Applying resolved-scale linearly forced isotropic turbulence in rational subgrid-scale modeling

  • Chuhan Wang
  • Mingwei GeEmail author
Research Paper


In previous attempts of rational subgrid-scale (SGS) modeling by employing the Kolmogorov equation of filtered (KEF) quantities, it was necessary to assume that the resolved-scale second-order structure function is stationary. Forced isotropic turbulence is often used as a framework for establishing and validating such SGS models based on stationary restrictions, for it generates statistical stationary samples. However, traditional forcing method at low wavenumbers cannot provide an analytic form of forcing term for a complete KEF in physical space, which has been illustrated to be essential in the modeling of such SGS models. Thus, an alternative forcing method giving an analytic forcing term in physical space is needed for rational SGS modeling. Giving an analytic linear driving term in physical space, linearly forced isotropic turbulence should be considered an ideal theoretical framework for rational SGS modeling. In this paper, we demonstrate the feasibility of establishing a rational SGS model with stationary restriction based on linearly forced isotropic turbulence. The performance of this rational SGS model is validated. We, therefore, propose the use of linearly forced isotropic turbulence as a complement to free-decaying isotropic turbulence and low-wavenumber forced isotropic turbulence for SGS model validations.


Homogeneous isotropic turbulence Large-eddy simulation Subgrid-scale model Forced turbulence 



We are grateful to Professor Le Fang for the scientific input. This work was supported by the National Natural Science Foundation of China (Grant 11772128) and the Fundamental Research Funds for the Central Universities (Grants 2017MS022 and 2018ZD09).


