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Recent Developments in Variational Multiscale Methods for Large-Eddy Simulation of Turbulent Flow

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Abstract

The variational multiscale method is reviewed as a framework for developing computational methods for large-eddy simulation of turbulent flow. In contrast to other articles reviewing this topic, which focused on large-eddy simulation of turbulent incompressible flow, this study covers further aspects of numerically simulating turbulent flow as well as applications beyond incompressible single-phase flow. The various concepts for subgrid-scale modeling within the variational multiscale method for large-eddy simulation proposed by researchers in this field to date are illustrated. These conceptions comprise (i) implicit large-eddy simulation, represented by residual-based and stabilized methods, (ii) functional subgrid-scale modeling via small-scale subgrid-viscosity models and (iii) structural subgrid-scale modeling via the introduction of multifractal subgrid scales. An overview on exemplary numerical test cases to which the reviewed methods have been applied in the past years is provided, including explicit computational results obtained from turbulent channel flow. Wall-layer modeling, passive and active scalar transport as well as developments for large-eddy simulation of turbulent two-phase flow and combustion are discussed to complete this exposition.

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References

  1. Ahmed N, Chacón Rebollo T, John V, Rubino S (2017) A review of variational multiscale methods for the simulation of turbulent incompressible flows. Arch Comput Methods Eng 24:115. doi: 10.1007/s11831-015-9161-0

    MathSciNet  MATH  Google Scholar 

  2. Akkerman I, Bazilevs Y, Kees CE, Farthing MW (2011) Isogeometric analysis of free-surface flow. J Comput Phys 230:4137–4152

    MathSciNet  MATH  Google Scholar 

  3. Araya G, Bohr E, Jansen K, Castillo L, Peterson K (2006) Generation of turbulent inlet conditions for thermal/velocity boundary layer simulations. AIAA Paper 2006-0699, Reno, NV

  4. Avila M, Codina R, Principe J (2014) Large eddy simulation of low Mach number flows using dynamic and orthogonal subgrid scales. Comput Fluids 99:44–66

    MathSciNet  Google Scholar 

  5. Balaras E, Benocci C, Piomelli U (1996) Two layer approximate boundary conditions for large-eddy simulations. AIAA J 34:1111–1119

    MATH  Google Scholar 

  6. Bardina J, Ferziger JH, Reynolds WC (1980) Improved subgrid models for large eddy simulation. AIAA Paper 1980-1357, Snowmass, CO

  7. Bardina J, Ferziger JH, Reynolds WC (1983) Improved turbulence models based on large eddy simulation of homogeneous, incompressible, turbulent flows. Technical Report TF-19, Thermosciences Division, Department of Mechanical Engineering, Stanford University

  8. Bauer G, Gamnitzer P, Gravemeier V, Wall WA (2013) An isogeometric variational multiscale method for large-eddy simulation of coupled multi-ion transport in turbulent flow. J Comput Phys 251:194–208

    MathSciNet  MATH  Google Scholar 

  9. Bazilevs Y, Calo VM, Cottrell JA, Hughes TJR, Reali A, Scovazzi G (2007) Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput Methods Appl Mech Eng 197:173–201

    MathSciNet  MATH  Google Scholar 

  10. Bazilevs Y, Hughes TJR (2007) Weak imposition of Dirichlet boundary conditions in fluid mechanics. Comput Fluids 36:12–26

    MathSciNet  MATH  Google Scholar 

  11. Bazilevs Y, Michler C, Calo VM, Hughes TJR (2010) Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes. Comput Methods Appl Mech Eng 199:780–790

    MathSciNet  MATH  Google Scholar 

  12. Becker R, Braack M (2001) A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38:173–199

    MathSciNet  MATH  Google Scholar 

  13. Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45:601–620

    MATH  Google Scholar 

  14. Berselli LC, Iliescu T, Layton WJ (2006) Mathematics of large eddy simulation of turbulent flows. Springer, Berlin

    MATH  Google Scholar 

  15. Bilger RW, Pope SB, Bray KNC, Driscoll JF (2005) Paradigms in turbulent combustion research. Proc Combust Inst 30:21–42

    Google Scholar 

  16. Bochev PB, Gunzburger MD, Lehoucq RB (2007) On stabilized finite element methods for the Stokes problem in the small time step limit. Int J Numer Methods Fluids 53:573–597

    MathSciNet  MATH  Google Scholar 

  17. Boris JP, Grinstein FF, Oran ES, Kolbe RL (1992) New insights into large-eddy simulation. Fluid Dyn Res 10:199–228

    Google Scholar 

  18. van der Bos F, Gravemeier V (2009) Numerical simulation of premixed combustion using an enriched finite element method. J Comput Phys 228:3605–3624

    MathSciNet  MATH  Google Scholar 

  19. van der Bos F, van der Vegt JJW, Geurts BJ (2007) A multi-scale formulation for compressible turbulent flows suitable for general variational discretization techniques. Comput Methods Appl Mech Eng 196:2863–2875

    MathSciNet  MATH  Google Scholar 

  20. Braack M, Burman E (2006) Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. Comput Methods Appl Mech Eng 43:2544–2566

    MathSciNet  MATH  Google Scholar 

  21. Braack M, Burman E, John V, Lube G (2007) Stabilized finite element methods for the generalized Oseen problem. Comput Methods Appl Mech Eng 196:853–866

    MathSciNet  MATH  Google Scholar 

  22. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface tension. J Comput Phys 100:335–354

