1932

Abstract

Space-time correlation is a staple method for investigating the dynamic coupling of spatial and temporal scales of motion in turbulent flows. In this article, we review the space-time correlation models in both the Eulerian and Lagrangian frames of reference, which include the random sweeping and local straining models for isotropic and homogeneous turbulence, Taylor's frozen-flow model and the elliptic approximation model for turbulent shear flows, and the linear-wave propagation model and swept-wave model for compressible turbulence. We then focus on how space-time correlations are used to develop time-accurate turbulence models for the large-eddy simulation of turbulence-generated noise and particle-laden turbulence. We briefly discuss their applications to two-point closures for Kolmogorov's universal scaling of energy spectra and to the reconstruction of space-time energy spectra from a subset of spatial and temporal signals in experimental measurements. Finally, we summarize the current understanding of space-time correlations and conclude with future issues for the field.

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2017-01-03
2024-06-18
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Literature Cited

  1. Armenio V, Piomelli U, Fiorotto V. 1999. Effect of the subgrid scales on particle motion. Phys. Fluids 11:3030–42 [Google Scholar]
  2. Bailly C, Lafon P, Candel S. 1997. Subsonic and supersonic jet noise predictions from statistical source models. AIAA J. 35:1688–96 [Google Scholar]
  3. Balachandar S, Eaton JK. 2010. Turbulent dispersed multiphase flow. Annu. Rev. Fluid Mech. 42:111–33 [Google Scholar]
  4. Balarac G, Le Sommer J, Meunier X, Vollant A. 2013. A dynamic regularized gradient model of the subgrid-scale scalar flux for large eddy simulations. Phys. Fluids 25:075107 [Google Scholar]
  5. Bardina J, Ferziger JH, Reynolds WC. 1980. Improved subgrid-scale models for large-eddy simulation Presented at AIAA Fluid Plasma Dyn. Conf., 13th, Snowmass, CO, AIAA Pap 1980–1357 [Google Scholar]
  6. Bertoglio JP. 1985. A stochastic subgrid model for sheared turbulence. Lect. Notes Phys. 230:100–19 [Google Scholar]
  7. Biferale L, Boffetta G, Celani A, Devenish BJ, Lanotte A, Toschi F. 2004. Multifractal statistics of Lagrangian velocity and acceleration in turbulence. Phys. Rev. Lett. 93:064502 [Google Scholar]
  8. Bodony DJ, Lele SK. 2003. A statistical subgrid scale noise model: formulation Presented at AIAA/CEAS Aeroacoust. Conf., 9th, Hilton Head, SC, AIAA Pap 2003–3252 [Google Scholar]
  9. Borgas MS. 1993. The multifractal Lagrangian nature of turbulence. Philos. Trans. R. Soc. Lond. A 342:379–411 [Google Scholar]
  10. Bos WJT, Bertoglio JP. 2013. Lagrangian Markovianized field approximation for turbulence. J. Turbul. 14:99–120 [Google Scholar]
  11. Cambon C, Scott JF. 1999. Linear and nonlinear models of anisotropic turbulence. Annu. Rev. Fluid Mech. 31:1–53 [Google Scholar]
  12. Carati D, Ghosal S, Moin P. 1995. On the representation of backscatter in dynamic localization models. Phys. Fluids 7:606–16 [Google Scholar]
  13. Cernick MJ, Tullis SW, Lightstone MF. 2015. Particle subgrid scale modelling in large-eddy simulations of particle-laden turbulence. J. Turbul. 16:101–35 [Google Scholar]
  14. Chen SY, Kraichnan RH. 1989. Sweeping decorrelation in isotropic turbulence. Phys. Fluids A 1:2019–24 [Google Scholar]
  15. Chevillard L, Roux SG, Levêque E, Mordant N, Pinton JF, Arneodo A. 2003. Lagrangian velocity statistics in turbulent flows: effects of dissipation. Phys. Rev. Lett. 91:214502 [Google Scholar]
