Yaglom Equations Research Articles

In-situ observations of the three-dimensional energy cascade rate and Yaglom flux in the Earth's magnetosheath

Abstract Measuring the energy cascade rate in space plasmas is a challenging task for several reasons. This quantity is (i) inherently three-dimensional (ii) scale-dependent, (iii) anisotropic in the interplanetary plasma, and (iv) requires measurements of plasma parameters in at least four points. Here, we show how three of these problems have been addressed by applying the novel Lag Polyhedra Derivative Ensemble (LPDE) technique to the Magnetospheric Multiscale (MMS) mission in the Earth's magnetosheath. In particular, this technique solves the full vectorial Yaglom equation, handling problem (i), does not require the assumption of isotropy, solving problem (iii), while the application to MMS addresses constraint (iv).

Weak formulation and scaling properties of energy fluxes in three-dimensional numerical turbulent Rayleigh–Bénard convection

We apply the weak formalism on the Boussinesq equations, to characterize scaling properties of the mean and the standard deviation of the potential, kinetic and viscous energy flux in very high resolution numerical simulations. The local Bolgiano-Oboukhov length $L_{BO}$ is investigated and it is found that its value may change of an order of magnitude through the domain, in agreement with previous results. We investigate the scale-by-scale averaged terms of the weak equations, which are a generalization of the Karman-Howarth-Monin and Yaglom equations. We have not found the classical Bolgiano-Oboukhov picture, but evidence of a mixture of Bolgiano-Oboukhov and Kolmogorov scalings. In particular, all the terms are compatible with a Bolgiano-Oboukhov local H\"older exponent for the temperature and a Kolmogorov 41 for the velocity. This behavior may be related to anisotropy and to the strong heterogeneity of the convective flow, reflected in the wide distribution of Bolgiano-Oboukhov local scales. The scale-by-scale analysis allows us also to compare the theoretical Bolgiano-Oboukhov length $L_{BO}$ computed from its definition with that empirically extracted through scalings obtained from weak analysis. The key result of the work is to show that the analysis of local weak formulation of the problem is powerful to characterize the fluctuation properties.

Resolved and subgrid dynamics of Rayleigh–Bénard convection

In this work we present and demonstrate the reliability of a theoretical framework for the study of thermally driven turbulence. It consists of scale-by-scale budget equations for the second-order velocity and temperature structure functions and their limiting cases, represented by the turbulent kinetic energy and temperature variance budgets. This framework represents an extension of the classical Kolmogorov and Yaglom equations to inhomogeneous and anisotropic flows, and allows for a novel assessment of the turbulent processes occurring at different scales and locations in the fluid domain. Two relevant characteristic scales, $\ell _{c}^{u}$ for the velocity field and $\ell _{c}^{\unicode[STIX]{x1D703}}$ for the temperature field, are identified. These variables separate the space of scales into a quasi-homogeneous range, characterized by turbulent kinetic energy and temperature variance cascades towards dissipation, and an inhomogeneity-dominated range, where the production and the transport in physical space are important. This theoretical framework is then extended to the context of large-eddy simulation to quantify the effect of a low-pass filtering operation on both resolved and subgrid dynamics of turbulent Rayleigh–Bénard convection. It consists of single-point and scale-by-scale budget equations for the filtered velocity and temperature fields. To evaluate the effect of the filter length $\ell _{F}$ on the resolved and subgrid dynamics, the velocity and temperature fields obtained from a direct numerical simulation are split into filtered and residual components using a spectral cutoff filter. It is found that when $\ell _{F}$ is smaller than the minimum values of the cross-over scales given by $\ell _{c,min}^{\unicode[STIX]{x1D703}\ast }=\ell _{c,min}^{\unicode[STIX]{x1D703}}Nu/H=0.8$, the resolved processes correspond to the exact ones, except for a depletion of viscous and thermal dissipations, and the only role of the subgrid scales is to drain turbulent kinetic energy and temperature variance to dissipate them. On the other hand, the resolved dynamics is much poorer in the near-wall region and the effects of the subgrid scales are more complex for filter lengths of the order of $\ell _{F}\approx 3\ell _{c,min}^{\unicode[STIX]{x1D703}}$ or larger. This study suggests that classic eddy-viscosity/diffusivity models employed in large-eddy simulation may suffer from some limitations for large filter lengths, and that alternative closures should be considered to account for the inhomogeneous processes at subgrid level. Moreover, the theoretical framework based on the filtered Kolmogorov and Yaglom equations may represent a valuable tool for future assessments of the subgrid-scale models.

