Olivier Debliquy

Energy transfers in forced MHD turbulence

The energy cascade in magnetohydrodynamics is studied using high resolution direct numerical simulations of forced isotropic turbulence. The magnetic Prandtl number is unity and the large scale forcing is a function of the velocity that injects a constant rate of energy without generating a mean flow. A shell decomposition of the velocity and magnetic fields is proposed and is extended to the Elsässer variables. The analysis of energy exchanges between these shell variables shows that the velocity and magnetic energy cascades are mainly local and forward, though non-local energy transfer does exist between the large, forced, velocity scales and the small magnetic structures. The possibility of splitting the shell-to-shell energy transfer into forward and backward contributions is also discussed.

Energy fluxes and shell-to-shell transfers in three-dimensional decaying magnetohydrodynamic turbulence

A spectral analysis of the energy cascade in magnetohydrodynamics (MHD) is presented using high-resolution direct numerical simulation of decaying isotropic turbulence. The Fourier representations of both the velocity and the magnetic fields are split into subsets that correspond to shells of wave vectors. A detailed study of the shell-to-shell interactions is performed and a comparison with theoretical prediction based on field-theoretic method is proposed. Two different definitions for the forward and backward energy transfers are suggested. They provide diagnostics that can be used in order to assess subgrid-scale modeling in large eddy simulation for turbulent MHD systems.

Subgrid-scale energy and pseudo pressure in large-eddy simulation

A dynamic model based on the Germano identity is proposed to evaluate the subgrid-scale energy in large-eddy simulations as a function of the large-scale velocity field only. The model is shown to allow the satisfactory reconstruction of the total energy contained in a direct numerical simulation from large-eddy simulations with different resolutions. The predicted subgrid-scale energy is given as a simple algebraic expression based on the Leonard tensor appearing in the dynamic procedure and does not require an additional transport equation. The model assumes a Kolmogorov spectrum and is implemented with and without the introduction of a dissipation cutoff in the high wave vector range.

Direct numerical simulation and large-eddy simulation of a shear-free mixing layer

High resolution direct numerical simulation (DNS) (512×1024×512) and large-eddy simulation (LES) of a shear-free mixing layer are presented. The geometry of the flow consists of two layers with different turbulence intensities that are in contact and interact through a fairly thin mixing layer. This geometry is used to explore the influence of inhomogeneities in the characteristic length scales, times scales and energy scales on the turbulence properties. Comparison of DNS results is made with the Veeravalli & Warhaft ( J. Fluid Mech. 207 , 191–229, 1989) experiment. The LES is performed on a 32×64×32 grid using an eddy-viscosity model. The use of such a model appears to be justified by the very weak departures from isotropy that are observed in the shear-free mixing layer. The LES predictions are compared with the filtered DNS data and show that the eddy viscosity model performs very well in predicting the energy profile as well as the deviation from Gaussianity in the turbulent velocity field statistics.

Local shell-to-shell energy transfer via nonlocal interactions in fluid turbulence

In this paper we analytically compute the strength of nonlinear interactions in a triad, and the energy exchanges between wave-number shells in incompressible fluid turbulence. The computation has been done using first-order perturbative field theory. In three dimensions, magnitude of triad interactions is large for nonlocal triads, and small for local triads. However, the shell-to-shell energy transfer rate is found to be local and forward. This result is due to the fact that the nonlocal triads occupy much less Fourier space volume than the local ones. The analytical results on three-dimensional shell-to-shell energy transfer match with their numerical counterparts. In two-dimensional turbulence, the energy transfer rates to the nearby shells are forward, but to the distant shells are backward; the cumulative effect is an inverse cascade of energy.

A Dynamic Subgrid-Scale Model Based on the Turbulent Kinetic Energy

A dynamic subgrid-scale model using explicitly the turbulent kinetic energy, k, is investigated. The evolution for k is predicted by a transport equation. This subgrid-scale model is tested on LES of decaying homogeneous turbulence in a cubic geometry with periodic boundary conditions.