    MathSciNet  MATH  Google Scholar 

  23. Bray KNC, Moss JB (1977) A unified statistical model of the premixed turbulent flame. Acta Astronaut 4:291–319

    Google Scholar 

  24. Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, New York

    MATH  Google Scholar 

  25. Brezzi F, Franca LP, Hughes TJR, Russo A (1997) \(b = \int g.\) Comput Methods Appl Mech Eng 145:329–339

  26. Brooks AN, Hughes TJR (1982) Streamline Upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 32:199–259

    MathSciNet  MATH  Google Scholar 

  27. Buch KA, Dahm WJA (1998) Experimental study of the fine-scale structure of conserved scalar mixing in turbulent shear flows. Part 2. \({\text Sc}\approx 1\). J Fluid Mech 364:1–29

    MATH  Google Scholar 

  28. Burman E, Fernández MA, Hansbo P (2006) Continuous interior penalty finite element method for Oseen’s equations. SIAM J Numer Anal 44:1248–1274

    MathSciNet  MATH  Google Scholar 

  29. Burman E, Hansbo P (2004) Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput Methods Appl Mech Eng 193:1437–1453

    MathSciNet  MATH  Google Scholar 

  30. Burman E, Hansbo P (2012) Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl Numer Math 62:328–341

    MathSciNet  MATH  Google Scholar 

  31. Burman E, Hansbo P (2014) Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem. ESAIM. Math Model Numer Anal 48:859–874

    MathSciNet  MATH  Google Scholar 

  32. Burman E, Zunino P (2012) Numerical approximation of large contrast problems with the unfitted Nitsche method. In: Blowey J, Jensen M (eds) Frontiers in numerical analysis. Lecture Notes in Computational Science and Engineering, vol 85. Springer, Berlin, pp 227–282

  33. Burton GC (2003) A multifractal subgrid-scale model for large-eddy simulation of turbulent flows. Dissertation, The University of Michigan

  34. Burton GC (2008) The nonlinear large-eddy simulation method applied to \({\text Sc} \approx 1 \, {\text and} \; {\text Sc} \gg 1\) passive-scalar mixing. Phys Fluids 20:035103

    MATH  Google Scholar 

  35. Borton GC, Dahm WJA (2005) Multifractal subgrid-scale modeling for large-eddy simulation. I. Model development and a priori testing. Phys Fluid 17:075111

    MathSciNet  MATH  Google Scholar 

  36. Burton GC, Dahm WJA (2005) Multifractal subgrid-scale modeling for large-eddy simulation. II. Backscatter limiting and a posteriori evaluation. Phys Fluids 17:075112

    MathSciNet  MATH  Google Scholar 

  37. Cabot W, Moin P (1999) Approximate wall boundary conditions in the large-eddy simulation of high Reynolds number flows. Flow Turbul Combust 63:269–291

    MATH  Google Scholar 

  38. Calo VM (2004) Residual-based multiscale turbulence modeling: Finite volume simulations of bypass transition. Dissertation, Stanford University

  39. Cant RS, Mastorakos E (2008) An introduction to turbulent reacting flows. Imperial College Press, London

    MATH  Google Scholar 

  40. Chacón Rebollo T, Gómez Mármol M, Rubino S (2015) Numerical analysis of a finite element projection-based VMS turbulence model with wall laws. Comput Methods Appl Mech Eng 285:379–405

    MathSciNet  Google Scholar 

  41. Chakravarthy VK, Menon S (2001) Large-eddy simulation of turbulent premixed flames in the flamelet regime. Combust Sci Technol 162:175–222

    Google Scholar 

  42. Chapman DR (1979) Computational aerodynamics development and outlook. AIAA J 17:1293–1313

    MATH  Google Scholar 

  43. Chessa J, Belytschko T (2003) An extended finite element method for two-phase fluids. J Appl Mech 70:10–17

    MathSciNet  MATH  Google Scholar 

  44. Choi H, Moin P (1994) Effects of the computational time step on numerical solutions of turbulent flow. J Comput Phys 113:1–4

    MATH  Google Scholar 

  45. Choi H, Moin P (2012) Grid-point requirements for large eddy simulation: Chapman‘s estimates revisited. J Comput Phys 24:011702

    Google Scholar 

  46. Chung J, Hulbert GM (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-\(\alpha\) method. J Appl Mech 60:371–375

    MathSciNet  MATH  Google Scholar 

  47. Codina R (2002) Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput Methods Appl Mech Eng 191:4295–4321

    MathSciNet  MATH  Google Scholar 

  48. Codina R, Principe J, Avila M (2010) Finite element approximation of turbulent thermally coupled incompressible flows with numerical sub-grid scale modelling. Int J Numer Methods Heat Fluid Flow 20:492–515

    MathSciNet  MATH  Google Scholar 

  49. Codina R, Principe J, Guasch O, Badia S (2007) Time dependent subscales in the stabilized finite element approximation of incompressible flow problems. Comput Methods Appl Mech Eng 196:2413–2430

    MathSciNet  MATH  Google Scholar 

  50. Collis SS (2002) The DG/VMS method for unified turbulence simulation. AIAA Paper 2002-3124, St. Louis, MO

  51. Collis SS (2001) Monitoring unresolved scales in multiscale turbulence modeling. Phys Fluids 13:1800–1806

    MATH  Google Scholar 

  52. Colomés O, Badia S, Codina R, Principe J (2015) Assessment of variational multiscale models for the large eddy simulation of turbulent incompressible flows. Comput Methods Appl Mech Eng 285:32–63

    MathSciNet  Google Scholar 

  53. Comerford A, Gravemeier V, Wall WA (2013) An algebraic variational multiscale-multigrid method for large-eddy simulation of turbulent pulsatile flows in complex geometries with detailed insight into pulmonary airway flow. Int J Numer Methods Fluids 71:1207–1225