Generalised scale-by-scale energy-budget equations and large-eddy simulations of anisotropic scalar turbulence at various Schmidt numbers

A necessary condition for the accurate prediction of turbulent flows using large-eddy simulation (LES) is the correct representation of energy transfer between the different scales of turbulence in the LES. For scalar turbulence, transfer of energy between turbulent length scales is described by a transport equation for the second moment of the scalar increment. For homogeneous isotropic turbulence, the underlying equation is the well-known Yaglom equation. In the present work, we study the turbulent mixing of a passive scalar with an imposed mean gradient by homogeneous isotropic turbulence. Both direct numerical simulations (DNS) and LES are performed for this configuration at various Schmidt numbers, ranging from 0.11 to 5.56. As the assumptions made in the derivation of the Yaglom equation are violated for the case considered here, a generalised Yaglom equation accounting for anisotropic effects, induced by the mean gradient, is derived in this work. This equation can be interpreted as a scale-by-scale energy-budget equation, as it relates at a certain scale r terms representing the production, turbulent transport, diffusive transport and dissipation of scalar energy. The equation is evaluated for the conducted DNS, followed by a discussion of physical effects present at different scales for various Schmidt numbers. For an analysis of the energy transfer in LES, a generalised Yaglom equation for the second moment of the filtered scalar increment is derived. In this equation, new terms appear due to the interaction between resolved and unresolved scales. In an a-priori test, this filtered energy-budget equation is evaluated by means of explicitly filtered DNS data. In addition, LES calculations of the same configuration are performed, and the energy budget as well as the different terms are thereby analysed in an a-posteriori test. It is shown that LES using an eddy viscosity model is able to fulfil the generalised filtered Yaglom equation for the present configuration. Further, the dependence of the terms appearing in the filtered energy-budget equation on varying Schmidt numbers is discussed.

Numerical model of a turbulent flow behind a heated grid in a wind tunnel

A closure of the Corrsin equation is performed using the gradient hypothesis connecting a third-order mixed correlation moment with a second-order two-point correlation function of a passive scalar field. A numerical model of the locally isotropic turbulence is constructed based on the closed system of the Kolmogorov and Yaglom equations. On the assumption of constant Loitsiansky and Corrsin invariants, a self-similar solution of the Corrsin equation is constructed corresponding to infinitely large Reynolds and Peclet numbers. A numerical model of the turbulence dynamics and temperature fluctuations behind a heated grid in a wind tunnel is constructed based on the closed Karman-Howarth and Corrsin equations.

On Corrsin equation closure

Corrsin equation closure is done using the gradient hypothesis relating a two-point third-order correlation moment to a two-point second-order correlation function of a passive scalar field. A numerical locally isotropic turbulence model based on a closed system of Kolmogorov and Yaglom equations is constructed. A similarity solution of the Corrsin equation, which corresponds to infinitely high Reynolds and Peclet numbers, is constructed under assumption of constancy of Corrsin and Loitsiansky invariants. A numerical model of turbulence dynamics and temperature fluctuations behind a heated grid in a wind tunnel, which is based on Karman-Howarth and Corrsin closed equations, is developed.

Anisotropy, inhomogeneity and inertial-range scalings in turbulent convection

This paper provides a detailed study of turbulent statistics and scale-by-scale budgets in turbulent Rayleigh–Bénard convection. It aims at testing the applicability of Kolmogorov and Bolgiano theories in the case of turbulent convection and at improving the understanding of the underlying inertial-range scalings, for which a general agreement is still lacking. Particular emphasis is laid on anisotropic and inhomogeneous effects, which are often observed in turbulent convection between two differentially heated plates. For this purpose, the SO(3) decomposition of structure functions and a method of description of inhomogeneities are used to derive inhomogeneous and anisotropic generalizations of Kolmogorov and Yaglom equations applying to Rayleigh–Bénard convection, which can be extended easily to other types of anisotropic and/or inhomogeneous flows. The various contributions to these equations are computed in and off the central plane of a convection cell using data produced by a direct numerical simulation of turbulent Boussinesq convection at , thus preventing Kolmogorov scalings from showing up in convective flows at lower Rayleigh numbers.

