Eddy-damped Quasi-normal Markovian Closure Research Articles

An EDQNM study of the dissipation rate in isotropic non-equilibrium turbulence

The energy cascade from large to small scales is a robust feature of three-dimensional turbulence. In statistically steady turbulence, the average dissipation is in equilibrium with the energy injected in the system. A global quantity measuring the deviations from such a flux equilibrium is the normalised dissipation rate , corresponding to the viscous dissipation, normalised by quantities associated with the largest scales of the system. Recent investigations have pointed out how this normalised dissipation rate varies in unsteady flows. We focus on two test-cases to assess non-equilibrium in isotropic turbulence. These cases are, respectively, turbulence in the presence of a large-scale periodic forcing and turbulence with reversed initial conditions. We show, using the Eddy-Damped Quasi-Normal Markovian closure, that for turbulence in the presence of periodic forcing, a scaling is reproduced ( indicating the Taylor-scale Reynolds number) when the forcing-frequency is adjusted to be of the order of the inverse of the integral time-scale. It is shown that the spectrum can be decomposed into an equilibrium spectrum, governed by Kolmogorov scaling in the inertial range, and a perturbation spectrum, proportional to , k being the wavenumber. For reversed turbulence, a novel procedure is introduced to prescribe initial conditions for the nonlinear transfer. Subsequently a clear transient with a scaling is observed in the dynamics.

Staircase scaling of short-time energy transfer in turbulence

It is illustrated that a sharply truncated initial kinetic energy spectrum evolves to a staircase-shaped spectrum at short times. This effect is directly associated with the triadic nature of the energy transfer. A rigorous analysis leads to predictions on the time-dependence of this effect, and these predictions are verified by both direct numerical simulations and eddy-damped quasi-normal Markovian closure integrations.

Nonlinear spectral model for rotating sheared turbulence

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.

Self-similar regimes of turbulence in weakly coupled plasmas under compression.

Turbulence in weakly coupled plasmas under compression can experience a sudden dissipation of kinetic energy due to the abrupt growth of the viscosity coefficient governed by the temperature increase. We investigate in detail this phenomenon by considering a turbulent velocity field obeying the incompressible Navier-Stokes equations with a source term resulting from the mean velocity. The system can be simplified by a nonlinear change of variable, and then solved using both highly resolved direct numerical simulations and a spectral model based on the eddy-damped quasinormal Markovian closure. The model allows us to explore a wide range of initial Reynolds and compression numbers, beyond the reach of simulations, and thus permits us to evidence the presence of a nonlinear cascade phase. We find self-similarity of intermediate regimes as well as of the final decay of turbulence, and we demonstrate the importance of initial distribution of energy at large scales. This effect can explain the global sensitivity of the flow dynamics to initial conditions, which we also illustrate with simulations of compressed homogeneous isotropic turbulence and of imploding spherical turbulent layers relevant to inertial confinement fusion.

Prandtl number effects in decaying homogeneous isotropic turbulence with a mean scalar gradient

ABSTRACTDecaying homogeneous isotropic turbulence with an imposed mean scalar gradient is investigated numerically, thanks to a specific eddy-damped quasi-normal Markovian closure developed recently for passive scalar mixing in homogeneous anisotropic turbulence (BGC). The present modelling is compared successfully with recent direct numerical simulations and other models, for both very large and small Prandtl numbers. First, scalings for the cospectrum and scalar variance spectrum in the inertial range are recovered analytically and numerically. Then, at large Reynolds numbers, the decay and growth laws for the scalar variance and mixed velocity–scalar correlations, respectively, derived in BGC, are shown numerically to remain valid when the Prandtl number strongly departs from unity. Afterwards, the normalised correlation ρwθ is found to decrease in magnitude at a fixed Reynolds number when Pr either increases or decreases, in agreement with earlier predictions. Finally, the small scales return to isotropy of the scalar second-order moments is found to depend not only on the Reynolds number, but also on the Prandtl number.