    MathSciNet  Google Scholar 

  54. De Mulder T (1998) The role of bulk viscosity in stabilized finite element formulations for incompressible flow: a review. Comput Methods Appl Mech Eng 163:1–10

    MathSciNet  MATH  Google Scholar 

  55. Deardorff JW (1970) A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J Fluid Mech 41:453–480

    MATH  Google Scholar 

  56. Domaradzki JA, Adams NA (2002) Direct modeling of subgrid-scales of turbulence in large eddy simulation. J Turbul 3:024

    Google Scholar 

  57. Domaradzki JA, Loh K (1999) The subgrid-scale estimation model in the physical space representation. Phys Fluids 11:2330–2342

    MathSciNet  MATH  Google Scholar 

  58. Elgeti S, Sauerland H (2016) Deforming fluid domains within the finite element method: five mesh-based tracking methods in comparison. Arch Comput Methods Eng 23:323–361

    MathSciNet  MATH  Google Scholar 

  59. Erlebacher G, Hussaini MY, Speziale CG, Zang TA (1992) Toward the large-eddy simulation of compressible turbulent flows. J Fluid Mech 238:155–185

    MATH  Google Scholar 

  60. Farhat C, Rajasekharan A, Koobus B (2006) A dynamic variational multiscale method for large eddy simulations on unstructured meshes. Comput Methods Appl Mech Eng 195:1667–1691

    MathSciNet  MATH  Google Scholar 

  61. Fedorchenko AT (1997) A model of unsteady subsonic flow with acoustics excluded. J Fluid Mech 334:135–155

    MathSciNet  MATH  Google Scholar 

  62. Franca LP, Hughes TJR (1988) Two classes of mixed finite element methods. Comput Methods Appl Mech Eng 69:89–129

    MathSciNet  MATH  Google Scholar 

  63. Franca LP, Nesliturk A (2001) On a two-level finite element method for the incompressible Navier-Stokes equations. Int J Numer Methods Eng 52:433–453

    MATH  Google Scholar 

  64. Frederiksen RD, Dahm WJA, Dowling DR (1997) Experimental assessment of fractal scale similarity in turbulent flows. Part 3. Multifractal scaling. J Fluid Mech 338:127–155

    Google Scholar 

  65. Fries TP, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84:253–304

    MathSciNet  MATH  Google Scholar 

  66. Fröhlich J, Rodi W (2002) Introduction to large eddy simulation of turbulent flows. In: Launder BE, Sandham ND (eds) Closure strategies for turbulent and transitional flows. Cambridge University Press, Cambridge, pp 267–298

    Google Scholar 

  67. Fröhlich J, von Terzi D (2008) Hybrid LES/RANS methods for the simulation of turbulent flows. Prog Aerosp Sci 44:349–377

    Google Scholar 

  68. Gamnitzer P, Gravemeier V, Wall WA (2010) Time-dependent subgrid scales in residual-based large eddy simulation of turbulent channel flow. Comput Methods Appl Mech Eng 199:819–827

    MathSciNet  MATH  Google Scholar 

  69. Gamnitzer P, Gravemeier V, Wall WA (2012) A mixed/hybrid Dirichlet formulation for wall-bounded flow problems including turbulent flow. Comput Methods Appl Mech Eng 245–246:22–35

    MathSciNet  MATH  Google Scholar 

  70. Garnier E, Adams N, Sagaut P (2009) Large eddy simulation for compressible flows. Springer, New York

    MATH  Google Scholar 

  71. Georgiadis NJ, Rizzetta DP, Fureby C (2010) Large-eddy simulation: current capabilities, recommended practices, and future research. AIAA J 48:1772–1784

    Google Scholar 

  72. Germano M (1992) Turbulence: the filtering approach. J Fluid Mech 238:325–336

    MathSciNet  MATH  Google Scholar 

  73. Germano M, Piomelli U, Moin P, Cabot WH (1991) A dynamic subgrid-scale eddy viscosity model. Phys Fluids A 3:1760–1765

    MATH  Google Scholar 

  74. Gerstenberger A, Wall WA (2010) An embedded Dirichlet formulation for 3D continua. Int J Numer Methods Eng 82:537–563

    MathSciNet  MATH  Google Scholar 

  75. Geurts BJ (2004) Elements of direct and large eddy simulation. R. T. Edwards, Philadelphia

    Google Scholar 

  76. Ghosal S, Lund TS, Moin P, Akselvoll K (1995) A dynamic localization model for large-eddy simulation of turbulent flows. J Fluid Mech 286:229–255

    MathSciNet  MATH  Google Scholar 

  77. Gicquel LYM, Staffelbach G, Poinsot T (2012) Large eddy simulation of gaseous flames in gas turbine combustion chambers. Prog Energy Combust Sci 38:782–817

    Google Scholar 

  78. Gravemeier V (2006) A consistent dynamic localization model for large eddy simulation of turbulent flows based on a variational formulation. J Comput Phys 218:677–701

    MathSciNet  MATH  Google Scholar 

  79. Gravemeier V (2006) Scale-separating operators for variational multiscale large eddy simulation of turbulent flows. J Comput Phys 212:400–435

    MathSciNet  MATH  Google Scholar 

  80. Gravemeier V (2006) The variational multiscale method for laminar and turbulent flow. Arch Comput Methods in Eng 13:249–324

    MathSciNet  MATH  Google Scholar 

  81. Gravemeier V (2007) Variational multiscale large eddy simulation of turbulent flow in a diffuser. Comput Mech 39:477–495

    MATH  Google Scholar 

  82. Gravemeier V, Comerford A, Yoshihara L, Ismail M, Wall WA (2012) A novel formulation for Neumann inflow boundary conditions in biomechanics. Int J Numer Methods Biomed Eng 28:560–573