Large eddy simulation of scalar turbulence using a new subgrid eddy diffusivity model

A new subgrid eddy diffusivity model is used for large eddy simulation of scalar turbulence. The new model is based on the modified Yaglom equation, i.e., the dynamical equation for the structure function of resolved-scale scalar fluctuations, by which the scalar transport between resolved grid scale and unresolved subgrid scale can be described exactly. The new model has been applied in the large eddy simulations of scalar transport in both decaying isotropic turbulence and fully developed turbulent channel flow. By the comparison with Smagorinsky model and the dynamical model, it is found that the new model can achieve similar accuracy as the dynamical model, and can reduce the computational cost about 20% in terms of CPU time relative to the dynamical model.

Dissipation rate correction methods

Dissipation rates of the turbulent kinetic energy and of the scalar variance are underestimated when the measurement resolution of the small scales of a turbulent flow field are insufficient. Results are presented of experiments conducted in a salt-stratified water tunnel (Schmidt number ∼700). Dissipation rates are determined to be underestimated, and thus correction techniques based on velocity structure functions and mixed-moment functions are proposed. Dissipation rates in laboratory experiments of shear-free, grid-generated turbulence are determined from balance calculations of the kinetic energy and scalar variance evolution equations. Comparisons between the structure function and balance estimates of dissipation show that the corrections are O(1) for the kinetic energy dissipation rate, and are O(100) for the scalar variance dissipation rate. This difference is due to the lack of resolution down to the Batchelor scales that is required for a high Schmidt number flow. Simple correction functions based on microscale Reynolds numbers are developed for both turbulent kinetic energy and scalar variance dissipation rates. Application of the technique to the results of laboratory experiments of density stratified turbulence, sheared turbulence, and sheared density stratified turbulence yields successful corrections. It is also demonstrated that the Karman–Howarth equality (and the analogous Yaglom equation) that relates second and third-order structure functions to dissipation rates is valid for both unstrained (decaying grid-generated turbulence) and density stratified and sheared turbulence at least up to the magnitudes of strains of the current experiments Nt∼10, St∼10, respectively. This is helpful for it allows the use of these equations in the analysis of turbulence even when the large scale background profiles of velocity and scalar are unknown.

Two-dimensional cascades and mixing: a physical space approach

We develop an analysis of the two-dimensional cascade of a tracer (passive or active Lagrangian-conserved scalar), locally in space and time, and establish connections with the modelling of turbulent mixing. We define a local scale-to-scale flux of tracer variance based on the dynamics of tracer increments. This flux reduces at small scales to the production or destruction rate of tracer gradients by stirring as a function of their local orientation with respect to the compressional axis of the strain-rate tensor. The local detailed budget of tracer variance on which this approach is based is compared to the global statistical budget expressed by Yaglom's equation. The spatial pattern of the local transfers produced by a numerical simulation as well as their statistical distribution are discussed.We then address the problem of the parameterization of turbulent mixing. We consider an anisotropic tensor diffusivity proportional to the velocity gradients. In this model the tracer dissipation involves the axes of the strain-rate tensor and we shall refer to it as strain diffusivity. We show analytically that it locally matches the scale-to-scale flux through the cutoff scale. This matching is studied numerically in decaying two-dimensional turbulence. A comparison is made with eddy diffusivity and hyperdiffusivity. The presence of a numerical instability and ways to suppress it are discussed from numerical and fundamental points of view.We consider the special case of vorticity, an active scalar in two dimensions. When applied to vorticity, models affect the energy budget. The two-dimensional inverse energy cascade requires that parameterizations conserve energy and we show that strain diffusivity conserves energy. We finally study the sensitivity of the large scales of the flow to the operator used on vorticity in forced stationary simulations. Strain diffusivity is found to produce more realistic spectral features than hyperviscosity.

Effect of gradient production on scalar fluctuations in decaying grid turbulence.

