Nonlocal Triads Research Articles

Physical/numerical duality of explicit/implicit subgrid-scale modelling

The development of implicit approaches has prompted debate on the actual usefulness of any explicit subgrid-scale modelling in large-eddy simulation. This question is addressed here by considering two generic turbulent flows: (i) the Taylor-Green vortex problem; (ii) the pipe flow. For both flow configurations, implicit modelling is found to overtake the very popular Smagorinsky model. To understand this robust observation, an analysis in the Fourier space is presented for the Taylor-Green vortex problem. The concept of spectral eddy viscosity, widely used in the pioneer work of Marcel Lesieur in two-point closure and subgrid-scale modelling, is revisited in a general framework based on explicit/implicit subgrid-scale modelling. In particular, the essentially anisotropic nature of implicit modelling is exhibited, as a favourable feature in terms of consistency with the computational mesh. Smagorinsky's model, considered as a generic explicit subgrid-scale model in the framework of Boussinesq's hypothesis, is found to be highly sensitive to numerical errors. Removing the latter is easy but makes computationally inefficient this type of explicit modelling. Comparisons between a priori and a posteriori spectral eddy viscosities show that neither Smagorinsky's model nor implicit modelling can mimic the expected spectral behaviour. Smagorinsky's model is observed to be weakly scale-selective with a poor ability to actually filter the solution. The feature of scale-selectivity is well replicated by implicit modelling which exhibits excellent capabilities for filtering. However, its lack of influence at the largest scales is against the expected behaviour for the spectral eddy viscosity at low wavenumber through the establishment of a non-zero plateau value. This lack of consistency of implicit LES could be overcome thanks to an extra explicit modelling but the attempt to mix Smagorinsky's model and implicit LES is not successful in this study. The potential of implicit large-eddy simulation is also exhibited for the accurate computation of near-wall turbulence inside a pipe flow despite the use of a regular Cartesian mesh with an immersed boundary method. Interestingly, the resulting coarse wall-normal resolution in the near-wall direction does not prevent the reliable prediction of statistical profiles up to the capture of subgrid-scale details. It is suggested that the regularisation associated with implicit modelling is a necessary condition to reach numerical accuracy. However, to faithfully represent the large-scale dynamics, present results confirm that non-local triad interactions must be taken into account as widely discussed in the inspiring textbook Lesieur [Turbulence in fluids. 4th ed. Springer; 2008] of Marcel Lesieur.

Fractally Fourier decimated homogeneous turbulent shear flow in noninteger dimensions.

Time evolution of the fully resolved incompressible homogeneous turbulent shear flow in noninteger Fourier dimensions is numerically investigated. The Fourier dimension of the flow field is extended from the integer value 3 to the noninteger values by projecting the Navier-Stokes equation on the fractal set of the active Fourier modes with dimensions 2.7≤d≤3.0. The results of this study revealed that the dynamics of both large and small scale structures are nontrivially influenced by changing the Fourier dimension d. While both turbulent production and dissipation are significantly hampered as d decreases, the evolution of their ratio is almost independent of the Fourier dimension. The mechanism of the energy distribution among different spatial directions is also impeded by decreasing d. Due to this deficient energy distribution, turbulent field shows a higher level of the large-scale anisotropy in lower Fourier dimensions. In addition, the persistence of the vortex stretching mechanism and the forward spectral energy transfer, which are three-dimensional turbulence characteristics, are examined at changing d, from the standard case d=3.0 to the strongly decimated flow field for d=2.7. As the Fourier dimension decreases, these forward energy transfer mechanisms are strongly suppressed, which in turn reduces both the small-scale intermittency and the deviation from Gaussianity. Besides the energy exchange intensity, the variations of d considerably modify the relative weights of local to nonlocal triadic interactions. It is found that the contribution of the nonlocal triads to the total turbulent kinetic energy exchange increases as the Fourier dimension increases.