    MathSciNet  MATH  Google Scholar 

  83. Gravemeier V, Gee MW, Kronbichler M, Wall WA (2010) An algebraic variational multiscale-multigrid method for large-eddy simulation of turbulent flow. Comput Methods Appl Mech Eng 199:853–864

    MathSciNet  MATH  Google Scholar 

  84. Gravemeier V, Gee MW, Wall WA (2009) An algebraic variational multiscale-multigrid method based on plain aggregation for convection-diffusion problems. Comput Methods Appl Mech Eng 198:3821–3835

    MathSciNet  MATH  Google Scholar 

  85. Gravemeier V, Kronbichler M, Gee MW, Wall WA (2011) An algebraic variational multiscale-multigrid method for large-eddy simulation: generalized-\(\alpha\) time integration, Fourier analysis and application to turbulent flow past a square-section cylinder. Comput Mech 47:217–233

    MathSciNet  MATH  Google Scholar 

  86. Gravemeier V, Lenz S, Wall WA (2008) Towards a taxonomy for multiscale methods in computational mechanics: building blocks of existing methods. Comput Mech 41:279–291

    MathSciNet  MATH  Google Scholar 

  87. Gravemeier V, Wall WA (2010) An algebraic variational multiscale-multigrid method for large-eddy simulation of turbulent variable-density flow at low Mach number. J Comput Phys 229:6047–6070

    MathSciNet  MATH  Google Scholar 

  88. Gravemeier V, Wall WA (2011) Residual-based variational multiscale methods for laminar, transitional and turbulent variable-density flow at low Mach number. Int J Numer Methods Fluids 65:1260–1278

    MathSciNet  MATH  Google Scholar 

  89. Gravemeier V, Wall WA (2011) Variational multiscale methods for premixed combustion based on a progress-variable approach. Combust Flame 158:1160–1170

    Google Scholar 

  90. Gravemeier V, Wall WA, Ramm E (2004) A three-level finite element method for the instationary incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 193:1323–1366

    MathSciNet  MATH  Google Scholar 

  91. Gravemeier V, Wall WA, Ramm E (2005) Large eddy simulation of turbulent incompressible flows by a three-level finite element method. Int J Numer Methods Fluids 48:1067–1099

    MathSciNet  MATH  Google Scholar 

  92. Gresho PM, Sani RL (2000) Incompressible flow and the finite element method, volume 1, advection-diffusion. Wiley, Chichester

    MATH  Google Scholar 

  93. Gresho PM, Sani RL (2000) Incompressible flow and the finite element method, volume 2, isothermal laminar flow. Wiley, Chichester

    MATH  Google Scholar 

  94. Guermond JL (1999) Stabilization of Galerkin approximations of transport equations by subgrid modeling. Math Model Numer Anal 33:1293–1316

    MathSciNet  MATH  Google Scholar 

  95. Guermond JL, Oden JT, Prudhomme S (2004) Mathematical perspectives on large eddy simulation models for turbulent flows. J Math Fluid Mech 6:194–248

    MathSciNet  MATH  Google Scholar 

  96. Hachem E, Rivaux B, Kloczko T, Digonnet H, Coupez T (2010) Stabilized finite element method for incompressible flows with high Reynolds number. J Comput Phys 229:8643–8665

    MathSciNet  MATH  Google Scholar 

  97. Hansbo A, Hansbo P (2002) An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput Methods Appl Mech Eng 191:5537–5552

    MathSciNet  MATH  Google Scholar 

  98. Hansbo P, Larson MG, Zahedi S (2014) A cut finite element method for a Stokes interface problem. Appl Numer Math 85:90–114

    MathSciNet  MATH  Google Scholar 

  99. Harari I, Hauke G (2007) Semidiscrete formulations for transient transport at small time steps. Int J Numer Methods Fluids 54:731–743

    MathSciNet  MATH  Google Scholar 

  100. Härtel C, Kleiser L, Unger F, Friedrich R (1994) Subgrid-scale energy transfer in the near-wall region of turbulent flows. Phys Fluids 6:3130–3143

    MATH  Google Scholar 

  101. Harten A (1996) Multiresolution representation of data: a general framework. SIAM J Numer Anal 33:1205–1256

    MathSciNet  MATH  Google Scholar 

  102. Henke F (2012) An extended finite element method for turbulent premixed combustion. Dissertation, Technische Universität München

  103. Henke F, Winklmaier M, Gravemeier V, Wall WA (2014) A semi-Lagrangean time-integration approach for extended finite element methods. Int J Numer Methods Eng 98:174–202

    MathSciNet  MATH  Google Scholar 

  104. Hickel S, Adams NA, Domaradzki JA (2006) An adaptive local deconvolution method for implicit LES. J Comput Phys 213:413–436

    MathSciNet  MATH  Google Scholar 

  105. Hsu TC, Akkerman I, Bazilevs Y (2011) High-performance computing of wind turbine aerodynamics using isogeometric analysis. Comput Fluids 49:93–100

    MathSciNet  MATH  Google Scholar 

  106. Hughes TJR (1995) Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput Methods Appl Mech Eng 127:387–401

    MathSciNet  MATH  Google Scholar 

  107. Hughes TJR, Feijóo GR, Mazzei L, Quincy JB (1998) The variational multiscale method—A paradigm for computational mechanics. Comput Methods Appl Mech Eng 166:3–24

    MathSciNet  MATH  Google Scholar 

  108. Hughes TJR, Franca LP, Balestra M (1986) A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolation. Comput Methods Appl Mech Eng 59:85–99

    MATH  Google Scholar 

  109. Hughes TJR, Franca LP, Hulbert M (1989) A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Comput Methods Appl Mech Eng 73:173–189

