11 results
Nonlinear spectral model for rotating sheared turbulence
- Ying Zhu, C. Cambon, F. S. Godeferd, A. Salhi
-
- Journal:
- Journal of Fluid Mechanics / Volume 866 / 10 May 2019
- Published online by Cambridge University Press:
- 06 March 2019, pp. 5-32
-
- Article
- Export citation
-
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.
Spectral modelling of high Reynolds number unstably stratified homogeneous turbulence
- A. Burlot, B.-J. Gréa, F. S. Godeferd, C. Cambon, J. Griffond
-
- Journal:
- Journal of Fluid Mechanics / Volume 765 / 25 February 2015
- Published online by Cambridge University Press:
- 15 January 2015, pp. 17-44
-
- Article
- Export citation
-
We study unconfined homogeneous turbulence with a destabilizing background density gradient in the Boussinesq approximation. Starting from initial isotropic turbulence, the buoyancy force induces a transient phase toward a self-similar regime accompanied by a rapid growth of kinetic energy and Reynolds number, along with the development of anisotropic structures in the flow in the direction of gravity. We model this with a two-point statistical approach using an axisymmetric eddy-damped quasi-normal Markovian (EDQNM) closure that includes buoyancy production. The model is able to match direct numerical simulations (DNS) in a parametric study showing the effect of initial Froude number and mixing intensity on the development of the flow. We further improve the model by including the stratification timescale in the characteristic relaxation time for triple correlations in the closure. It permits the computation of the long-term evolution of unstably stratified turbulence at high Reynolds number. This agrees with recent theoretical predictions concerning the self-similar dynamics and brings new insight into the spectral energy distribution and anisotropy of the flow.
Non-geostrophic instabilities of an equilibrium baroclinic state
- Alexandre B. Pieri, F. S. Godeferd, C. Cambon, A. Salhi
-
- Journal:
- Journal of Fluid Mechanics / Volume 734 / 10 November 2013
- Published online by Cambridge University Press:
- 14 October 2013, pp. 535-566
-
- Article
- Export citation
-
We consider non-geostrophic homogeneous baroclinic turbulence without solid boundaries, and we focus on its energetics and dynamics. The homogeneous turbulent flow is therefore submitted to both uniform vertical shear $S$ and stable vertical stratification, parametrized by the Brunt–Väisälä frequency $N$, and placed in a rotating frame with Coriolis frequency $f$. Direct numerical simulations show that the threshold of baroclinic instability growth depends mostly on two dimensionless numbers, the gradient Richardson number $\mathit{Ri}= {N}^{2} / {S}^{2} $ and the Rossby number $\mathit{Ro}= S/ f$, whereas linear theory predicts a threshold that depends only on $\mathit{Ri}$. At high Rossby numbers the nonlinear limit is found to be $\mathit{Ri}= 0. 2$, while in the limit of low $\mathit{Ro}$ the linear stability bound $\mathit{Ri}= 1$ is recovered. We also express the stability results in terms of background potential vorticity, which is an important quantity in baroclinic flows. We show that the linear symmetric instability occurs from the presence of negative background potential vorticity. The possibility of simultaneous existence of symmetric and baroclinic instabilities is also investigated. The dominance of symmetric instability over baroclinic instability for $\mathit{Ri}\ll 1$ is confirmed by our direct numerical simulations, and we provide an improved understanding of the dynamics of the flow by exploring the details of energy transfers for moderate Richardson numbers.
