Interscale Energy Transfer Research Articles

Interscale energy transfer in decaying turbulence and vorticity–strain-rate dynamics in grid-generated turbulence

For decaying homogeneous turbulence, we present two assumptions about the energy spectrum and one on the dissipation rate coefficient Cϵ which, in a high inlet/initial Reynolds number limit, imply that a wide range of wavenumbers exists where the interscale energy flux is dependent on inlet/initial Reynolds number, is negative (kinetic energy is on average transferred from small to high wavenumbers) and is independent of wavenumber but not necessarily of viscosity. Our assumptions about the energy spectrum are not unusual, one concerns the finite nature of the energy and the other its time dependence, and our assumption about Cϵ is inspired by recent wind tunnel and water channel measurements of turbulence generated by fractal and regular grids. We then present a direct numerical simulation of fractal-generated turbulence where the second-order structure function in time exhibits a well-defined 2/3 power law over more than a decade at a position close to the grid where the local Reynolds number Reλ is only about 30 and where there is neither average production of enstrophy nor of strain rate. The Q–R and Qs–Rs diagrams do not have their usual appearance at this position but develop it gradually as the flow progresses downstream and the wide 2/3 power law of the second-order structure function is eroded. It is believed that this is the first time that the spatial development of Q, R, Qs and Rs statistics is obtained for a spatially developing turbulent flow.

Nonlinear gravity-wave interactions in stratified turbulence

To investigate the dynamics of gravity waves in stratified Boussinesq flows, a model is derived that consists of all three-gravity-wave-mode interactions (the GGG model), excluding interactions involving the vortical mode. The GGG model is a natural extension of weak turbulence theory that accounts for exact three-gravity-wave resonances. The model is examined numerically by means of random, large-scale, high-frequency forcing. An immediate observation is a robust growth of the so-called vertically sheared horizontal flow (VSHF). In addition, there is a forward transfer of energy and equilibration of the nonzero-frequency (sometimes called “fast”) gravity-wave modes. These results show that gravity-wave-mode interactions by themselves are capable of systematic interscale energy transfer in a stratified fluid. Comparing numerical simulations of the GGG model and the full Boussinesq system, for the range of Froude numbers (Fr) considered (0.05 ≤ Fr ≤ 1), in both systems the VSHF is hardest to resolve. When adequately resolved, VSHF growth is more vigorous in the GGG model. Furthermore, a VSHF is observed to form in milder stratification scenarios in the GGG model than the full Boussinesq system. Finally, fully three-dimensional nonzero-frequency gravity-wave modes equilibrate in both systems and their scaling with vertical wavenumber follows similar power-laws. The slopes of the power-laws obtained depend on Fr and approach −2 (from above) at Fr = 0.05, which is the strongest stratification that can be properly resolved with our computational resources.

The role of directionality on the structure and dynamics of strongly anisotropic turbulent flows

Strong anisotropy in turbulent flows may be induced by body forces, Coriolis, buoyancy, Lorentz, and/or by large-scale gradients. These effects combined to the redistribution pressure terms are first identified by an angle dependence of the wave vector in Fourier space, the directionality. The resulting anisotropic structure is not taken into account in classical phenomenological theory, using essentially ‘isotropised’ dimensional analysis. Besides, it is generally hidden in practical engineering models by means of tuned constants, which may vary if the flow changes of nature. In this paper, different examples of anisotropic turbulence are revisited and compared to each other in order to shed light on fundamental aspects of this specific turbulence. To begin with, flows without energy production like rotating turbulence are considered. In this case, isotropy is broken by mean of third-order correlations in the equations. These correlations quantify the interscale energy transfer, and must be investigated at three-point, or triad by triad in Fourier space. This allows to account for the role of typical anisotropic frequency 2Ωcos θ k , with θ k the angle of the wave vector to the axis of rotation, and to simultaneously restore the role of phase coherence. We pursue the discussion with a second flow case, with production, quasi-static magnetohydrodynamics. This illustrates turbulence forced towards two-dimensional structure by an explicit Ohmic dissipation term. Linear dynamics displays an angle (called Moreau, or Shebalin) capable of reflecting the basic anisotropy in models as simple as . In the final phase of transition towards 2D structure, however, dynamics are essentially driven by third-order velocity correlations, and both successive linear and nonlinear phases yield counter-intuitive anisotropic results. The last case considered here is the turbulent mixing induced by a Rayleigh-Taylor instability. It is shown that anisotropy plays a central role in the dynamics of the mixing zone by means of an angular dimensionality parameter similar to the Moreau angle but for the density field, and appearing in a global model of buoyancy-drag equation.