    MathSciNet  MATH  Google Scholar 

  110. Hughes TJR, Mazzei L, Jansen KE (2000) Large eddy simulation and the variational multiscale method. Comput Vis Sci 3:47–59

    MATH  Google Scholar 

  111. Hughes TJR, Mazzei L, Oberai AA, Wray AA (2001) The multiscale formulation of large eddy simulation: decay of homogeneous isotropic turbulence. Phys Fluids 13:505–512

    MATH  Google Scholar 

  112. Hughes TJR, Wells GN (2005) Conservation properties for the Galerkin and stabilised forms of the advection-diffusion and incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 194:1141–1159

    MathSciNet  MATH  Google Scholar 

  113. Janicka J, Sadiki A (2005) Large eddy simulation of turbulent combustion systems. Proc Combust Inst 30:537–547

    Google Scholar 

  114. Jansen KE, Tejada-Martínez AE (2002) An evaluation of the variational multiscale model for large-eddy simulation while using a hierarchical basis. AIAA Paper 2002-0283, Reno, NV

  115. Jansen KE, Whiting CH, Hulbert GM (2000) A generalized-\(\alpha\) method for integrating the filtered Navier-Stokes equations with a stabilized finite element method. Comput Methods Appl Mech Eng 190:305–319

    MathSciNet  MATH  Google Scholar 

  116. Jeanmart H, Winckelmans GS (2007) Investigation of eddy-viscosity models modified using discrete filters: a simplified “regularized variational multiscale model” and an “enhanced field model”. Phys Fluids 19:055110

    MATH  Google Scholar 

  117. John V (2004) Large eddy simulation of turbulent incompressible flows. Springer, Berlin

    Google Scholar 

  118. John V (2006) On large eddy simulation and variational multiscale methods in the numerical simulation of turbulent incompressible flows. Appl Math 51:321–353

    MathSciNet  MATH  Google Scholar 

  119. John V, Kaya S (2005) A finite element variational multiscale method for the Navier-Stokes equations. SIAM J Sci Comput 26:1485–1503

    MathSciNet  MATH  Google Scholar 

  120. John V, Kaya S (2008) Finite element error analysis for a projection-based variational multiscale method with nonlinear eddy viscosity. J Math Anal Appl 344:627–641

    MathSciNet  MATH  Google Scholar 

  121. John V, Kindl A (2010) Numerical studies of finite element variational multiscale methods for turbulent flow simulations. Comput Methods Appl Mech Eng 199:841–852

    MathSciNet  MATH  Google Scholar 

  122. John V, Kindl A (2010) A variational multiscale method for turbulent flow simulation with adaptive large scale space. J Comput Phys 229:301–312

    MathSciNet  MATH  Google Scholar 

  123. Johnson C, Nävert U, Pitkäranta J (1984) Finite element methods for linear hyperbolic problems. Comput Methods Appl Mech Eng 45:285–312

    MathSciNet  MATH  Google Scholar 

  124. Kamran K, Rossi R, Oñate E (2015) A locally extended finite element method for the simulation of multi-fluid flows using the particle level set method. Comput Methods Appl Mech Eng 294:1–18

    MathSciNet  Google Scholar 

  125. Kawamura H, Ohsaka K, Abe H, Yamamoto K (1998) DNS of turbulent heat transfer in channel flow with low to medium-high Prandtl number fluid. Int J Heat Fluid Flow 19:482–491

    Google Scholar 

  126. Kees CE, Akkerman I, Farthing MW, Bazilevs Y (2011) A conservative level set method suitable for variable-order approximations and unstructured meshes. J Comput Phys 230:4536–4558

    MathSciNet  MATH  Google Scholar 

  127. Knaepen B, Debliquy O, Carati D (2005) Large-eddy simulation without filter. J Comput Phys 205:98–107

    MathSciNet  MATH  Google Scholar 

  128. Kolmogorov AN (1991) The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. In: Proceedings of the Royal Society of London A: Mathematical, Physical & Engineering Sciences, vol 434, pp 9–13 (republished English translation of Doklady Akademii Nauk SSSR, vol 30, pp 299–303, 1941 in Russian)

  129. Koobus B, Farhat C (2004) A variational multiscale method for the large eddy simulation of compressible turbulent flows on unstructured meshes - application to vortex shedding. Comput Methods Appl Mech Eng 193:1367–1383

    MathSciNet  MATH  Google Scholar 

  130. Krank B, Wall WA (2016) A new approach to wall modeling in LES of incompressible flow via function enrichment. J Comput Phys 316:94–116

    MathSciNet  MATH  Google Scholar 

  131. Labourasse E, Lacanette D, Toutant A, Lubin P, Vincent S, Lebaigue O, Caltagirone JP, Sagaut P (2007) Towards large eddy simulation of isothermal two-phase flows: governing equations and a priori tests. Int J Multiph Flow 33:1–39

    Google Scholar 

  132. Lallemand MH, Steve H, Dervieux A (1992) Unstructured multigridding by volume agglomeration: current status. Comput Fluids 21:397–433

    MATH  Google Scholar 

  133. Larsson J, Kawai S, Bodart J, Bermejo-Moreno I (2016) Large eddy simulation with modeled wall-stress: recent progress and future directions. Mech Eng Rev 3:15–00418

    Google Scholar 

  134. Layton W (1999) Weak imposition of “no-slip” conditions in finite element methods. Comput Math Appl 38:129–142

    MathSciNet  MATH  Google Scholar 

  135. Layton W, Röhe L, Tran H (2011) Explicitly uncoupled VMS stabilization of fluid flow. Comput Methods Appl Mech Eng 200:3183–3199

