Energy Cascade In Two-dimensional Turbulence Research Articles

Korrelyatsii zavikhrennosti vnutri kogerentnogo vikhrya

We investigate fluctuations of vorticity inside a coherent vortex generated by the inverse energy cascade in two-dimensional turbulence. Temporal and spatial correlations can be characterized by the pair correlation function. The interaction of fluctuations leads to a nonzero third moment of vorticity. We analyze the pair correlation function and the third moment using a model in which the pumping is short-correlated in time and derive explicit expressions for the Gaussian spatial correlation function for the pumping force.

Statistics of clustered vortices in the inverse energy cascade of two-dimensional turbulence

Strong, long-lived vortices play a prominent role in two-dimensional (2D) turbulent flows, and understanding their statistics is key to unravelling 2D turbulence. Here it is shown that clustered same-sign vortices in forced 2D turbulence have a scale-invariant distribution of areas and are described by a mean-field theory. Larger clusters come apart more quickly, and the flow surrounding the remaining clusters becomes progressively more randomized and neutral, while vortex merger concentrates coherent enstrophy into a smaller and smaller fraction of the domain.

Dissipation of coherent structures in confined two-dimensional turbulence

Since the seminal article “Inertial ranges in two-dimensional turbulence” by Kraichnan in 1967, our understanding of the dynamics and transport properties of two-dimensional turbulence is largely built on the assumption of homogeneity and isotropy of statistically steady or decaying turbulence. In the last two decades, more attention has been paid to the presence of lateral walls, either with stress-free or no-slip boundary conditions, and also considering a variety of geometries such as square, rectangular, or circular domains. The impact of confining boundaries on the dynamics of two-dimensional turbulence is important. This is in sharp contrast with three-dimensional turbulence, where homogeneity and isotropy are locally restored due to the cascade process in the inertial range. The impact of confining boundaries is therefore limited in three-dimensional turbulence. The presence of an inverse energy cascade in two-dimensional turbulence, however, will continuously generate large-scale energy-containing eddies, and their vigorous interaction with, in particular, no-slip walls generates large amounts of vorticity and contributes significantly to the dissipation of kinetic energy of the flow. The dissipation is even strongly enhanced compared with the unbounded case. In this review, we will focus on one of the elementary structures observed in two-dimensional turbulent flows: the dipolar vortex. With its self-induced velocity, it propagates easily through the domain and is hence likely to interact with domain boundaries. Standard vortex generation mechanisms allow to create well-defined dipoles and to investigate the collision of such structures with rigid domain boundaries in detail. Relevant aspects of the collision process concern the dynamics and stability of the generated boundary layers, the vorticity, and vorticity gradients contained in these boundary layers, and the dissipation of kinetic energy when dipoles collide with walls. Some of these aspects will be discussed in this review. Moreover, we are interested in the Reynolds-number dependence of these processes, including exploration of the vanishing viscosity limit.

Detached eddy simulation of shallow mixing layer development between parallel streams

Results of a high resolution Detached Eddy Simulation (DES) are used to characterize the evolution of a shallow mixing layer developing between two parallel streams in a long open channel with a smooth flat bed and medium size dunes. The study discusses the vertical non-uniformity in the mixing layer structure and provides a quantitative characterization of the growth of the large-scale quasi two-dimensional (2D) coherent structures with the distance from the splitter plate. Results show that in streamwise sections situated between 75D (D is the channel depth) and 150D from the splitter plate, the width of the mixing layer close to the free surface is 20–30% more than the width in the near-bed region. This is mostly because of the tilting of the mixing layer interface on the low-speed side toward the low speed stream as the free surface is approached. Power spectra of the horizontal velocity components near the free surface show the presence of a −3 subrange, corresponding to inverse energy cascade in two-dimensional turbulence, at streamwise locations situated more than 10D from the splitter plate, consistent with the presence of large-scale quasi 2D horizontal eddies and the transfer of energy (inverse energy cascade) from the smaller scales toward these eddies. Consistent with visualizations of the mass transport of a passive scalar within the mixing layer, close to the free surface, the estimated streamwise length of the quasi 2D mixing layer eddies is about 2.5–3.0 times larger than the local width of the mixing layer. The presence of large-scale roughness elements in the form of an array of two-dimensional dunes with a maximum height of 0.25D (D is the channel depth) induces a much more rapid and larger shift of the centerline of the mixing layer due to the increased influence of the bottom roughness.

Two-point vorticity statistics in the inverse cascade of two-dimensional turbulence

A statistical analysis of the two-point vorticity statistics in the inverse energy cascade of two-dimensional turbulence is presented in terms of probability density functions (PDFs). Evolution equations for the PDFs are derived in the framework of the Lundgren–Monin–Novikov hierarchy, and the unclosed terms are studied with the help of direct numerical simulations (DNS). Furthermore, the unclosed terms are evaluated in a Gaussian approximation and compared to the DNS results. It turns out that the statistical equations can be interpreted in terms of the dynamics of screened vortices. The two-point statistics is related to the dynamics of two point vortices with screened velocity field, where an effective relative motion of the two point vortices originating from the turbulent surroundings is identified to be a major characteristics of the dynamics underlying the inverse cascade.