Enstrophy inertial range dynamics in generalized two-dimensional turbulence

The enstrophy inertial range dynamics of generalized two-dimensional turbulence is investigated using the Eddy-Damped Quasi-Normal Markovianized approximation equation. The transition to ${k}^{-1}$ spectrum is analytically derived and the non-local triad interactions are shown to be responsible for the transition.

Spectral non-locality, absolute equilibria and Kraichnan–Leith–Batchelor phenomenology in two-dimensional turbulent energy cascades

AbstractWe study the degree to which Kraichnan–Leith–Batchelor (KLB) phenomenology describes two-dimensional energy cascades in$\alpha $turbulence, governed by$\partial \theta / \partial t+ J(\psi , \theta )= \nu {\nabla }^{2} \theta + f$, where$\theta = {(- \Delta )}^{\alpha / 2} \psi $is generalized vorticity, and$\hat {\psi } (\boldsymbol{k})= {k}^{- \alpha } \hat {\theta } (\boldsymbol{k})$in Fourier space. These models differ in spectral non-locality, and include surface quasigeostrophic flow ($\alpha = 1$), regular two-dimensional flow ($\alpha = 2$) and rotating shallow flow ($\alpha = 3$), which is the isotropic limit of a mantle convection model. We re-examine arguments for dual inverse energy and direct enstrophy cascades, including Fjørtoft analysis, which we extend to general$\alpha $, and point out their limitations. Using an$\alpha $-dependent eddy-damped quasinormal Markovian (EDQNM) closure, we seek self-similar inertial range solutions and study their characteristics. Our present focus is not on coherent structures, which the EDQNM filters out, but on any self-similar and approximately Gaussian turbulent component that may exist in the flow and be described by KLB phenomenology. For this, the EDQNM is an appropriate tool. Non-local triads contribute increasingly to the energy flux as$\alpha $increases. More importantly, the energy cascade is downscale in the self-similar inertial range for$2. 5\lt \alpha \lt 10$. At$\alpha = 2. 5$and$\alpha = 10$, the KLB spectra correspond, respectively, to enstrophy and energy equipartition, and the triad energy transfers and flux vanish identically. Eddy turnover time and strain rate arguments suggest the inverse energy cascade should obey KLB phenomenology and be self-similar for$\alpha \lt 4$. However, downscale energy flux in the EDQNM self-similar inertial range for$\alpha \gt 2. 5$leads us to predict that any inverse cascade for$\alpha \geq 2. 5$will not exhibit KLB phenomenology, and specifically the KLB energy spectrum. Numerical simulations confirm this: the inverse cascade energy spectrum for$\alpha \geq 2. 5$is significantly steeper than the KLB prediction, while for$\alpha \lt 2. 5$we obtain the KLB spectrum.

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.

Infrared Reynolds number dependency of the two-dimensional inverse energy cascade

High-resolution simulations of forced two-dimensional turbulence reveal that the inverse cascade range is sensitive to an infrared Reynolds number, Reα = kf/kα, where kf is the forcing wavenumber and kα is a frictional wavenumber based on linear friction. In the limit of high Reα, the classic k−5/3 scaling is lost and we obtain steeper energy spectra. The sensitivity is traced to the formation of vortices in the inverse energy cascade range. Thus, it is hypothesized that the dual limit Reα → ∞ and Reν = kd/kf → ∞, where kd is the small-scale dissipation wavenumber, will lead to a steeper energy spectrum than k−5/3 in the inverse energy cascade range. It is also found that the inverse energy cascade is maintained by non-local triad interactions.

Scale Locality of Magnetohydrodynamic Turbulence

We investigate the scale locality of cascades of conserved invariants at high kinetic and magnetic Reynold's numbers in the "inertial-inductive range" of magnetohydrodynamic (MHD) turbulence, where velocity and magnetic field increments exhibit suitable power-law scaling. We prove that fluxes of total energy and cross helicity-or, equivalently, fluxes of Elsässer energies-are dominated by the contributions of local triads. Flux of magnetic helicity may be dominated by nonlocal triads. The magnetic stretching term may also be dominated by nonlocal triads, but we prove that it can convert energy only between velocity and magnetic modes at comparable scales. We explain the disagreement with numerical studies that have claimed conversion nonlocally between disparate scales. We present supporting data from a 1024{3} simulation of forced MHD turbulence.