    MathSciNet  MATH  Google Scholar 

  136. Layton WJ (2002) A connection between subgrid scale eddy viscosity and mixed methods. Appl Math Comput 133:147–157

    MathSciNet  MATH  Google Scholar 

  137. Leonard A (1974) Energy cascade in large eddy simulation of turbulent fluid flow. Adv Geophys A 18:237–248

    Google Scholar 

  138. Lesieur M, Métais O (1996) New trends in large-eddy simulations of turbulence. Annu Rev Fluid Mech 28:45–82

    MathSciNet  Google Scholar 

  139. Lessani B, Papalexandris MV (2006) Time-accurate calculation of variable density flows with strong temperature gradients and combustion. J Comput Phys 212:218–246

    MathSciNet  MATH  Google Scholar 

  140. Lilly DK (1992) A proposed modification of the Germano subgrid-scale closure method. Phys Fluids 4:633–635

    Google Scholar 

  141. Lin PT, Sala M, Shadid JN, Tuminaro RS (2006) Performance of fully coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport. Int J Numer Methods Eng 67:208–225

    MATH  Google Scholar 

  142. Lins EF, Elias RN, Fuerra GM, Rochinha FA, Coutinho ALGA (2009) Edge-based finite element implementation of the residual-based variational multiscale method. Int J Numer Methods Fluids 61:1–22

    MathSciNet  MATH  Google Scholar 

  143. Liu S, Meneveau C, Katz J (1994) On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet. J Fluid Mech 215:83–119

    Google Scholar 

  144. Liu W (2009) A triple level finite element method for large eddy simulations. J Comput Phys 228:2690–2706

    MathSciNet  MATH  Google Scholar 

  145. Majda A, Sethian J (1985) The derivation and numerical solution of the equations for zero Mach number combustion. Combust Sci Technol 42:185–205

    Google Scholar 

  146. Martin MP, Piomelli U, Candler GV (2000) Subgrid-scale models for compressible large-eddy simulations. Theor Comput Fluid Dyn 13:361–376

    MATH  Google Scholar 

  147. Masud A, Calderer R (2011) A variational multiscale method for incompressible turbulent flows: bubble functions and fine scale fields. Comput Methods Appl Mech Eng 200:2577–2593

    MathSciNet  MATH  Google Scholar 

  148. Mavriplis DJ, Venkatakrishnan V (1996) A 3D agglomeration multigrid solver for the Reynolds-averaged Navier-Stokes equations on unstructured meshes. Int J Numer Methods Fluids 23:527–544

    MATH  Google Scholar 

  149. Meneveau C (2012) Germano identity-based subgrid-scale modeling: a brief survey of variations on a fertile theme. Phys Fluids 24:121301

    Google Scholar 

  150. Meneveau C, Katz J (2000) Scale-invariance and turbulence models for large-eddy simulation. Annu Rev Fluid Mech 32:1–32

    MathSciNet  MATH  Google Scholar 

  151. Meneveau C, Sreenivasan KR (1991) The multifractal nature of turbulent energy dissipation. J Fluid Mech 224:429–484

    MATH  Google Scholar 

  152. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150

    MATH  Google Scholar 

  153. Moin P (2002) Advances in large eddy simulation methodology for complex flows. Int J Heat Fluid Flow 23:710–720

    Google Scholar 

  154. Moin P, Squires K, Cabot W, Lee S (1991) A dynamic subgrid-scale model for compressible turbulence and scalar transport. Phys Fluids 3:2746–2757

    MATH  Google Scholar 

  155. Moser RD, Kim J, Mansour NN (1999) Direct numerical simulation of turbulent channel flow up to Re\(_\tau\) = 590. Phys Fluids 11:943–945

    MATH  Google Scholar 

  156. Müller B (1998) Low-Mach-number asymptotics of the Navier-Stokes equations. J Eng Math 34:97–109

    MathSciNet  MATH  Google Scholar 

  157. Mullin JA, Dahm WJA (2006) Dual-plane stereo particle image velocimetry measurements of velocity gradient tensor fields in turbulent shear flow. II. Experimental results. Phys Fluids 18:035102

    Google Scholar 

  158. Munts EA, Hulshoff SJ, de Borst R (2007) A modal-based multiscale method for large eddy simulation. J Comput Phys 224:389–402

    MathSciNet  MATH  Google Scholar 

  159. Nagrath S, Jansen KE, Lahey RT Jr (2005) Computation of incompressible bubble dynamics with a stabilized finite element level set method. Comput Methods Appl Mech Eng 194:4565–4587

    MathSciNet  MATH  Google Scholar 

  160. Nitsche J (1971) Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 36:9–15

    MathSciNet  MATH  Google Scholar 

  161. Oñate E (1998) Derivation of stabilized equations for numerical solution of advective-diffusive transport and fluid flow problems. Comput Methods Appl Mech Eng 151:233–265

    MathSciNet  MATH  Google Scholar 

  162. Oñate E, Valls A, Garcia J (2007) Computation of turbulent flows using a finite calculus-finite element formulation. Int J Numer Methods Fluids 54:609–637

    MathSciNet  MATH  Google Scholar 

  163. Oberai AA, Liu J, Sondak D, Hughes TJR (2014) A residual based eddy viscosity model for the large eddy simulation of turbulent flows. Comput Methods Appl Mech Eng 282:54–70

    MathSciNet  Google Scholar 

  164. Oberai AA, Wanderer J (2005) Variational formulation of the Germano identity for the Navier-Stokes equations. J Turbul 6:1–17

    MathSciNet  MATH  Google Scholar 

  165. Olshanskii M, Lube G, Heister T, Löwe J (2009) Grad-div stabilization and pressure models for the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 198:3975–3988