Dissipation scales and anomalous sinks in steady two-dimensional turbulence

In previous papers I have argued that the fusion rules hypothesis, which was originally introduced by L'vov and Procaccia in the context of the problem of three-dimensional turbulence, can be used to gain a deeper insight in understanding the enstrophy cascade and inverse energy cascade of two-dimensional turbulence. In the present paper, we show that the fusion rules hypothesis, combined with nonperturbative locality, itself a consequence of the fusion rules hypothesis, dictates the location of the boundary separating the inertial range from the dissipation range. In so doing, the hypothesis that there may be an anomalous enstrophy sink at small scales and an anomalous energy sink at large scales emerges as a consequence of the fusion rules hypothesis. More broadly, we illustrate the significance of viewing inertial ranges as multi-dimensional regions where the fully unfused generalized structure functions of the velocity field are self-similar, by considering, in this paper, the simplified projection of such regions in a two-dimensional space, involving a small scale r and a large scale R, which we call, in this paper, the (r,R)-plane. We see, for example, that the logarithmic correction in the enstrophy cascade, under standard molecular dissipation, plays an essential role in inflating the inertial range in the (r,R) plane to ensure the possibility of local interactions. We have also seen that increasingly higher orders of hyperdiffusion at large scales or hypodiffusion at small scales make the predicted sink anomalies more resilient to possible violations of the fusion rules hypothesis.

Physical mechanism of the inverse energy cascade of two-dimensional turbulence: a numerical investigation

We report an investigation of inverse energy cascade in steady-state two-dimensional turbulence by direct numerical simulation (DNS) of the two-dimensional Navier–Stokes equation, with small-scale forcing and large-scale damping. We employed several types of damping and dissipation mechanisms in simulations up to 20482resolution. For all these simulations we obtained a wavenumber range for which the mean spectral energy flux is a negative constant and the energy spectrum scales ask−5/3, consistent with the predictions of Kraichnan (Phys. Fluids, vol. 439, 1967, p. 1417). To gain further insight, we investigated the energy cascade in physical space, employing a local energy flux defined by smooth filtering. We found that the inverse energy cascade is scale local, but that the strongly local contribution vanishes identically, as argued by Kraichnan (J. Fluid Mech., vol. 47, 1971, p. 525). The mean flux across a length scale ℓ was shown to be due mainly to interactions with modes two to eight times smaller. A major part of our investigation was devoted to identifying the physical mechanism of the two-dimensional inverse energy cascade. One popular idea is that inverse energy cascade proceeds via merger of like-sign vortices. We made a quantitative study employing a precise topological criterion of merger events. Our statistical analysis showed that vortex mergers play a negligible direct role in producing mean inverse energy flux in our simulations. Instead, we obtained with the help of other works considerable evidence in favour of a ‘vortex thinning’ mechanism, according to which the large-scale strains do negative work against turbulent stress as they stretch out the isolines of small-scale vorticity. In particular, we studied a multi-scale gradient (MSG) expansion developed by Eyink (J. Fluid Mech., vol. 549, 2006a, p. 159) for the turbulent stress, whose contributions to inverse cascade can all be explained by ‘thinning’. The MSG expansion up to second order in space gradients was found to predict well the magnitude, spatial structure and scale distribution of the local energy flux. The majority of mean flux was found to be due to the relative rotation of strain matrices at different length scales, a first-order effect of ‘thinning’. The remainder arose from two second-order effects, differential strain rotation and vorticity gradient stretching. Our findings give strong support to vortex thinning as the fundamental mechanism of two-dimensional inverse energy cascade.

Robert Harry Kraichnan

Robert Harry Kraichnan

On the universality of large-scale turbulence

Large-scale three-dimensional turbulence is a challenge to researchers, being one of the most complicated aspects of turbulence studies. The universal large scales behavior connected with the inverse energy cascade in two-dimensional turbulence has been known for about 35 years and studied experimentally and numerically. We have revealed the universality of three-dimensional large-scale turbulence properties experimentally and described it theoretically. A rigorous solution of this problem in the presence of an energy source gives scaling exponents −4/3 for velocity correlations and 1/3 for energy spectra of the large-scale turbulence. Such spectra are also observed in atmospheric air flows under different conditions—stable, convective, in cirrus clouds. The revealed physical phenomenon is important for the development of turbulence theory complementing the results obtained for its smaller scales.

Sensitivity of horizontal flows to forcing geometry

We investigate the horizontal flow produced by source–sink forcing in a stably stratified fluid. The forcing jets are kept laminar and are placed along the boundary of a square domain. We find that the resultant flow patterns are extremely sensitive to the forcing geometry. The single dominant vortex pattern, interpreted as the result of inverse energy cascade of two-dimensional turbulence in our previous work (Boubnov, Dalziel & Linden 1994), turns out to be a special case. We show that some of the steady patterns resemble the eigenmodes of the Helmholtz equation as the inviscid vorticity equation. Although there are significant discrepancies in the streamfunction vs. vorticity relations between the observed flows and the analytical solutions, we identify the differences as a result of viscous diffusion of vorticity from the source flows. We also study the transition from forced to decaying flow. The flow assumes the properties of Stokes flow at quite large Reynolds number, indicating transformation into patterns with small advective acceleration.

