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

2012 ◽  
Vol 23 (04) ◽  
pp. 1250032 ◽  
Author(s):  
MASAHARU ISOBE
Keyword(s):  
Reynolds Number ◽  
Energy Spectrum ◽  
Large Scale ◽  
Strong Relationship ◽  
Two Dimensions ◽  
Condensed State ◽  
Einstein Condensation ◽  
Two Dimensional ◽  
Developed Turbulence ◽  
Granular Gas

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.

Generation mechanism of a hierarchy of vortices in a turbulent boundary layer

2019 ◽  
Vol 865 ◽  
pp. 1085-1109 ◽  
Author(s):  
Yutaro Motoori ◽  
Susumu Goto
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Energy Spectrum ◽  
Turbulent Boundary Layer ◽  
Large Scale ◽  
Generation Mechanism ◽  
Small Scale ◽  
Generation Process ◽  
Mean Flow ◽  
The Mean

To understand the generation mechanism of a hierarchy of multiscale vortices in a high-Reynolds-number turbulent boundary layer, we conduct direct numerical simulations and educe the hierarchy of vortices by applying a coarse-graining method to the simulated turbulent velocity field. When the Reynolds number is high enough for the premultiplied energy spectrum of the streamwise velocity component to show the second peak and for the energy spectrum to obey the$-5/3$power law, small-scale vortices, that is, vortices sufficiently smaller than the height from the wall, in the log layer are generated predominantly by the stretching in strain-rate fields at larger scales rather than by the mean-flow stretching. In such a case, the twice-larger scale contributes most to the stretching of smaller-scale vortices. This generation mechanism of small-scale vortices is similar to the one observed in fully developed turbulence in a periodic cube and consistent with the picture of the energy cascade. On the other hand, large-scale vortices, that is, vortices as large as the height, are stretched and amplified directly by the mean flow. We show quantitative evidence of these scale-dependent generation mechanisms of vortices on the basis of numerical analyses of the scale-dependent enstrophy production rate. We also demonstrate concrete examples of the generation process of the hierarchy of multiscale vortices.

Turbulent resistivity in wavy two-dimensional magnetohydrodynamic turbulence

2008 ◽  
Vol 595 ◽  
pp. 173-202 ◽  
Author(s):  
SHANE R. KEATING ◽  
P. H. DIAMOND
Keyword(s):  
Large Scale ◽  
Mean Field ◽  
Magnetic Reynolds Number ◽  
Two Dimensions ◽  
Magnetic Potential ◽  
Turbulence Theory ◽  
Two Dimensional ◽  
Restoring Force ◽  
Constrained System ◽  
Magnetohydrodynamic Turbulence

The theory of turbulent resistivity in ‘wavy’ magnetohydrodynamic turbulence in two dimensions is presented. The goal is to explore the theory of quenching of turbulent resistivity in a regime for which the mean field theory can be rigorously constructed at large magnetic Reynolds number Rm. This is achieved by extending the simple two-dimensional problem to include body forces, such as buoyancy or the Coriolis force, which convert large-scale eddies into weakly interacting dispersive waves. The turbulence-driven spatial flux of magnetic potential is calculated to fourth order in wave slope – the same order to which one usually works in wave kinetics. However, spatial transport, rather than spectral transfer, is the object here. Remarkably, adding an additional restoring force to the already tightly constrained system of high Rm magnetohydrodynamic turbulence in two dimensions can actually increase the turbulent resistivity, by admitting a spatial flux of magnetic potential which is not quenched at large Rm, although it is restricted by the conditions of applicability of weak turbulence theory. The absence of Rm-dependent quenching in this wave-interaction-driven flux is a consequence of the presence of irreversibility due to resonant nonlinear three-wave interactions, which are independent of collisional resistivity. The broader implications of this result for the theory of mean field electrodynamics are discussed.

scholarly journals Self-similarity of time-evolving plane wakes

1998 ◽  
Vol 367 ◽  
pp. 255-289 ◽  
Author(s):  
ROBERT D. MOSER ◽  
MICHAEL M. ROGERS ◽  
DANIEL W. EWING
Keyword(s):  
Reynolds Number ◽  
Large Scale ◽  
Initial Conditions ◽  
Spreading Rate ◽  
Momentum Thickness ◽  
Two Dimensional ◽  
Self Similarity ◽  
Order Of Magnitude ◽  
The Difference ◽  
Velocity Spectra

Direct numerical simulations of three time-developing turbulent plane wakes have been performed. Initial conditions for the simulations were obtained using two realizations of a direct simulation from a turbulent boundary layer at momentum-thickness Reynolds number 670. In addition, extra two-dimensional disturbances were added in two of the cases to mimic two-dimensional forcing. The wakes are allowed to evolve long enough to attain approximate self-similarity, although in the strongly forced case this self-similarity is of short duration. For all three flows, the mass-flux Reynolds number (equivalent to the momentum-thickness Reynolds number in spatially developing wakes) is 2000, which is high enough for a short k−5/3 range to be evident in the streamwise one-dimensional velocity spectra.The spreading rate, turbulence Reynolds number, and turbulence intensities all increase with forcing (by nearly an order of magnitude for the strongly forced case), with experimental data falling between the unforced and weakly forced cases. The simulation results are used in conjunction with a self-similar analysis of the Reynolds stress equations to develop scalings that approximately collapse the profiles from different wakes. Factors containing the wake spreading rate are required to bring profiles from different wakes into agreement. Part of the difference between the various cases is due to the increased level of spanwise-coherent (roughly two-dimensional) energy in the forced cases. Forcing also has a significant impact on flow structure, with the forced flows exhibiting more organized large-scale structures similar to those observed in transitional wakes.

scholarly journals Controlling the dual cascade of two-dimensional turbulence

2010 ◽  
Vol 668 ◽  
pp. 202-222 ◽  
Author(s):  
M. M. FARAZMAND ◽  
N. K.-R. KEVLAHAN ◽  
B. PROTAS
Keyword(s):  
Reynolds Number ◽  
Energy Spectrum ◽  
Three Dimensional ◽  
Careful Analysis ◽  
Space Structure ◽  
Inertial Range ◽  
Two Dimensional ◽  
Control Approach ◽  
Time Space ◽  
Moderate Reynolds Number

The Kraichnan–Leith–Batchelor (KLB) theory of statistically stationary forced homogeneous isotropic two-dimensional turbulence predicts the existence of two inertial ranges: an energy inertial range with an energy spectrum scaling of k−5/3, and an enstrophy inertial range with an energy spectrum scaling of k−3. However, unlike the analogous Kolmogorov theory for three-dimensional turbulence, the scaling of the enstrophy range in the two-dimensional turbulence seems to be Reynolds-number-dependent: numerical simulations have shown that as Reynolds number tends to infinity, the enstrophy range of the energy spectrum converges to the KLB prediction, i.e. E ~ k−3. The present paper uses a novel optimal control approach to find a forcing that does produce the KLB scaling of the energy spectrum in a moderate Reynolds number flow. We show that the time–space structure of the forcing can significantly alter the scaling of the energy spectrum over inertial ranges. A careful analysis of the optimal forcing suggests that it is unlikely to be realized in nature, or by a simple numerical model.

Large scales in the developing mixing layer

1976 ◽  
Vol 76 (1) ◽  
pp. 127-144 ◽  
Author(s):  
F. K. Browand ◽  
P. D. Weidman
Keyword(s):  
Reynolds Number ◽  
Large Scale ◽  
Vortical Structure ◽  
Mixing Layer ◽  
Two Dimensional ◽  
Conditional Sampling ◽  
Pairing Interaction ◽  
Vortical Structures ◽  
Developing Flow ◽  
Moderate Reynolds Number

A new experimental technique is described for the study of the interactions between the large-scale vortical features in the two-dimensional mixing layer. Detector probes above and below the mixing layer are used to monitor the large-scale structure. Conditional sampling is performed in a moderate Reynolds number developing flow, by using phase and amplitude information from these detector probes. It is shown that significant Reynolds-stress production is associated with the pairing interaction in which two vortical structures combine to form a single, larger vortical structure.