Mixed-derivative skewness for high Prandtl and Reynolds numbers in homogeneous isotropic turbulence

The mixed-derivative skewness Suθ of a passive scalar field in high Reynolds and Prandtl numbers decaying homogeneous isotropic turbulence is studied numerically using eddy-damped quasi-normal Markovian closure, for Reλ ≥ 103 up to Pr = 105. A convergence of Suθ for Pr ≥ 103 is observed for any high enough Reynolds number. This asymptotic high Pr regime can be interpreted as a saturation of the mixing properties of the flow at small scales. The decay of the derivative skewnesses from high to low Reynolds numbers and the influence of large scales initial conditions are investigated as well.

Spectral modelling for passive scalar dynamics in homogeneous anisotropic turbulence

The present work aims at developing a spectral model for a passive scalar field and its associated scalar flux in homogeneous anisotropic turbulence. This is achieved using the paradigm of eddy-damped quasi-normal Markovian (EDQNM) closure extended to anisotropic flows. In order to assess the validity of this approach, the model is compared to several detailed direct numerical simulations (DNS) and experiments of shear-driven flows and isotropic turbulence with a mean scalar gradient at moderate Reynolds numbers. This anisotropic modelling is then used to investigate the passive scalar dynamics at very high Reynolds numbers. In the framework of homogeneous isotropic turbulence submitted to a mean scalar gradient, decay and growth exponents for the cospectrum and scalar energies are obtained analytically and assessed numerically thanks to EDQNM closure. With the additional presence of a mean shear, the scaling of the scalar flux and passive scalar spectra in the inertial range are investigated and confirm recent theoretical predictions. Finally, it is found that, in shear-driven flows, the small scales of the scalar second-order moments progressively return to isotropy when the Reynolds number increases.

Decay and growth laws in homogeneous shear turbulence

ABSTRACTHomogeneous anisotropic turbulence has been widely studied in the past decades, both numerically and experimentally. Shear flows have received a particular attention because of the numerous physical phenomena they exhibit. In the present paper, both the decay and growth of anisotropy in homogeneous shear flows at high Reynolds numbers are revisited thanks to a recent eddy-damped quasi-normal Markovian closure adapted to homogeneous anisotropic turbulence. The emphasis is put on several aspects: an asymptotic model for the slow part of the pressure–strain tensor is derived for the return to isotropy process when mean velocity gradients are released. Then, a general decay law for purely anisotropic quantities in Batchelor turbulence is proposed. At last, a discussion is proposed to explain the scattering of global quantities obtained in DNS and experiments in sustained shear flows: the emphasis is put on the exponential growth rate of the kinetic energy and on the shear parameter.

Anomalous eddy viscosity for two-dimensional turbulence

We study eddy viscosity for generalized two-dimensional (2D) fluids. The governing equation for generalized 2D fluids is the advection equation of an active scalar q by an incompressible velocity. The relation between q and the stream function ψ is given by q = − (−∇2)α/2ψ. Here, α is a real number. When the evolution equation for the generalized enstrophy spectrum Qα(k) is truncated at a wavenumber kc, the effect of the truncation of modes with larger wavenumbers than kc on the dynamics of the generalized enstrophy spectrum with smaller wavenumbers than kc is investigated. Here, we refer to the effect of the truncation on the dynamics of Qα(k) with k < kc as eddy viscosity. Our motivation is to examine whether the eddy viscosity can be represented by normal diffusion. Using an asymptotic analysis of an eddy-damped quasi-normal Markovian (EDQNM) closure approximation equation for the enstrophy spectrum, we show that even if the wavenumbers of interest k are sufficiently smaller than kc, the eddy viscosity is not asymptotically proportional to k2Qα(k), i.e., a normal diffusion, but to k4−αQα(k) for α > 0 and k4Qα(k) for α < 0, i.e., an anomalous diffusion. This indicates that the eddy viscosity as normal diffusion is asymptotically realized only for α = 2 (Navier–Stokes system). The proportionality constant, the eddy viscosity coefficient, is asymptotically negative. These results are confirmed by numerical calculations of the EDQNM closure approximation equation and direct numerical simulations of the governing equation for forced and dissipative generalized 2D fluids. The negative eddy viscosity coefficient is explained using Fjørtoft’s theorem and a spreading hypothesis for the spectrum.