Localness of energy cascade in hydrodynamic turbulence. II. Sharp spectral filter

We investigate the scale-locality of subgrid-scale (SGS) energy flux and interband energy transfers defined by the sharp spectral filter. We show by rigorous bounds, physical arguments, and numerical simulations that the spectral SGS flux is dominated by local triadic interactions in an extended turbulent inertial range. Interband energy transfers are also shown to be dominated by local triads if the spectral bands have constant width on a logarithmic scale. We disprove in particular an alternative picture of “local transfer by nonlocal triads,” with the advecting wavenumber mode at the energy peak. Although such triads have the largest transfer rates of all individual wavenumber triads, we show rigorously that, due to their restricted number, they make an asymptotically negligible contribution to energy flux and log-banded energy transfers at high wavenumbers in the inertial range. We show that it is only the aggregate effect of a geometrically increasing number of local wavenumber triads which can sustain an energy cascade to small scales. Furthermore, nonlocal triads are argued to contribute even less to the space-average energy flux than is implied by our rigorous bounds, because of additional cancellations from scale-decorrelation effects. We can thus recover the −4/3 scaling of nonlocal contributions to spectral energy flux predicted by Kraichnan’s abridged Lagrangian-history direct-interaction approximation and test-field model closures. We support our results with numerical data from a 5123 pseudospectral simulation of isotropic turbulence with phase-shift dealiasing. We also discuss a rigorous counterexample of Eyink [Physica D 78, 222 (1994)], which showed that nonlocal wavenumber triads may dominate in the sharp spectral flux (but not in the SGS energy flux for graded filters). We show that this mathematical counterexample fails to satisfy reasonable physical requirements for a turbulent velocity field, which are employed in our proof of scale locality. We conclude that the sharp spectral filter has a firm theoretical basis for use in large-eddy simulation modeling of turbulent flows.

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.

Interacting scales and triad enstrophy transfers in generalized two-dimensional turbulence

The local and nonlocal characteristics of triad enstrophy transfer in the enstrophy inertial range of generalized two-dimensional turbulence, so-called alpha turbulence, are investigated using direct numerical simulations, with a special emphasis on alpha=1 , 2, and 3. The enstrophy transfer via nonlocal triad interactions dominates the transfer dynamics in the enstrophy inertial range, irrespective of alpha . However, the contributions from more local interactions to the total enstrophy transfer increase as alpha decreases. The results are discussed in connection with the local and nonlocal transition of the enstrophy transfer at alpha=2 expected from the phenomenological scaling theory. The specific nature of the enstrophy transfer in surface quasigeostrophic turbulence (alpha=1) is also discussed.

Nonlocal modulation of the energy cascade in broadband-forced turbulence

Classically, large-scale forced turbulence is characterized by a transfer of energy from large to small scales via nonlinear interactions. We have investigated the changes in this energy transfer process in broadband forced turbulence where an additional perturbation of flow at smaller scales is introduced. The modulation of the energy dynamics via the introduction of forcing at smaller scales occurs not only in the forced region but also in a broad range of length scales outside the forced bands due to nonlocal triad interactions. Broadband forcing changes the energy distribution and energy transfer function in a characteristic manner leading to a significant modulation of the turbulence. We studied the changes in this transfer of energy when changing the strength and location of the small-scale forcing support. The energy content in the larger scales was observed to decrease, while the energy transport power for scales in between the large and small scale forcing regions was enhanced. This was investigated further in terms of the detailed transfer function between the triad contributions and observing the long-time statistics of the flow. The energy is transferred toward smaller scales not only by wave numbers of similar size as in the case of large-scale forced turbulence, but by a much wider extent of scales that can be externally controlled.