Coarse-graining study of homogeneous and isotropic Hall magnetohydrodynamics turbulence

Inter-scale magnetic energy transfer associated with the Hall effects in homogeneous and isotropic magnetohydrodynamics (MHD) turbulence and the applicability of a Smagorinsky-type diffusive sub-grid-scale (SGS) model to Hall MHD turbulence are studied numerically. Low-pass filter analysis on magnetic energy transfer functions shows that their profiles change considerably when the cut-off wave number is comparable to the Taylor-scale wave number. It is shown that the Smagorinsky-type model is applicable to Hall MHD turbulence as a basic SGS model, while the Hall effects tend to overestimate the grid-scale magnetic energy transfer.

The Similarity of Non-local Triad Interactions in Energy Transfer of Isotropic Turbulence

The inter-scale energy transfer process is one of the key issues in the multi-scale nature of turbulent flows. A direct numerical simulation of homogeneous isotropic turbulent flows is conducted and the energy transfer is studied in detail in spectral space. The band-to-band energy transfer function involving non-local triad interactions with large scale modes peaks sharply. The similarity of this non-local energy transfer function is observed in both the shape and the amplitude. In addition, the similarity is satisfied in both the inertial subrange and dissipation range, implying that it is a general property of turbulence. The amplitude is determined by wave numbers p and q independently in a power law, with the power rates −3 and −2, respectively.

A subgrid-scale model for large-eddy simulation based on the physics of interscale energy transfer in turbulence

The scale-similarity model in large-eddy simulation (LES) leads to an attractive, functionally simple expression for the subgrid-scale (SGS) stress tensor. It is well known, however, that the similarity model fails to accurately predict some of the most fundamental quantities in turbulent flows, perhaps the most important being the global energy transfer and the associated subgrid-scale dissipation. To address this, additional dissipative terms are usually added to the similarity model to improve its performance. In the present paper, considerations of interscale energy transfer have been used to identify sources of the observed deficiencies of the similarity model, specifically its inadequate balancing of terms contributing energy to the smallest scales and its duplication of terms producing effects in the largest scales. These considerations provide guidance in the development of a new model, which shows more favorable characteristics of energy transfer while preserving the functional simplicity of the scale-similarity model. Partial nonlinear terms are used to decompose the nonlinear transfer present in LES and to formulate a model expression capable of balancing small-scale production terms depositing energy near the LES cutoff. The proposed model is formulated in the same vein as the scale-similarity model, consisting of test filtered velocities and their products, but offers clear improvements in predictions of mean flow quantities and the global energy flux from the resolved to subgrid scales without the need for additional terms to augment subgrid-scale energy dissipation. The application of the new interscale transfer model in LES of wall-bounded flows leads to predictions of mean and RMS flow quantities comparable to those obtained for other, established SGS models.

Nonlinear inertia-gravity wave-mode interactions in three dimensional rotating stratified flows

To investigate the dynamics of gravity waves in stratified Boussinesq flows, a model is derived that consists of all three-gravity-wave-mode interactions (the GGG model), excluding interactions involving the vortical mode. The GGG model is a natural extension of weak turbulence theory that accounts for exact three-gravity-wave resonances. The model is examined numerically by means of random, large-scale, high-frequency forcing. An immediate observation is a robust growth of the so-called vertically sheared horizontal flow (VSHF). In addition, there is a forward transfer of energy and equilibration of the nonzero-frequency (sometimes called “fast”) gravity-wave modes. These results show that gravity-wave-mode interactions by themselves are capable of systematic interscale energy transfer in a stratified fluid. Comparing numerical simulations of the GGG model and the full Boussinesq system, for the range of Froude numbers (Fr) considered (0.05 ≤ Fr ≤ 1), in both systems the VSHF is hardest to resolve. When adequately resolved, VSHF growth is more vigorous in the GGG model. Furthermore, a VSHF is observed to form in milder stratification scenarios in the GGG model than the full Boussinesq system. Finally, fully three-dimensional nonzero-frequency gravity-wave modes equilibrate in both systems and their scaling with vertical wavenumber follows similar power-laws. The slopes of the power-laws obtained depend on Fr and approach −2 (from above) at Fr = 0.05, which is the strongest stratification that can be properly resolved with our computational resources