Numerical investigation on the partial return to isotropy of freely decaying homogeneous axisymmetric turbulence

The return to isotropy of freely decaying homogeneous axisymmetric turbulence is numerically studied using the Eddy-Damped Quasi-Normal Markovian closure. The model, whose classical formulation has been extended to moderately anisotropic flows by Cambon, Jeandel, and Mathieu [“Spectral modelling of homogeneous non-isotropic turbulence,” J. Fluid Mech. 104, 247–262 (1981)], allows for an accurate description of the turbulence decay for very long times. More specifically, the observed results encompass both the high and low Reynolds number asymptotic regimes. Such an analysis escapes both wind tunnel experiments and direct numerical simulations possibilities at the present time. Anisotropy generation mechanisms considered in the present paper do not affect the nature of nonlinear interactions and the related energy cascade mechanisms. Their influence on the decay regime is quantified by the investigation of the features of the initial three dimensional kinetic energy spectrum. The initial anisotropy level is always chosen in agreement with experimental grid turbulence observations. The present results show that the relaxation towards an isotropic state is observed in the inertial range of the energy spectrum E(k, t) during the initial high Reynolds regime, while the large scales conserve the imposed anisotropy level if the permanence of large eddies hypothesis is verified. A direct consequence is that the ratio between axial and transverse kinetic energies γ increases when the low Reynolds asymptotic regime is reached. Saturation effects, i.e., effects related to the boundedness of the physical domain under consideration, are also investigated. Isotropic saturation (same cut off scale in every direction) leads to the relaxation towards a fully isotropic state in all cases, whereas anisotropic saturation (unbounded domain in the axial direction) leads to an amplification of the initial anisotropy.

Eddy damped quasinormal Markovian simulations of superfluid turbulence in helium II

A two fluid (the normal fluid and the superfluid) spectral model based on the HVBK equations is used to compute the energy spectra of developed turbulence in superfluid helium (helium II) at different temperatures and different values of mutual friction coefficients, looking for both asymptotic and realistic helium II solutions. An original computational model is proposed, based on the model suggested by L’vov, Nazarenko, and Skrbek [JLTP 145, 125 (2006)] for mutual friction term and an eddy damped quasinormal Markovian closure for the energy flux term. On the one hand, the four asymptotic solutions predicted by theoretical analysis are confirmed. On the other hand, we present results for normal and superfluid energy spectra in realistic cases using experimental values of the mutual friction coefficients corresponding to helium II. Good qualitative agreement in the small scale range is found with the direct numerical simulation results obtained by Roche, Barenghi, and Leveque [Europhys. Lett. 87, 54006 (2009)].

Reynolds number dependency of the scalar flux spectrum in isotropic turbulence with a uniform scalar gradient

In this paper, the eddy-damped quasi-normal Markovian closure is used to study the behavior of the scalar flux spectrum in isotropic turbulence as the Reynolds number Reλ varies in a range between 30 and 107. The different contributions to the evolution equation of the scalar flux spectrum are studied. One-dimensional spectra are in good agreement with direct numerical simulation (DNS) and experiments at moderate Reλ. The closure shows that at high Reynolds numbers, a K−7∕3 scaling is found for the scalar flux spectrum, in agreement with Lumley’s prediction [Phys. Fluids 10, 855 (1967)], but enormous Reλ are needed before it can be clearly observed. In the range of wind tunnel experiments, the spectral exponent for the scalar flux is closer to −2 in agreement with existing measurements [Mydlarski and Warhaft, J. Fluid Mech. 358, 135 (1998)]. The results for the molecular dissipation of scalar flux are in agreement with the DNS results of Overholt and Pope [Phys. Fluids A 8, 3128 (1996)]. The large Reλ behavior of this quantity is also addressed.

