A three-level finite element method for the instationary incompressible Navier?Stokes equations

We present a two- and a three-level finite element method for the numerical simulation of incompressible flow governed by the Navier-Stokes equations. Within the theoretical framework of the variational multiscale method and exploiting the concept of residual-free bubbles we propose a separation of three different scale groups, namely large (resolved) scales, small (resolved) scales, and unresolved scales. The resolution of the large and small scales takes place on levels 1 and 2 with the aid of diverse approaches. The dynamic calculation of a subgrid viscosity representing the effect of the unresolved scales constitutes level 3 of our algorithm. The proposed algorithm is tested for various laminar flow situations and compared to the results obtained via an unusual stabilized finite element method. It is supposed to be the basic scheme also for turbulent flow in a subsequent study.

[1]  Thomas J. R. Hughes,et al.  A space-time formulation for multiscale phenomena , 1996 .

[2]  S. Collis,et al.  Monitoring unresolved scales in multiscale turbulence modeling , 2001 .

[3]  Antonini Macedo,et al.  Two Level Finite Element Method and its Application to the Helmholtz Equation , 1997 .

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

[5]  J. Z. Zhu,et al.  The finite element method , 1977 .

[6]  Numerical Analysis 1999 , 2000 .

[7]  Alessandro Russo,et al.  Further considerations on residual-free bubbles for advective - diffusive equations , 1998 .

[8]  P. Sagaut.  Large Eddy Simulation for Incompressible Flows , 2001 .

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

[10]  Walter Tollmien,et al.  Über die ausgebildete Turbulenz , 1961 .

[11]  M. Lesieur,et al.  New Trends in Large-Eddy Simulations of Turbulence , 1996 .

[12]  T. Hughes,et al.  Stabilized finite element methods. I: Application to the advective-diffusive model , 1992 .

[13]  S. Mittal,et al.  Incompressible flow past a circular cylinder: dependence of the computed flow field on the location of the lateral boundaries , 1995 .

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

[15]  Jukka Saranen,et al.  Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations , 1986 .

[16]  R. Codina.  Stabilized finite element approximation of transient incompressible flows using orthogonal subscales , 2002 .

[17]  R. Sani,et al.  Incompressible Flow and the Finite Element Method, Volume 1, Advection-Diffusion and Isothermal Laminar Flow , 1998 .

[18]  Claes Johnson.  Numerical solution of partial differential equations by the finite element method , 1988 .

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

[20]  Thomas J. R. Hughes,et al.  What are C and h ?: inequalities for the analysis and design of finite element methods , 1992 .

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

[22]  U. Piomelli.  Large-Eddy Simulation of turbulent flows , 1998 .

[23]  C. Norberg.  An experimental investigation of the flow around a circular cylinder: influence of aspect ratio , 1994, Journal of Fluid Mechanics.

[24]  Martin Stynes,et al.  On the Stability of Residual-Free Bubbles for Convection-Diffusion Problemsand their Approximation by a Two-Level Finite Element Method. , 1997 .

[25]  Peter Hansbo,et al.  A velocity pressure streamline diffusion finite element method for Navier-Stokes equations , 1990 .

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

[27]  Antonio Huerta,et al.  Time-accurate solution of stabilized convection-diffusion-reaction equations: I-time and space discretization , 2002 .

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

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

[30]  O. Zienkiewicz,et al.  The hierarchical concept in finite element analysis , 1983 .

[31]  Thierry Dubois,et al.  Dynamic multilevel methods and the numerical simulation of turbulence , 1999 .

[32]  C. Farhat,et al.  Bubble Functions Prompt Unusual Stabilized Finite Element Methods , 1994 .

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

[34]  Peter Hansbo,et al.  A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equation , 1990 .

[35]  Eugenio Oñate,et al.  Recent developments in finite element analysis , 1994 .

[36]  I. Babuska,et al.  Finite Element Analysis , 2021 .

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

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

[39]  Franco Brezzi,et al.  $b=\int g$ , 1997 .

