Finite Element Approximation of an Unsteady Projection-Based VMS Turbulence Model with Wall Laws

In this work we present the numerical analysis and study the performance of a finite element projection-based Variational MultiScale (VMS) turbulence model that includes general non-linear wall laws. We introduce Lagrange finite element spaces adapted to approximate the slip condition. The sub-grid effects are modeled by an eddy diffusion term that acts only on a range of small resolved scales. Moreover, high-order stabilization terms are considered, with the double aim to guarantee stability for coarse meshes, and help to counter-balance the accumulation of sub-grid energy together with the sub-grid eddy viscosity term. We prove stability and convergence for solutions that only need to bear the natural minimal regularity, in unsteady regime. We also study the asymptotic energy balance of the system. We finally include some numerical tests to assess the performance of the model described in this work.

[1]  Thomas J. R. Hughes,et al.  Large eddy simulation of turbulent channel flows by the variational multiscale method , 2001 .

[2]  L. Prandtl.  7. Bericht über Untersuchungen zur ausgebildeten Turbulenz , 1925 .

[3]  Frédéric Hecht,et al.  Numerical approximation of the Smagorinsky turbulence model applied to the primitive equations of the ocean , 2014, Math. Comput. Simul..

[4]  R. Verfürth.  Finite element approximation of steady Navier-Stokes equations with mixed boundary conditions , 1985 .

[5]  Volker John,et al.  Analysis of a full space-time discretization of the Navier-Stokes equations by a local projection stabilization method , 2015 .

[6]  Cornelius O. Horgan,et al.  Korn's Inequalities and Their Applications in Continuum Mechanics , 1995, SIAM Rev..

[7]  Roger Lewandowski,et al.  Mathematical and Numerical Foundations of Turbulence Models and Applications , 2014 .

[8]  E. Boschi.  Recensioni: J. L. Lions - Quelques méthodes de résolution des problémes aux limites non linéaires. Dunod, Gauthier-Vi;;ars, Paris, 1969; , 1971 .

[9]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .

[10]  Volker Gravemeier,et al.  Scale-separating operators for variational multiscale large eddy simulation of turbulent flows , 2006, J. Comput. Phys..

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

[12]  T. Hughes,et al.  Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows , 2007 .

[13]  T. Hughes,et al.  Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes , 2010 .

[14]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[15]  R. Verfürth.  Finite element approximation on incompressible Navier-Stokes equations with slip boundary condition , 1987 .

[16]  Macarena Gómez Mármol,et al.  Numerical analysis of a finite element projection-based VMS turbulence model with wall laws , 2015 .

[17]  Thomas J. R. Hughes,et al.  The multiscale formulation of large eddy simulation: Decay of homogeneous isotropic turbulence , 2001 .

[18]  Christine Bernardi,et al.  Discr'etisations variationnelles de probl`emes aux limites elliptiques , 2004 .

[19]  J. Smagorinsky,et al.  GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS , 1963 .

[20]  P. Moin,et al.  Effects of the Computational Time Step on Numerical Solutions of Turbulent Flow , 1994 .

[21]  Vivette Girault,et al.  A high order term-by-term stabilization solver for incompressible flow problems , 2013 .

[22]  Victor M. Calo,et al.  Weak Dirichlet Boundary Conditions for Wall-Bounded Turbulent Flows , 2007 .

[23]  Carlos Paré,et al.  Existence, uniqueness and regularity of solution of the equations of a turbulence model for incompressible fluids , 1992 .

[24]  Tomás Chacón Rebollo.  An analysis technique for stabilized finite element solution of incompressible flows , 2001 .

[25]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[26]  T. Chacón Rebollo,et al.  Numerical Analysis of Penalty Stabilized Finite Element Discretizations of Evolution Navier–Stokes Equations , 2015, J. Sci. Comput..

[27]  P. Moin,et al.  Numerical investigation of turbulent channel flow , 1981, Journal of Fluid Mechanics.

[28]  D. Spalding.  A Single Formula for the “Law of the Wall” , 1961 .

[29]  T. C. Rebollo,et al.  A variational finite element model for large-eddy simulations of turbulent flows , 2013, 1302.2753.

[30]  Volker John,et al.  Numerical Studies of Finite Element Variational Multiscale Methods for Turbulent Flow Simulations , 2010 .

[31]  I. Akkerman.  Adaptive Variational Multiscale Formulations using the Discrete Germano , 2007 .

[32]  T. Kármán,et al.  Mechanische Ahnlichkeit und Turbulenz , 1930 .

[33]  R. Codina.  Comparison of some finite element methods for solving the diffusion-convection-reaction equation , 1998 .

[34]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[35]  E. R. V. Driest.  On Turbulent Flow Near a Wall , 1956 .

[36]  Li-Bin Liu,et al.  A Robust Adaptive Grid Method for a System of Two Singularly Perturbed Convection-Diffusion Equations with Weak Coupling , 2014, J. Sci. Comput..

[37]  Volker John,et al.  Finite element error analysis for a projection-based variational multiscale method with nonlinear eddy viscosity , 2008 .

[38]  John Kim,et al.  DIRECT NUMERICAL SIMULATION OF TURBULENT CHANNEL FLOWS UP TO RE=590 , 1999 .