Eddy-damped Quasi-normal Markovian Model Research Articles

Entropy budget and coherent structures associated with a spectral closure model of turbulence

We ‘derive’ the eddy-damped quasi-normal Markovian model (EDQNM) by a method that replaces the exact equation for the Fourier phases with a solvable stochastic model, and we analyse the entropy budget of the EDQNM. We show that a quantity that appears in the probability distribution of the phases may be interpreted as the rate at which entropy is transferred from the Fourier phases to the Fourier amplitudes. In this interpretation, the decrease in phase entropy is associated with the formation of structures in the flow, and the increase of amplitude entropy is associated with the spreading of the energy spectrum in wavenumber space. We use Monte Carlo methods to sample the probability distribution of the phases predicted by our theory. This distribution contains a single adjustable parameter that corresponds to the triad correlation time in the EDQNM. Flow structures form as the triad correlation time becomes very large, but the structures take the form of vorticity quadrupoles that do not resemble the monopoles and dipoles that are actually observed.

Challenging Mix Models on Transients to Self-Similarity of Unstably Stratified Homogeneous Turbulence

The present work aims at expanding the set of buoyancy-driven unstable reference flows—a critical ingredient in the development of turbulence models—by considering the recently introduced “Unstably Stratified Homogeneous Turbulence” (USHT) in both its self-similar and transient regimes. The previously established accuracy of an anisotropic Eddy-Damped Quasi-Normal Markovian Model (EDQNM) on the USHT has allowed us to: (i) build a data set of well defined transient flows from Homogeneous Isotropic Turbulence (HIT) to late-time self-similar USHT and (ii) on this basis, calibrate, validate, and compare three common Reynolds-Averaged Navier–Stokes (RANS) mixing models (two-equation, Reynolds stress, and two-fluid). The model calibrations were performed on the self-similar flows constrained by predefined long range correlations (Saffman or Batchelor type). Then, with fixed constants, validations were carried out over the various transients defined by the initial Froude number and mixing intensity. Significant differences between the models are observed, but none of them can accurately capture all of the transient regimes at once. Closer inspection of the various model responses hints at possible routes for their improvement.

Self-truncation and scaling in Euler-Voigt-α and related fluid models.

A generalization of the 3D Euler-Voigt-α model is obtained by introducing derivatives of arbitrary order β (instead of 2) in the Helmholtz operator. The β→∞ limit is shown to correspond to Galerkin truncation of the Euler equation. Direct numerical simulations (DNS) of the model are performed with resolutions up to 2048(3) and Taylor-Green initial data. DNS performed at large β demonstrate that this simple classical hydrodynamical model presents a self-truncation behavior, similar to that previously observed for the Gross-Pitaeveskii equation in Krstulovic and Brachet [Phys. Rev. Lett. 106, 115303 (2011)]. The self-truncation regime of the generalized model is shown to reproduce the behavior of the truncated Euler equation demonstrated in Cichowlas et al. [Phys. Rev. Lett. 95, 264502 (2005)]. The long-time growth of the self-truncation wave number k(st) appears to be self-similar. Two related α-Voigt versions of the eddy-damped quasinormal Markovian model and the Leith model are introduced. These simplified theoretical models are shown to reasonably reproduce intermediate time DNS results. The values of the self-similar exponents of these models are found analytically.

An adaptive numerical method for solving EDQNM equations for the analysis of long-time decay of isotropic turbulence

A new numerical formulation for the Eddy-Damped Quasi-Normal Markovian (EDQNM) model is proposed in the present paper. This formulation is based on an adaptive procedure that progressively modifies the spectral mesh at the large scales, forcing a resolution requirement set by the user.The resulting adaptive numerical method for the EDQNM model has been systematically tested by comparison with the classical model, covering a wide range of initial conditions. In particular, the sensitivity of the results to the initial Reynolds number Reλ and to the slope of the energy spectrum at the large scales σ (E(k→0)∝kσ) has been investigated. For all the initial conditions prescribed, the adaptive numerical method recovers exactly the same solution of the classical version. This result has been observed in the analysis of the main statistical quantities of interest, such as the turbulent kinetic energy K and the energy dissipation rate ε. The same conclusions have been drawn by the direct comparison of the energy spectra E.An extension of the adaptive numerical method for the analysis of sheared/rotating turbulence is as well proposed and successfully assessed.The numerical algorithm proves to be progressively more efficient when long-time simulations are performed. In particular, a reduction up to one order of magnitude in the computational resources required is observed. Due to its efficiency and precision, the adaptive formulation of the EDQNM model promises to be an optimal tool to be blended with the methods of the Uncertainty Quantification (UQ) theory, in order to provide new insights about isotropic turbulence decay.