Energy Fluxes and Shell-to-Shell Transfers in MHD Turbulence

A spectral analysis of the energy cascade in magnetohydrodynamics (MHD) is presented using high resolution direct numerical simulations of both forced and decaying isotropic turbulence. The triad interactions between velocity and magnetic field modes are averaged into shell interactions between similar length scales phenomena. This is achieved by combining all the velocity Fourier modes that correspond to wave vectors with similar amplitude into a shell velocity variable. The same procedure is adopted for the magnetic field. The analysis of the interactions between these shell variables gives a global picture of the energy transfers between different length scales, as well as between the velocity and the magnetic fields. Also, two different attempts to separate the shell-to-shell interactions into forward and backward energy transfers are proposed. They provide diagnostics that can be used in order to assess subgrid-scale modelling in large-eddy simulation for turbulent MHD systems.

Modelling of MHD Turbulence

Numerical simulations of turbulent phenomena in fluids have made considerable progress with the emergence of large parallel computers. For simple geometries, very efficient numerical methods have been developed to provide accurate numerical solutions to the equations of fluid dynamics. These approaches are referred to as direct numerical simulation (DNS) and their predictions are often regarded as almost as reliable as the experimental data. If the propagation of sound waves can be neglected and if thermal effects are not considered, turbulence in non-conducting fluids is described by the incompressible Navier–Stokes equations. For a given geometry, the turbulence properties are then fully determined by a single dimensionless parameter: the Reynolds number Re. There is however a strong limitation to the use of DNS. Indeed, the number of degrees of freedom required to characterize a velocity field ui that corresponds to a turbulent flow is known to increase as Re in three-dimensional (3D) turbulent systems [1]. DNS of the Navier– Stokes equations are thus limited to moderately small Reynolds number flows. For electrically conducting fluids, the situation is similar, though even more complex. The flow may be strongly affected by the coupling between the velocity and magnetic fields and is described by the magnetohydrodynamic (MHD) equations. The number of degrees of freedom is expected to be even higher in MHD and the limitations to the use of DNS techniques are at least as severe for MHD as they are for non-conducting fluids. There is thus an interest in developing modelling techniques in which only a fraction of the total number of degrees of freedom is actually simulated. Among these techniques, large eddy simulation (LES) has attracted much interest in the past few decades [2]. LES can be defined as a computer experiment in which the large scales are simulated directly while the small scales are modelled. This technique has been first developed for simulating Navier– Stokes problems at high Reynolds numbers and has been recently adapted to MHD flow simulations [3–7]. It will be presented in § 2 and tested in § 3.1 using

POD of the Autonomous Near-Wall Region in a Minimal Channel Flow

We propose to examine the structure and the dynamics of the empirical eigenfunctions of the autonomous near-wall region in a turbulent minimal channel flow. This simplified and artificial situation consists of a severe restriction of the interactions between the outer flow and the structures of the viscous and buffer layers so that it is possible to maintain turbulence in the near-wall region without any input from the outer flow. This is achieved in a DNS of a pressure-gradient driven turbulent channel flow for a minimal flow unit by damping artificially fluctuations in the outer flow while the near-wall region survives indefinitely. The use of empirical eigenfunctions (determined by the Karhuenen-Loeve procedure or Proper Orthogonal Decomposition) reveals being an adequate tool for analysing the low-dimensional dynamics of the flow.

High Resolution DNS of a Shear-Free Mixing Layer and LES

Results from a high resolution (512 × 1024 × 512) DNS of a shear-free mixing layer are investigated. The mixing layer consists of the transition between two regions of homogeneous turbulence characterized by different turbulent intensities and energy spectra. The simulation has been initialised in order to reproduce the conditions of laboratory investigations of the same flow and an extensive comparison between the DNS and the experimental results is proposed. Also, the possibility to reproduce the main features of this flow using LES is explored.