    MathSciNet  MATH  Google Scholar 

  166. Peters N (2000) Turbulent combustion. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  167. Piomelli U (1999) Large-eddy simulation: Achievements and challenges. Prog Aerosp Sci 35:335–362

    Google Scholar 

  168. Piomelli U (2008) Wall-layer models for large-eddy simulations. Prog Aerosp Sci 44:437–446

    Google Scholar 

  169. Piomelli U, Balaras E (2002) Wall-layer models for large-eddy simulations. Annu Rev Fluid Mech 34:349–374

    MathSciNet  MATH  Google Scholar 

  170. Piomelli U, Cabot WH, Moin P, Lee S (1991) Subgrid-scale backscatter in turbulent and transitional flows. Phys Fluids A 3:1766–1771

    MATH  Google Scholar 

  171. Pitsch H (2006) Large-eddy simulation of turbulent combustion. Annu Rev Fluid Mech 38:453–482

    MathSciNet  MATH  Google Scholar 

  172. Poinsot T, Veynante D (2005) Theoretical and numerical combustion. R.T. Edwards, Philadelphia

    Google Scholar 

  173. Pope SB (2000) Turbulent flows. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  174. Prasad RR, Meneveau C, Sreenivasan KR (1988) Multifractal nature of the dissipation field of passive scalars in fully turbulent flows. Phys Rev Lett 61:74–77

    Google Scholar 

  175. Ramakrishnan S, Collis SS (2006) Partition selection in multiscale turbulence modeling. Phys Fluids 18:075105

    MathSciNet  MATH  Google Scholar 

  176. Rasquin M, Smith C, Chitale K, Seol ES, Matthews BA, Martin JL, Sahni O, Loy RM, Shephard MS, Jansen KE (2014) Scalable implicit flow solver for realistic wing simulations with flow control. Comput Sci Eng 16:13–21

    Google Scholar 

  177. Rasthofer U (2015) Computational multiscale methods for turbulent single and two-phase flows. Dissertation, Technische Universität München

  178. Rasthofer U, Burton GC, Wall WA, Gravemeier V (2014) An algebraic variational multiscale-multigrid-multifractal method (AVM\(^4\)) for large-eddy simulation of turbulent variable-density flow at low Mach number. Int J Numer Meth Fluids 76:416–449

    MathSciNet  Google Scholar 

  179. Rasthofer U, Burton GC, Wall WA, Gravemeier V (2014) Multifractal subgrid-scale modeling within a variational multiscale method for large-eddy simulation of passive-scalar mixing in turbulent flow at low and high Schmidt numbers. Phys Fluids 26:055108

    Google Scholar 

  180. Rasthofer U, Gravemeier V (2013) Multifractal subgrid-scale modeling within a variational multiscale method for large-eddy simulation of turbulent flow. J Comput Phys 234:79–107

    MathSciNet  MATH  Google Scholar 

  181. Rasthofer U, Henke F, Wall WA, Gravemeier V (2011) An extended residual-based variational multiscale method for two-phase flow including surface tension. Comput Methods Appl Mech Eng 200:1866–1876

    MathSciNet  MATH  Google Scholar 

  182. Rasthofer U, Wall WA, Gravemeier V (2016) An extended algebraic variational multiscale-multigrid-multifractal method (XAVM4) for large-eddy simulation of turbulent two-phase flow (under review)

  183. Rehm RG, Baum HR (1978) The equations of motion for thermally driven, buoyant flows. J Res Natl Bur Sci 83:297–308

    MATH  Google Scholar 

  184. Rodriguez JM, Sahni O, Lahey RT Jr, Jansen KE (2013) A parallel adaptive mesh method for the numerical simulation of multiphase flows. Comput Fluids 87:115–131

    MathSciNet  MATH  Google Scholar 

  185. Rogallo RS, Moin P (1984) Numerical simulation of turbulent flows. Annu Rev Fluid Mech 16:99–137

    MATH  Google Scholar 

  186. Röhe L, Lube G (2010) Analysis of a variational multiscale method for large-eddy simulation and its application to homogeneous isotropic turbulence. Comput Methods Appl Mech Eng 199:2331–2342

    MathSciNet  MATH  Google Scholar 

  187. Russo A (1996) Bubble stabilization of finite element methods for the linearized incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 132:335–343

    MathSciNet  MATH  Google Scholar 

  188. Sagaut P (2006) Large eddy simulation for incompressible flows. Springer, Berlin

    MATH  Google Scholar 

  189. Sagaut P, Ciardi M (2006) A finite-volume variational multiscale method coupled with a discrete interpolation filter for large-eddy simulation of isotropic turbulence and fully developed channel flow. Phys of Fluids 18:115101

    MATH  Google Scholar 

  190. Sagaut P, Deck S, Terracol M (2006) Multiscale and multiresolution approaches in turbulence. Imperial College Press, London

    MATH  Google Scholar 

  191. Sauerland H, Fries TP (2011) The extended finite element method for two-phase and free-surface flows: a systematic study. J Comput Phys 230:3369–3390

    MathSciNet  MATH  Google Scholar 

  192. Scardovelli R, Zaleski S (1999) Direct numerical simulation of free-surface and interfacial flow. Annu Rev Fluid Mech 31:567–603

    MathSciNet  Google Scholar 

  193. Schott B, Rasthofer U, Gravemeier V, Wall WA (2015) A face-oriented stabilized Nitsche-type extended variational multiscale method for incompressible two-phase flow. Int J Numer Methods Eng 104:721–748

    MathSciNet  MATH  Google Scholar 

  194. Schott B, Wall WA (2014) A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 276:233–265

    MathSciNet  Google Scholar 

  195. Schumann U (1975) Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J Comput Phys 18:376–404

    MATH  Google Scholar 

  196. Smagorinsky J (1963) General circulation experiments with the primitive equations. I. The basic experiment. Mon Weather Rev 91:99–164