Local shell-to-shell energy transfer via nonlocal interactions in fluid turbulence

In this paper we analytically compute the strength of nonlinear interactions in a triad, and the energy exchanges between wave-number shells in incompressible fluid turbulence. The computation has been done using first-order perturbative field theory. In three dimensions, magnitude of triad interactions is large for nonlocal triads, and small for local triads. However, the shell-to-shell energy transfer rate is found to be local and forward. This result is due to the fact that the nonlocal triads occupy much less Fourier space volume than the local ones. The analytical results on three-dimensional shell-to-shell energy transfer match with their numerical counterparts. In two-dimensional turbulence, the energy transfer rates to the nearby shells are forward, but to the distant shells are backward; the cumulative effect is an inverse cascade of energy.

Phase shift of long-wavelength mode due to drift-wave turbulence

The phase shift of long-wavelength modes due to drift-wave turbulence is studied in the Hasegawa–Mima model. Although the growth rate has been the main focus, it is equally crucial to understand the frequency shift from the linear characteristics. The test-field model is applied in order to properly treat the sweeping effect of the large scales on a small scale. The frequency shift is found to be small compared with the growth rate and the non-local triad mode-interaction time turns out to be determined by the energy of the long-wavelength modes.

Infrared Dynamics of Decaying Two-Dimensional Turbulence Governed by the Charney-Hasegawa-Mima Equation

The low wave number range of decaying turbulence governed by the Charney-Hasegawa-Mima (CHM) equation is examined theoretically and by direct numerical simulation. Here, the low wave number range is defined as values of the wave number k below the wave number kE corresponding to the peak of the energy spectrum, or alternatively the centroid wave number of the energy spectrum. The energy spectrum in the low wave number range in the infrared regime (k →0) is theoretically derived to be E(k) ∼k5, using a quasinormal Markovianized model of the CHM equation. This result is verified by direct numerical simulation of the CHM equation. The wave number triads (k,p,q) responsible for the formation of the low wave number spectrum are also examined. It is found that the energy flux Π(k) for k k. Furthermore, the contribution of nonlocal triad interactions to the energy flux is found to be predominant in the range log (k/kE)<-0.5, where the nonlocal interactions are defined to be those triad interactions for which the ratio of the largest leg of the triad to the smallest leg is larger than four. ©2001 The Physical Society of Japan

Effect of System Rotation on Energy Transfer of Homogeneous Decaying Turbulence.

The effect of system rotation on energy spectrum, energy transfer function and triadic energy transfer function is investigated using the DNS data of homogeneous decaying turbulence subjected to system rotation. The DNS is performed with 128 Fourier modes at various rotation rate up to 50[rad/s]. The initial Taylor micro-scale Reynolds number is 53.6. The results of this study are summarized as follows : (1) The profile of the energy spectrum approaches the linealized solution with an increase in the rotation. That is, the decay of the energy spectrum is inhibited and promoted in low and high wave number regions, respectively, by the rotation. The absolute value of the energy transfer function decreases with an increase in the rotation. This phenomena is explained by scrambling effect. (2) The rotation gives the directional anisotropy on the energy spectrum and energy transfer function. However the effect of the directional anisotropy on the energy spectrum becomes negligible when the rotation rate is too high. (3) The triadic energy transfer is also prevented by the rotation. The local energy transfer is dominant in the non-local triad for both rotating and non-rotating systems. The rotation also gives the directional anisotropy on the triadic energy transfer function.

Multi-scalar triadic interactions in differential diffusion with and without mean scalar gradients