  16. Chibbaro S, Marchioli C, Salvetti MV, Soldati A. 2014. Particle tracking in LES flow fields: conditional Lagrangian statistics of filtering error. J. Turbul. 15:22–33 [Google Scholar]
  17. Clark RA, Ferziger JH, Reynolds WC. 1979. Evaluation of sub-grid-scale models using an accurately simulated turbulent flow. J. Fluid Mech. 91:1–16 [Google Scholar]
  18. Corrsin S. 1959. Progress report on some turbulent diffusion research. Adv. Geophys. 6:161–64 [Google Scholar]
  19. Davies POA, Fisher MJ, Barratt MJ. 1963. The characteristics of the turbulence in the mixing region of a round jet. J. Fluid Mech. 15:337–67 [Google Scholar]
  20. de Kat R, Ganapathisubramani B. 2015. Frequency-wavenumber mapping in turbulent shear flows. J. Fluid Mech. 783:166–90 [Google Scholar]
  21. del Álamo JC, Jiménez J. 2009. Estimation of turbulent convection velocities and corrections to Taylor's approximation. J. Fluid Mech. 640:5–26 [Google Scholar]
  22. Domaradzki JA, Adams NA. 2002. Direct modelling of subgrid scales of turbulence in large eddy simulations. J. Turbul. 3:N24 [Google Scholar]
  23. Dong YH, Sagaut P. 2008. A study of time correlations in lattice Boltzmann-based large-eddy simulation of isotropic turbulence. Phys. Fluids 20:035105 [Google Scholar]
  24. Fadai-Ghotbi A, Friess C, Manceau R, Gatski TB, Borée J. 2010. Temporal filtering: a consistent formalism for seamless hybrid RANS-LES modeling in inhomogeneous turbulence. Int. J. Heat Fluid Flow 31:378–89 [Google Scholar]
  25. Favier B, Godeferd FS, Cambon C. 2010. On space and time correlations of isotropic and rotating turbulence. Phys. Fluids 22:015101 [Google Scholar]
  26. Fede P, Simonin O. 2006. Numerical study of the subgrid fluid turbulence effects on the statistics of heavy colliding particles. Phys. Fluids 18:045103 [Google Scholar]
  27. Fede P, Simonin O, Villedieu P, Squires KD. 2006. Stochastic modeling of the turbulent subgrid fluid velocity along inertial particle trajectories. Proc. Summer Program 2006247–58 Stanford, CA: Stanford Univ. Cent. Turbul. Res. [Google Scholar]
  28. Flohr P, Vassilicos JC. 2000. A scalar subgrid model with flow structure for large-eddy simulations of scalar variances. J. Fluid Mech. 407:315–49 [Google Scholar]
  29. Fox RO. 2012. Large-eddy-simulation tools for multiphase flows. Annu. Rev. Fluid Mech. 44:47–76 [Google Scholar]
  30. Frisch U. 1995. Turbulence: The Legacy of A.N. Kolmogorov. Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  31. Fung JCH, Hunt JCR, Malik NA, Perkins RJ. 1992. Kinematic simulation of homogeneous turbulence by unsteady random Fourier modes. J. Fluid Mech. 236:281–318 [Google Scholar]
  32. Geng CH, He GW, Wang YS, Xu CX, Lozano-Duran A, Wallace JM. 2015. Taylor's hypothesis in turbulent channel flow considered using a transport equation analysis. Phys. Fluids 27:025111 [Google Scholar]
  33. Ghate AS, Lele SK. 2015. A modeling framework for wind farm analysis: wind turbine wake interactions Presented at AIAA SciTech 2015, Wind Energy Symp., 33rd, Kissimmee, FL, AIAA Pap. 2015-0725 [Google Scholar]
  34. Gotoh T, Rogallo RS, Herring JR, Kraichnan RH. 1993. Lagrangian velocity correlations in homogeneous isotropic turbulence. Phys. Fluids A 5:2846–64 [Google Scholar]
  35. Grossmann S, Lohse D. 2001. Thermal convection for large Prandtl numbers. Phys. Rev. Lett. 86:3316–19 [Google Scholar]
  36. Guo L, Li D, Zhang X, He GW. 2012. LES prediction of space-time correlations in turbulent shear flows. Acta Mech. Sin. 28:993–98 [Google Scholar]
