Spectral modeling of turbulent flows and the role of helicity

Dynamic Phase Alignment in Navier-Stokes Turbulence

In Navier-Stokes turbulence, energy and helicity injected at large scales are subject to a joint direct cascade, with both quantities exhibiting a spectral scaling ∝k^{-5/3}. We demonstrate via direct numerical simulations that the two cascades are compatible due to the existence of a strong scale-dependent phase alignment between velocity and vorticity fluctuations, with the phase alignment angle scaling as cosα_{k}∝k^{-1}.

Self-Similar Subgrid-Scale Models for Inertial Range Turbulence and Accurate Measurements of Intermittency

A class of spectral subgrid models based on a self-similar and reversible closure is studied with the aim to minimize the impact of subgrid scales on the inertial range of fully developed turbulence. In this manner, we improve the scale extension where anomalous exponents are measured by roughly 1 order of magnitude when compared to direct numerical simulations or to other popular subgrid closures at the same resolution. We find a first indication that intermittency for high-order moments is not captured by many of the popular phenomenological models developed so far.

Hindered Energy Cascade in Highly Helical Isotropic Turbulence

The conventional approach to the turbulent energy cascade, based on Richardson-Kolmogorov phenomenology, ignores the topology of emerging vortices, which is related to the helicity of the turbulent flow. It is generally believed that helicity can play a significant role in turbulent systems, e.g., supporting the generation of large-scale magnetic fields, but its impact on the energy cascade to small scales has never been observed. We suggest, for the first time, a generalized phenomenology for isotropic turbulence with an arbitrary spectral distribution of the helicity. We discuss various scenarios of direct turbulent cascades with new helicity effect, which can be interpreted as a hindering of the spectral energy transfer. Therefore, the energy is accumulated and redistributed so that the efficiency of nonlinear interactions will be sufficient to provide a constant energy flux. We confirm our phenomenology by high Reynolds number numerical simulations based on a shell model of helical turbulence. The energy in our model is injected at a certain large scale only, whereas the source of helicity is distributed over all scales. In particular, we found that the helical bottleneck effect can appear in the inertial interval of the energy spectrum.

Disentangling the triadic interactions in Navier-Stokes equations

We study the role of helicity in the dynamics of energy transfer in a modified version of the Navier-Stokes equations with explicit breaking of the mirror symmetry. We select different set of triads participating in the dynamics on the basis of their helicity content. In particular, we remove the negative helically polarized Fourier modes at all wave numbers except for those falling on a localized shell of wave number, |k| ~ k(m). Changing k(m) to be above or below the forcing scale, k(f), we are able to assess the energy transfer of triads belonging to different interaction classes. We observe that when the negative helical modes are present only at a wave number smaller than the forced wave numbers, an inverse energy cascade develops with an accumulation of energy on a stationary helical condensate. Vice versa, when negative helical modes are present only at a wave number larger than the forced wave numbers, a transition from backward to forward energy transfer is observed in the regime when the minority modes become energetic enough.

Nonhelical inverse transfer of a decaying turbulent magnetic field

In the presence of magnetic helicity, inverse transfer from small to large scales is well known in magnetohydrodynamic (MHD) turbulence and has applications in astrophysics, cosmology, and fusion plasmas. Using high resolution direct numerical simulations of magnetically dominated self-similarly decaying MHD turbulence, we report a similar inverse transfer even in the absence of magnetic helicity. We compute for the first time spectral energy transfer rates to show that this inverse transfer is about half as strong as with helicity, but in both cases the magnetic gain at large scales results from velocity at similar scales interacting with smaller-scale magnetic fields. This suggests that both inverse transfers are a consequence of universal mechanisms for magnetically dominated turbulence. Possible explanations include inverse cascading of the mean squared vector potential associated with local near two dimensionality and the shallower k^{2} subinertial range spectrum of kinetic energy forcing the magnetic field with a k^{4} subinertial range to attain larger-scale coherence. The inertial range shows a clear k^{-2} spectrum and is the first example of fully isotropic magnetically dominated MHD turbulence exhibiting weak turbulence scaling.