Quasi-static magnetohydrodynamic turbulence at high Reynolds number
- B. FAVIER, F. S. GODEFERD, C. CAMBON, A. DELACHE, W. J. T. BOS
-
- Journal:
- Journal of Fluid Mechanics / Volume 681 / 25 August 2011
- Published online by Cambridge University Press:
- 21 June 2011, pp. 434-461
-
- Article
- Export citation
-
We analyse the anisotropy of homogeneous turbulence in an electrically conducting fluid submitted to a uniform magnetic field, for low magnetic Reynolds number, in the quasi-static approximation. We interpret contradictory earlier predictions between linearized theory and simulations: in the linear limit, the kinetic energy of transverse velocity components, normal to the magnetic field, decays faster than the kinetic energy of the axial component, along the magnetic field (Moffatt, J. Fluid Mech., vol. 28, 1967, p. 571); whereas many numerical studies predict a final state characterized by dominant energy of transverse velocity components. We investigate the corresponding nonlinear phenomenon using direct numerical simulation (DNS) of freely decaying turbulence, and a two-point statistical spectral closure based on the eddy-damped quasi-normal Markovian (EDQNM) model. The transition from the three-dimensional turbulent flow to a ‘two-and-a-half-dimensional’ flow (Montgomery & Turner, Phys. Fluids, vol. 25, 1982, p. 345) is a result of the combined effects of short-time linear Joule dissipation and longer time nonlinear creation of polarization anisotropy. It is this combination of linear and nonlinear effects which explains the disagreement between predictions from linearized theory and results from numerical simulations. The transition is characterized by the elongation of turbulent structures along the applied magnetic field, and by the strong anisotropy of directional two-point correlation spectra, in agreement with experimental evidence. Inertial equatorial transfers in both DNS and the model are presented to describe in detail the most important equilibrium dynamics. Spectral scalings are maintained in high-Reynolds-number turbulence attainable only with the EDQNM model, which also provides simplified modelling of the asymptotic state of quasi-static magnetohydrodynamic (MHD) turbulence.
Wave turbulence in rapidly rotating flows
- F. BELLET, F. S. GODEFERD, J. F. SCOTT, C. CAMBON
-
- Journal:
- Journal of Fluid Mechanics / Volume 562 / 10 September 2006
- Published online by Cambridge University Press:
- 14 August 2006, pp. 83-121
-
- Article
- Export citation
-
An asymptotic quasi-normal Markovian (AQNM) model is developed in the limit of small Rossby number $Ro$ and high Reynolds number, i.e. for rapidly rotating turbulent flow. Based on the ‘slow’ amplitudes of inertial waves, the kinetic equations are close to those that would be derived from Eulerian wave-turbulence theory. However, for their derivation we start from an EDQNM statistical closure model in which the velocity field is expanded in terms of the eigenmodes of the linear wave regime. Unlike most wave-turbulence studies, our model accounts for the detailed anisotropy as the angular dependence in Fourier space. Nonlinear equations at small Rossby number are derived for the set $e$, $Z$, $h$ – energy, polarization anisotropy, helicity – of spectral quantities which characterize second-order two-point statistics in anisotropic turbulence, and which generate every quadratic moment of inertial wave amplitudes. In the simplest symmetry consistent with the background equations, i.e. axisymmetry without mirror symmetry, $e$, $Z$ and $h$ depend on both the wavevector modulus $k$ and its orientation $\theta$ to the rotation axis. We put the emphasis on obtaining accurate numerical simulations of a generalized Lin equation for the angular-dependent energy spectrum $e(k, \theta, t)$, in which the energy transfer reduces to integrals over surfaces given by the triadic resonant conditions of inertial waves. Starting from a pure three-dimensional isotropic state in which $e$ depends only on $k$ and $Z\,{=}\,h\,{=}\,0$, the spectrum develops an inertial range in the usual fashion as well as angular anisotropy. After the development phase, we observe the following features:
A $k^{-3}$ power law for the spherically averaged energy spectrum. However, this is the average of power laws whose exponents vary with the direction of the wavevector from $k^{-2}$ for wavevectors near the plane perpendicular to the rotation axis, to $k^{-4}$ for parallel wavevectors.
The spectral evolution is self-similar. This excludes the possibility of a purely two-dimensional large-time limit.
The energy density is very large near the perpendicular wavevector plane, but this singularity is integrable. As a result, the total energy has contributions from all directions and is not dominated by this singular contribution.
The kinetic energy decays as $t^{-0.8}$, an exponent which is about half that one without rotation.