Pressure statistics in self-similar freely decaying isotropic turbulence

AbstractThe time evolution of pressure statistics in freely decaying homogeneous isotropic turbulence (HIT) is investigated via eddy-damped quasi-normal Markovian (EDQNM) computations. The present results show that the time decay rate of pressure-based statistical quantities, such as pressure variance and pressure gradient variance, are sensitive to the breakdown of permanence of large eddies. New formulae for the associated time-decay exponents are proposed, which extend previous relations proposed in Lesieur, Ossia & Metais (Phys. Fluids, vol. 11, 1999, p. 1535). Particular attention is paid to finite-Reynolds-number (FRN) effects on the pressure spectrum and pressure statistics. The results also suggest that $R{e}_{\lambda } = O(1{0}^{4} )$ must be considered to observe a one-decade inertial range in the pressure spectrum with Kolmogorov $- 7/ 3$ scaling. This threshold value is larger than almost all existing direct numerical simulation (DNS) and experimental data, justifying the discussion about other possible scaling laws. The $- 5/ 3$ slope reported in some DNS data is also recovered by the EDQNM model, but it is observed to be a low-Reynolds-number effect. Another important result is that FRN effects yield a departure from asymptotic theoretical behaviours which appear similar to some effects attributed to intermittency by most authors. This is exemplified by the ratio between pressure-based and velocity-based Taylor microscales. Therefore, high-Reynolds-number DNS or experiments such that $R{e}_{\lambda } = O(1{0}^{4} )$ would be required in order to remove FRN effects and to analyse pure intermittency effects.

Reynolds number effect on the velocity increment skewness in isotropic turbulence

Second and third order longitudinal structure functions and wavenumber spectra of isotropic turbulence are computed using the eddy-damped quasi-normal Markovian model (EDQNM) and compared to results of the multifractal formalism. It is shown that both the multifractal model and EDQNM give power-law corrections to the inertial range scaling of the velocity increment skewness. For the multifractal formalism, this is an intermittency correction that persists at any high Reynolds number. For EDQNM, this correction is a finite Reynolds number effect, and it is shown that very high Reynolds numbers are needed for this correction to become insignificant with respect to intermittency corrections. Furthermore, the two approaches yield realistic behavior of second and third order statistics of the velocity fluctuations in the dissipative and near-dissipative ranges. Similarities and differences are highlighted, in particular, the Reynolds number dependence.

Quasi-static magnetohydrodynamic turbulence at high Reynolds number

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.

Stochastic Shell Model for Turbulent Mixing of Multiple Scalars with Mean Gradients and Differential Diffusion

In this paper, we develop a shell model for the velocity and scalar concentrations that, by design, is consistent with the eddy damped quasi-normal Markovian (EDQNM) model for multiple mixing scalars. We review the realizable form of the EDQNM model derived by Ulitsky and Collins (J Fluid Mech 412:303–329, 2000), which forms the basis for the shell model. The equations governing the velocity and scalar within each shell are stochastic ordinary differential equations with drift and diffusion terms chosen so that the velocity variance, velocity–scalar cross correlations, and scalar–scalar cross correlations within each shell precisely match the EDQNM model predictions. Consequently, shell averages can be thought of as a representation of the discrete three-dimensional spectrum. An advantage the shell model has over the original EDQNM equations is that the sum of each realization over the shells is a model for the fine-grained, joint velocity/scalar probability density function (PDF). Indeed, this provides some of the motivation for the development of the model. We cannot exploit this feature in the present study of the mixing of two scalars with uniform mean gradients, as the PDF is a joint Gaussian throughout (and hence the correlation matrix completely defines the distribution). The model is capable of predicting Lagrangian correlation functions for the scalar, scalar dissipation and velocity. We find the predictions of the model are in good qualitative agreement with direct numerical simulations by Yeung (J Fluid Mech 427:241–274, 2001). Eventually we will apply the shell model to scalars that are initially highly non-Gaussian (e.g., double delta function) and observe the relaxation towards a Gaussian. As the shell model contains information on the spectral distribution of the scalar field, the relaxation rate will depend upon the length and time scales of the turbulence and the scalar fields, as well as the molecular diffusivities of the species. The full capabilities of the PDF predictions of the model will be the subject of a future publication.