Joint-constraint model for large-eddy simulation of helical turbulence

A three-term mixed subgrid-scale (SGS) stress model is proposed for large-eddy simulation (LES) of helical turbulence. The new model includes a Smagorinsky-Lilly term, a velocity gradient term, and a symmetric vorticity gradient term. The model coefficients are determined by minimizing the mean square error between the realistic and modeled Leonard stresses under a joint constraint of kinetic energy and helicity fluxes. The model formulated as such is referred to as joint-constraint dynamic three-term model (JCD3TM). First, the new model is evaluated a priori using the direct numerical simulation (DNS) data of homogeneous isotropic turbulence with helical forcing. It is shown that the SGS dissipation fractions from all three terms in JCD3TM have the properties of length-scale invariance in inertial subrange. JCD3TM can predict the SGS stresses, energy flux, and helicity flux more accurately than the dynamic Smagorinsky model (DSM) and dynamic mixed helical model (DMHM) in both pointwise and statistical senses. Then, the performance of JCD3TM is tested a posteriori in LESs of both forced and freely decaying helical isotropic turbulence. It is found that JCD3TM possesses certain features of superiority over the other two models in predicting the energy spectrum, helicity spectrum, high-order statistics, etc. It is also noteworthy that JCD3TM is capable of simulating the evolutions of both energy and helicity spectra more precisely than other models in decaying helical turbulence. We claim that the present SGS model can capture the main helical features of turbulent motions and may serve as a useful tool for LES of helical turbulent flows.

Refined subgrid-scale model for large-eddy simulation of helical turbulence

A refined two-term helical subgrid-scale (SGS) stress model with respect to that suggested by Li et al. [Phys. Rev. E 74, 026310 (2006)] is designed for large-eddy simulation (LES) of helical turbulence. The model coefficients in the new model are verified a priori to be scale invariant in inertial range, which proves that our model is local in scale. A dynamic method based on minimizing the residual resolved energy and helicity dissipations is suggested to simultaneously evaluate the coefficients of the mixed SGS model as the simulation progresses. In addition, an SGS helicity dissipation (or helicity flux) constraint condition is proposed to optimize the mixed two-term model. Both techniques are first tested and validated in the LES of forced isotropic helical turbulence. The statistical results are analyzed and compared with those obtained from the dynamic Smagorinsky model, the traditional dynamic mixed model, and the direct numerical simulation. It is found that the introduction of this dynamic procedure can help overcome the drawback of the traditional dynamic method which can not capture the negative helicity fluxes and SGS dissipations. The probability density functions of the energy flux and the conditioned helicity flux and SGS stress demonstrate that the helicity flux constrained dynamic SGS model can effectively predict the real SGS helical effects on the resolved scales, such as backscatters of energy and helicity, accurate helicity dissipation rate, and so on. The present models are also applied to the simulation of freely decaying isotropic turbulence with no apparent improvement observed in comparison with the traditional SGS models. The underlying reasons for these issues are addressed in detail.

Decay of Batchelor and Saffman rotating turbulence

The decay rate of isotropic and homogeneous turbulence is known to be affected by the large-scale spectrum of the initial perturbations, associated with at least two canonical self-preserving solutions of the von Kármán-Howarth equation: the so-called Batchelor and Saffman spectra. The effect of long-range correlations in the decay of anisotropic flows is less clear, and recently it has been proposed that the decay rate of rotating turbulence may be independent of the large-scale spectrum of the initial perturbations. We analyze numerical simulations of freely decaying rotating turbulence with initial energy spectra ∼k^{4} (Batchelor turbulence) and ∼k^{2} (Saffman turbulence) and show that, while a self-similar decay can not be identified for the total energy, the decay is indeed affected by long-range correlations. The decay of two- and three-dimensional modes follows distinct power laws in each case, which are consistent with predictions derived from the anisotropic von Kármán-Howarth equation, and with conservation of anisotropic integral quantities by the flow evolution.

Anisotropy and nonuniversality in scaling laws of the large-scale energy spectrum in rotating turbulence

Rapidly rotating turbulent flow is characterized by the emergence of columnar structures that are representative of quasi-two-dimensional behavior of the flow. It is known that when energy is injected into the fluid at an intermediate scale Lf, it cascades towards smaller as well as larger scales. In this paper we analyze the flow in the inverse cascade range at a small but fixed Rossby number, Rof≈0.05. Several numerical simulations with helical and nonhelical forcing functions are considered in periodic boxes with unit aspect ratio. In order to resolve the inverse cascade range with reasonably large Reynolds number, the analysis is based on large eddy simulations which include the effect of helicity on eddy viscosity and eddy noise. Thus, we model the small scales and resolve explicitly the large scales. We show that the large-scale energy spectrum has at least two solutions: one that is consistent with Kolmogorov-Kraichnan-Batchelor-Leith phenomenology for the inverse cascade of energy in two-dimensional (2D) turbulence with a ∼k⊥-5/3 scaling, and the other that corresponds to a steeper ∼k⊥-3 spectrum in which the three-dimensional (3D) modes release a substantial fraction of their energy per unit time to the 2D modes. The spectrum that emerges depends on the anisotropy of the forcing function, the former solution prevailing for forcings in which more energy is injected into the 2D modes while the latter prevails for isotropic forcing. In the case of anisotropic forcing, whence the energy goes from the 2D to the 3D modes at low wave numbers, large-scale shear is created, resulting in a time scale τsh, associated with shear, thereby producing a ∼k-1 spectrum for the total energy with the horizontal energy of the 2D modes still following a ∼k⊥-5/3 scaling.