Localness of energy cascade in hydrodynamic turbulence. I. Smooth coarse graining

We introduce a novel approach to scale decomposition of the fluid kinetic energy (or other quadratic integrals) into band-pass contributions from a series of length scales. Our decomposition is based on a multiscale generalization of the “Germano identity” for smooth, graded filter kernels. We employ this method to derive a budget equation that describes the transfers of turbulent kinetic energy both in space and in scale. It is shown that the interscale energy transfer is dominated by local triadic interactions, assuming only the scaling properties expected in a turbulent inertial range. We derive rigorous upper bounds on the contributions of nonlocal triads, extending the work of Eyink [Physica D 207, 91 (2005)] for low-pass filtering. We also propose a physical explanation of the differing exponents for our rigorous upper bounds and for the scaling predictions of Kraichnan [Phys. Fluids 9, 1728 (1966); J. Fluid Mech. 47, 525 (1971)]. The faster decay predicted by Kraichnan is argued to be the consequence of additional cancellations in the signed contributions to transfer from nonlocal triads after averaging over space. This picture is supported by data from a 5123 pseudospectral simulation of Navier–Stokes turbulence with phase-shift dealiasing.

The role of coherent structures in subgrid-scale energy transfer within the log layer of wall turbulence

The present effort documents the relationship between dominant subgrid-scale energy transfer events and coherent motions within the log layer of wall turbulence. Instantaneous velocity fields in the streamwise–wall-normal plane of a zero-pressure-gradient turbulent boundary layer acquired by particle-image velocimetry at Reτ≡u*δ∕ν=2350 are spatially filtered to generate an ensemble of resolved-scale velocity fields in the spirit of large-eddy simulation. The relationship between subgrid-scale dissipation and embedded coherent structures is then studied using instantaneous realizations and conditional averaging techniques. This analysis reveals that strong forward- and backward-scatter events occur spatially coincident to individual hairpin vortices and their larger-scale organization into vortex packets. In particular, large-scale regions of forward scatter are observed along the inclined interface of the packets, coincident with strong ejections induced by the individual vortices. The most intense forward-scatter events are found to occur when these ejections are opposed by sweep motions. Strong backward scatter of energy is observed at the trailing edge of the vortex packets and weaker backscatter is also noted locally around the individual heads of the hairpin structures. The collective observations presented herein demonstrate that hairpin vortices and their organization into larger-scale packets are important contributors to interscale energy transfer in the log layer of wall turbulence.

Experimental analysis of a confined transitional plume with respect to subgrid-scale modelling

Free convection flow of water induced by a small prismatic heat source in a confined space is studied in the present paper. The forcing of the flow is in the range of Rayleigh numbers where a spatial transition from laminar to turbulent flow can be observed.Velocity measurements were performed by means of Particle Tracking Velocimetry (PTV ) . From the velocity data a resolved and a subgrid field is obtained by means of spatial filtering. These data are analysed with respect to modelling consequences in Large-Eddy Simulation (LES) , constituting an a priori test.Astatistically steady flow has been obtained with a converged temporal mean and standard deviation as function of space. A low correlation of time mean model stresses with exact stresses is found. In the transitional region the inter-scale kinetic energy transfer, taken over a wavenumber corresponding to the laminar plume width, is found to possess a relative large standard deviation. In this region backscatter and forward scatter of kinetic energy, with respect to the mentioned wavenumber, are of equal importance. Application of a dynamic model to the filtered data yields a qualitative good representation of the exact inter-scale kinetic energy transfer.

A comparative study of subgrid scale models in homogeneous isotropic turbulence

Recently, a number of studies have indicated that Large Eddy Simulation (LES) models are fairly insensitive to the adopted Subgrid Scale (SGS) models. In order to study this and to gain further insight into LES, simulations of forced and decaying homogeneous isotropic turbulence have been performed for Taylor Re numbers between 35 and 248 using various SGS models, representative of the contemporary state of the art. The predictive capability of the LES concept is analyzed by comparison with DNS data and with results obtained from a theoretical model of the energy spectrum. The resolved flow is examined by visualizing the morphology and by analyzing the distribution of resolved enstrophy, rate of strain, stretching, SGS kinetic energy, and viscosity. Furthermore, the correlation between eigenvalues of the resolved rate of strain tensor and the vorticity is investigated. Although the gross features of the flow appear independent of the SGS model, pronounced differences between the models become apparent when the SGS kinetic energy and the interscale energy transfer are investigated.