The spectral mechanisms of the differential diffusion of pairs of passive scalars with different molecular diffusivities are studied in stationary isotropic turbulence, using direct numerical simulation data at Taylor-scale Reynolds number up to 160 on 1283 and 2563 grids. Of greatest interest are the roles of nonlinear triadic interactions between different scale ranges of the velocity and scalar fields in the evolution of spectral coherency between the scalars, and the effects of mean scalar gradients.Analysis of single-scalar spectral transfer (extending the results of a previous study) indicates a robust local forward cascade behaviour at high wavenumbers, which is strengthened by both high diffusivity and mean gradients. This cascade is driven primarily by moderately non-local interactions in which two small-scale scalar modes are coupled via a lower-wavenumber velocity mode near the peak of the energy dissipation spectrum. This forward cascade is coherent, tending to increase the coherency between different scalars at high wavenumbers but to decrease it at lower wavenumbers. However, at early times coherency evolution at high wavenumbers is dominated by de-correlating effects due to a different type of non-local triad consisting of two scalar modes with a moderate scale separation and a relatively high-wavenumber velocity mode. Consequently, although the small-scale motions play little role in spectral transfer, they are responsible for the rapid de-correlation observed at early times. At later times both types of competing triadic interactions become important over a wider wavenumber range, with increased relative strength of the coherent cascade, so that the coherency becomes slow-changing. When uniform mean scalar gradients are present, a stationary state develops in the coherency spectrum as a result of a balance between a coherent mean gradient contribution (felt within about 1 eddy-turnover time) and the net contribution from scale interactions. The latter is made less de-correlating because of a strengthened coherent forward cascade, which is in turn caused by uniform mean gradients acting as a primarily low-wavenumber source of scalar fluctuations with the same spectral content as the velocity field.

Degrees of locality of energy transfer in the inertial range

Recent interpretations of direct numerical simulation (DNS) measurements have led some authors to suggest that energy is largely transferred downscale locally, supporting a basic concept of the Kolmogorov phenomenology that leads to the universal inertial subrange. However, these authors conclude that the local energy transfer results from nonlocal triad interactions. This claim brings into question the validity of the assumption of the statistical independence of the large- and small-scale motions in the Kolmogorov universal theory of turbulence. In this Letter, the measured raw transfer interactions have been summed in a way that directly indicates the scale disparity (s) of contributions to the net energy flux across the spectrum. It is found that the dependence upon s closely follows the s−4/3 form predicted by classical arguments. As a result, it is concluded that DNS measurements, in fact, lend support to the classical Kolmogorov phenomenology of local interactions and local transfer in an inertial range.

Nonlocal triad interactions and the dissipation range of isotropic turbulence

Detailed transfer functions T(k‖p,q), which express turbulent energy transfer rate to modes k caused by their nonlinear interactions with modes p and q, are analyzed using results of direct numerical simulations of homogeneous turbulence. A previously found phenomenological scaling for the functions T(k‖p,q), which brings them into a self-similar form, is used to deduce the form of the energy spectrum in the dissipation range proportional to k−2 exp(−ak) and the transfer spectrum proportional to exp(−ak). A physical mechanism of the energy transfer process consistent with the self-similarity scaling is proposed.

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.

The response of isotropic turbulence to isotropic and anisotropic forcing at the large scales

The nonlinear interscale couplings in a turbulent flow are studied through direct numerical simulations of the response of isotropic turbulence to isotropic and anisotropic forcing applied at the large scales. Specifically, forcing is applied to the energy-containing wave-number range for about two eddy turnover times to fully developed isotropic turbulence at Taylor-scale Reynolds number 32 on an 1283 grid. When forced isotropically, the initially isotropic turbulence remains isotropic at all wave numbers. However, anisotropic forcing applied through an array of counter-rotating rectilinear vortices induces high levels of anisotropy at the small scales. At low wave numbers the force term feeds energy directly into two velocity components in the plane of the forced vortices. In contrast, at high wave numbers the third (spanwise) component receives the most energy, producing small-scale anisotropy very different from that at the large scales. Detailed analysis shows that the development of small-scale anisotropy is caused primarily by nonlocal wave-vector triads with one leg in the forced low-wave-number range. This latter result is particularly significant because asymptotic analysis of the Fourier-transformed Navier–Stokes equations shows that distant triadic interactions coupling the energy-containing and dissipative scales persist at asymptotically high Reynolds numbers, suggesting that the structural couplings between large and small scales in these moderate Reynolds number simulations would also exist in high Reynolds number forced turbulence. The results therefore imply a departure from the classical hypothesis of statistical independence between large- and small-scale structure and local isotropy.