Simulations Of Two-dimensional Turbulence Research Articles

On the non-local nature of turbulent fluxes of passive scalars

The parameterization of fluxes associated with representing unresolved dynamics in turbulent flows, especially in the atmosphere and ocean (which have a vast range of scales), remains a challenging task. This is especially true for Earth system models including complex biogeochemistry and requiring very long simulations. The problem of representing the dependence of the mean flux of a passive tracer in terms of the mean has a very long history; in this study, we take a somewhat different approach. We use a formalism showing that the mean flux will be a functional of the mean gradients, a formalism that can be used to calculate the structure of the functional which is non-local in both space and time. Two-dimensional turbulent simulations are used to explore the weight of nearby (in space or time) gradients. We also use stochastic velocities and iterated maps to show that the results are similar. The functional formalism provides an understanding of when non-locality needs to be considered and when a local eddy diffusivity can be a reasonably good approximation. Furthermore, the formalism provides guidance for the development of data-driven parameterizations.

Surface tension as the destabiliser of a vortical interface

We study the dynamics of an initially flat interface between two immiscible fluids, with a vortex situated on it. We show how surface tension causes vorticity generation at a general curved interface. This creates a velocity jump across the interface which increases quadratically in time, and causes the Kelvin–Helmholtz instability. Surface tension thus acts as a destabiliser by vorticity creation, winning over its own tendency to stabilise by smoothing out interfacial perturbations to reduce surface energy. We further show that this instability is manifested within the vortex core at times larger than ${\sim}(k We)^{1/4}$ for a Weber number $We$ and perturbation wavenumber $k$ , destroying the flow structure. The vorticity peels off into small-scale structures away from the interface. Using energy balance we provide the growth of total interface length in time. A density difference between the fluids produces additional instabilities outside the vortex core due to centrifugal effects. We demonstrate the importance of this mechanism in two-dimensional turbulence simulations with a prescribed initial interface.

A priori analysis on deep learning of subgrid-scale parameterizations for Kraichnan turbulence

In the present study, we investigate different data-driven parameterizations for large eddy simulation of two-dimensional turbulence in the \emph{a priori} settings. These models utilize resolved flow field variables on the coarser grid to estimate the subgrid-scale stresses. We use data-driven closure models based on localized learning that employs multilayer feedforward artificial neural network (ANN) with point-to-point mapping and neighboring stencil data mapping, and convolutional neural network (CNN) fed by data snapshots of the whole domain. The performance of these data-driven closure models is measured through a probability density function and is compared with the dynamic Smagorinksy model (DSM). The quantitative performance is evaluated using the cross-correlation coefficient between the true and predicted stresses. We analyze different frameworks in terms of the amount of training data, selection of input and output features, their characteristics in modeling with accuracy, and training and deployment computational time. We also demonstrate computational gain that can be achieved using the intelligent eddy viscosity model that learns eddy viscosity computed by the DSM instead of subgrid-scale stresses. We detail the hyperparameters optimization of these models using the grid search algorithm.

Numerical simulation of two-dimensional turbulence based on Immersed Boundary Lattice Boltzmann method

The two-dimensional turbulence is numerically investigated using Immersed Boundary Lattice Boltzmann Method. The 2D turbulence should be considered as 2D channel flow where the flow is forced by the arrays of cylinders vertically placed in the inlet of 2D channel. The inverse cascade in two cases, the normal wall boundary case and rough wall boundary case, is obtained in the inertial range. The scaling behavior of energy spectrum in the inverse cascade is k−5/3, which is according with the Kraichnan theory of 2D turbulence. It is found that the time-evolving vortex number density distribution n(A)∼t−1A−1, based on the Rortex vortex definition criterion, which is in fair agreement with the theory of Burgess and Scott at intermediate scales sufficiently far from the forcing and the largest vortex scale. The energy flux of the normal wall boundary case cascades to the large scale. The energy flux of the rough wall boundary case cascades to large scale when the scale l < D. The energy flux at l > D cascades to the small scale. The energy dissipation filed εl coarse grained at the scale l has the pow-law behaviors with the scale. The intermittency measured by PDF exists in the inverse inertial range of 2D turbulence. On the basis of the vortex scaling law of Burgess and Scott and our simulation results, the universal scaling law of velocity structure function in the inverse cascade is obtained by extending the 3D turbulent S-L intermittency model to the inverse inertial range of 2D turbulence.

Effects of Thermophysical Variable Properties on Liquid-Sodium Convective Flows in a Square Enclosure

The influence of the variable properties on the buoyancy-driven flows of liquid sodium established in a square enclosure including two inner heated plates is numerically investigated. Two-dimensional turbulent simulations are obtained, considering uniform wall temperature heating conditions. The low-Reynolds k–ω turbulence model is employed. The average Nusselt number and the dimensionless mass-flow rate evaluated between the inner plates are obtained for a wide range of the Rayleigh number, varying from 103 to 1012. Several practical correlations are presented. The results obtained for different heating intensities are analyzed and compared. The expected decay in the heat transfer coefficients is less relevant than that previously obtained for airflows. The thermal structure of the flow into the enclosure is also shown.

Study on Direct Numerical Simulation of Two-dimensional Turbulence

Turbulence is one of the unsolved problems in fluid mechanics. The study of 3D turbulence is very complex and difficult to achieve; while 2D turbulence is similar to 3D turbulence in many properties, and the related study is easier to carry out. Therefore, the study of 2D turbulence is significant to understand the characteristics of 3D turbulence. In this paper, the vorticity-stream function method is applied to the direct numerical simulation of 2D turbulence without any turbulence model. The third order Runge Kutta-Crank Nicholson method is used as the time integration method, and finite difference method of second order accuracy is for space discretization with central difference scheme and staggered grid. Turbulence conditions are used as boundary conditions and initial conditions. The vorticity distribution under different Reynolds number and different grid resolutions are obtained by numerical simulation. When the Reynolds number is low, the numerical discretization method with low order accuracy can resolve all vortex structures in space with higher grid resolution. But as the Reynolds number increases, the higher accuracy of the numerical discretization method is the best choice compared with the grid resolution enhancement.

A critical comparison of Lagrangian methods for coherent structure detection.

We review and test twelve different approaches to the detection of finite-time coherent material structures in two-dimensional, temporally aperiodic flows. We consider both mathematical methods and diagnostic scalar fields, comparing their performance on three benchmark examples: the quasiperiodically forced Bickley jet, a two-dimensional turbulence simulation, and an observational wind velocity field from Jupiter's atmosphere. A close inspection of the results reveals that the various methods often produce very different predictions for coherent structures, once they are evaluated beyond heuristic visual assessment. As we find by passive advection of the coherent set candidates, false positives and negatives can be produced even by some of the mathematically justified methods due to the ineffectiveness of their underlying coherence principles in certain flow configurations. We summarize the inferred strengths and weaknesses of each method, and make general recommendations for minimal self-consistency requirements that any Lagrangian coherence detection technique should satisfy.

Optimal initial condition of passive tracers for their maximal mixing in finite time

The efficiency of a fluid mixing device is often limited by fundamental laws and/or design constraints, such that a perfectly homogeneous mixture cannot be obtained in finite time. Here, we address the natural corollary question: Given the best available mixer, what is the optimal initial tracer pattern that leads to the most homogeneous mixture after a prescribed finite time? For ideal passive tracers, we show that this optimal initial condition coincides with the right singular vector (corresponding to the smallest singular value) of a suitably truncated Perron-Frobenius (PF) operator. The truncation of the PF operator is made under the assumption that there is a small length-scale threshold $\ell_\nu$ under which the tracer blobs are considered, for all practical purposes, completely mixed. We demonstrate our results on two examples: a prototypical model known as the sine flow and a direct numerical simulation of two-dimensional turbulence. Evaluating the optimal initial condition through this framework only requires the position of a dense grid of fluid particles at the final instance and their preimages at the initial instance of the prescribed time interval. As such, our framework can be readily applied to flows where such data is available through numerical simulations or experimental measurements.

Comparison of numerical advection schemes in two-dimensional turbulence simulation

Abstract The influence of numerical approximations on statistical characteristics of modelled two-dimensional turbulence sustained by a stochastic external forcing is studied. The ability of various finite-difference and semi-Lagrangian schemes to reproduce reliably the dual energy and enstrophy cascades for coarse spatial resolution is tested. It is also studied how the requirement of preserving invariants inherent to a two-dimensional ideal fluid is important relative to numerical schemes. The results of calculations with high spatial resolution were taken as a reference solution. The choice of studied schemes was motivated by their use in atmosphere and ocean numerical models, in particular, in the Institute of Numerical Mathematics climate model (INMCM) and semi-Lagrangian absolute vorticity (SLAV) model of medium-range weather forecast. The importance of conservation laws for integral vorticity and enstrophy is revealed in the numerical experiments with a small-scale external forcing.