  37. He GW, Ansari HP, Jin GD, Mani A. 2012a. A Lagrangian filtering approach for large-eddy simulation of particle-laden turbulence. Proc. Summer Program 2012375–83 Stanford, CA: Stanford Univ. Cent. Turbul. Res. [Google Scholar]
  38. He GW, Jin GD, Zhao X. 2009. Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence. Phys. Rev. E 80:066313 [Google Scholar]
  39. He GW, Rubinstein R, Wang LP. 2002. Effects of subgrid-scale modeling on time correlations in large eddy simulation. Phys. Fluids 14:2186–93 [Google Scholar]
  40. He GW, Wang M, Lele SK. 2004. On the computation of space-time correlations by large-eddy simulation. Phys. Fluids 16:3859–67 [Google Scholar]
  41. He GW, Zhang JB. 2006. Elliptic model for space-time correlations in turbulent shear flows. Phys. Rev. E 73:055303 [Google Scholar]
  42. He XZ, Bodenschatz E, Ahlers G. 2016. Azimuthal diffusion of the large-scale-circulation plane, and absence of significant non-Boussinesq effects, in turbulent convection near the ultimate-state transition. J. Fluid Mech. 791:R3–13 [Google Scholar]
  43. He XZ, Funfschilling D, Nobach H, Bodenschatz E, Ahlers G. 2012b. Transition to the ultimate state of turbulent Rayleigh-Bénard convection. Phys. Rev. Lett. 108:024502 [Google Scholar]
  44. He XZ, He GW, Tong P. 2010. Small-scale turbulent fluctuations beyond Taylor's frozen-flow hypothesis. Phys. Rev. E 81:065303 [Google Scholar]
  45. He XZ, van Gils DPM, Bodenschatz E, Ahlers G. 2014. Logarithmic spatial variations and universal f−1power spectra of temperature fluctuations in turbulent Rayleigh-Bénard convection. Phys. Rev. Lett. 112:174501 [Google Scholar]
  46. He XZ, van Gils DPM, Bodenschatz E, Ahlers G. 2015. Reynolds numbers and the elliptic approximation near the ultimate state of turbulent Rayleigh-Bénard convection. New J. Phys. 17:063028 [Google Scholar]
  47. Heskestad G. 1965. A generalized Taylor hypothesis with application for high Reynolds number turbulent shear flows. J. Appl. Mech. 32:735–39 [Google Scholar]
  48. Hogg J, Ahlers G. 2013. Reynolds-number measurements for low-Prandtl-number turbulent convection of large-aspect-ratio samples. J. Fluid Mech. 725:664–80 [Google Scholar]
  49. Homann H, Kamps O, Friedrich R, Grauer R. 2009. Bridging from Eulerian to Lagrangian statistics in 3D hydro- and magnetohydrodynamic turbulent flows. New J. Phys. 11:073020 [Google Scholar]
  50. Jin GD, He GW, Wang LP. 2010. Large-eddy simulation of turbulent collision of heavy particles in isotropic turbulence. Phys. Fluids 22:055106 [Google Scholar]
  51. Jung J, Yeo K, Lee C. 2008. Behavior of heavy particles in isotropic turbulence. Phys. Rev. E 77:016307 [Google Scholar]
  52. Kaneda Y. 1981. Renormalized expansions in the theory of turbulence with the use of the Lagrangian position function. J. Fluid Mech. 107:131–45 [Google Scholar]
  53. Kim J, Hussain F. 1993. Propagation velocity of perturbations in turbulent channel flow. Phys. Fluids A 5:695–706 [Google Scholar]
  54. Kraichnan RH. 1959. The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5:497–543 [Google Scholar]
  55. Kraichnan RH. 1962. Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5:1374–89 [Google Scholar]
  56. Kraichnan RH. 1964. Kolmogorov hypotheses and Eulerian turbulence theory. Phys. Fluids 7:1723–34 [Google Scholar]
  57. Kraichnan RH. 1968. Lagrangian-history statistical theory for Burgers equation. Phys. Fluids 11:265–77 [Google Scholar]
  58. Kraichnan RH. 1977. Lagrangian velocity covariance in helical turbulence. J. Fluid Mech. 81:385–98 [Google Scholar]