  1. 1.
    Yang, X., Sadique, J., Mitta, R., et al.: Integral wall model for large eddy simulations of wall-bounded turbulent flows. Phys. Fluids 27, 025112 (2015)CrossRefGoogle Scholar
  2. 2.
    Ge, M.W., Wu, Y., Liu, Y.Q., et al.: A two-dimensional model based on the expansion of physical wake boundary for wind-turbine wakes. Appl. Energy 233–234, 975–984 (2019)CrossRefGoogle Scholar
  3. 3.
    Yang, X., Sadique, J., Mittal, R., et al.: Exponential roughness layer and analytical model for turbulent boundary layer flow over rectangular-prism roughness elements. J. Fluid Mech. 789, 127–165 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Yang, X., Meneveau, C.: Large eddy simulations and parameterisation of roughness element orientation and flow direction effects in rough wall boundary layers. J. Turbul. 17, 1072–1085 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Yang, X., Park, G., Moin, P.: Log-layer mismatch and modeling of the fluctuating wall stress in wall-modeled large-eddy simulations. Phys. Rev. Fluids 2, 104601 (2017)CrossRefGoogle Scholar
  6. 6.
    Sagaut, P.: Large Eddy Simulation for Imcompressible Flows. Springer, Paris (2006)zbMATHGoogle Scholar
  7. 7.
    Meneveau, C., Katz, J.: Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid Mech. 32, 1–32 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fang, L., Shao, L., Bertoglio, J.P.: Recent understanding on the subgrid-scale modeling of large-eddy simulation in physical space. Sci. China Phys. Mech. 57, 2188–2193 (2014)CrossRefGoogle Scholar
  9. 9.
    Lu, H., Rutland, C.J.: Structural subgrid-scale modeling for large-eddy simulation: a review. Acta Mech. Sin. 32, 567–578 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fang, L., Ge, M.W.: Mathematical constraints in multiscale subgrid-scale modeling of nonlinear systems. Chin. Phys. Lett. 34, 030501 (2017)CrossRefGoogle Scholar
  11. 11.
    Fang, L., Sun, X.Y., Liu, Y.W.: A criterion of orthogonality on the assumption and restrictions in subgrid-scale modelling of turbulence. Phys. Lett. A 380, 3988–3992 (2016)CrossRefGoogle Scholar
  12. 12.
    Spalart, P.R.: Strategies for turbulence modelling and simulations. Int. J. Heat Fluid Flow 21, 252–263 (2000)CrossRefGoogle Scholar
  13. 13.
    Cui, G.X., Zhou, H.B., Zhang, Z.S., et al.: A new dynamic subgrid eddy viscosity model with application to turbulent channel flow. Phys. Fluids 16, 2835–2842 (2004)CrossRefzbMATHGoogle Scholar
  14. 14.
    Cui, G.X., Xu, C.X., Fang, L., et al.: A new subgrid eddy-viscosity model for large-eddy simulation of anisotropic turbulence. J. Fluid Mech. 582, 377–397 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fang, L., Shao, L., Bertoglio, J.P., et al.: An improved velocity increment model based on Kolmogorov equation of filtered velocity. Phys. Fluids 21, 065108 (2009)CrossRefzbMATHGoogle Scholar
  16. 16.
    Fang, L., Shao, L., Bertoglio, J.P., et al.: The rapid-slow decomposition of the subgrid flux in inhomogeneous scalar turbulence. J. Turbul. 12, 1–23 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yang, Z.X., Cui, G.X., Zhang, Z.S., et al.: A modified nonlinear sub-grid scale model for large eddy simulation with application to rotating turbulent channel flows. Phys. Fluids 24, 075113 (2012)CrossRefGoogle Scholar
  18. 18.
    Yang, Z.X., Cui, G.X., Xu, C.X., et al.: Large eddy simulation of rotating turbulent channel flow with a new dynamic global-coefficient nonlinear subgrid stress model. J. Turbul. 13, N48 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Smagorinsky, J.: General circulation experiments with primitive equation. Mon. Weather Rev. 91, 99–164 (1963)CrossRefGoogle Scholar
  20. 20.
    Bardina, J., Ferziger, J., Reynolds, W.C.: Improved subgrid-scale models for large-eddy simulation. In: 13th Fluid and PlasmaDynamics Conference Snowmass, CO, USA, p. 1357 (1987)Google Scholar
  21. 21.
    Fang, L., Li, B., Lu, L.P.: Scaling law of resolved-scale isotropic turbulence and its application in large-eddy simulation. Acta Mech. Sin. 30, 339–350 (2014)CrossRefGoogle Scholar
  22. 22.
    Fang, L., Bos, W.J.T., Zhou, X.Z., et al.: Corrections to the scaling of the second-order structure function in isotropic turbulence. Acta Mech. Sin. 26, 151–157 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wu, J.Z., Fang, L., Shao, L., et al.: Theories and applications of second-order correlation of longitudinal velocity increments at three points in isotropic turbulence. Phys. Lett. A 382, 1665–1671 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Germano, M., Piomelli, U., Moin, P., et al.: A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A 3, 1760–1765 (1991)CrossRefzbMATHGoogle Scholar
  25. 25.
    Meneveau, C.: Statistics of turbulence subgrid-scale stresses: necessary conditions and experimental tests. Phys. Fluids 6, 815–833 (1994)CrossRefzbMATHGoogle Scholar
  26. 26.
    Brun, C., Friedrich, R., Da Silva, C.B.: A non-linear SGS model based on the spatial velocity increment. Theor. Comput. Fluid Dyn. 20, 1–21 (2006)CrossRefzbMATHGoogle Scholar
  27. 27.
    Shao, L., Zhang, Z.S., Cui, G.X., et al.: Subgrid modeling of anisotropic rotating homogeneous turbulence. Phys. Fluids 17, 115106 (2005)CrossRefzbMATHGoogle Scholar
  28. 28.
    Fang, L., Zhu, Y., Liu, Y.W., et al.: Spectral non-equilibrium property in homogeneous isotropic turbulence and its implication in subgrid-scale modeling. Phys. Lett. A 379, 2331–2336 (2015)CrossRefzbMATHGoogle Scholar
  29. 29.
    Orszag, S.A., Patterson, G.S.: Numerical simulation of three-dimensional homogeneous isotropic turbulence. Phys. Rev. Lett. 28, 76–79 (1972)CrossRefGoogle Scholar
