Residual-based variational multiscale methods for turbulent flows and fluid-structure interaction

Physical problems are often characterised by a two-way interaction of fluid flow and elastic structural deformation. The fluid part will involve fully turbulent, incompressible flow in many applications of engineering interest. The present work is on residual-based variational multiscale modelling of turbulence for such problems. In this thesis, the residual-based large-eddy approach is stated in an Arbitrary-Lagrangean-Eulerian setting. It is studied thoroughly, especially with respect to the time-dependency of the subgrid scales and the numerical dissipation of the model terms. In addition, the influence of isogeometric representations and weak boundary conditions is evaluated, and a successful application of the approach in an example computation is provided.

[1]  P Jamet,et al.  Numerical solution of the eulerian equations of compressible flow by a finite element method which follows the free boundary and the interfaces , 1975 .

[2]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[3]  C. Truesdell,et al.  The Non-Linear Field Theories Of Mechanics , 1992 .

[4]  U. Frisch.  Turbulence: The Legacy of A. N. Kolmogorov , 1996 .

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

[6]  Charbel Farhat,et al.  A Variational Multiscale Method for the Large Eddy Simulation of Compressible Turbulent Flows on Unstructured Meshes - Application to vortex shedding , 2004 .

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

[8]  Erich Rothe,et al.  Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben , 1930 .

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

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

[11]  Gerald Farin,et al.  NURBS: From Projective Geometry to Practical Use , 1999 .

[12]  D FalgoutRobert.  An Introduction to Algebraic Multigrid , 2006 .

[13]  Kenneth E. Jansen,et al.  Spatial test filters for dynamic model large‐eddy simulation with finite elements , 2002 .

[14]  David F. Rogers,et al.  An Introduction to NURBS , 2000 .

[15]  Charbel Farhat,et al.  A three-dimensional torsional spring analogy method for unstructured dynamic meshes , 2002 .

[16]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[17]  Ramon Codina,et al.  Dynamic subscales in the finite element approximation of thermally coupled incompressible flows , 2007 .

[18]  John A. Evans,et al.  Isogeometric analysis using T-splines , 2010 .

[19]  Volker John,et al.  On large Eddy simulation and variational multiscale methods in the numerical simulation of turbulent incompressible flows , 2006 .

[20]  Jurijs Bazilevs,et al.  Isogeometric analysis of turbulence and fluid -structure interaction , 2006 .

[21]  N. Adams,et al.  An approximate deconvolution procedure for large-eddy simulation , 1999 .

[22]  Thomas J. R. Hughes,et al.  Patient-specific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device , 2009 .

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

[24]  R. Codina.  Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods , 2000 .

[25]  B. Geurts,et al.  Large-eddy simulation of the turbulent mixing layer , 1997, Journal of Fluid Mechanics.

[26]  Peter Deuflhard,et al.  Numerische Mathematik II , 1994 .

[27]  J. Huber,et al.  Grundlagen der Wahrscheinlichkeitsrechnung für iterative Decodierverfahren , 2002 .

[28]  Victor M. Calo,et al.  Residual-based multiscale turbulence modeling: Finite volume simulations of bypass transition , 2005 .

[29]  P. Wriggers.  Computational contact mechanics , 2012 .

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

[31]  Peter Knabner,et al.  Numerik partieller Differentialgleichungen , 2000 .

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

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

[34]  Julian F. Scott,et al.  An Introduction to Turbulent Flow , 2000 .

[35]  Anath Fischer,et al.  New B‐Spline Finite Element approach for geometrical design and mechanical analysis , 1998 .

[36]  C. W. Hirt,et al.  An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds , 1997 .

[37]  Y. Notay,et al.  Efficient iterative solution of constrained finite element analyses , 1998 .

[38]  S. Rebay,et al.  High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations , 1997 .

[39]  Volker John,et al.  Simulations of the turbulent channel flow at Reτ = 180 with projection-based finite element variational multiscale methods , 2007 .

[40]  Nikolaus A. Adams,et al.  An adaptive local deconvolution method for implicit LES , 2005, J. Comput. Phys..

[41]  Wulf G. Dettmer,et al.  An analysis of the time integration algorithms for the finite element solutions of incompressible Navier-Stokes equations based on a stabilised formulation , 2003 .

[42]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[43]  Lewis F. Richardson,et al.  Weather Prediction by Numerical Process , 1922 .

[44]  Kumbakonam R. Rajagopal,et al.  An Introduction to the Mechanics of Fluids , 1999 .

[45]  Roland Wüchner,et al.  Isogeometric shell analysis with Kirchhoff–Love elements , 2009 .

[46]  L. Franca,et al.  Stabilized Finite Element Methods , 1993 .

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

[48]  B. Simeon,et al.  Adaptive isogeometric analysis by local h-refinement with T-splines , 2010 .

