Two-dimensional Navier-Stokes Turbulence Research Articles

Coherent structure detection and the inverse cascade mechanism in two-dimensional Navier–Stokes turbulence

Coherent structures in two-dimensional Navier–Stokes turbulence are ubiquitously observed in nature, experiments and numerical simulations. The present study conducts a comparison between several structure detection schemes based on the Okubo–Weiss criterion, the vorticity magnitude and Lagrangian coherent structures (LCSs), focusing on the inverse cascade in two-dimensional hydrodynamic turbulence. A recently introduced vortex scaling phenomenology (Burgess & Scott, J. Fluid Mech., vol. 811, 2017, pp. 742–756) allows the quantification of the respective thresholds required by these methods based on physical properties of the flow. The resulting improved comparability allows us to identify characteristic relative differences in the detection sensitivity between the employed structure detection techniques. With respect to the inverse cascade of energy, coherent structures contribute, as expected, substantially less to the cross-scale flux than the residual incoherent parts of the flow although the energetically dominant coherent structures lead to an important large-scale deformation of the energy spectrum. This cascade inactivity can be understood by an increased misalignment of strain-rate and subgrid stress tensors within coherent structures. At the same time, the structures exhibit strong and localised nonlinear cross-scale interactions that appear to stabilise them. We quantify and interpret the resulting shape preservation of coherent structures in terms of a multi-scale gradient approach (Eyink, J. Fluid Mech., vol. 549, 2006, pp. 191–214) as the depletion of strain rotation and vorticity gradient stretching whereas the dynamics of the residual fluctuations are consistent with the vortex thinning picture.

Multilocality and fusion rules on the generalized structure functions in two-dimensional and three-dimensional Navier-Stokes turbulence.

Using the fusion-rules hypothesis for three-dimensional and two-dimensional Navier-Stokes turbulence, we generalize a previous nonperturbative locality proof to multiple applications of the nonlinear interactions operator on generalized structure functions of velocity differences. We call this generalization of nonperturbative locality to multiple applications of the nonlinear interactions operator "multilocality." The resulting cross terms pose a new challenge requiring a new argument and the introduction of a new fusion rule that takes advantage of rotational symmetry. Our main result is that the fusion-rules hypothesis implies both locality and multilocality in both the IR and UV limits for the downscale energy cascade of three-dimensional Navier-Stokes turbulence and the downscale enstrophy cascade and inverse energy cascade of two-dimensional Navier-Stokes turbulence. We stress that these claims relate to nonperturbative locality of generalized structure functions on all orders and not the term-by-term perturbative locality of diagrammatic theories or closure models that involve only two-point correlation and response functions.

Energy and potential enstrophy flux constraints in quasi-geostrophic models

We investigate an inequality constraining the energy and potential enstrophy flux spectra in two-layer and multi-layer quasi-geostrophic models. Its physical significance is that it can diagnose whether any given multi-layer model that allows co-existing downscale cascades of energy and potential enstrophy can allow the downscale energy flux to become large enough to yield a mixed energy spectrum where the dominant k−3 scaling is overtaken by a subdominant k−5/3 contribution beyond a transition wavenumber kt situated in the inertial range. The validity of the flux inequality implies that this scaling transition cannot occur within the inertial range, whereas a violation of the flux inequality beyond some wavenumber kt implies the existence of a scaling transition near that wavenumber. This flux inequality holds unconditionally in two-dimensional Navier–Stokes turbulence, however, it is far from obvious that it continues to hold in multi-layer quasi-geostrophic models, because the dissipation rate spectra for energy and potential enstrophy no longer relate in a trivial way, as in two-dimensional Navier–Stokes. We derive the general form of the energy and potential enstrophy dissipation rate spectra for a generalized symmetrically coupled multi-layer model. From this result, we prove that in a symmetrically coupled multi-layer quasi-geostrophic model, where the dissipation terms for each layer consist of the same Fourier-diagonal linear operator applied on the streamfunction field of only the same layer, the flux inequality continues to hold. It follows that a necessary condition to violate the flux inequality is the use of asymmetric dissipation where different operators are used on different layers. We explore dissipation asymmetry further in the context of a two-layer quasi-geostrophic model and derive upper bounds on the asymmetry that will allow the flux inequality to continue to hold. Asymmetry is introduced both via an extrapolated Ekman term, based on a 1980 model by Salmon, and via differential small-scale dissipation. The results given are mathematically rigorous and require no phenomenological assumptions about the inertial range. Sufficient conditions for violating the flux inequality, on the other hand, require phenomenological hypotheses, and will be explored in future work.