  59. Kuerten JGM. 2006. Subgrid modeling in particle-laden channel flow. Phys. Fluids 18:025108 [Google Scholar]
  60. Kuerten JGM, Vreman AW. 2005. Can turbophoresis be predicted by large-eddy simulation?. Phys. Fluids 17:011701 [Google Scholar]
  61. LaBryer A, Attar PJ, Vedula P. 2015. A framework for large eddy simulation of Burgers turbulence based upon spatial and temporal statistical information. Phys. Fluids 27:035116 [Google Scholar]
  62. Langford JA, Moser RD. 1999. Optimal LES formulations for isotropic turbulence. J. Fluid Mech. 398:321–46 [Google Scholar]
  63. Lee S, Lele SK, Moin P. 1992. Simulation of spatially evolving turbulence and the applicability of Taylor's hypothesis in compressible flow. Phys. Fluids A 4:1521–30 [Google Scholar]
  64. Leib SJ, Goldstein ME. 2011. Hybrid source model for predicting high-speed jet noise. AIAA J. 49:1324–35 [Google Scholar]
  65. Leith CE. 1990. Stochastic backscatter in a subgrid-scale model: plane shear mixing layer. Phys. Fluids A 2:297–99 [Google Scholar]
  66. Li D, Zhang X, He GW. 2013. Temporal decorrelations in compressible isotropic turbulence. Phys. Rev. E 88:021001 [Google Scholar]
  67. Lighthill MJ. 1952. On sound generated aerodynamically. I. General theory. Proc. R. Soc. Lond. A 211:564–87 [Google Scholar]
  68. Lilley GM. 1994. The radiated noise from isotropic turbulence. Theor. Comput. Fluid Dyn. 6:281–301 [Google Scholar]
  69. Lin CC. 1953. On Taylor's hypothesis and the acceleration terms in the Navier-Stokes equations. Q. Appl. Math. 10:295–306 [Google Scholar]
  70. Lohse D, Xia KQ. 2010. Small-scale properties of turbulent Rayleigh-Bénard convection. Annu. Rev. Fluid Mech. 42:335–64 [Google Scholar]
  71. Loucks RB, Wallace JM. 2012. Velocity and velocity gradient based properties of a turbulent plane mixing layer. J. Fluid Mech. 699:280–319 [Google Scholar]
  72. Lumley JL. 1962. The mathematical nature of the problem of relating Langrangian and Eulerian statistical functions in turbulence. Mécanique de la Turbulence17–26 Paris: CNRS [Google Scholar]
  73. Lumley JL. 1965. Interpretation of time spectra measured in high-intensity shear flows. Phys. Fluids 8:1056–62 [Google Scholar]
  74. Marstorp L, Brethouwer G, Johansson AV. 2007. A stochastic subgrid model with application to turbulent flow and scalar mixing. Phys. Fluids 19:035107 [Google Scholar]
  75. Martin MP. 2005. Preliminary study of the SGS time scales for compressible boundary layers using DNS data Presented at AIAA Aerosp. Sci. Meet. Exhib., 43rd, Reno, NV, AIAA Pap. 2005-0665 [Google Scholar]
  76. Mason PJ, Thomson DJ. 1992. Stochastic backscatter in large-eddy simulations of boundary layers. J. Fluid Mech. 242:51–78 [Google Scholar]
  77. Mazzitelli IM, Toschi F, Lanotte AS. 2014. An accurate and efficient Lagrangian sub-grid model. Phys. Fluids 26:095101 [Google Scholar]
  78. Meneveau C, Katz J. 2000. Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid Mech. 32:1–32 [Google Scholar]
  79. Moin P. 2009. Revisiting Taylor's hypothesis. J. Fluid Mech. 640:1–4 [Google Scholar]
  80. Morris PJ, Farassat F. 2002. Acoustic analogy and alternative theories for jet noise prediction. AIAA J. 40:671–80 [Google Scholar]
  81. O'Brien J, Urzay J, Ihme M, Moin P, Saghafian A. 2014. Subgrid-scale backscatter in reacting and inert supersonic hydrogen-air turbulent mixing layers. J. Fluid Mech. 743:554–84 [Google Scholar]
  82. Orszag SA. 1977. Lectures on the statistical theory of turbulence. Fluid Dynamics: Lecture Notes of the Les Houches Summer School 1973 R Balian, JL Peube 235–374 New York: Gordon & Breach [Google Scholar]
