai Anitescu,et al.  IMPLICIT FOR LOCAL EFFECTS AND EXPLICIT FOR NONLOCAL EFFECTS IS UNCONDITIONALLY STABLE , 2004 .

[26]  Victor M. Calo,et al.  Improving stability of stabilized and multiscale formulations in flow simulations at small time steps , 2010 .

[27]  Luigi C. Berselli,et al.  Analytical and Numerical Results for the Rational Large Eddy Simulation Model , 2007 .

[28]  Victor M. Calo,et al.  YZβ discontinuity capturing for advection‐dominated processes with application to arterial drug delivery , 2007 .

[29]  W. Layton,et al.  NUMERICAL ANALYSIS OF A HIGHER ORDER TIME RELAXATION MODEL OF FLUIDS , 2007 .

[30]  Volker Gravemeier,et al.  The variational multiscale method for laminar and turbulent flow , 2006 .

[31]  Essential Spectrum of the Linearized 2D Euler Equation and Lyapunov–Oseledets Exponents , 2003, math-ph/0306026.

[32]  W. Layton,et al.  A connection between subgrid scale eddy viscosity and mixed methods , 2002, Appl. Math. Comput..

[33]  S. Corrsin,et al.  Simple Eulerian time correlation of full-and narrow-band velocity signals in grid-generated, ‘isotropic’ turbulence , 1971, Journal of Fluid Mechanics.

[34]  G. Taylor,et al.  LXXV. On the decay of vortices in a viscous fluid , 1923 .

[35]  M. Stynes,et al.  Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems , 1996 .

[36]  J. Guermond.  Stabilization of Galerkin approximations of transport equations by subgrid modelling , 1999 .

[37]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[38]  K. Lilly.  The representation of small-scale turbulence in numerical simulation experiments , 1966 .