Transition of the scaling law in inverse energy cascade range caused by a nonlocal excitation of coherent structures observed in two-dimensional turbulent fields

We numerically investigate the inverse energy cascade range of two-dimensional Navier-Stokes turbulence. Our focus is on the universality of the Kolmogorov's phenomenology. In our direct numerical simulations, two types of forcing processes, the random forcing and the deterministic forcing, are employed besides the systematically varied numerical parameters. We first calculate the two-dimensional Navier-Stokes equations and confirm that results in the quasi steady state are consistent with the classical phenomenology for both types of forcing processes. It is also found that the difference in forcing process appears after the inverse energy cascade range reaches the system size; the dipole coherent vortices emerge and grow only when the random forcing is adopted. Then we add a large-scale drag term to the Navier-Stokes equations to obtain the statistically stationary state. When the random forcing is used, the scaling exponent of the energy spectrum in the stationary state starts to differ from the predicted -5/3 in the inverse energy cascade range as the infrared Reynolds number Re(d) increases, where Re(d) is defined as k(f)/k(d) with the forcing wave number k(f) and the large-scale drag wave number k(d). That can be interpreted as a transition phenomenon in which the local maximum vorticity grows like an order parameter caused by excitation of strong coherent vortices. Strong coherent vortices emerge and grow after the quasi steady state and destroy the scaling law when Re(d) is over a critical value. These coherent vortices are not due to the finite-size effect, unlike the dipole coherent vortices. On the other hand, when the deterministic forcing is adopted, strong coherent vortices are hardly seen and the -5/3 scaling law holds independently of Re(d). We examine the cases of the combination of both types of forcing processes and find that formation of such coherent vortices is sensitive to the mechanism of the external forcing process as well as the numerical parameters. Several types of large-scale drag terms are also tested and their insignificant influence on these qualitative properties is revealed.

GRANULAR TURBULENCE IN TWO DIMENSIONS: MICROSCALE REYNOLDS NUMBER AND FINAL CONDENSED STATES

Granular gases from the viewpoint of "two-dimensional turbulence" are investigated. In the quasi-elastic and thermodynamic limit, we obtained clear evidence for an enstrophy (square of vorticity) cascade and -3 exponent in the Kraichnan–Leith–Bachelor energy spectrum by performing large-scale (N ~ 16.8 million number of disks) event-driven molecular dynamics simulations. In these calculations, the enstrophy dissipation rate showed a strong relationship with the evolution of the exponent in the energy spectrum. The growth of the Reynolds number based on the microscale confirmed that the enstrophy cascade regime was that of fully developed turbulence. Moreover, a condensed state resembling Bose–Einstein condensation in decaying two-dimensional Navier–Stokes turbulence also appeared as the final attractor of the evolving granular gas in the long time limit.

Dynamics of saturated energy condensation in two-dimensional turbulence

In two-dimensional forced Navier-Stokes turbulence, energy cascades to the largest scales in the system to form a pair of coherent vortices known as the Bose condensate. We show, both numerically and analytically, that the energy condensation saturates and the system reaches a statistically stationary state. The time scale of saturation is inversely proportional to the viscosity and the saturation energy level is determined by both the viscosity and the force. We further show that, without sufficient resolution to resolve the small-scale enstrophy spectrum, numerical simulations can give a spurious result for the saturation energy level. We also find that the movement of the condensate is similar to the motion of an inertial particle with an effective drag force. Furthermore, we show that the profile of the saturated coherent vortices can be described by a Gaussian core with exponential wings.