Turbulent diffusion in rapidly rotating flows with and without stable stratification
- C. CAMBON, F. S. GODEFERD, F. C. G. A. NICOLLEAU, J. C. VASSILICOS
-
- Journal:
- Journal of Fluid Mechanics / Volume 499 / 25 January 2004
- Published online by Cambridge University Press:
- 27 January 2004, pp. 231-255
-
- Article
- Export citation
-
In this work, three different approaches are used for evaluating some Lagrangian properties of homogeneous turbulence containing anisotropy due to the application of a stable stratification and a solid-body rotation. The two external frequencies are the magnitude of the system vorticity $2\Omega$, chosen vertical here, and the Brunt–Väisälä frequency $N$, which gives the strength of the vertical stratification. Analytical results are derived using linear theory for the Eulerian velocity correlations (single-point, two-time) in the vertical and the horizontal directions, and Lagrangian ones are assumed to be equivalent, in agreement with an additional Corrsin assumption used by Kaneda (2000). They are compared with results from the kinematic simulation model (KS) by Nicolleau & Vassilicos (2000), which also incorporates the wave–vortex dynamics inherited from linear theory, and directly yields Lagrangian correlations as well as Eulerian ones. Finally, results from direct numerical simulations (DNS) are obtained and compared for the rotation-dominant case $B\,{=}\,2\Omega/N\,{=}\,10$, the stratification-dominant case $B\,{=}\,1/10$, the non-dispersive case $B\,{=}\,1$, and pure stratification $B\,{=}\,0$ and pure rotation $N\,{=}\,0$. The last situation is shown to be singular with respect to the mixed stratified/rotating ones. We address the question of the validity of Corrsin's simplified hypothesis, which states the equivalence between Eulerian and Lagrangian correlations. Vertical correlations are found to follow this postulate, but not the horizontal ones. Consequences for the vertical and horizontal one-particle dispersion are examined. In the analytical model, the squared excursion lengths are calculated by time integrating the Lagrangian (equal to the Eulerian) two-time correlations, according to Taylor's procedure. These quantities are directly computed from fluctuating trajectories by both KS and DNS. In the case of pure rotation, the analytical procedure allows us to relate Brownian $t$-asymptotic laws of dispersion in both the horizontal and vertical directions to the angular phase-mixing properties of the inertial waves. If stratification is present, the inertia–gravity wave dynamics, which affects the vertical motion, yields a suppressed vertical diffusivity, but not a suppressed horizontal diffusivity, since part of the horizontal velocity field escapes wavy motion.
Statistical modelling and direct numerical simulations of decaying stably stratified turbulence. Part 2. Large-scale and small-scale anisotropy
- F. S. GODEFERD, C. STAQUET
-
- Journal:
- Journal of Fluid Mechanics / Volume 486 / 10 June 2003
- Published online by Cambridge University Press:
- 24 June 2003, pp. 115-159
-
- Article
- Export citation
-
Stably stratified freely decaying homogeneous turbulence is investigated by means of direct numerical simulations (DNS) and a two-point closure statistical model of the EDQNM type; a careful comparison with laboratory experiments is also made. Several aspects of anisotropy in the flow are studied, both at large and small scales. DNS and EDQNM approaches give very similar results up to the finest indicators of the flow, namely anisotropic spectra of velocity fields. Hence the statistical model predicts the structure of the flow at all scales.
Large-scale anisotropy appears in the Reynolds stress components and in the directional integral length scales. The well-known collapse of vertical turbulent motion, which yields the organization of the flow into quasi-horizontal vertically decorrelated vortex structures, is retrieved and quantified. Thus, the thickness of the vortex structures is shown to be set by their Froude number being of order one, in agreement with a previous dimensional analysis for an inviscid flow. Small-scale anisotropy is quantified from the components of the velocity and temperature gradients, whereby models for the dissipation rate of kinetic energy and available potential energy are discussed. The mixing properties of the flow are also investigated: the counter-gradient heat flux that exists at small scales appears to inhibit mixing when diffusivity is low enough and the Cox number varies linearly with the parameter $\epsilon/\nu N^2$.
All results agree very well with laboratory experiments on stably stratified grid turbulence, though the initial condition of our computations is different from the flow just behind the grid. This suggests a relative independence of decaying stably stratified turbulence of initial conditions.
Two-point closures and their applications: report on a workshop
- F. S. GODEFERD, C. CAMBON, J. F. SCOTT
-
- Journal:
- Journal of Fluid Mechanics / Volume 436 / 10 June 2001
- Published online by Cambridge University Press:
- 22 June 2001, pp. 393-407
-
- Article
- Export citation
-
This international scientific workshop was organized in Lyon, France, from 10 to 12 May 2000. Its focus was ‘Two-point closures and their applications’, with the understanding that the analysis and design of such models requires expert knowledge coming from a wide range of areas in turbulence research, e.g. experiments, numerical simulations, asymptotic models, etc.
