Skip to main content Accessibility help
×
Home

A dissipative random velocity field for fully developed fluid turbulence

  • Rodrigo M. Pereira (a1) (a2), Christophe Garban (a3) and Laurent Chevillard (a1)

Abstract

We investigate the statistical properties, based on numerical simulations and analytical calculations, of a recently proposed stochastic model for the velocity field (Chevillard et al., Europhys. Lett., vol. 89, 2010, 54002) of an incompressible, homogeneous, isotropic and fully developed turbulent flow. A key step in the construction of this model is the introduction of some aspects of the vorticity stretching mechanism that governs the dynamics of fluid particles along their trajectories. An additional further phenomenological step aimed at including the long range correlated nature of turbulence makes this model dependent on a single free parameter, ${\it\gamma}$ , that can be estimated from experimental measurements. We confirm the realism of the model regarding the geometry of the velocity gradient tensor, the power-law behaviour of the moments of velocity increments (i.e. the structure functions) including the intermittent corrections and the existence of energy transfer across scales. We quantify the dependence of these basic properties of turbulent flows on the free parameter ${\it\gamma}$ and derive analytically the spectrum of exponents of the structure functions in a simplified non-dissipative case. A perturbative expansion in power of ${\it\gamma}$ shows that energy transfer, at leading order, indeed take place, justifying the dissipative nature of this random field.