[49]  N. Adams,et al.  Implicit subgrid-scale modeling by adaptive deconvolution , 2004 .

[50]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[51]  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 .

[52]  P. Wesseling.  Principles of Computational Fluid Dynamics , 2000 .

[53]  Assad A. Oberai,et al.  A dynamic approach for evaluating parameters in a numerical method , 2005 .

[54]  A. Huerta,et al.  Arbitrary Lagrangian–Eulerian Methods , 2004 .

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

[56]  E. Hairer,et al.  Solving Ordinary Differential Equations I , 1987 .

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

[58]  A. Oberai,et al.  Spectral analysis of the dissipation of the residual-based variational multiscale method , 2010 .

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

[60]  Rolf Rannacher,et al.  On the Numerical Solution of the Incompressible Navier‐Stokes Equations , 1993 .

[61]  Giancarlo Alfonsi,et al.  Reynolds-Averaged Navier–Stokes Equations for Turbulence Modeling , 2009 .

[62]  Alexander Linke,et al.  Collision in a cross-shaped domain - A steady 2d Navier-Stokes example demonstrating the importance of mass conservation in CFD , 2009 .

[63]  Emanuel Parzen,et al.  Stochastic Processes , 1962 .

[64]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[65]  Yong Jung Kim.  A MATHEMATICAL INTRODUCTION TO FLUID MECHANICS , 2008 .

[66]  Chi-Wang Shu,et al.  High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems , 2009, SIAM Rev..

[67]  Thomas J. R. Hughes,et al.  NURBS-based isogeometric analysis for the computation of flows about rotating components , 2008 .

[68]  A. Kolmogorov.  The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[69]  Thomas J. R. Hughes,et al.  Conservation properties for the Galerkin and stabilised forms of the advection–diffusion and incompressible Navier–Stokes equations , 2005 .

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

[71]  A. R. Humphries,et al.  Dynamical Systems And Numerical Analysis , 1996 .

[72]  R. Bird,et al.  Constitutive equations for polymeric liquids , 1995 .

[73]  G. Böhme,et al.  Strömungsmechanik nichtnewtonscher Fluide , 2000 .

[74]  R. Peyret.  Spectral Methods for Incompressible Viscous Flow , 2002 .

[75]  C. Farhat,et al.  Torsional springs for two-dimensional dynamic unstructured fluid meshes , 1998 .

[76]  P. Hansbo,et al.  Nitsche's method combined with space–time finite elements for ALE fluid–structure interaction problems☆ , 2004 .

[77]  Masud Behnia,et al.  Reynolds averaged simulation of unsteady separated flow , 2003 .

[78]  Christiane Förster,et al.  Robust methods for fluid-structure interaction with stabilised finite elements , 2007 .

[79]  E. D. Obasaju,et al.  An experimental study of pressure fluctuations on fixed and oscillating square-section cylinders , 1982, Journal of Fluid Mechanics.

[80]  P Jamet,et al.  Numerical computation of the free boundary for the two-dimensional Stefan problem by space-time finite elements , 1977 .

[81]  C F Curtiss,et al.  Integration of Stiff Equations. , 1952, Proceedings of the National Academy of Sciences of the United States of America.

[82]  Pb Bornemann.  Time integration algorithms for the steady states of dissipative non-linear dynamic systems , 2003 .

[83]  Erik Burman,et al.  Stabilized finite element methods for the generalized Oseen problem , 2007 .

[84]  Volker Gravemeier,et al.  The variational multiscale method for laminar and turbulent incompressible flow , 2003 .

[85]  Jintai Chung,et al.  A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method , 1993 .

[86]  Antonio Huerta,et al.  Imposing essential boundary conditions in mesh-free methods , 2004 .

[87]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[88]  Ted Belytschko,et al.  Quasi-Eulerian Finite Element Formulation for Fluid-Structure Interaction , 1980 .

[89]  T. Tezduyar,et al.  A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. I: The concept and the preliminary numerical tests , 1992 .

[90]  W. Rodi.  Comparison of LES and RANS calculations of the flow around bluff bodies , 1997 .

[91]  R. Lahey,et al.  Direct numerical simulation of turbulent channel flows using a stabilized finite element method , 2009 .

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

[93]  T. A. Zang,et al.  Spectral Methods: Fundamentals in Single Domains , 2010 .

[94]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[95]  Kai Lai Chung,et al.  A Course in Probability Theory , 1949 .

[96]  Rainald Löhner,et al.  Improved ALE mesh velocities for moving bodies , 1996 .

[97]  R. D. Wood,et al.  Nonlinear Continuum Mechanics for Finite Element Analysis , 1997 .

[98]  R. Courant,et al.  Über die partiellen Differenzengleichungen der mathematischen Physik , 1928 .