In the global challenge of turbulence modelling, two-point closures prove useful in many ways. Two-point correlations and spectra are useful measures of the distortion of the eddy structure of turbulence by stratification, large-scale strains, rotation, etc. In some cases, e.g. near boundaries, spectra can be drastically changed. In addition to the accurate characterization of turbulence, the explicit computation of two-point correlations or spectra shows how the internal dynamics of the various scales of motion are affected by such distortion, especially the cascade process on which the production/dissipation relationship depends. Distortion can be the cause of large departures from isotropic homogeneous turbulence, pulling turbulent flows far away from the local equilibrium that is often assumed. A rather weak departure can allow the use of linearized theories such as rapid distortion theory, for the applicability of which rational bounds may be estimated by comparisons with weakly nonlinear calculations. A different approach is necessary when dealing with larger departures, for instance due to growth of instabilities. In that case new physical or similarity arguments have to be employed to obtain a satisfactory description of the modification to the cascade process, which can even undergo reversal in the limit when three-dimensional turbulence becomes two-dimensional. Of course, significant changes in spectra have direct implications for one-point measures of turbulence – which can be explicitly derived by integration of two-point correlations – used in most industrial closure schemes. Such one-point models consequently need to be adapted when turbulence is strongly affected by distortion.
Direct numerical simulations of turbulence with confinement and rotation
- F. S. GODEFERD, L. LOLLINI
-
- Journal:
- Journal of Fluid Mechanics / Volume 393 / 25 August 1999
- Published online by Cambridge University Press:
- 25 August 1999, pp. 257-308
-
- Article
- Export citation
-
The goal of this work is to analyse how solid body rotation affects forced turbulence enclosed within solid boundaries, and to compare it to results of the experiment performed by Hopfinger et al. (1982). In order to identify various mechanisms associated with rotation, confinement, and forcing, a numerical pseudo-spectral code is used for performing direct numerical simulations. The geometry is simplified with respect to the experimental one. First, we are able to reproduce the linear regime, as propagating inertial waves that undergo reflections at the walls. Second, the Ekman pumping phenomenon, proportional to the rotation rate, is identified in freely decaying turbulence, for which the evolution of the flow bounded by walls is compared to the evolution of unbounded homogeneous turbulence. Finally we introduce a local forcing on a plane in physical space, for simulating the effect of an oscillating grid, so that diffusive turbulence is created, and we examine the structuring of the flow under the combination of the linear and nonlinear mechanisms. A transition to an almost two-dimensional state is shown to occur between the region close to the forcing and an outer region in which vortices appear, the number of which depends on the Reynolds and Rossby numbers. In this region, the anisotropy of turbulence is examined, and the numerical predictions are shown to reproduce many of the most important features present in the experimental flow.
Statistical modelling and direct numerical simulations of decaying stably stratified turbulence. Part 1. Flow energetics
- C. STAQUET, F. S. GODEFERD
-
- Journal:
- Journal of Fluid Mechanics / Volume 360 / 10 April 1998
- Published online by Cambridge University Press:
- 10 April 1998, pp. 295-340
-
- Article
- Export citation
-
The dynamics of a homogeneous turbulent flow subjected to a stable stratification are studied by means of direct numerical simulations (DNS) and by a two-point closure statistical EDQNM model, adapted for anisotropic flows by Cambon (1989). The purpose of this work is to investigate the validity of the anisotropic statistical model, which we refer to as the EDQNM2 model. The numerical simulations are of high resolution, 2563, which permits Reynolds numbers comparable to those of recent laboratory experiments. Thus, detailed comparisons with the wind-tunnel experiments of Lienhardt & Van Atta (1990) and Yoon & Warhaft (1990) are also presented.