Large-eddy simulation of very large kinetic and magnetic Reynolds number isotropic magnetohydrodynamic turbulence using a spectral subgrid model

A spectral eddy viscosity and magnetic resistivity subgrid-scale model based on the eddy-damped quasi-normal Markovian (EDQNM) kinetic and magnetic energy transfers is used in large-eddy simulation (LES) of asymptotically large kinetic and magnetic Reynolds number magnetohydrodynamic (MHD) turbulence. The model is assessed a posteriori on three-dimensional, incompressible, isotropic, nonhelical, freely decaying MHD turbulence. Using LES initialized with spectra such that the Alfvén ratio of kinetic to magnetic energy equals unity, it is shown that the kinetic energy and magnetic energy spectra exhibit k−5∕3 Kolmogorov inertial subrange scalings consistent with the EDQNM model.

A spectral study of differential diffusion of passive scalars in isotropic turbulence

In a companion paper, Ulitsky & Collins (2000) applied the eddy-damped quasi-normal Markovian (EDQNM) turbulence theory to the mixing of two inert passive scalars with different diffusivities in stationary isotropic turbulence. Their paper showed that a rigorous application of the EDQNM approximation leads to a scalar covariance spectrum that violates the Cauchy–Schwartz inequality over a range of wavenumbers. The violation results from the improper functionality of the inverse diffusive time scales that arise from the Markovianization of the time evolution of the triple correlations. The modified inverse time scale they proposed eliminates this problem and allows meaningful predictions of the scalar covariance spectrum both with and without a uniform mean gradient.This study uses the modified EDQNM model to investigate the spectral dynamics of differential diffusion. Consistent with recent DNS results by Yeung (1996), we observe that whereas spectral transfer is predominantly from low to high wavenumbers, spectral incoherence, being of molecular origin, originates at high wavenumbers and is transferred in the opposite direction by the advective terms. Quantitative comparisons between the EDQNM model and the DNS show good agreement. In addition, the model is shown to give excellent estimates for the dissipation coefficient defined by Yeung (1998).We show that the EDQNM scalar covariance spectrum for two scalars with different molecular diffusivities can be approximated by the EDQNM autocorrelation spectrum for a scalar with molecular diffusivity equal to the arithmetic mean of the first two scalars. The result is exact for the case of an isotropic scalar and is shown to be a very good approximation for the scalar with a uniform mean gradient. We then exploit this relationship to derive a simple formula for the correlation coefficient of two differentially diffusing scalars as a function of their two Schmidt numbers and the turbulent Reynolds number. A comparison of the formula with the EDQNM model shows the model predicts the correct Reynolds number scaling and qualitatively predicts the dependence on the Schmidt numbers.To investigate the degree of local versus non-local transfer of the scalar covariance spectrum, we divided the energy spectrum into three ranges corresponding to the energy-containing eddies, the inertial range, and the dissipation range. Then, by conditioning the scalar transfer on the energy contained within each of the three ranges, we have determined that the transfer process is dominated first by local interactions (local transfer) followed by non-local interactions leading to local transfer. Non-local interactions leading to non-local transfer are found to be significant at the higher wavenumbers. This result has important implications for defining simpler spectral models that potentially can be applied to more complex engineering flows.

Comparison between a spectral and probability density function model for turbulent reacting flows

This study compares the performance of a newly developed spectral model based on the eddy damped quasi-normal Markovian (EDQNM) theory with a standard probability density function (PDF) model for the case of two initially unmixed reactants undergoing a finite-rate bimolecular reaction. The two models were chosen because they involve complementary treatments of the nonlinearities and mixing terms. That is, nonlinearities are exactly treated in the PDF and mixing is modeled, whereas the opposite is true for EDQNM. The predictions of the two models are compared to direct numerical simulations. The results show that the PDF model is capable of describing the mixing of the major species reasonably well, but fails to describe the correlations between the reactants and the products even qualitatively. This suggests that the mixing model in the PDF is adequate for describing mixing between major species, but is incapable of describing mixing of the more spatially segregated product species. The EDQNM model does a slightly better job of describing the mixing of reactant species and a much better job of describing mixing of the product species. Presumably the improvement is associated with the more accurate description of the interscale dynamics that are especially important for the segregated products. The implication is that a model that combines the strengths of the EDQNM for describing mixing and the PDF for describing the nonlinearities would yield the best of both worlds.

Energy transfer in rotating turbulence

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.

Remarks on statistical mechanics of discrete and continuous media with application to the magnetohydrodynamic dynamo

In this paper commemorating the 60th birthday of David Montgomery, issues tantalizing to me that evolved from my collaborations with him are highlighted: sustainment of the magnetohydrodynamic dynamo, relationships between discrete and continuous models of media, and closures for nonlinear theories of discrete and continuous media. In particular, it is speculated that a tractable statistical model of a magnetohydrodynamic dynamo may arise from a proper formulation of a solenoidal expansion basis for the magnetic and velocity fields in the context of an appropriate closure, such as the eddy-damped quasinormal Markovian model.