  83. Park GI, Urzay J, Bassenne M, Moin P. 2015. A dynamic subgrid-scale model based on differential filters for LES of particle-laden turbulent flows. Annu. Res. Briefs 201517–26 Stanford, CA: Stanford Univ. Cent. Turbul. Res. [Google Scholar]
  84. Park N, Yoo JY, Choi H. 2005. Toward improved consistency of a priori tests with a posteriori tests in large eddy simulation. Phys. Fluids 17:015103 [Google Scholar]
  85. Piomelli U, Balint JL, Wallace JM. 1989. On the validity of Taylor's hypothesis for wall-bounded flows. Phys. Fluids A 1:609–11 [Google Scholar]
  86. Piomelli U, Cabot WH, Moin P, Lee SS. 1991. Subgrid-scale backscatter in turbulent and transitional flows. Phys. Fluids A 3:1766–71 [Google Scholar]
  87. Pozorski J, Apte SV. 2009. Filtered particle tracking in isotropic turbulence and stochastic modeling of subgrid-scale dispersion. Int. J. Multiphase Flow 35:118–28 [Google Scholar]
  88. Pruett CD, Gatski TB, Grosch CE, Thacker WD. 2003. The temporally filtered Navier-Stokes equations: properties of the residual stress. Phys. Fluids 15:2127–40 [Google Scholar]
  89. Ray B, Collins LR. 2011. Preferential concentration and relative velocity statistics of inertial particles in Navier-Stokes turbulence with and without filtering. J. Fluid Mech. 680:488–510 [Google Scholar]
  90. Ray B, Collins LR. 2013. Investigation of sub-Kolmogorov inertial particle pair dynamics in turbulence using novel satellite particle simulations. J. Fluid Mech. 720:192–211 [Google Scholar]
  91. Ray B, Collins LR. 2014. A subgrid model for clustering of high-inertia particles in large-eddy simulations of turbulence. J. Turbul. 15:366–85 [Google Scholar]
  92. Renard N, Deck S. 2015. On the scale-dependent turbulent convection velocity in a spatially developing flat plate turbulent boundary layer at Reynolds number . J. Fluid Mech. 775:105–48 [Google Scholar]
  93. Rubinstein R, Zhou Y. 1999. Effects of helicity on Lagrangian and Eulerian time correlations in turbulence. Phys. Fluids 11:2288–90 [Google Scholar]
  94. Rubinstein R, Zhou Y. 2000. The frequency spectrum of sound radiated by isotropic turbulence. Phys. Lett. A 267:379–83 [Google Scholar]
  95. Rubinstein R, Zhou Y. 2002. Characterization of sound radiation by unresolved scales of motion in computational aeroacoustics. Eur. J. Mech. B 21:105–11 [Google Scholar]
  96. Seror C, Sagaut P, Bailly C, Juve D. 2000. Subgrid-scale contribution to noise production in decaying isotropic turbulence. AIAA J. 38:1795–803 [Google Scholar]
  97. Shotorban B, Mashayek F. 2006. A stochastic model for particle motion in large-eddy simulation. J. Turbul. 7:N18 [Google Scholar]
  98. Shotorban B, Zhang KKQ, Mashayek F. 2007. Improvement of particle concentration prediction in large-eddy simulation by defiltering. Int. J. Heat Mass Transfer 50:3728–39 [Google Scholar]
  99. Smith FB, Hay JS. 1961. The expansion of clusters of particles in the atmosphere. Q. J. R. Meteorol. Soc. 87:82–101 [Google Scholar]
  100. Smith LM, Woodruff SL. 1998. Renormalization-group analysis of turbulence. Annu. Rev. Fluid Mech. 30:275–310 [Google Scholar]
  101. Smits AJ, McKeon BJ, Marusic I. 2011. High–Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43:353–75 [Google Scholar]
  102. Squires KD, Eaton JK. 1991. Preferential concentration of particles by turbulence. Phys. Fluids A 3:1169–79 [Google Scholar]
  103. Stolz S, Adams NA. 1999. An approximate deconvolution procedure for large-eddy simulation. Phys. Fluids 11:1699–701 [Google Scholar]