[99]  Clive Fletcher,et al.  Computational techniques for fluid dynamics., 2nd Edition , 1991 .

[100]  Lilia Krivodonova,et al.  High-order accurate implementation of solid wall boundary conditions in curved geometries , 2006 .

[101]  Pavel B. Bochev,et al.  On stabilized finite element methods for the Stokes problem in the small time step limit , 2007 .

[102]  Ramon Codina,et al.  The dissipative structure of variational multiscale methods for incompressible flows , 2010 .

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

[104]  Dominik Schötzau,et al.  hp -finite element simulations for Stokes flow—stable and stabilized , 1999 .

[105]  Tayfun E. Tezduyar,et al.  Finite element stabilization parameters computed from element matrices and vectors , 2000 .

[106]  K. Jansen,et al.  A dynamic Smagorinsky model with dynamic determination of the filter width ratio , 2004 .

[107]  Ulrich Küttler,et al.  Effiziente Lösungsverfahren für Fluid-Struktur-Interaktions-Probleme , 2009 .

[108]  Rolf Stenberg,et al.  On some techniques for approximating boundary conditions in the finite element method , 1995 .

[109]  C. Meneveau,et al.  Scale-Invariance and Turbulence Models for Large-Eddy Simulation , 2000 .

[110]  J. P. Boris,et al.  New insights into large eddy simulation , 1992 .

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

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

[113]  M. Gurtin,et al.  An introduction to continuum mechanics , 1981 .

[114]  T. N. Stevenson,et al.  Fluid Mechanics , 2021, Nature.

[115]  T. Hughes,et al.  Isogeometric Fluid–structure Interaction Analysis with Applications to Arterial Blood Flow , 2006 .

[116]  G. Hulbert,et al.  A generalized-α method for integrating the filtered Navier–Stokes equations with a stabilized finite element method , 2000 .

[117]  Pavel B. Bochev,et al.  On the Finite Element Solution of the Pure Neumann Problem , 2005, SIAM Rev..

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

[119]  J. M. Watt.  Numerical Initial Value Problems in Ordinary Differential Equations , 1972 .

[120]  Joel H. Ferziger,et al.  Computational methods for fluid dynamics , 1996 .

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

[122]  Maxim A. Olshanskii,et al.  Grad–div stabilization and subgrid pressure models for the incompressible Navier–Stokes equations , 2009 .

[123]  A. Kolmogorov.  Dissipation of energy in the locally isotropic turbulence , 1941, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[124]  A. Huerta,et al.  Finite Element Methods for Flow Problems , 2003 .

[125]  Charles R. Doering.  The 3D Navier-Stokes Problem , 2009 .

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

[127]  M. Benzi.  Preconditioning techniques for large linear systems: a survey , 2002 .

[128]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[129]  Wolfgang A. Wall,et al.  Enhancement of fixed‐grid methods towards complex fluid–structure interaction applications , 2008 .

[130]  Peter Deuflhard,et al.  Numerische Mathematik. I , 2002 .

[131]  Thomas J. R. Hughes,et al.  Isogeometric shell analysis: The Reissner-Mindlin shell , 2010 .

[132]  Leonhard Kleiser,et al.  Subgrid‐scale energy transfer in the near‐wall region of turbulent flows , 1994 .

[133]  L. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communications.

[134]  L. Davidson,et al.  Large Eddy Simulation of Flow Past a Square Cylinder: Comparison of Different Subgrid Scale Models , 2000 .

[135]  Thomas J. R. Hughes,et al.  Weak imposition of Dirichlet boundary conditions in fluid mechanics , 2007 .

[136]  Marek Behr,et al.  Stabilized finite element methods for incompressible flows with emphasis on moving boundaries and interfaces , 1992 .

[137]  Joel H. Ferziger,et al.  Status of Large Eddy Simulation: Results of a Workshop , 1997 .

[138]  Thomas J. R. Hughes,et al.  Finite element modeling of blood flow in arteries , 1998 .

[139]  B. Lee.  The effect of turbulence on the surface pressure field of a square prism , 1975, Journal of Fluid Mechanics.

[140]  G. Batchelor,et al.  An Introduction to Fluid Dynamics , 1968 .

[141]  Spencer J. Sherwin,et al.  Spectral/hp discontinuous Galerkin methods for modelling 2D Boussinesq equations , 2006, J. Comput. Phys..

[142]  T. R. Hughes,et al.  Mathematical foundations of elasticity , 1982 .

[143]  K. Jansen,et al.  On the interaction between dynamic model dissipation and numerical dissipation due to streamline upwind/Petrov–Galerkin stabilization , 2005 .

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

[145]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[146]  Santiago Badia,et al.  Analysis of a Stabilized Finite Element Approximation of the Transient Convection-Diffusion Equation Using an ALE Framework , 2006, SIAM J. Numer. Anal..

[147]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .