Revisiting two-dimensional turbulence and mesoscale spectra

<p>Two-dimensional (2D) turbulence is not only a basic research topic that needs further investigations, it is also relevant for wind energy applications as today’s wind farm clusters can be as large as thousands of kilometers squared and individual turbines hundreds of meters tall. This challenges the use of classical turbulence models applicable for scales smaller than ~1 h, or as denoted in Högström et al. (2002) the Kolmogoroff inertial subrange, the shear production range, and for ranges the spectral gap region.</p><p>This study revisits some key characteristics of 2D turbulence and interpretation of the physics behind it, including general literatures as well as a series of our studies in recent years (Larsén et al. 2013, 2016, 2021). This includes</p><ul><li>The respective frequency/wave number range and the spectral behaviours for the wind speed: the synoptic scales where the spectral slope is -3, the mesoscale where the spectral slope is typically -5/3, and the gap region. We analyze at what scales the spectra meet and merge, and how the spectra are affected by weather types, seasons and stability.</li> <li>The 2D-isotropy characteristics. We analyze how the longitudinal and lateral velocities correlate across the scales.</li> <li>The application of stationarity. The validity of an assumption of stationary time series decides to how large scales we can perform the analysis of the longitudinal and lateral velocity components, the Taylor frozen hypothesis and 2D-isotropy.</li> </ul><p>The primary datasets are from several met stations over Denmark and the North Sea region, including both 10-min and sonic measurements from about 10 m up to 240 m.</p><p> </p><p>References:</p><p>Högström U, Hunt J, Smedman AS (2002) Theory and measurements for turbulence spectra and variances in the atmospheric neutral surface layer. Boundary-Layer Meteorol 103:101–124</p><p>Larsén, X. G., Larsen, S. E., Petersen, E. L., & Mikkelsen, T. K. (2021). A Model for the Spectrum of the Lateral Velocity Component from Mesoscale to Microscale and Its Application to Wind-Direction Variation. Boundary-Layer Meteorology, 178, 415-434. https://doi.org/10.1007/s10546-020-00575-0</p><p>Larsén X. Larsen S. and Petersen E. (2016): Full-scale spectrum of the boundary layer wind. Boundary-Layer Meteorology, Vol 159, p 349-371</p><p>Larsén X., Vincent C. and Larsen S.E. (2013): Spectral structure of mesoscale winds over the water, Q. J. R. Meteorol. Soc., DOI:10.1002/qj.2003, 139, 685-700.</p>

Three dimensional flows beneath a thin layer of 2D turbulence induced by Faraday waves

Faraday waves occur on a fluid being subject to vertical shaking. Although it is well known that form and shape of the wave pattern depend on driving amplitude and frequency, only recent studies discovered the existence of a horizontal velocity field at the surface, called Faraday flow. This flow exhibits attributes of two-dimensional turbulence and is replicated in this study. Despite the increasing attention towards the inverse energy flux in the Faraday flow and other not strictly two-dimensional (2D) systems, little is known about the velocity fields developing beneath the fluid surface. In this study, planar velocity fields are measured by means of particle image velocimetry with high spatio-temporal resolution on the water surface and at different depths below it. A sudden drop in velocity and turbulent kinetic energy is observed at half a Faraday wavelength below the surface revealing that the surface flow is the main source of turbulent fluid motion. The flow structures below the surface comprise much larger spatial scales than those on the surface leading to very long-tailed temporal and spatial velocity (auto-) correlation functions. The three-dimensionality of the flow is estimated by the compressibility, which increases strongly with depth while the divergence changes its appearance from intermittent and single events to a large scale pattern resembling 2D cut-planes of convection rolls. Our findings demonstrate that the overall fluid flow beneath the surface is highly three-dimensional and that an inverse cascade and aspects of a confined 2D turbulence can coexist with a three-dimensional flow. Graphic abstract

Renormalized Onsager functions and merging of vortex clusters

This paper is devoted to an heuristic discussion of the merging mechanism between two clusters of point vortices, supported by some numerical simulations. A concept of renormalized Onsager function is introduced, elaboration of the solutions of the mean field equation. It is used to understand the shape of the single cluster observed as a result of the merging process. Potential implications for the inverse cascade 2D turbulence are discussed.

Dynamics of a long chain in turbulent flows: impact of vortices

We show and explain how a long bead–spring chain, immersed in a homogeneous isotropic turbulent flow, preferentially samples vortical flow structures. We begin with an elastic, extensible chain which is stretched out by the flow, up to inertial-range scales. This filamentary object, which is known to preferentially sample the circular coherent vortices of two-dimensional (2D) turbulence, is shown here to also preferentially sample the intense, tubular, vortex filaments of three-dimensional (3D) turbulence. In the 2D case, the chain collapses into a tracer inside vortices. In the 3D case on the contrary, the chain is extended even in vortical regions, which suggests that the chain follows axially stretched tubular vortices by aligning with their axes. This physical picture is confirmed by examining the relative sampling behaviour of the individual beads, and by additional studies on an inextensible chain with adjustable bending-stiffness. A highly flexible, inextensible chain also shows preferential sampling in three dimensions, provided it is longer than the dissipation scale, but not much longer than the vortex tubes. This is true also for 2D turbulence, where a long inextensible chain can occupy vortices by coiling into them. When the chain is made inflexible, however, coiling is prevented and the extent of preferential sampling in two dimensions is considerably reduced. In three dimensions, on the contrary, bending stiffness has no effect, because the chain does not need to coil in order to thread a vortex tube and align with its axis. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

A new data-driven subgrid 2d turbulence parameterization and comparison with conventional kinetic energy backscatter parameterizations in NEMO ocean model

<p>Kinetic energy backscatter (KEB) parameterizations of subgrid 2d turbulence have shown their efficiency in ocean models as they restore activity of mesoscale eddies. Modern KEBs utilize only two properties of badly resolved inverse energy cascade: KEB tendency should be larger than turbulent viscosity in spatial scale and amount of returning energy should compensate energy loss due to eddy viscosity. Typical operators for KEB tendency are Laplace operator with negative viscosity coefficient and stochastic process. Application of artificial neural networks (ANN) to approximate subgrid forces may give rise to new KEB models. The main challenge in this direction is to preprocess subgrid forces in such a way to reveal a part corresponding to returning of energy from subgrid scales. In this work, we propose to define subgrid forces as a term nudging a coarse-resolution model toward high-resolution model. This force is energy-generating and may be approximated with ANN. Conventional KEBs and ANN model are compared in Double-Gyre configuration of NEMO ocean model.</p>

Vortex statistics in two-dimensional turbulence based on IB-LBM

In this paper, the two-dimensional (2D) turbulence perturbed by arrays of cylinders placed both horizontally and vertically is investigated by Immersed Boundary Lattice Boltzmann Method (IB-LBM). The energy spectrum reveals the coexistence of the inverse and direct cascades in 2D grid turbulence. By observing at the distribution of fluxes in space, the energy and enstrophy fluxes have explained the physical mechanism of the double cascades where the two Kolmogorov laws for structure functions are simultaneously observed. The results of vortex statistics by the conditional analysis, which are based on a new and accurate vortex identification criteria called Liutex, show that the algebraic number density [Formula: see text], where [Formula: see text] is vortex area. The time-evolving vortex number density distribution constructs a theoretical framework involving a three-part: [Formula: see text]; [Formula: see text]; [Formula: see text], which is satisfied with the prediction well. The relationship between the vortex circulation [Formula: see text] and vortex area [Formula: see text] is [Formula: see text] and the one between the kinetic energy of vortex [Formula: see text] and [Formula: see text] is [Formula: see text] in the range where [Formula: see text]. Moreover, it has been found that vortices contain about 30% of the total energy of the flow by studying the energy ratio of all vortices to the entire flow field. What is more, it is an interesting phenomenon is that there is only a range where [Formula: see text] in the energy spectrum for the coherent structure field which is obtained by using Liutex as the extraction of vortices. The probability density function (PDF) of the fluctuations of longitudinal velocity shows that an indication of small intermittency in the direct cascade and the absence of intermittency in the inverse cascade range. On the other hand, the scaling exponents [Formula: see text] of the structure function for the inverse cascade are consistent with Kr67 model, which shows the absence of intermittency. While the measured intermittency parameters are [Formula: see text] and [Formula: see text], which explains that there is a very weak intermittent correction in the direct cascade, and ESS has verified the existence of intermittency in our 2D turbulence.