Skip to main content Accessibility help

Pressure statistics in self-similar freely decaying isotropic turbulence

  • Marcello Meldi (a1) and Pierre Sagaut (a1)


The time evolution of pressure statistics in freely decaying homogeneous isotropic turbulence (HIT) is investigated via eddy-damped quasi-normal Markovian (EDQNM) computations. The present results show that the time decay rate of pressure-based statistical quantities, such as pressure variance and pressure gradient variance, are sensitive to the breakdown of permanence of large eddies. New formulae for the associated time-decay exponents are proposed, which extend previous relations proposed in Lesieur, Ossia & Metais (Phys. Fluids, vol. 11, 1999, p. 1535). Particular attention is paid to finite-Reynolds-number (FRN) effects on the pressure spectrum and pressure statistics. The results also suggest that $R{e}_{\lambda } = O(1{0}^{4} )$ must be considered to observe a one-decade inertial range in the pressure spectrum with Kolmogorov $- 7/ 3$ scaling. This threshold value is larger than almost all existing direct numerical simulation (DNS) and experimental data, justifying the discussion about other possible scaling laws. The $- 5/ 3$ slope reported in some DNS data is also recovered by the EDQNM model, but it is observed to be a low-Reynolds-number effect. Another important result is that FRN effects yield a departure from asymptotic theoretical behaviours which appear similar to some effects attributed to intermittency by most authors. This is exemplified by the ratio between pressure-based and velocity-based Taylor microscales. Therefore, high-Reynolds-number DNS or experiments such that $R{e}_{\lambda } = O(1{0}^{4} )$ would be required in order to remove FRN effects and to analyse pure intermittency effects.


Corresponding author

Email address for correspondence:


Hide All
Batchelor, G. K. 1951 Pressure fluctuations in isotropic turbulence. Proc. Camb. Phil. Soc. 47, 359374.
Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Bos, W. J. T., Chevillard, L., Scott, J. F. & Rubinstein, R. 2012 Reynolds number effect on the velocity increment skewness in isotropic turbulence. Phys. Fluids 24, 015108.
Cao, N., Chen, S. & Doolen, G. D. 1999 Statistics and structures of pressure in isotropic turbulence. Phys. Fluids 11, 2235.
Comte-Bellot, G. & Corrsin, S. 1966 The use of a contraction to improve the isotropy of grid-generated turbulence. J. Fluid Mech. 25, 657682.
Davidson, P. A. 2004 Turbulence. An Introduction for Scientists and Engineers. Oxford University Press.
Donzis, D. A., Sreenivasan, K. R. & Yeung, P. K. 2012 Some results on the Reynolds number scaling of pressure statistics in isotropic turbulence. Physica D 241, 164168.
Eyink, G. L. & Thomson, D. J. 2000 Free decay of turbulence and breakdown of self-similarity. Phys. Fluids 12, 477479.
Gotoh, T. & Fukayama, D. 2001 Pressure spectrum in homogeneous turbulence. Phys. Rev. Lett. 86, 37753778.
Gotoh, T. & Rogallo, R. S. 1999 Intermittency and scaling of pressure at small scales in forced isotropic turbulence. J. Fluid Mech. 396, 257285.
Heisenberg, W. 1948 Zur statischen Theorie der Turbulenz. Z. Phys. 124, 628657.
Hill, R. J. & Wilczak, J. M. 1995 Pressure structure functions and spectra for locally isotropic turbulence. J. Fluid Mech. 296, 247269.
Hinze, J. O. 1975 Turbulence. McGraw-Hill.
Kim, J. & Antonia, R. A. 1993 Isotropy of the small scales of turbulence at low Reynolds number. J. Fluid Mech. 251, 219238.
Kolmogorov, A. N. 1941 On the degeneration of isotropic turbulence in an incompressible viscous fluid. Dokl. Akad. Nauk SSSR 31, 538541.
Lesieur, M. 2008 Turbulence in Fluids. Springer.
Lesieur, M., Ossia, S. & Metais, O. 1999 Infrared pressure spectra in two- and three-dimensional isotropic incompressible turbulence. Phys. Fluids 11, 1535.
Meldi, M. & Sagaut, P. 2012 On non-self similar regimes in homogeneous isotropic turbulence decay. J. Fluid Mech. 711, 364393.
Meldi, M., Sagaut, P. & Lucor, D. 2011 A Stochastic view of isotropic turbulence decay. J. Fluid Mech. 668, 351362.
Orszag, S. A. 1970 Analytical theories of turbulence. J. Fluid Mech. 41, 363386.
Pearson, B. R. & Antonia, R. A. 2001 Reynolds-number dependence of turbulent velocity and pressure increments. J. Fluid Mech. 444, 343382.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
Pumir, A. 1994 A numerical study of pressure fluctuations in three-dimensional, incompressible, homogeneous, isotropic turbulence. Phys. Fluids 6, 2071.
Sagaut, P. & Cambon, C. 2008 Homogeneous Turbulence Dynamics. Cambridge University Press.
Schumann, U. & Patterson, G. S. 1978 Numerical study of pressure and velocity fluctuations in nearly isotropic turbulence. J. Fluid Mech. 88, 685709.
Taylor, G. I. 1935 Statistical Theory of Turbulence. Proc. R. Soc. Lond. A 151, 421444.
Tchoufag, J., Sagaut, P. & Cambon, C. 2012 Spectral approach to finite Reynolds number effects on Kolmogorov $4/ 5$ law in isotropic turbulence. Phys. Fluids 24, 015107.
Tsuji, Y. & Ishihara, T. 2003 Similarity scaling of pressure fluctuation in turbulence. Phys. Rev. E 68, 026309.
Uberoi, M. S. & Corrsin, S. 1953 Diffusion of heat from a line source in isotropic turbulence. NACA Rep. 1142.
Vedula, P. & Yeung, P. K. 1999 Similarity scaling of acceleration and pressure statistics in numerical simulations of isotropic turbulence. Phys. Fluids 11, 1208.
Yeung, P. K., Donzis, D. A. & Sreenivasan, K. R. 2012 Dissipation, enstrophy and pressure statistics in turbulence simulations at high Reynolds numbers. J. Fluid Mech. 700, 515.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed