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Nonlinear spectral model for rotating sheared turbulence

  • Ying Zhu (a1), C. Cambon (a1), F. S. Godeferd (a1) and A. Salhi (a1) (a2)


We propose a statistical model for homogeneous turbulence undergoing distortions, which improves and extends the MCS model by Mons, Cambon & Sagaut (J. Fluid Mech., vol. 788, 2016, 147–182). The spectral tensor of two-point second-order velocity correlations is predicted in the presence of arbitrary mean-velocity gradients and in a rotating frame. For this, we numerically solve coupled equations for the angle-dependent energy spectrum ${\mathcal{E}}(\boldsymbol{k},t)$ that includes directional anisotropy, and for the deviatoric pseudo-scalar  $Z(\boldsymbol{k},t)$ , that underlies polarization anisotropy ( $\boldsymbol{k}$  is the wavevector, $t$ the time). These equations include two parts: (i) exact linear terms representing the viscous spectral linear theory (SLT) when considered alone; (ii) generalized transfer terms mediated by two-point third-order correlations. In contrast with MCS, our model retains the complete angular dependence of the linear terms, whereas the nonlinear transfer terms are closed by a reduced anisotropic eddy damped quasi-normal Markovian (EDQNM) technique similar to MCS, based on truncated angular harmonics expansions. And in contrast with most spectral approaches based on characteristic methods to represent mean-velocity gradient terms, we use high-order finite-difference schemes (FDSs). The resulting model is applied to homogeneous rotating turbulent shear flow with several Coriolis parameters and constant mean shear rate. First, we assess the validity of the model in the linear limit. We observe satisfactory agreement with existing numerical SLT results and with theoretical results for flows without rotation. Second, fully nonlinear results are obtained, which compare well to existing direct numerical simulation (DNS) results. In both regimes, the new model improves significantly the MCS model predictions. However, in the non-rotating shear case, the expected exponential growth of turbulent kinetic energy is found only with a hybrid model for nonlinear terms combining the anisotropic EDQNM closure and Weinstock’s return-to-isotropy model.


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André, J. C. & Lesieur, M. 1977 Influence of helicity on the evolution of isotropic turbulence at high Reynolds number. J. Fluid Mech. 81, 187207.
Balbus, S. A. & Hawley, J. F. 1998 Instability, turbulence, and enhanced transport in accretion disks. Rev. Mod. Phys. 70 (1), 153.
Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Batchelor, G. K. & Proudman, I. 1954 The effect of rapid distortion of a fluid in turbulent motion. Q. J. Mech. Appl. Maths 7 (1), 83103.
Bellet, F., Godeferd, F. S., Scott, J. F. & Cambon, C. 2006 Wave turbulence in rapidly rotating flows. J. Fluid Mech. 562, 83121.
Bradshaw, P. 1969 The analogy between streamline curvature and buoyancy in turbulent shear flow. J. Fluid Mech. 36 (01), 177191.
Brethouwer, G. 2005 The effect of rotation on rapidly sheared homogeneous turbulence and passive scalar transport. Linear theory and direct numerical simulation. J. Fluid Mech. 542, 305342.
Briard, A., Gréa, B.-J., Mons, V., Cambon, C., Gomez, T. & Sagaut, P. 2018 Advanced spectral anisotropic modelling for shear flows. J. Turbul. 19 (7), 570599.
Burlot, A., Gréa, B.-J., Godeferd, F. S., Cambon, C. & Griffond, J. 2015 Spectral modelling of high Reynolds number unstably stratified homogeneous turbulence. J. Fluid Mech. 765, 1744.
Cambon, C. & Jacquin, L. 1989 Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295317.
Cambon, C., Jeandel, D. & Mathieu, J. 1981 Spectral modelling of homogeneous non-isotropic turbulence. J. Fluid Mech. 104, 247262.
Cambon, C., Mansour, N. N. & Godeferd, F. S. 1997 Energy transfer in rotating turbulence. J. Fluid Mech. 337, 303332.
Cambon, C., Mons, V., Gréa, B.-J. & Rubinstein, R. 2017 Anisotropic triadic closures for shear-driven and buoyancy-driven turbulent flows. Comput. Fluids 151, 7384.
Cambon, C. & Rubinstein, R. 2006 Anisotropic developments for homogeneous shear flows. Phys. Fluids 18 (8), 085106.
Canuto, V. M., Dubovikov, M. S., Cheng, Y. & Dienstfrey, A. 1996 Dynamical model for turbulence. III. Numerical results. Phys. Fluids 8 (2), 599613.
Canuto, V. M. & Dubovikov, M. S. 1996a A dynamical model for turbulence. I. General formalism. Phys. Fluids 8 (2), 571586.
Canuto, V. M. & Dubovikov, M. S. 1996b A dynamical model for turbulence. II. Shear-driven flows. Phys. Fluids 8 (2), 587598.
Clark, T. T., Kurien, S. & Rubinstein, R. 2018 Generation of anisotropy in turbulent flows subjected to rapid distortion. Phys. Rev. E 97 (1), 013112.
Craya, A.1957 Contribution à l’analyse de la turbulence associée à des vitesses moyennes. PhD thesis, Université de Grenoble.
Dong, C., McWilliams, J. C. & Shchepetkin, A. F. 2007 Island wakes in deep water. J. Phys. Oceanogr. 37 (4), 962981.
Gréa, B.-J. 2013 The rapid acceleration model and the growth rate of a turbulent mixing zone induced by Rayleigh–Taylor instability. Phys. Fluids 25 (1), 015118.
Hanazaki, H. & Hunt, J. C. R. 2004 Structure of unsteady stably stratified turbulence with mean shear. J. Fluid Mech. 507, 142.
Herring, J. R. 1974 Approach of axisymmetric turbulence to isotropy. Phys. Fluids 17 (5), 859872.
Hiwatashi, K., Alfredsson, P. H., Tillmark, N. & Nagata, M. 2007 Experimental observations of instabilities in rotating plane Couette flow. Phys. Fluids 19 (4), 048103.
Johnston, J. P., Halleent, R. M. & Lezius, D. K. 1972 Effects of spanwise rotation on the structure of two-dimensional fully developed turbulent channel flow. J. Fluid Mech. 56 (03), 533557.
Kassinos, S. C., Reynolds, W. C. & Rogers, M. M. 2001 One-point turbulence structure tensors. J. Fluid Mech. 428, 213248.
Launder, B. E., Reece, G. Jr. & Rodi, W. 1975 Progress in the development of a Reynolds-stress turbulence closure. J. Fluid Mech. 68 (3), 537566.
Leblanc, S. & Cambon, C. 1998 Effects of the coriolis force on the stability of stuart vortices. J. Fluid Mech. 356, 353379.
Lesur, G. & Longaretti, P. Y. 2005 On the relevance of subcritical hydrodynamic turbulence to accretion disk transport. Astron. Astrophys. 444, 2544.
Mishra, A. A. & Girimaji, S. S. 2017 Toward approximating non-local dynamics in single-point pressure–strain correlation closures. J. Fluid Mech. 811, 168188.
Moffatt, H. K. 1967 The interaction of turbulence with strong wind shear. In Proceedings of the International Colloquium on Atmospheric Turbulence and Radio Wave Propagation (ed. Yaglom, A. M. & Tatarsky, V. I.), pp. 139156. Nauka.
Mons, V., Cambon, C. & Sagaut, P. 2016 A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically averaged descriptors. J. Fluid Mech. 788, 147182.
Orszag, S. A. 1969 Analytical theories of turbulence. J. Fluid Mech. 41, 363386.
Perret, G., Stegner, A., Farge, M. & Pichon, T. 2006 Cyclone-anticyclone asymmetry of large-scale wakes in the laboratory. Phys. Fluids 18 (3), 036603.
Plunian, F. & Stepanov, R. 2007 A non-local shell model of hydrodynamic and magnetohydrodynamic turbulence. New J. Phys. 9 (8), 294.
Pouquet, A., Lesieur, M., André, J. C. & Basdevant, C. 1975 Evolution of high Reynolds number two-dimensional turbulence. J. Fluid Mech. 72, 305319.
Rogallo, R. S.1981 Numerical experiments in homogeneous turbulence. NASA Tech. Mem. 81315.
Rotta, J. 1951 Statistische theorie nichthomogener turbulenz i. Z. Phys. 129 (6), 547572.
Sagaut, P. & Cambon, C. 2018 Homogeneous Turbulence Dynamics, 2nd edn. Springer.
Salhi, A. & Cambon, C. 1997 An analysis of rotating shear flow using linear theory and DNS and LES results. J. Fluid Mech. 347, 171195.
Salhi, A. & Cambon, C. 2010 Stability of rotating stratified shear flow: an analytical study. Phys. Rev. E 81 (2), 026302.
Salhi, A., Cambon, C. & Speziale, C. G. 1997 Linear stability analysis of plane quadratic flows in a rotating frame. Phys. Fluids 9 (8), 23002309.
Salhi, A., Jacobitz, F. G., Schneider, K. & Cambon, C. 2014 Nonlinear dynamics and anisotropic structure of rotating sheared turbulence. Phys. Rev. E 89 (1), 013020.
Waleffe, F. 1992 The nature of triad interactions in homogeneous turbulence. Phys. Fluids A 4 (2), 350363.
Weinstock, J. 1982 Theory of the pressure–strain rate. Part 2. Diagonal elements. J. Fluid Mech. 116, 129.
Weinstock, J. 2013 Analytical theory of homogeneous mean shear turbulence. J. Fluid Mech. 727, 256281.
Zhu, Y.2019 Modelling and calculation for shear-driven rotating turbulence, with multiscale and directional approach. PhD thesis, École centrale de Lyon.
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