Copyright

Corresponding author

Email address for correspondence: rodrigo.pereira@ens-lyon.fr

References

Hide All
Adrian, R. & Moin, P. 1988 Stochastic estimation of organized turbulent structure: homogeneous shear flow. J. Fluid Mech. 190, 531559.
Antonia, R. A., Phan Thien, N. & Satyaprakash, B. R. 1981 Autocorrelation and spectrum of dissipation fluctuations in a turbulent jet. Phys. Fluids 24, 554555.
Arneodo, A., Bacry, E. & Muzy, J.-F. 1998 Random cascades on wavelet dyadic tree. J. Math. Phys. 39, 41424164.
Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Benzi, R., Biferale, L., Crisanti, A., Paladin, G., Vergassola, M. & Vulpiani, A. 1993 A random process for the construction of multiaffine fields. Physica D 65, 352358.
Biferale, L., Boffetta, G., Celani, A., Crisanti, A. & Vulpiani, A. 1998 Mimicking a turbulent signal: sequential multiaffine processes. Phys. Rev. E 57 (6), 62616264.
Çağlar, M. 2007 Velocity fields with power-law spectra for modeling turbulent flows. Appl. Math. Model. 31, 19341946.
Chen, S., Sreenivasan, K., Nelkin, M. & Cao, N. 1997 Refined similarity hypothesis for transverse structure functions in fluid turbulence. Phys. Rev. Lett. 79, 22532256.
Chevillard, L.2015 A random painting of fluid turbulence. Habilitation à diriger des recherches, ENS Lyon https://tel.archives-ouvertes.fr/tel-01212057.
Chevillard, L., Castaing, B., Arneodo, A., Lévêque, E., Pinton, J.-F. & Roux, S. 2012 A phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows. C. R. Physique 13, 899928.
Chevillard, L., Lévêque, E., Taddia, F., Meneveau, C., Yu, H. & Rosales, C. 2011 Local and non local pressure Hessian effects in real and synthetic fluid turbulence. Phys. Fluids 23, 095108.
Chevillard, L., Rhodes, R. & Vargas, V. 2013 Gaussian multiplicative chaos for symmetric isotropic matrices. J. Stat. Phys. 150, 678703.
Chevillard, L., Robert, R. & Vargas, V. 2010 A stochastic representation of the local structure of turbulence. Europhys. Lett. 89, 54002.
Constantin, P. 1994 Geometric statistics in turbulence. SIAM Rev. 36, 7398.
Dhruva, B., Tsuji, Y. & Sreenivasan, K. 1997 Transverse structure functions in high-Reynolds-number turbulence. Phys. Rev. E 56, R4928R4930.
Duchon, J. & Robert, R. 2000 Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations. Nonlinearity 13, 249255.
Eyink, G. & Sreenivasan, K. 2006 Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78, 87135.
Frigo, M. & Johnson, S. G. 2005 The design and implementation of fftw3. Proc. IEEE 93 (2), 216231.
Frisch, U. 1995 Turbulence, The Legacy of A. N. Kolmogorov. Cambridge University Press.
Gagne, Y. & Hopfinger, E. J. 1979 High order dissipation correlations and structure functions in an axisymmetric jet and a plane channel flow. In 2nd Symposium on Turbulent Shear Flows, Imperial College, London, 11.7–11.2.
Grauer, R., Homann, H. & Pinton, J.-F. 2012 Longitudinal and transverse structure functions in high-Reynolds-number turbulence. New J. Phys. 14, 063016.
Hedevang, E. & Schmiegel, A. 2014 A Lévy based approach to random vector fields: with a view towards turbulence. Intl J. Nonlinear Sci. Numer. Simul. 15, 411435.
Hill, R. 2001 Equations relating structure functions of all orders. J. Fluid Mech. 434, 379388.
Juneja, A., Lathrop, D. P., Sreenivasan, K. R. & Stolovitzky, G. 1994 Synthetic turbulence. Phys. Rev. E 49 (6), 51795194.
Kahane, J.-P. 1985 Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9, 105150.
Kolmogorov, A. N. 1941 The local structure of turbulence in a incompressible viscous fluid for very large Reynolds number. Dokl. Akad. Nauk SSSR 30, 299303.
Kolmogorov, A. N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 8285.
Langford, J. & Moser, R. 1999 Optimal LES formulations for isotropic turbulence. J. Fluid Mech. 398, 321346.
Majda, A. & Bertozzi, A. 2002 Vorticity and Incompressible Flow. Cambridge University Press.
Mandelbrot, B. B. 1972 Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. In Statistical Models and Turbulence (ed. Rosenblatt, M. & Van Atta, C.), Lecture Notes in Physics, vol. 12, pp. 333351. Springer.
Mandelbrot, B. B. & Van Ness, J. W. 1968 Fractional Brownian motion, fractional noises and applications. SIAM Rev. 10, 422437.
Meneveau, C. 2011 Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech. 43, 219245.
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics, vol. 1 & 2. MIT Press.
Nawroth, A. P. & Peinke, J. 2006 Multiscale reconstruction of time series. Phys. Lett. A 360, 234237.
Obukhov, A. M. 1962 Some specific features of atmospheric turbulence. J. Fluid Mech. 13, 7781.
Papoulis, A. 1991 Probability, Random Variables and Stochastic Processes, 3rd edn. McGraw-Hill International Editions.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
Rhodes, R. & Vargas, V. 2014 Gaussian multiplicative chaos and applications: A review. Probability Surveys 11, 315392.
Robert, R. & Vargas, V. 2008 Hydrodynamic turbulence and intermittent random fields. Comm. Math. Phys. 284, 649673.
Rosales, C. & Meneveau, C. 2008 Anomalous scaling and intermittency in three-dimensional synthetic turbulence. Phys. Rev. E 78, 016313.
Schmiegel, J., Cleve, J., Eggers, H., Pearson, B. & Greiner, M. 2004 Stochastic energy-cascade model for (1+1)-dimensional fully developed turbulence. Phys. Lett. A 320, 247253.
Sidje, R. B. 1998 Expokit: A software package for computing matrix exponentials. ACM Trans. Math. Softw. 24 (1), 130156.
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.
Tsinober, A. 2001 An Informal Introduction to Turbulence. Kluwer.
Vieillefosse, P. 1982 Local interaction between vorticity and shear in a perfect incompressible fluid. J. Phys. (Paris) 43, 837842.
Wallace, J. 2009 Twenty years of experimental and direct numerical simulation access to the velocity gradient tensor: What have we learned about turbulence? Phys. Fluids 21, 021301.
Wilczek, M. & Meneveau, C. 2014 Pressure Hessian and viscous contributions to velocity gradient statistics based on Gaussian random fields. J. Fluid Mech. 756, 191225.
Yakhot, V. 2001 Mean-field approximation and a small parameter in turbulence theory. Phys. Rev. E 63, 026307.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Related content

Powered by UNSILO

A dissipative random velocity field for fully developed fluid turbulence

  • Rodrigo M. Pereira (a1) (a2), Christophe Garban (a3) and Laurent Chevillard (a1)

Metrics

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.