Predictability of the inverse energy cascade in 2D turbulence

The predictability problem in the inverse energy cascade of two-dimensional turbulence is addressed by means of high resolution direct numerical simulations. The growth rate as a function of the error level is determined by means of a finite size extension of the Lyapunov exponent. For errors within the inertial range, the linear growth of the error energy, predicted by dimensional argument, is verified with great accuracy. Our numerical findings quantitatively confirm the results of the classical TFM closure approximation.

Inverse cascade in film flows

Developed interfacial dynamics of thin film flows with moderate Reynolds numbers exhibits a remarkable feature: the evolution is dominated by solitary-like pulses with a natural large wavelength between them. This phenomenon is robust and resembles the inverse energy cascade in two-dimensional turbulence. A new simple evolution equation is proposed to describe the film flow dynamics which captures such an inverse cascade. The equation combines the simplest kinematic nonlinearity with the exact linear term. The spectral kernel of the linear term is found from the numerical solution of the associated linear stability problem.

Inverse energy cascade in two-dimensional turbulence: deviations from gaussian behavior

High-resolution numerical simulations of stationary inverse energy cascade in two-dimensional turbulence are presented. Deviations from Gaussian behavior of velocity differences statistics are quantitatively investigated. The level of statistical convergence is pushed enough to permit reliable measurement of the asymmetries in the probability distribution functions of longitudinal increments and odd-order moments, which bring the signature of the inverse energy flux. No measurable intermittency corrections could be found in their scaling laws. The seventh order skewness increases by almost two orders of magnitude with respect to the third, thus becoming of order unity.

Derivation of the relative dispersion law in the inverse energy cascade of two-dimensional turbulence

We present a numerical study of relative dispersion processes in a stationary two-dimensional turbulence. A simple analytical derivation of the Richardson relative dispersion law in the inverse energy cascade is proposed. A comparative study is performed using Lagrangian numerical experiments.

Two-layer geostrophic vortex dynamics. Part 2. Alignment and two-layer V-states

The process of alignment, a new fundamental interaction between vortices in a stratified and rapidly rotating fluid, is defined and studied in detail in the context of the two-layer quasi-geostrophic model. Alignment occurs when two vortices in different density layers coalesce by reducing their horizontal separation. It is found that only vortices whose radii are comparable with or larger than the Rossby deformation radius can align. In the same way as the merger process (in a single two-dimensional layer) is related to the reverse energy cascade of two-dimensional turbulence, geostrophic potential vorticity alignment is related the barotropic-to-baroclinic energy cascade of geostrophic turbulence in two layers. It is also shown how alignment is intimately connected with the existence of two-layer doubly connected geostrophic potential vorticity equilibria (V-states), for which the analysis of the geometry of the stream function in the corotating frame is found to be a crucial diagnostic. The finite-area analogues of the hetons of Hogg & Stommel (1985) are also determined: they consist of a propagating pair of opposite-signed potential vorticity patches located in different layers.

Spectral laws for the energy cascade in a two-dimensional turbulence

We give a Von Karman-type model for the inertial transfer to develop a generalized spectral law for the inverse energy cascade in a forced two-dimensional turbulence that is in agreement with the experimental observations (Sommeria, 1986) of the inverse energy cascade in a statistically steady two-dimensional turbulence. We also demonstrate that the transfer of turbulent kinetic energy at small wave numbers can be modeled as a stationary continuous spectral cascading process. The equipartition spectrum for two-dimensional turbulence is demonstrated to be viscous in character and to differ radically from the equipartition spectrum for three-dimensional turbulence which is inviscid in character.

Inverse energy cascade in two-dimensional turbulence

The closure method developed in earlier papers [J. Qian, Phys. Fluids 26, 2098 (1983); 27, 2412 (1984)] is applied to the study of the inverse energy cascade in two-dimensional turbulence. The resultant inertial-range energy spectrum is E(k)=2.58g0.244ε2/3k−5/3. Here g is a localization factor and ε is the rate of energy cascade. This result is compatible with the numerical experiments by Lilly [D. K. Lilly, Phys. Fluids Suppl. II 12, 24 (1969)], Siggia and Aref [E. D. Siggia and H. Aref, Phys. Fluids 24, 171 (1981)], and Frisch and Sulem [U. Frisch and P. L. Sulem, Phys. Fluids 27, 1921 (1984)].

Point-vortex simulation of the inverse energy cascade in two-dimensional turbulence

The conventional cloud-in-cell vortex representation of the two-dimensional Euler equations is modified to allow energy injection from small scales. A simulation of the inverse cascade yields an energy spectrum consistent with k−5/3 and a Kolmogoroff constant of 14.