Spectral Energy Dynamics in Magnetohydrodynamic Turbulence

Spectral direct numerical simulations of incompressible MHD turbulence at a resolution of up to 1024(3) collocation points are presented for a statistically isotropic system as well as for a setup with an imposed strong mean magnetic field. The spectra of residual energy, E(R)k=|E(M)k - E(K)k|, and total energy, Ek=E(K)k+E(M)k, are observed to scale self-similarly in the inertial range as E(R)k approximately k(-7/3), E(k)approximately k(-5/3) (isotropic case) and E(R)(k(perpendicular) approximately k(-2)(perpendicular), E(k(perpendicular))approximately k(-3/2)(perpendicular) (anisotropic case, perpendicular to the mean field direction). A model of dynamic equilibrium between kinetic and magnetic energy, based on the corresponding evolution equations of the eddy-damped quasinormal Markovian closure approximation, explains the findings. The assumed interplay of turbulent dynamo and Alfvén effect yields E(R)k approximately kE2(k), which is confirmed by the simulations.

Langevin models of turbulence: Renormalization group, distant interaction algorithms or rapid distortion theory?

A new dynamical turbulence model is validated by comparisons of its numerical simulations with fully resolved, direct numerical simulations (DNS) of the Navier–Stokes equations in three-dimensional, isotropic, homogeneous conditions. In this model the small-scale velocities are computed using a Langevin, linear, inhomogeneous, stochastic equation that is derived from a quasi-linear approximation of the Navier–Stokes equations, in the spirit of rapid distortion theory (RDT). The values of the turbulent viscosity involved in our Langevin model are compared with a theoretical prescription based on the renormalization group and the distant interaction algorithms (DSTA) model. We show that the empirical turbulent viscosities derived from simulations of the Langevin model are in good quantitative agreement with the DSTA predictions. Finally, Langevin simulations are compared with DNS and large eddy simulations based on the eddy-damped quasi-normal Markovian closure. The Langevin RDT model is able to reproduce the correct spectrum shape, intermittency statistics, and coherent flow structures for both the resolved and the largest sub-grid scales. It also predicts the evolution of the resolved scales better than the alternative models.

Eddy-damped quasinormal Markovian closure: a closure for magnetohydrodynamic turbulence?

A set of nonlinear differential equations are developed that are analogous to the spectral evolution equations of incompressible magnetohydrodynamics (MHD). Because these equations possess little detail of MHD, apart from salient symmetry properties, they provide a toy model in which aspects of turbulent MHD can be understood readily. In the context of this model, the eddy-damped quasinormal Markovian (EDQNM) closure often used in Navier–Stokes turbulence is demonstrated to provide physically realizable spectra for magnetohydrodynamic turbulence, if the eddy-damping functions are chosen to satisfy certain symmetry properties. The requirements of physical realizability are more demanding in MHD than in fluid turbulence. In the absence of mean fields, this model demonstrates that the components of not only the turbulent kinetic energy spectrum, but also the magnetic energy spectra, never become negative. Another condition for realizability possessed by this model is that the components of the turbulent cross-helicity spectrum always satisfy a Schwarz inequality with respect to the corresponding components of the kinetic and magnetic energy spectra.

On constructing realizable, conservative mixed scalar equations using the eddy-damped quasi-normal Markovian theory