Entropic Multi-Relaxation Models for Simulation of Fluid Turbulence

A recently introduced family of lattice Boltzmann (LB) models (Karlin, Bosch, Chikatamarla, Phys. Rev. E, 2014; Ref [22]) is studied in detail for incompressible two-dimensional flows. A framework for developing LB models based on entropy considerations is laid out extensively. Second order rate of convergence is numerically confirmed and it is demonstrated that these entropy based models recover the Navier-Stokes solution in the hydrodynamic limit. Comparison with the standard Bhatnagar-Gross-Krook (LBGK) and the entropic lattice Boltzmann method (ELBM) demonstrates the superior stability and accuracy for several benchmark flows and a range of grid resolutions and Reynolds numbers. High Reynolds number regimes are investigated through the simulation of two-dimensional turbulence, particularly for under-resolved cases. Compared to resolved LBGK simulations, the presented class of LB models demonstrate excellent performance and capture the turbulence statistics with good accuracy.

How turbulence regulates biodiversity in systems with cyclic competition.

Cyclic, nonhierarchical interactions among biological species represent a general mechanism by which ecosystems are able to maintain high levels of biodiversity. However, species coexistence is often possible only in spatially extended systems with a limited range of dispersal, whereas in well-mixed environments models for cyclic competition often lead to a loss of biodiversity. Here we consider the dispersal of biological species in a fluid environment, where mixing is achieved by a combination of advection and diffusion. In particular, we perform a detailed numerical analysis of a model composed of turbulent advection, diffusive transport, and cyclic interactions among biological species in two spatial dimensions and discuss the circumstances under which biodiversity is maintained when external environmental conditions, such as resource supply, are uniform in space. Cyclic interactions are represented by a model with three competitors, resembling the children's game of rock-paper-scissors, whereas the flow field is obtained from a direct numerical simulation of two-dimensional turbulence with hyperviscosity. It is shown that the space-averaged dynamics undergoes bifurcations as the relative strengths of advection and diffusion compared to biological interactions are varied.

Hybrid Vlasov-Maxwell simulations of two-dimensional turbulence in plasmas

Turbulence in plasmas is a very challenging problem since it involves wave-particle interactions, which are responsible for phenomena such as plasma dissipation, acceleration mechanisms, heating, temperature anisotropy, and so on. In this work, a hybrid Vlasov-Maxwell numerical code is employed to study local kinetic processes in a two-dimensional turbulent regime. In the present model, ions are treated as a kinetic species, while electrons are considered as a fluid. As recently reported in [S. Servidio, Phys. Rev. Lett. 108, 045001 (2012)], nearby regions of strong magnetic activity, kinetic effects manifest through a deformation of the ion velocity distribution function that consequently departs from the equilibrium Maxwellian configuration. Here, the structure of turbulence is investigated in detail in phase space, by evaluating the high-order moments of the particle velocity distribution, i.e., temperature, skewness, and kurtosis. This analysis provides quantitative information about the non-Maxwellian character of the system dynamics. This departure from local thermodynamic equilibrium triggers several processes commonly observed in many astrophysical and laboratory plasmas.

Computing Lagrangian coherent structures from their variational theory

Using the recently developed variational theory of hyperbolic Lagrangian coherent structures (LCSs), we introduce a computational approach that renders attracting and repelling LCSs as smooth, parametrized curves in two-dimensional flows. The curves are obtained as trajectories of an autonomous ordinary differential equation for the tensor lines of the Cauchy-Green strain tensor. This approach eliminates false positives and negatives in LCS detection by separating true exponential stretching from shear in a frame-independent fashion. Having an explicitly parametrized form for hyperbolic LCSs also allows for their further in-depth analysis and accurate advection as material lines. We illustrate these results on a kinematic model flow and on a direct numerical simulation of two-dimensional turbulence.

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.

The enstrophy cascade in forced two-dimensional turbulence

We carry out direct numerical simulations of two-dimensional turbulence with forcing at different wavenumbers and resolutions up to 327682 grid points. In the absence of large-scale drag, a state is reached where enstrophy is quasi-stationary while energy is growing. In the enstrophy cascade range the energy spectrum has the form E(k) = εω2/3k−3, without any logarithmic correction, where εω is the enstrophy dissipation and is of the order of unity. However, is varying between different simulations and is thus not a perfect constant. This variation can be understood as a consequence of large-scale dissipation intermittency, following the argument by Landau (Landau &amp; Lifshitz, Fluid Mechanics, 1959, Pergamon). In the presence of a large-scale drag, we obtain a slightly steeper spectrum. When forcing is applied at a scale which is somewhat smaller than the computational domain, no vortices are formed, and the statistics remain close to Gaussian in the enstrophy cascade range. When forcing is applied at a smaller scale, long-lived coherent vortices form at larger scales than the forcing scale, and intermittency measures become very large at all scales, including the scales of the enstrophy cascade. We conclude that the enstrophy cascade with a k−3-spectrum is a robust feature of the two-dimensional Navier–Stokes equations. However, there is a complete lack of universality of higher-order statistics of vorticity increments in the enstrophy cascade range.