Influence of flow topology on Lagrangian statistics in two-dimensional turbulence

The influence of flow topology on Lagrangian statistics in fluid turbulence is investigated. The Weiss criterion provides a tool to split the flow into topologically different regions: elliptic (vortex dominated), hyperbolic (deformation dominated), and intermediate (turbulent background) regions. The flow corresponds to forced two-dimensional Navier-Stokes turbulence in a double periodic or a circular domain, the latter with no-slip boundary conditions. A Lagrangian approach is adopted by tracking large ensembles of passively advected tracers. The pdfs (probability density functions) of the residence time in the topologically different regions are computed introducing the Lagrangian Weiss field, i.e., the Weiss field computed along the particles' trajectories. In elliptic and hyperbolic regions, the pdfs of the residence time have self-similar algebraic decaying tails. In contrast, in the intermediate regions the pdf shows exponentially decaying tails. Furthermore, the conditional pdfs of the Lagrangian acceleration with respect to the flow topology confirm that both strong elliptic and hyperbolic regions contribute to the large values of Lagrangian acceleration.

Effects of friction on forced two-dimensional Navier-Stokes turbulence

Large-scale dissipation mechanisms have been routinely employed in numerical simulations of two-dimensional turbulence to absorb energy at large scales, presumably mimicking the quasisteady picture of Kraichnan in an unbounded fluid. Here, "side effects" of such a mechanism--mechanical friction--on the small-scale dynamics of forced two-dimensional Navier-Stokes turbulence are elaborated by both theoretical and numerical analysis. Given a positive friction coefficient α, viscous dissipation of enstrophy has been known to vanish in the inviscid limit ν→0. This effectively renders the scale-neutral friction the only mechanism responsible for enstrophy dissipation in that limit. The resulting dynamical picture is that the classical enstrophy inertial range becomes a dissipation range in which the dissipation of enstrophy by friction mainly occurs. For each α>0, there exists a critical viscosity ν(c), which depends on physical parameters, separating the regimes of predominant viscous and frictional dissipation of enstrophy. It is found that ν(c)=[η'(1/3)/(Ck(f)(2))]exp[-η'(1/3)/(Cα)], where η' is half the enstrophy injection rate, k(f) is the forcing wave number, and C is a nondimensional constant (the Kraichnan-Batchelor constant). The present results have important theoretical and practical implications. Apparently, mechanical friction is a poor choice in numerical attempts to address fundamental issues concerning the direct enstrophy transfer in two-dimensional Navier-Stokes turbulence. Furthermore, as relatively strong friction naturally occurs on the surfaces and at lateral boundaries of experimental fluids as well as at the interfaces of shallow layers in geophysical fluid models, the frictional effects discussed in this study are crucial in understanding the dynamics of these systems.

Vorticity statistics in the direct cascade of two-dimensional turbulence

For the direct cascade of steady two-dimensional (2D) Navier-Stokes turbulence, we derive analytically the probability of strong vorticity fluctuations. When ϖ is the vorticity coarse-grained over a scale R, the probability density function (PDF), P(ϖ), has a universal asymptotic behavior lnP~-ϖ/ϖ(rms) at ϖ≫ϖ(rms)=[Hln(L/R)](1/3), where H is the enstrophy flux and L is the pumping length. Therefore, the PDF has exponential tails and is self-similar, that is, it can be presented as a function of a single argument, ϖ/ϖ(rms), in distinction from other known direct cascades.

Lagrangian statistics and flow topology in forced two-dimensional turbulence

A study of the relationship between Lagrangian statistics and flow topology in fluid turbulence is presented. The topology is characterized using the Weiss criterion, which provides a conceptually simple tool to partition the flow into topologically different regions: elliptic (vortex dominated), hyperbolic (deformation dominated), and intermediate (turbulent background). The flow corresponds to forced two-dimensional Navier-Stokes turbulence in doubly periodic and circular bounded domains, the latter with no-slip boundary conditions. In the double periodic domain, the probability density function (pdf) of the Weiss field exhibits a negative skewness consistent with the fact that in periodic domains the flow is dominated by coherent vortex structures. On the other hand, in the circular domain, the elliptic and hyperbolic regions seem to be statistically similar. We follow a Lagrangian approach and obtain the statistics by tracking large ensembles of passively advected tracers. The pdfs of residence time in the topologically different regions are computed introducing the Lagrangian Weiss field, i.e., the Weiss field computed along the particles' trajectories. In elliptic and hyperbolic regions, the pdfs of the residence time have self-similar algebraic decaying tails. In contrast, in the intermediate regions the pdf has exponential decaying tails. The conditional pdfs (with respect to the flow topology) of the Lagrangian velocity exhibit Gaussian-like behavior in the periodic and in the bounded domains. In contrast to the freely decaying turbulence case, the conditional pdfs of the Lagrangian acceleration in forced turbulence show a comparable level of intermittency in both the periodic and the bounded domains. The conditional pdfs of the Lagrangian curvature are characterized, in all cases, by self-similar power-law behavior with a decay exponent of order -2.