The initial condition is chosen so as to test the anisotropic closure assumption of the EDQNM2 model. This choice yields a ratio of kinetic to potential energy of 2[ratio ]1. This important amount of initial potential energy drives the flow dynamics during the first Brunt–Väisälä period. Because stronger transfer rates of potential energy than of kinetic energy occur toward small scales, the heat flux is (persistently) counter gradient at those small scales. The loss of potential energy at large scales is partly made up for by conversion of vertical kinetic energy, and this sets up a down-gradient heat flux at those scales, as if no or little potential energy were present at the initial time. Thus, common features with wind-tunnel experiments (in which there is relatively little potential energy just behind the grid) are found. Interestingly, only one quantity displays a similarity law in the DNS, in the EDQNM2 model and in the experiments of Lienhardt & Van Atta (1990) and Yoon & Warhaft (1990) as well: this is the ratio of the vertical heat flux to the dissipation rate of kinetic energy, which can also be interpreted as an instantaneous mixing efficiency. Thus, this parameter seems to be independent of initial flow conditions.
Our calculations simulate a longer evolution of the flow dynamics than laboratory experiments (in which the flow develops for at most one Brunt–Väisälä period). We find that the flow dynamics change from about 1.5 Brunt–Väisälä periods. At that time, the heat flux collapses while the dissipation rate of kinetic energy displays a self-similarity law attesting that this quantity becomes driven by buoyancy forces. This permits us to link the collapse of the largest scales of the flow with the smallest scales being influenced by the buoyancy force. We finally discuss the influence of a geometrical confinement effect upon the above results.
The EDQNM2 model compares remarkably well with the DNS, with respect to previous statistical models of stably stratified turbulent flows. Insufficient decorrelation between the vertical velocity and the temperature fluctuations is however observed, but with no dynamical significance. The vortex part of the flow is also overestimated by the EDQNM2 model, but the relative difference between the model prediction and the DNS does not exceed 15% after 6 Brunt–Väisälä periods. The EDQNM2 model offers interesting perpectives because of its ability to predict the dynamics of stratified flows at high Reynolds numbers. Knowledge about small-scale behaviour will be especially useful, to build up parameterization of the subgrid scales for instance.
Energy transfer in rotating turbulence
- CLAUDE CAMBON, N. N. MANSOUR, F. S. GODEFERD
-
- Journal:
- Journal of Fluid Mechanics / Volume 337 / 25 April 1997
- Published online by Cambridge University Press:
- 25 April 1997, pp. 303-332
-
- Article
- Export citation
-
The influence of rotation on the spectral energy transfer of homogeneous turbulence is investigated in this paper. Given the fact that linear dynamics, e.g. the inertial waves regime found in an RDT (rapid distortion theory) analysis, cannot affect a homogeneous isotropic turbulent flow, the study of nonlinear dynamics is of prime importance in the case of rotating flows. Previous theoretical (including both weakly nonlinear and EDQNM theories), experimental and DNS (direct numerical simulation) results are collected here and compared in order to give a self-consistent picture of the nonlinear effects of rotation on turbulence.
The inhibition of the energy cascade, which is linked to a reduction of the dissipation rate, is shown to be related to a damping of the energy transfer due to rotation. A model for this effect is quantified by a model equation for the derivative-skewness factor, which only involves a micro-Rossby number Roω=ω′/(2Ω) – ratio of r.m.s. vorticity and background vorticity – as the relevant rotation parameter, in accordance with DNS and EDQNM results.
In addition, anisotropy is shown also to develop through nonlinear interactions modified by rotation, in an intermediate range of Rossby numbers (RoL<1 and Roω>1), which is characterized by a macro-Rossby number RoL based on an integral lengthscale L and the micro-Rossby number previously defined. This anisotropy is mainly an angular drain of spectral energy which tends to concentrate energy in the wave-plane normal to the rotation axis, which is exactly both the slow and the two-dimensional manifold. In addition, a polarization of the energy distribution in this slow two-dimensional manifold enhances horizontal (normal to the rotation axis) velocity components, and underlies the anisotropic structure of the integral length-scales. Finally a generalized EDQNM (eddy damped quasi-normal Markovian) model is used to predict the underlying spectral transfer structure and all the subsequent developments of classic anisotropy indicators in physical space. The results from the model are compared to recent LES results and are shown to agree well. While the EDQNM2 model was developed to simulate ‘strong’ turbulence, it is shown that it has a strong formal analogy with recent weakly nonlinear approaches to wave turbulence.