  30. 30.
    Ghosal, S., Lund, T.S., Moin, P., et al.: A dynamic localization model for large-eddy simulation of turbulent flows. J. Fluid Mech. 286, 229–255 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Jeffrey, R., Chasnov, J.R.: Simulation of the kolmogorov inertial subrange using an improved subgrid model. Phys. Fluids A 3, 188–200 (1991)CrossRefzbMATHGoogle Scholar
  32. 32.
    Seror, C., Sagaut, P., Bailly, C., et al.: On the radiated noise computed by large-eddy simulation. Phys. Fluids 13, 476–487 (2001)CrossRefzbMATHGoogle Scholar
  33. 33.
    Sullivan, N.P., Shankar, M.: Deterministic forcing of homogeneous, isotropic turbulence. Phys. Fluids 6, 1612–1614 (1994)CrossRefGoogle Scholar
  34. 34.
    Métais, O., Lesieur, M.: Spectral large-eddy simulation of isotropic and stably stratified turbulence. J. Fluid Mech. 239, 157–194 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lundgren, T.S.: Linearly forces isotropic turbulence. In: Annual Research Briefs. Center for Turbulence Research, Stanford (2003)Google Scholar
  36. 36.
    Rosales, C., Meneveau, C.: Linear forcing in numerical simulations of isotropic turbulence: physical space implementations and convergence properties. Phys. Fluids 17, 095106 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Petersen, M.R., Livescu, D.: Forcing for statistically stationary compressible isotropic turbulence. Phys. Fluids 22, 116101 (2010)CrossRefGoogle Scholar
  38. 38.
    Carroll, P.L., Verma, S., Blanquart, G.: Characteristics of linearly-forced scalar mixing in homogeneous, isotropic turbulence. In: ICCFD7, Hawaii, USA (2012)Google Scholar
  39. 39.
    Akylas, E.E., Kassinos, S.C., Rouson, D.W.I.: Accelerating stationarity in linearly forced isotropic turbulence. In: The Sixth International Symposium on Turbulence and Shear Flow Phenomena, Seoul, Korea (2009)Google Scholar
  40. 40.
    Bassenne, M., Urzay, J., Park, G.I., et al.: Constant-energetics physical-space forcing methods for improved convergence to homogeneous-isotropic turbulence with application to particle-laden flows. Phys. Fluids 28, 035114 (2016)CrossRefGoogle Scholar
  41. 41.
    Carroll, P.L., Blanquart, G.: A proposed modification to lundgren’s physical space velocity forcing method for isotropic turbulence. Phys. Fluids 25, 105114 (2013)CrossRefGoogle Scholar
  42. 42.
    Goldstein, D.E., Vasilyev, O.V.: Stochastic coherent adaptive large eddy simulation method. Phys. Fluids 16, 2497–2513 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    De Stefano, G., Vasilyev, O.V.: Stochastic coherent adaptive large eddy simulation of forced isotropic turbulence. J. Fluid Mech. 646, 453–470 (2010)CrossRefzbMATHGoogle Scholar
  44. 44.
    de Laage de Meux, B., Audebert, B., Manceau, R., et al.: Anisotropic linear forcing for synthetic turbulence generation in large eddy simulation and hybrid RANS/LES modeling. Phys. Fluids 27, 035115 (2015)CrossRefGoogle Scholar
  45. 45.
    Monin, A.S., Yaglom, A.M., Lumley, J., et al.: Statistical Fluid Mechanics. Mechanics of Turbulence. MIT Press, Cambridge (1975)Google Scholar
  46. 46.
    Hill, R.J.: Exact second-order structure function relationship. J. Fluid Mech. 468, 317–326 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Fang, L., Bos, W.J.T., Shao, L., et al.: Time reversibility of Navier–Stokes turbulence and its implication for subgrid scale models. J. Turbul. 13, N3 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Yao, S.Y., Fang, L., Lv, J.M., et al.: Multiscale three-point velocity increment correlation in turbulent flows. Phys. Lett. A 378, 886–891 (2014)CrossRefzbMATHGoogle Scholar
  49. 49.
    Bos, W.T.S., Rubinstein, R., Fang, L.: Reduction of mean-square advection in turbulent passive scalar mixing. Phys. Fluids 24, 075104 (2012)CrossRefGoogle Scholar
  50. 50.
    Fang, L., Bos, W.J.T., Jin, G.D.: Short-time evolution of Lagrangian velocity gradient correlations in isotropic turbulence. Phys. Fluids 27, 125102 (2015)CrossRefGoogle Scholar
  51. 51.
    Fang, L., Zhang, Y.J., Fang, J., et al.: Relation of the fourth-order statistical invariants of velocity gradient tensor in isotropic turbulence. Phys. Rev. E 94, 023114 (2016)CrossRefGoogle Scholar
  52. 52.
    Qin, Z.C., Fang, L., Fang, J.: How isotropic are turbulent flows generated by using periodic conditions in a cube? Phys. Lett. A 380, 1310–1317 (2016)CrossRefGoogle Scholar
  53. 53.
    Fang, L.: Background scalar-level anisotropy caused by low-wave-number truncation in turbulent flows. Phys. Rev. E 95, 033102 (2017)CrossRefGoogle Scholar
  54. 54.
    Yang, Z.X., Wang, B.C.: On the topology of the eighenframe of the subgrid-scale stress tensor. J. Fluid Mech. 798, 598–627 (2016)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Oskouie, S.N., Yang, Z.X., Wang, B.C.: Study of passive plume mixing due to two line source emission in isotropic turbulence. Phys. Fluids 30, 075105 (2018)CrossRefGoogle Scholar
  56. 56.
    He, G.W., Rubinstein, R., Wang, L.P.: Effects of subgrid-scale modeling on time correlations in large eddy simulation. Phys. Fluids 14, 2186–2193 (2002)CrossRefzbMATHGoogle Scholar
  57. 57.
    He, G.W., Jin, G.D., Yang, Y.: Space-time correlations and dynamic coupling in turbulent flows. Annu. Rev. Fluid Mech. 49, 51–70 (2017)CrossRefzbMATHGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Renewable EnergyNorth China Electric Power UniversityBeijingChina
  2. 2.LMP, Ecole Centrale de PékinBeihang UniversityBeijingChina

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