In this investigation, a data-driven turbulence closure framework is introduced and deployed for the sub-grid modelling of Kraichnan turbulence. The novelty of the proposed method lies in the fact that snapshots from high-fidelity numerical data are used to inform artificial neural networks for predicting the turbulence source term through localized gridresolved information. In particular, our proposed methodology successfully establishes a map between inputs given by stencils of the vorticity and the streamfunction along with information from two well-known eddy-viscosity kernels. Through this we predict the sub-grid vorticity forcing in a temporally and spatially dynamic fashion. Our study is both a-priori and a-posteriori in nature. In the former, we present an extensive hyper-parameter optimization analysis in addition to learning quantification through probability density function based validation of sub-grid predictions. In the latter, we analyse the performance of our framework for flow evolution in a classical decaying twodimensional turbulence test case in the presence of errors related to temporal and spatial discretization. Statistical assessments in the form of angle-averaged kinetic energy spectra demonstrate the promise of the proposed methodology for sub-grid quantity inference. In addition, it is also observed that some measure of a-posteriori error must be considered during optimal model selection for greater accuracy. The results in this article thus represent a promising development in the formalization of a framework for generation of heuristic-free turbulence closures from data.

The scaling properties of one- and two-point statistics of the acceleration, pressure, and pressure gradient are studied in incompressible isotropic turbulence by direct numerical simulation. Ensemble-averaged Taylor-scale Reynolds numbers (Rλ) are up to about 230 on grids from 323 to 5123. From about Rλ 40 onwards the acceleration variance normalized by Kolmogorov variables is found to increase as Rλ1/2. This nonuniversal behavior is traced to the dominant irrotational pressure gradient contributions to the acceleration (whereas the much weaker solenoidal viscous part is universal). Longitudinal and transverse two-point correlations of the pressure gradient differ according to kinematic constraints, but both (especially the latter) extend over distances of intermediate scale size large compared to the Kolmogorov scale. These extended-range properties essentially provide the Eulerian mechanism whereby (as found in recent work) the accelerations of a pair of fluid particles can remain significantly correlated for relatively long periods of time even as they move apart from each other. Although a limited inertial range is attained in the energy spectrum, little evidence for classical inertial scaling is found in acceleration correlations and pressure structure functions. The probability density function (PDF) of pressure fluctuations has negatively skewed tails that exhibit a stretched-exponential form. Pressure gradient statistics show a rapid increase in intermittency with Reynolds number, characterized by widening tails in the PDF and large flatness factors. The practicality of computing acceleration correlations from velocity structure functions is also assessed using direct numerical simulations (DNS); within some resolution limitations good agreement is obtained with experimental data in grid turbulence at comparable Reynolds number.