Two-Dimensional Navier–Stokes Turbulence in Bounded Domains

In this review we will discuss recent experimental and numerical results of quasi-two-dimensional decaying and forced Navier–Stokes turbulence in bounded domains. We will give a concise overview of developments in two-dimensional turbulence research, with emphasis on the progress made during the past 10 years. The scope of this review concerns the self-organization of two-dimensional Navier–Stokes turbulence, the quasi-stationary final states in domains with no-slip boundaries, the role of the lateral no-slip walls on two-dimensional turbulence, and their role on the possible destabilization of domain-sized vortices. The overview of the laboratory experiments on quasi-two-dimensional turbulence is restricted to include only those carried out in thin electromagnetically forced shallow fluid layers and in stratified fluids. The effects of the quasi-two-dimensional character of the turbulence in the laboratory experiments will be discussed briefly. As a supplement, the main results from numerical simulations of forced and decaying two-dimensional turbulence in rectangular and circular domains, thus explicitly taking into account the lateral sidewalls, will be summarized and compared with the experimental observations.

Constraints on inertial range scaling laws in forced two-dimensional Navier–Stokes turbulence

This study examines the dual transfer and spectral scaling laws in forced two-dimensional Navier–Stokes turbulence. The k−5∕3 energy inertial range emerges from this analysis as a minimally steep range that is barely able to support an inverse energy cascade. In other words, an energy inertial range shallower than k−5∕3 is incapable of sustaining a persistent transfer of energy to the large scales, and steeper inertial ranges are possible. The vorticity is found to diverge in the limit of large Reynolds numbers or small viscosity, allowing the classical k−3 enstrophy inertial range to be the only one that may remain bounded in this limit. Steeper (unbounded) enstrophy inertial ranges cannot be ruled out.

Nonrobustness of the two-dimensional turbulent inverse cascade

The inverse energy cascade in two-dimensional Navier-Stokes turbulence is examined in the quasisteady regime, with small-scale, band-limited forcing at scale kf-1, with particular attention to the influence of forcing Reynolds number Re on the energy distribution at large scales. The strength of the inverse energy cascade, or fraction of energy input that is transferred to larger scales, increases monotonically toward unity with increasing Re proportional, variantkmax2kf2, where kmax is the maximum resolved wave number. Moreover, as Re increases beyond a critical value, for which a direct enstrophy cascade to small scales is first realized, the energy spectrum in the energy-cascading range steepens from a k-53 to k-2 dependence. The steepening is interpreted as the result of a greater tendency for coherent vortex formation in cases when forcing scales are adequately resolved. In spectral space, it is associated with nonlocality of the inverse energy transfer.

Inverse Turbulent Cascades and Conformally Invariant Curves

We offer a new example of conformal invariance (local scale invariance) far from equilibrium-the inverse cascade of surface quasigeostrophic (SQG) turbulence. We show that temperature isolines are statistically equivalent to curves that can be mapped into a one-dimensional Brownian walk (called Schramm-Loewner evolution or SLEkappa). The diffusivity is close to kappa=4, that is, isotemperature curves belong to the same universality class as domain walls in the O(2) spin model. Several statistics of temperature clusters and isolines are shown to agree with the theoretical expectations for such a spin system at criticality. We also show that the direct cascade in two-dimensional Navier-Stokes turbulence is not conformal invariant. The emerging picture is that conformal invariance may be expected for inverse turbulent cascades of strongly interacting systems.