[40]  Franco Brezzi,et al.  Virtual bubbles and Galerkin-least-squares type methods (Ga.L.S.) , 1993 .

[41]  C. Ross Ethier,et al.  Exact fully 3D Navier–Stokes solutions for benchmarking , 1994 .

[42]  Alessandro Russo,et al.  CHOOSING BUBBLES FOR ADVECTION-DIFFUSION PROBLEMS , 1994 .

[43]  D. Wilcox.  Turbulence modeling for CFD , 1993 .

[44]  Eugene O'Riordan,et al.  Special meshes for finite difference approximations to an advection-diffusion equation with parabolic layers , 1995 .

[45]  M. V. Dyke,et al.  An Album of Fluid Motion , 1982 .

[46]  Jean-Luc Guermond,et al.  Subgrid stabilization of Galerkin approximations of linear monotone operators , 2001 .

[47]  Ugo Piomelli,et al.  Large-eddy simulation: achievements and challenges , 1999 .

[48]  Parviz Moin,et al.  ADVANCES IN LARGE EDDY SIMULATION METHODOLOGY FOR COMPLEX FLOWS , 2002, Proceeding of Second Symposium on Turbulence and Shear Flow Phenomena.

[49]  L. Franca,et al.  Refining the submesh strategy in the two‐level finite element method: application to the advection–diffusion equation , 2002 .

[50]  Liang Cheng,et al.  Spanwise length effects on three-dimensional modelling of flow over a circular cylinder , 2001 .

[51]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[52]  K. Bathe,et al.  On upwind methods for parabolic finite elements in incompressible flows , 2000 .

[53]  Christian H. Whiting,et al.  STABILIZED FINITE ELEMENT METHODS FOR FLUID DYNAMICS USING A HIERARCHICAL BASIS , 1999 .

[54]  L. Franca,et al.  On an Improved Unusual Stabilized Finite Element Method for theAdvective-Reactive-Diffusive Equation , 1999 .

[55]  Charbel Farhat,et al.  Unusual Stabilized Finite Element Methods and Residual-Free-Bubbles , 1996 .

[56]  W. Rodi,et al.  Closure Strategies for Turbulent and Transitional Flows: Introduction to Large Eddy Simulation of Turbulent Flows , 2002 .

[57]  Endre Süli,et al.  Modeling subgrid viscosity for advection–diffusion problems , 2000 .

[58]  P. Moin,et al.  DIRECT NUMERICAL SIMULATION: A Tool in Turbulence Research , 1998 .

[59]  C. Williamson.  Vortex Dynamics in the Cylinder Wake , 1996 .

[60]  S. Dey,et al.  Hierarchical basis for stabilized finite element methods for compressible flows , 2003 .

[61]  F. Brezzi,et al.  A relationship between stabilized finite element methods and the Galerkin method with bubble functions , 1992 .

[62]  P. Moin,et al.  Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations , 1984 .

[63]  P. Moin,et al.  A dynamic subgrid‐scale eddy viscosity model , 1990 .

[64]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[65]  T. Hughes,et al.  The Galerkin/least-squares method for advective-diffusive equations , 1988 .

[66]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuscka-Brezzi condition: A stable Petrov-Galerkin formulation of , 1986 .

[67]  Claes Johnson,et al.  Finite element methods for linear hyperbolic problems , 1984 .

[68]  Wolfgang A. Wall.  Fluid-Struktur-Interaktion mit stabilisierten Finiten Elementen , 1999 .

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

[70]  Leopoldo P. Franca,et al.  On a two‐level finite element method for the incompressible Navier–Stokes equations , 2000 .

[71]  Alessandro Russo,et al.  Bubble stabilization of finite element methods for the linearized incompressible Navier-Stokes equations , 1996 .

[72]  Kenneth E. Jansen,et al.  A stabilized finite element method for the incompressible Navier–Stokes equations using a hierarchical basis , 2001 .

[73]  H. Schlichting.  Boundary Layer Theory , 1955 .