  104. Tam CKW, Auriault L. 1999. Jet mixing noise from fine-scale turbulence. AIAA J. 37:145–53 [Google Scholar]
  105. Tam CKW, Pastuchenko NN, Viswanathan K. 2005. Fine-scale turbulence noise from hot jets. AIAA J. 43:1675–83 [Google Scholar]
  106. Taylor GI. 1938. The spectrum of turbulence. Proc. R. Soc. Lond. A 164:476–90 [Google Scholar]
  107. Tennekes H. 1975. Eulerian and Lagrangian time microscales in isotropic turbulence. J. Fluid Mech. 67:561–67 [Google Scholar]
  108. Tsinober A, Vedula P, Yeung PK. 2001. Random Taylor hypothesis and the behavior of local and convective accelerations in isotropic turbulence. Phys. Fluids 13:1974–84 [Google Scholar]
  109. Urzay J, Bassenne M, Park G, Moin P. 2014. Characteristic regimes of subgrid-scale coupling in LES of particle-laden turbulent flows. Annu. Res. Briefs 20143–13 Stanford, CA: Stanford Univ. Cent. Turbul. Res. [Google Scholar]
  110. Viswanathan K, Underbrink JR, Brusniak L. 2011. Space-time correlation measurements in near fields of jets. AIAA J. 49:1577–99 [Google Scholar]
  111. Wallace JM. 2014. Space-time correlations in turbulent flow: a review. Theor. Appl. Mech. Lett. 4:022003 [Google Scholar]
  112. Wang LP, Maxey MR. 1993. Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 256:27–68 [Google Scholar]
  113. Wang M, Freund JB, Lele SK. 2006. Computational prediction of flow-generated sound. Annu. Rev. Fluid Mech. 38:483–512 [Google Scholar]
  114. Wang QZ, Squires KD. 1996. Large eddy simulation of particle-laden turbulent channel flow. Phys. Fluids 8:1207–23 [Google Scholar]
  115. Wang W, Guan XL, Jiang N. 2014. TRPIV investigation of space-time correlation in turbulent flows over flat and wavy walls. Acta Mech. Sin. 30:468–79 [Google Scholar]
  116. Wei GX, Vinkovic I, Shao L, Simoëns S. 2006. Scalar dispersion by a large-eddy simulation and a Lagrangian stochastic subgrid model. Phys. Fluids 18:095101 [Google Scholar]
  117. Wilczek M, Narita Y. 2012. Wave-number-frequency spectrum for turbulence from a random sweeping hypothesis with mean flow. Phys. Rev. E 86:066308 [Google Scholar]
  118. Wilczek M, Stevens RJAM, Meneveau C. 2015. Spatio-temporal spectra in the logarithmic layer of wall turbulence: large-eddy simulations and simple models. J. Fluid Mech. 769:R1–12 [Google Scholar]
  119. Wills JAB. 1964. On convection velocities in turbulent shear flows. J. Fluid Mech. 20:417–32 [Google Scholar]
  120. Xu HT, Ouellette NT, Bodenschatz E. 2006. Multifractal dimension of Lagrangian turbulence. Phys. Rev. Lett. 96:114503 [Google Scholar]
  121. Yang Y, He GW, Wang LP. 2008. Effects of subgrid-scale modeling on Lagrangian statistics in large-eddy simulation. J. Turbul. 9:N8 [Google Scholar]
  122. Yao HD, He GW. 2009. A kinematic subgrid scale model for large-eddy simulation of turbulence-generated sound. J. Turbul. 10:N19 [Google Scholar]
  123. Zaman KBMQ, Hussain AKMF. 1981. Taylor hypothesis and large-scale coherent structures. J. Fluid Mech. 112:379–96 [Google Scholar]
  124. Zhao X, He GW. 2009. Space-time correlations of fluctuating velocities in turbulent shear flows. Phys. Rev. E 79:046316 [Google Scholar]
  125. Zhou Q, Li CM, Lu ZM, Liu YL. 2011. Experimental investigation of longitudinal space-time correlations of the velocity field in turbulent Rayleigh-Bénard convection. J. Fluid Mech. 683:94–111 [Google Scholar]
  126. Zhou Y. 2010. Renormalization group theory for fluid and plasma turbulence. Phys. Rep. 488:1–49 [Google Scholar]
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