Effect of subgrid models on the computed interscale energy transfer in isotropic turbulence

Direct and large eddy simulations of forced and decaying isotropic turbulence have been performed to investigate the behavior of subgrid models. Various subgrid models have been analyzed (i.e. Smagorinsky's eddy viscosity model, dynamic eddy viscosity model, dynamic one-equation model for the subgrid kinetic energy and scale-similarity model). A priori analysis showed that the subgrid stress and the subgrid energy flux predicted by the scale similarity model, and subgrid kinetic energy model (with fixed coefficients) correlate reasonably well with exact data, while the Smagorinsky's eddy viscosity model showed relatively poor agreement. However, the correlation for the scale similarity model decreased much more rapidly with decrease in grid resolution when compared to the subgrid kinetic energy model. The subgrid models were then used to carry out large-eddy simulations for a range of Reynolds number. When dynamic evaluation was incorporated, the correlation improved significantly. The dynamic subgrid kinetic energy model showed, consistently, a higher correlation for a range of Reynolds number when compared to the dynamic eddy viscosity model. These results demonstrate the capabilities of the dynamic one-equation model.

Subgrid-scale energy transfer and near-wall turbulence structure

Conditional averages of the velocity field, subgrid-scale (SGS) stresses and SGS dissipation are calculated using the velocity fields obtained from the DNS of plane channel flow. The detection criteria isolate the coherent turbulent structures that contribute most strongly to the energy transfer between the large, resolved scales and the subgrid, unresolved, ones. Separate averages are computed for forward and backward scatter. The interscale energy transfer is found to be strongly correlated with the presence of the turbulent structures typical of wall-bounded flows: quasi-streamwise and hairpin vortices, sweeps and ejections. In the buffer layer, strong SGS dissipation is observed near lifted shear layers; the forward scatter is associated with ejections, the backscatter with sweeps. Both backward and forward scatter occur in close proximity to longitudinal vortices that form a very shallow angle to the wall. Further away from the solid boundary, in the logarithmic region and beyond, both forward and backward energy transfer are associated prevalently with ejections. Eddy viscosity models do not predict the three-dimensional structure of these events adequately, while scale-similar models reproduce the correlation between the large-scale coherent structures and the SGS events more accurately.

The nature of triad interactions in homogeneous turbulence

Nonlinear interactions in homogeneous turbulence are investigated using a decomposition of the velocity field in terms of helical modes. There are two helical modes per wave vector and thus eight fundamental triad interactions. These eight elementary interactions fit in only two classes, ‘‘R’’ (for ‘‘reverse’’) and ‘‘F’’ (for ‘‘forward’’), depending on whether the small-scale helical modes have helicities of the same or of the opposite sign. In a single-triad interaction, the large scale is unstable when the small-scale helical modes have helicities of opposite signs (class ‘‘F’’), and the medium scale is unstable otherwise (class ‘‘R’’). It is proposed that, on average, the triple correlations in a turbulent flow correspond to these unstable states. In the limit of nonlocal triads, where one leg is much smaller than the other two, the triadic interscale energy transfer is largest for interactions of class ‘‘R.’’ In that case, most of the energy flows locally in wave number, from the medium scale to the smallest, with a comparatively small feedback on the large scale. However, integrating over all scales in an inertial range, the net effect of nonlocal interactions of class ‘‘R’’ is a reverse energy cascade from small to large scales. All other interactions transfer energy upward in wave number. In local triads, this upward energy transfer occurs primarily between modes with helicities of the same sign, through catalytic interactions with a mode whose helicity has the opposite sign. The class ‘‘F’’ interactions transfer energy to the small scales and exist only in three dimensions. The physical processes associated to both classes of interactions are discussed. It is shown that the large local transfers due to nonlocal ‘‘R’’ interactions appear in pairs of opposite signs that nearly cancel each other and the net effect corresponds to an advection in wave space.