A generalized form of Yaglom's equation, in which the effect of turbulent production of temperature fluctuations is taken into account, is derived. The modified Yaglom equation (which extends the work of Danaila et al. [J. Fluid Mech. 391, 359 (1999)] to incorporate the effects of decay and production) is compared with experimental data. The flow under consideration is decaying grid turbulence with an imposed mean temperature gradient (Mydlarski and Warhaft [J. Fluid Mech. 358, 135 (1998)]). The agreement between theory and experiment is good and the effects of nonhomogeneities in the flow are small.

Planar isotropy of passive scalar turbulent mixing with a mean perpendicular gradient.

A recently proposed evolution equation [Vaienti et al., Physica D 85, 405 (1994)] for the probability density functions (PDF's) of turbulent passive scalar increments obtained under the assumptions of fully three-dimensional homogeneity and isotropy is submitted to validation using direct numerical simulation (DNS) results of the mixing of a passive scalar with a nonzero mean gradient by a homogeneous and isotropic turbulent velocity field. It is shown that this approach leads to a quantitatively correct balance between the different terms of the equation, in a plane perpendicular to the mean gradient, at small scales and at large Péclet number. A weaker assumption of homogeneity and isotropy restricted to the plane normal to the mean gradient is then considered to derive an equation describing the evolution of the PDF's as a function of the spatial scale and the scalar increments. A very good agreement between the theory and the DNS data is obtained at all scales. As a particular case of the theory, we derive a generalized form for the well-known Yaglom equation (the isotropic relation between the second-order moments for temperature increments and the third-order velocity-temperature mixed moments). This approach allows us to determine quantitatively how the integral scale properties influence the properties of mixing throughout the whole range of scales. In the simple configuration considered here, the PDF's of the scalar increments perpendicular to the mean gradient can be theoretically described once the sources of inhomogeneity and anisotropy at large scales are correctly taken into account.

A generalization of Yaglom's equation which accounts for the large-scale forcing in heated decaying turbulence

In most real or numerically simulated turbulent flows, the energy dissipated at small scales is equal to that injected at very large scales, which are anisotropic. Despite this injection-scale anisotropy, one generally expects the inertial-range scales to be locally isotropic. For moderate Reynolds numbers, the isotropic relations between second-order and third-order moments for temperature (Yaglom's equation) or velocity increments (Kolmogorov's equation) are not respected, reflecting a non-negligible correlation between the scales responsible for the injection, the transfer and the dissipation of energy. In order to shed some light on the influence of the large scales on inertial-range properties, a generalization of Yaglom's equation is deduced and tested, in heated grid turbulence (Rλ=66). In this case, the main phenomenon responsible for the non-universal inertial-range behaviour is the non-stationarity of the second-order moments, acting as a negative production term.

Predictions of Small-Scale Statistics for a Passive Scalar in Turbulent Mixing

This Letter presents some quantitative predictions for the small-scale statistics for a passive scalar mixed in a fully turbulent flow. Our method uses an equation [S. Vaienti et al. Physica (Amsterdam) 73D, 99 (1994)] which controls the evolution through the scales of the probability density function of temperature increments. This equation, which needs some closure as well as initial data, uses results from direct numerical simulations or from experiments. For the dissipative and small inertial range scales, our predictions are in excellent agreement with the measured or numerically calculated probability density functions. The associated Yaglom equation is also well verified for the same range of scales.

Refined similarity hypotheses for passive scalars mixed by turbulence

In analogy with Kolmogorov's refined similarity hypotheses for the velocity field, two hypotheses are stated for passive scalar fields mixed by high-Reynolds-number turbulence. A ‘refined’ Yaglom equation is derived under the new assumption of local isotropy in pure ensembles, which is stronger than the usual assumption of local isotropy but weaker than the isotropy of the large scale. The new theoretical result is shown to be consistent with the hypotheses of refined similarity for passive scalars. These hypotheses are approximately verified by experimental data on temperature fluctuations obtained (in air) at moderate Reynolds numbers in the wake of a heated cylinder. The fact that the refined similarity hypotheses are stated for high Reynolds (and Péclet) numbers, but verified at moderate Reynolds (and Péclet) numbers suggests that these hypotheses are not sufficiently sensitive tests of universality. It is conjectured that possible departures from universality are hidden by the process of taking conditional expectations.