Vanishing enstrophy dissipation in two-dimensional Navier–Stokes turbulence in the inviscid limit

Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) developed a theory of two-dimensional turbulence based on the assumption that the dissipation of enstrophy (mean-square vorticity) tends to a finite non-zero constant in the limit of infinite Reynolds number Re. Here, by assuming power-law spectra, including the one predicted by Batchelor's theory, we prove that the maximum dissipation of enstrophy is in fact zero in this limit. Specifically, as Re → ∞, the dissipation approaches zero no slower than (lnRe) -1/2 . The physical reason behind this result is that the decrease of viscosity enhances the production of both palinstrophy (mean-square vorticity gradients) and its dissipation - but in such a way that the net growth of palinstrophy is less rapid than the decrease of viscosity, resulting in vanishing enstrophy dissipation. This result generalizes to a rich class of quasi-geostrophic models as well as to the case of a passive tracer in layerwise-two-dimensional turbulent flows having bounded enstrophy.

Enstrophy dissipation in freely evolving two-dimensional turbulence

Freely decaying two-dimensional Navier-Stokes turbulence is studied. The conservation of vorticity by advective nonlinearities renders a class of Casimirs that decays under viscous effects. A rigorous constraint on the palinstrophy production by nonlinear transfer is derived, and an upper bound for the enstrophy dissipation is obtained. This bound depends only on the decaying Casimirs, thus allowing the enstrophy dissipation to be bounded from above in terms of initial data of the flows. An upper bound for the enstrophy dissipation wave number is derived and the new result is compared with the classical dissipation wave number.

Numerical study of the decay of enstrophy in a two-dimensional Navier–Stokes fluid in the limit of very small viscosities

The rate of enstrophy decay in two-dimensional Navier–Stokes turbulence is explored numerically with high resolution pseudospectral simulations. Initial conditions and all other physical parameters are held fixed while viscosity is decreased through about three orders of magnitude, while increasing the numerical resolution as required to resolve the smallest scales that develop. The attempt is to discern whether or not a regime is being approached in which the behavior of the vorticity decay could be approximated as ideal and nondissipative. Although the enstrophy decay shows a (logarithmic) tendency to go to zero as the viscosity is decreased, at fixed times, the extrapolated values of viscosity to approach a zero enstrophy decay regime lie many orders of magnitude out of our currently available numerical resolution.

An easily implemented task-based parallel scheme for the Fourier pseudospectral solver applied to 2D Navier–Stokes turbulence

An efficient parallel scheme is proposed for performing direct numerical simulation (DNS) of two-dimensional Navier–Stokes turbulence at high Reynolds numbers. We illustrate the resulting numerical code by displaying relaxation to states close to those that have been predicted by statistical–mechanical methods which start from ideal (Euler) fluid mechanics. The validation of these predictions by DNS requires unusually long computation times on single-cpu workstations, and suggests the use of parallel computation. The performance of our MPI Fortran 90 code on the SGI Origin 3800 is reported, together with its comparison with another parallel method. A few computational results that illustrate tests of the statistical–mechanical predictions are presented.

Eulerian statistically preserved structures in passive scalar advection.

We analyze numerically the time-dependent linear operators that govern the dynamics of Eulerian correlation functions of a decaying passive scalar advected by a stationary, forced two-dimensional Navier-Stokes turbulence. We show how to naturally discuss the dynamics in terms of effective compact operators that display Eulerian statistically preserved structures which determine the anomalous scaling of the correlation functions. In passing we point out a bonus of the present approach, in providing analytic predictions for the time-dependent correlation functions in decaying turbulent transport.

Intermittency in two-dimensional Ekman-Navier-Stokes turbulence.

We study the statistics of the vorticity field in two-dimensional Navier-Stokes turbulence with linear Ekman friction. We show that the small-scale vorticity fluctuations are intermittent, as conjectured by Bernard [Europhys. Lett. 50, 333 (2000)] and Nam et al. [Phys. Rev. Lett. 84, 5134 (2000)]. The small-scale statistics of vorticity fluctuations coincide with that of a passive scalar with finite lifetime transported by the velocity field itself.