EDQNM model of a passive scalar with a uniform mean gradient

Dynamic equations for the scalar autocorrelation and scalar-velocity cross correlation spectra have been derived for a passive scalar with a uniform mean gradient using the Eddy Damped Quasi Normal Markovian (EDQNM) theory. The presence of a mean gradient in the scalar field makes all correlations involving the scalar axisymmetric with respect to the axis pointing in the direction of the mean gradient. Equivalently, all scalar spectra will be functions of the wave number k and the cosine of the azimuthal angle designated as μ. In spite of this complication, it is shown that the cross correlation vector can be completely characterized by a single scalar function Q(k). The scalar autocorrelation spectrum, in contrast, has an unknown dependence on μ. However, this dependency can be expressed as an infinite sum of Legendre polynomials of μ, as first suggested by Herring [Phys. Fluids 17, 859 (1974)]. Furthermore, since the scalar field is initially zero, terms beyond the second order of the Legendre expansion are shown to be exactly zero. The energy, scalar autocorrelation, and scalar-velocity cross correlation were solved numerically from the EDQNM equations and compared to results from direct numerical simulations. The results show that the EDQNM theory is effective in describing single-point and spectral statistics of a passive scalar in the presence of a mean gradient.

Decaying stratified turbulence: comparison between a two-point closure EDQNM model and direct numerical simulations

A new statistical two-point closure EDQNM (Eddy Damped Quasi Normal Markovian) model for axisymmetric stratified turbulence is presented. This model takes into account the detailed anisotropic structure of the flow, including angular dependence, and its closure assumption may involve explicit effects of the stratification. In the present paper, a simplified closure assumption is used, in which these explicit effects are not taken into account. The model is tested against three-dimensional direct numerical simulations (DNS) of decaying homogeneous stratified turbulence. A very good agreement is found between DNS results and EDQNM predictions, when energy transfers between the different modes of motion are considered. However, transfers among wavenumbers are insufficiently damped by the EDQNM model, because of the simplified closure assumption, thus yielding a somewhat less satisfactory agreement. An interesting prediction of the EDQNM model is the existence of a ‘directional’ anisotropy at all scales of motion, to the smallest: the level of the total (kinetic + potential) energy density spectrum E ( k , θ k ) varies with the angle θ k that the wavenumber vector k makes with the vertical.

Stellar turbulent convection - A new model and applications

Improvements of the mixing-length theory (MLT) of turbulent convection in stellar atmospheres are developed theoretically. It is pointed out that inaccuracies are introduced into MLT by the approximating assumptions of a single large eddy (rather than many eddies of different sizes) and of incompressibility. In the proposed new model, the full spectrum of turbulent eddies is determined using more recent turbulence models (e.g., the eddy-damped quasi-normal Markovian model of Orszag, 1977), and a new formula for the convective flux is obtained which gives values up to 10 times greater than those of the MLT at high convective efficiencies. The problem of compressibility is addressed by adding one of two new expressions (one with no free parameters) for the mixing length. Numerical results from simulations of a solar-type star and a 0.8-solar-mass globular-cluster star are presented in tables and graphs and discussed in detail; the agreement with observations is found to be better than with the MLT.

Simulations of scalar mixing in grid turbulence using an eddy-damped closure model

A physical-space eddy-damped quasinormal Markovian model for the two-point scalar correlation function is used to simulate wind tunnel experiments on scalar mixing. The simulations closely approximate experimentally observed trends of scalar variance and length scales in both heated-grid and heated-screen experiments. The simplicity of the model should enable its effective use in modeling the analogous turbulent mixing terms in the two-point scalar probability density function equations.

Subgrid-Scale Modeling of Enstrophy Transfer in Two-Dimensional Turbulence

Abstract A method of parameterization of the energy and enstropy transfers involving subgrid scales is proposed in order to predict the time evolution of the energy spectrum at larger scales. The method is based on the strong non-localness of enstrophy transfer within the enstrophy inertial range of turbulence in two dimensions. It is applied first to a master equation derived from an eddy-damped quasi-normal Markovian model. Comparisons with reference calculations including all the scales up to the dissipation range show that the method simulates the enstrophy inertial range accurately up to the cutoff wavenumber. It is then generalized to direct simulation of the two-dimensional Navier-Stokes equation: the inertial range is again accurately simulated.