The physical mechanisms underlying the dynamics of the dissipation of passive scalar fluctuations with a uniform mean gradient in stationary isotropic turbulence are studied using data from direct numerical simulations (DNS), at grid resolutions up to 512 3 . The ensemble-averaged Taylor-scale Reynolds number is up to about 240 and the Schmidt number is from ⅛ to 1. Special attention is given to statistics conditioned upon the energy dissipation rate because of their important role in the Lagrangian spectral relaxation (LSR) model of turbulent mixing. In general, the dominant physical processes are those of nonlinear amplification by strain rate fluctuations, and destruction by molecular diffusivity. Scalar dissipation tends to form elongated structures in space, with only a limited overlap with zones of intense energy dissipation. Scalar gradient fluctuations are preferentially aligned with the direction of most compressive strain rate, especially in regions of high energy dissipation. Both the nature of this alignment and the timescale of the resulting scalar gradient amplification appear to be nearly universal in regard to Reynolds and Schmidt numbers. Most of the terms appearing in the budget equation for conditional scalar dissipation show neutral behaviour at low energy dissipation but increased magnitudes at high energy dissipation. Although homogeneity requires that transport terms have a zero unconditional average, conditional molecular transport is found to be significant, especially at lower Reynolds or Schmidt numbers within the simulation data range. The physical insights obtained from DNS are used for a priori testing and development of the LSR model. In particular, based on the DNS data, improved functional forms are introduced for several model coefficients which were previously taken as constants. Similar improvements including new closure schemes for specific terms are also achieved for the modelling of conditional scalar variance. The physical mechanisms underlying the dynamics of the dissipation of passive scalar fluctuations with a uniform mean gradient in stationary isotropic turbulence are studied using data from direct numerical simulations (DNS), at grid resolutions up to 5123 . The ensemble-averaged Taylor-scale Reynolds number is up to about 240 and the Schmidt number is from 1 8 to 1. Special attention is given to statistics conditioned upon the energy dissipation rate because of their important role in the Lagrangian spectral relaxation (LSR) model of turbulent mixing. In general, the dominant physical processes are those of nonlinear amplification by strain rate fluctuations, and destruction by molecular diffusivity. Scalar dissipation tends to form elongated structures in space, with only a limited overlap with zones of intense energy dissipation. Scalar gradient fluctuations are preferentially aligned with the direction of most compressive strain rate, especially in regions of high energy dissipation. Both the nature of this alignment and the timescale of the resulting sc...

The properties of acceleration fluctuations in isotropic turbulence are studied in direct numerical simulations ͑DNS͒ by decomposing the acceleration as the sum of local and convective contributions ͑a L ϭ‫ץ‬u/‫ץ‬t and a C ϭu•ٌu͒, or alternatively as the sum of irrotational and solenoidal contributions ͓a I ϭϪٌ( p/) and a S ϭٌ 2 u͔. The main emphasis is on the nature of the mutual cancellation between a L and a C which must occur in order for the acceleration ͑a͒ to be small as predicted by the ''random Taylor hypothesis'' ͓Tennekes, J. Fluid Mech. 67, 561 ͑1975͔͒ of small eddies in turbulent flow being passively ''swept'' past a stationary Eulerian observer. Results at Taylor-scale Reynolds number up to 240 show that the random-Taylor scenario ͗a 2 ͘Ӷ͗a C 2 ͘ Ϸ͗a L 2 ͘, accompanied by strong antialignment between the vectors a L and a C , is indeed increasingly valid at higher Reynolds number. Mutual cancellation between a L and a C also leads to the solenoidal part of a being small compared to its irrotational part. Results for spectra in wave number space indicate that, at a given Reynolds number, the random Taylor hypothesis has greater validity at decreasing scale sizes. Finally, comparisons with DNS data in Gaussian random fields show that the mutual cancellation between a L and a C is essentially a kinematic effect, although the Reynolds number trends are made stronger by the dynamics implied in the Navier-Stokes equations.

In this paper we study acceleration statistics from laboratory measurements and direct numerical simulations in three-dimensional turbulence at Taylor-scale Reynolds numbers ranging from 38 to 1000. Using existing data, we show that at present it is not possible to infer the precise behavior of the unconditional acceleration variance in the large Reynolds number limit, since empirical functions satisfying both the Kolmogorov and refined Kolmogorov theories appear to fit the data equally well. We also present entirely new data for the acceleration covariance conditioned on the velocity, showing that these conditional statistics are strong functions of velocity, but that when scaled by the unconditional variance they are only weakly dependent on Reynolds number. For large values of the magnitude u of the conditioning velocity we speculate that the conditional covariance behaves like u 6 and show that this is qualitatively consistent with the stretched exponential tails of the unconditional acceleration probability density function ͑pdf͒. The conditional pdf is almost identical in shape to the unconditional pdf. From these conditional covariance data, we are able to calculate the conditional mean rate of change of the acceleration, and show that it is consistent with the drift term in second-order Lagrangian stochastic models of turbulent transport. We also calculate the correlation between the square of the acceleration and the square of the velocity, showing that it is small but not negligible.