    Google Scholar 

  197. Spalart PR (2009) Detached-eddy simulation. Annu Rev Fluid Mech 41:181–202

    MATH  Google Scholar 

  198. Spalart PR, Deck S, Shur ML, Squires KD, Strelets MK, Travin A (2006) A new version of detached-eddy simulation, resistant to ambiguous grid densities. Theor Comput Fluid Dyn 20:181–195

    MATH  Google Scholar 

  199. Spalart PR, Jou WH, Strelets M, Allmaras SR (1997) Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. In: Liu C, Liu Z (eds) Advances in DNS/LES. Greyden Press, Columbus, pp 137–147

    Google Scholar 

  200. Spalding DB (1961) A single formula for the law of the wall. J Appl Mech 28:444–458

    MATH  Google Scholar 

  201. Sreenivasan KR (1991) Fractals and multifractals in fluid turbulence. Annu Rev Fluid Mech 23:539–600

    MathSciNet  Google Scholar 

  202. Sreenivasan KR, Stolovitzky G (1995) Turbulent cascades. J Stat Phys 78:311–333

    MATH  Google Scholar 

  203. Stenberg R (1995) On some techniques for approximating boundary conditions in the finite element method. J Comput Appl Math 63:139–148

    MathSciNet  MATH  Google Scholar 

  204. Stolz S, Adams NA (1999) An approximate deconvolution procedure for large-eddy simulation. Phys Fluids 11:1699–1701

    MATH  Google Scholar 

  205. Stolz S, Schlatter P, Kleiser L (2005) High-pass filtered eddy-viscosity models for large-eddy simulations of transitional and turbulent flow. Phys Fluids 17:065103

    MATH  Google Scholar 

  206. Taylor CA, Hughes TJR, Zarins CK (1998) Finite element modeling of blood flow in arteries. Comput Methods Appl Mech Eng 158:155–196

    MathSciNet  MATH  Google Scholar 

  207. Tennekes H, Lumley JL (1972) A first course in turbulence. MIT Press, Cambridge

    MATH  Google Scholar 

  208. Tezduyar TE, Mittal S, Ray SE, Shih R (1992) Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements. Comput Methods Appl Mech Eng 95:221–242

    MATH  Google Scholar 

  209. Tezduyar TE, Osawa Y (2000) Finite element stabilization parameters computed from element matrices and vectors. Comput Methods Appl Mech Eng 190:411–430

    MATH  Google Scholar 

  210. Tryggvason G, Scardovelli R, Zaleski S (2011) Direct numerical simulations of gas-liquid multiphase flows. Cambridge University Press, New York

    MATH  Google Scholar 

  211. Tuminaro R, Tong C (2000) Parallel smoothed aggregation multigrid: aggregation strategies on massively parallel machines. In: J. Donnelley (ed.) Super computing 2000 proceedings

  212. Vaněk P, Mandel J, Brezina M (1996) Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56:179–196

    MathSciNet  MATH  Google Scholar 

  213. Veynante D, Vervisch L (2002) Turbulent combustion modeling. Prog Energy Combust Sci 28:192–266

    Google Scholar 

  214. Vreman AW (2003) The filtering analog of the varaitional multiscale method in large-eddy simulation. Phys Fluids 15:L61–L64

    MathSciNet  Google Scholar 

  215. Vreman AW (2004) An eddy-viscosity subgrid-scale model for turbulent shear flow: algebraic theory and applications. Phys Fluids 16:3670–3681

    MATH  Google Scholar 

  216. Wanderer J, Oberai AA (2008) A two-parameter variational multiscale method for large eddy simulation. Phys Fluids 20:085107

    MATH  Google Scholar 

  217. Wang M, Moin P (2002) Dynamic wall modeling for large-eddy simulation of complex turbulent flows. Phys Fluids 14:2043–2051

    MathSciNet  MATH  Google Scholar 

  218. Warnatz J, Maas U, Dibble RW (2001) Combustion: physical and chemical fundamentals, modeling and simulation, experiments, pollutant formation. Springer, Berlin

    MATH  Google Scholar 

  219. Wasberg CE, Gjesdal T, Reif BAP, Andreassen O (2009) Variational multiscale turbulence modelling in a high order spectral element method. J Comput Phys 228:7333–7356

    MathSciNet  MATH  Google Scholar 

  220. Whiting CH, Jansen KE (2001) A stabilized finite element method for the incompressible Navier-Stokes equations using a hierarchical basis. Int J Numer Methods Fluids 35:93–116

    MATH  Google Scholar 

  221. Williams FA (1985) Combustion theory. Perseus Books, Reading

    Google Scholar 

  222. Zang Y, Street RL, Koseff JR (1993) A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. Phys Fluids 5:3186–3196

    MATH  Google Scholar 

  223. Zienkiewicz OC, Taylor RL, Zhu JZ (2005) The finite element method, Volume 1, its basis & fundamentals. Butterworth-Heinemann, Oxford

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  1. Computational Science and Engineering Laboratory, ETH Zurich, Clausiusstr. 33, 8092, Zurich, Switzerland

    Ursula Rasthofer

  2. Institute for Computational Mechanics, Technical University of Munich, Boltzmannstr. 15, 85748, Garching, Germany

    Volker Gravemeier

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Rasthofer, U., Gravemeier, V. Recent Developments in Variational Multiscale Methods for Large-Eddy Simulation of Turbulent Flow. Arch Computat Methods Eng 25, 647–690 (2018). https://doi.org/10.1007/s11831-017-9209-4

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