The eddy-damped quasi-normal Markovian (EDQNM) turbulence theory has been applied to the covariance spectrum of two passive isotropic scalars with different diffusivities in stationary isotropic turbulence. A rigorous application of EDQNM, which introduces no new modelling assumptions or constants, is shown to yield a covariance spectrum that violates the Cauchy–Schwartz inequality over some of the wavenumbers. One approach to this problem is to derive a model based on a stochastic differential equation, as its presence guarantees realizability. For an isotropic scalar, it is possible to construct a Langevin equation for the Fourier transform of the scalar concentrations that is consistent with each EDQNM scalar autocorrelation spectrum. The Langevin equations can then be used to construct a model for the covariance spectrum that is realizable. However, the resulting covariance transfer term does not properly conserve the scalar covariance, and so the model is still not satisfactory. The problem can be traced to the Markovianization step, which leads to the presence of the scalar diffusivities in the transfer functions in an unphysical fashion. A simple fix is described which reconciles the two approaches and gives conservative, realizable results for all time.Next, we apply the EDQNM theory to a more general system involving the mixing of anisotropic scalars. Anisotropy in this case results from a uniform mean gradient of the two scalar concentrations in one direction. As with the isotropic scalars, direct application of the EDQNM closure results in a covariance spectrum that violates the Cauchy–Schwartz inequality; however, in this case it is not as simple to construct a Langevin model that reproduces all of the spectral interactions that result from the EDQNM procedure. Nevertheless, we show that the same modification of the inverse time scale as is done for the isotropic scalar results in an anisotropic scalar covariance spectrum that is realizable for all times.

An analytical model of velocity-derivative skewness of rotating homogeneous turbulence

The velocity-derivative skewness of rotating homogeneous turbulence is analytically derived from the turbulent kinetic energy dissipation rate equation by assuming a simple energy spectrum function. It predicts the cut-off of spectral energy at large rotation rate, and the predicted reduction of skewness due to rotation agrees well with the results of direct numerical simulation and eddy-damped quasi-normal Markovian closure.

Renormalized dissipation in the nonconservatively forced Burgers equation

A previous calculation [P. H. Diamond and T.-S. Hahm, Phys. Plasmas 2, 3640 (1995)] of the renormalized dissipation in the nonconservatively forced one-dimensional Burgers equation, which encountered a catastrophic long-wavelength divergence ∼kmin−3, is reconsidered. In the absence of velocity shear, analysis of the eddy-damped quasi-normal Markovian closure predicts only a benign logarithmic dependence on kmin. The original divergence is traced to an inconsistent resonance-broadening type of diffusive approximation, which fails in the present problem. Ballistic scaling of renormalized pulses is retained, but such scaling does not, by itself, imply a paradigm of self-organized criticality. An improved scaling formula for a model with velocity shear is also given.

Macroscopic structures of inhomogeneous, Navier–Stokes turbulence

Macroscopic properties of inhomogeneous, Navier–Stokes turbulence are calculated using a fundamental theoretical approach. Starting from a set of solenoidal basis vectors suitable for describing a turbulence bounded by infinite, parallel free-slip planes, invoking a random-phase approximation, and using the eddy-damped quasinormal Markovian closure, various quantities of dynamical interest are calculated. Quantities that affect the evolution of both the momentum density and the kinetic energy density of the turbulently evolving fluid are studied. These quantities include the ensemble-averaged values of gradients of the triple-velocity correlation and the pressure–velocity correlation. When the energy spectrum in wave-number space lacks certain reflection symmetries, a nontrivial mean flow velocity will be seen to emerge out of the turbulent eddies. In turn, these mean flow velocities will affect the evolution of the turbulence. Of course, such physically interesting mean-field structures cannot arise in dynamics suffused by the assumption of statistical homogeneity.

The realizable Markovian closure. I. General theory, with application to three-wave dynamics

A type of eddy-damped quasinormal Markovian (EDQNM) closure is shown to be potentially nonrealizable in the presence of linear wave phenomena. This statistical closure results from the application of a fluctuation–dissipation (FD) ansatz to the direct-interaction approximation (DIA); unlike in phenomenological formulations of the EDQNM, both the frequency and the damping rate are renormalized. A violation of realizability can have serious physical consequences, including the prediction of negative or even divergent energies. A new statistical approximation, the realizable Markovian closure (RMC), is proposed as a remedy. An underlying Langevin equation that makes no assumption of white-noise statistics is exhibited. Even in the wave-free case the RMC, which is based on a nonstationary version of the FD ansatz, provides a better representation of the true dynamics than does the EDQNM closure. The closure solutions are compared numerically against the exact ensemble dynamics of three interacting waves.