Nanostructuring of a silicon surface by laser redeposition of Si vapor

We report on the surface nanostructuring of silicon wafer by self-organization of redeposited Si nanoparticles, at various energy levels, in the vaporization regime of laser-matter interaction. By using the semiconfined configuration, a quasi-two-dimensional turbulent Si vapor field with gradients of pressure and temperature is formed. The turbulent field evolves into point vortices which condense into Si nanodroplets. At a low laser energy of ∼1.2 J (0.23 GW/cm2), the inertial instability of nanodroplets under gradients of pressure and temperature, cause their intermittent accumulation in the low-pressure regions of turbulent field. The solidification of Si nanodroplets into particles and their redeposition, cause a simple two-dimensional low density nanostructuring of Si wafer in the near periphery region, and a high density nanostructuring in the periphery region of the spot. The pattern of redeposited Si nanoparticles in these regions is equivalent to the pattern of point vortices in a two-dimensional turbulent field. Such a pattern of point vortices is obtained by numerical simulation from the two-dimensional Navier–Stokes equation for forced turbulence. The self-organization of the coherent point vortex pattern is generated by numerical simulation of the solitary turbulence model based on the nonlinear Schrödinger equation. At the high laser energy of ∼1.5 and ∼2.0 J (∼0.42 and ∼0.52 GW/cm2, respectively), the transition from simple intermittent two-dimensional nanoparticle organization into a continuous and more complex one takes place. The nanostructured pattern shows a continuous distribution of Si particles, whose size increases from the periphery toward the center without spatial intermittency, showing a gradient of particle size. In addition, the open and closed loops chain clusters appear, with morphology and fractal dimension similar to the chain clusters which grow according to the Meakin–Jullien model of cluster-cluster aggregation. At the higher power density of ∼0.52 GW/cm2, the chain clusters become connected and tend to compactification. They form a network similar to the one obtained by numerical simulation of two-dimensional turbulence at small Stokes numbers. The silicon surface nanostructured by recondensation in this case comprises only the nanometer sized particles.

A large-eddy-based lattice Boltzmann model for turbulent flow simulation

In this paper a novel and simple large-eddy-based lattice Boltzmann model is proposed to simulate two-dimensional turbulence. Unlike existing lattice Boltzmann models for turbulent flow simulation, which were based on primitive-variables Navier–Stokes equations, the target macroscopic equations of the present model are vorticity-streamfunction equations. Thanks to the intrinsic features of vorticity-streamfunction equations, the present model is efficient, stable and simple for two-dimensional turbulence simulation. The advantages of the present model are validated by numerical experiments.

Modeling temporal fluctuations in avalanching systems

We demonstrate how to model the toppling activity in avalanching systems by stochastic differential equations (SDEs). The theory is developed as a generalization of the classical mean-field approach to sandpile dynamics by formulating it as a generalization of Itô's SDE. This equation contains a fractional Gaussian noise term representing the branching of an avalanche into small active clusters and a drift term reflecting the tendency for small avalanches to grow and large avalanches to be constricted by the finite system size. If one defines avalanching to take place when the toppling activity exceeds a certain threshold, the stochastic model allows us to compute the avalanche exponents in the continuum limit as functions of the Hurst exponent of the noise. The results are found to agree well with numerical simulations in the Bak-Tang-Wiesenfeld and Zhang sandpile models. The stochastic model also provides a method for computing the probability density functions of the fluctuations in the toppling activity itself. We show that the sandpiles do not belong to the class of phenomena giving rise to universal non-Gaussian probability density functions for the global activity. Moreover, we demonstrate essential differences between the fluctuations of total kinetic energy in a two-dimensional turbulence simulation and the toppling activity in sandpiles.

Lagrangian statistics in forced two-dimensional turbulence

We report on simulations of two-dimensional turbulence in the inverse energy cascade regime. Focusing on the statistics of Lagrangian tracer particles, scaling behavior of the probability density functions of velocity fluctuations is investigated. The results are compared to the three-dimensional case. In particular an analysis in terms of compensated cumulants reveals the transition from a strong non-Gaussian behavior with large tails to Gaussianity. The reported computation of correlation functions for the acceleration components sheds light on the underlying dynamics of the tracer particles.

Energy and enstrophy fluxes in the double cascade of two-dimensional turbulence

High-resolution direct numerical simulations of two-dimensional turbulence in stationary conditions are presented. The development of an energy–enstrophy double cascade is investigated and its statistics found to be compatible with the classical Kraichnan theory in the limit of extended inertial ranges. The analysis of the joint distribution of energy and enstrophy fluxes in physical space reveals a small value of cross-correlation. This result supports many experimental and numerical studies where only one cascade is generated.