The interplay between helicity and rotation in turbulence: implications for scaling laws and small-scale dynamics

Invariance properties of physical systems govern their behaviour: energy conservation in turbulence drives a wide distribution of energy among modes, as observed in geophysical or astrophysical flows. In ideal hydrodynamics, the role of the invariance of helicity (correlation between velocity and its curl, measuring departures from mirror symmetry) remains unclear since it does not alter the energy spectrum. However, in the presence of rotation, significant differences emerge between helical and non-helical turbulent flows. We first briefly outline some of the issues such as the partition of energy and helicity among modes. Using massive numerical simulations, we then show that small-scale structures and their intermittency properties differ according to whether helicity is present or not, in particular with respect to the emergence of Beltrami core vortices that are laminar helical vertical updraft vortices. These results point to the discovery of a small parameter besides the Rossby number, a fact that would relate the problem of rotating helical turbulence to that of critical phenomena, through the renormalization group and weak-turbulence theory. This parameter can be associated with the adimensionalized ratio of the energy to helicity flux to small scales, the three-dimensional energy cascade being weak and self-similar.

Lack of universality in decaying magnetohydrodynamic turbulence

Using computations of three-dimensional magnetohydrodynamic (MHD) turbulence with a Taylor-Green flow, whose inherent time-independent symmetries are implemented numerically, and in the absence of either a forcing function or an imposed uniform magnetic field, we show that three different inertial ranges for the energy spectrum may emerge for three different initial magnetic fields, the selecting parameter being the ratio of nonlinear eddy to Alfvén time. Equivalent computational grids range from 128(3) to 2048(3) points with a unit magnetic Prandtl number and a Taylor Reynolds number of up to 1500 at the peak of dissipation. We also show a convergence of our results with Reynolds number. Our study is consistent with previous findings of a variety of energy spectra in MHD turbulence by studies performed in the presence of both a forcing term with a given correlation time and a strong, uniform magnetic field. However, in contrast to the previous studies, here the ratio of characteristic time scales can only be ascribed to the intrinsic nonlinear dynamics of the paradigmatic flows under study.

Cascades, thermalization, and eddy viscosity in helical Galerkin truncated Euler flows

The dynamics of the truncated Euler equations with helical initial conditions are studied. Transient energy and helicity cascades leading to Kraichnan helical absolute equilibrium at small scales, including a linear scaling of the relative helicity spectrum are obtained. Strong helicity effects are found using initial data concentrated at high wave numbers. Using low-wave-number initial conditions, the results of Cichowlas et.al. [Phys. Rev. Lett. 95, 264502 (2005)] are extended to helical flows. Similarities between the turbulent transient evolution of the ideal (time-reversible) system and viscous helical flows are found. Using an argument in the manner of Frisch et. al. [Phys. Rev. Lett. 101, 144501 (2008)], the excess of relative helicity found at small scales in the viscous run is related to the thermalization of the ideal flow. The observed differences in the behavior of truncated Euler and (constant viscosity) Navier-Stokes are qualitatively understood using the concept of eddy viscosity. The large scales of truncated Euler equations are then shown to follow quantitatively an effective Navier-Stokes dynamics based on a variable (scale dependent) eddy viscosity.

Spectral modeling of magnetohydrodynamic turbulent flows

We present a dynamical spectral model for large-eddy simulation of the incompressible magnetohydrodynamic (MHD) equations based on the eddy damped quasinormal Markovian approximation. This model extends classical spectral large-eddy simulations for the Navier-Stokes equations to incorporate general (non-Kolmogorovian) spectra as well as eddy noise. We derive the model for MHD flows and show that the introduction of an eddy damping time for the dynamics of spectral tensors, in the absence of equipartition between the velocity and magnetic fields, leads to better agreement with direct numerical simulations, an important point for dynamo computations.