Quasi-two-dimensional properties of a single shallow-water vortex with high initial Reynolds numbers

2010 ◽  
Vol 665 ◽  
pp. 274-299 ◽  
Author(s):  
D.-G. SEOL ◽  
G. H. JIRKA
Keyword(s):  
Reynolds Number ◽  
Shallow Water ◽  
Coherent Structures ◽  
Large Scale ◽  
Reynolds Numbers ◽  
Surface Velocity ◽  
Mode Analysis ◽  
Two Dimensional ◽  
Vortex System ◽  
Decay Model

The evolution and dynamics of a shallow-water vortex system with high initial Reynolds numbers are investigated experimentally without background rotation. A single vortex is generated by rotating a water mass at the centre of an experimental tank using a bottomless cylinder with internal sectors. The surface velocity field is observed via particle image velocimetry. The experimentally observed vorticity fields indicate that strong shallowness (the ratio of the cylinder diameter to the water depth) and high Reynolds number contribute to the formation of large-scale coherent structures in the form of a tripolar vortex system. The shallow-water vortices with high initial Reynolds numbers experience the transition from turbulent to laminar regimes in their decay process. The proposed first-order vortex decay model predicts that a shallow-water vortex decays as t−1 in the initial turbulent stage and as e−t in the later laminar stage due to horizontal diffusion and bottom friction. The estimated transition time scale from the turbulent to laminar stage increases with initial vortex Reynolds number and with shallowness. By taking the vortex expansion into consideration, the second-order vortex decay model is also presented. The azimuthally ensemble-averaged data elucidate effects of the vortex instabilities and of turbulent energy transfer on the formation of large-scale coherent flow structures. Normal mode analysis of the vortex systems is conducted to study the effect of shallowness and Reynolds number on the generation of two-dimensional large-scale coherent structures. The results show that the perturbation wavenumber of mode 2 is the fastest-growing instability in shallow-water conditions, and its effect depends on initial Reynolds number and shallowness.

scholarly journals A Generalization of Lorenz’s Model for the Predictability of Flows with Many Scales of Motion

2008 ◽  
Vol 65 (3) ◽  
pp. 1063-1076 ◽  
Author(s):  
R. Rotunno ◽  
C. Snyder
Keyword(s):  
Energy Spectrum ◽  
Large Scale ◽  
Doubling Time ◽  
Two Dimensional ◽  
Synoptic Scale ◽  
Lorenz Model ◽  
Seminal Paper ◽  
Kinetic Energy Spectrum ◽  
Scale Error ◽  
Quasigeostrophic Equations

Abstract In a seminal paper, E. N. Lorenz proposed that flows with many scales of motion in which smaller-scale error spreads to larger scales and in which the error-doubling time decreases with decreasing scale have a finite range of predictability. Although the Lorenz theory of limited predictability is widely understood and accepted, the model upon which the theory is based is less so. The primary objection to the model is that it is based on the two-dimensional vorticity equation (2DV) while simultaneously emphasizing results using a basic turbulent flow with a “−5/3” energy spectrum in the atmospheric synoptic scale instead of those using a more theoretically and observationally consistent “−3” spectrum. The present work generalizes the Lorenz model so that it may apply to the surface quasigeostrophic equations (SQGs), which are mathematically very similar to 2DV but are known to have a −5/3 kinetic energy spectrum downscale from a large-scale forcing. This generalized Lorenz model is applied here to both 2DV (with a −3 spectrum) and SQG (with a −5/3 spectrum), producing examples of flows with unlimited and limited predictability, respectively. Comparative analysis of the two models allows for the identification of the distinctive attributes of a many-scaled flow with limited predictability.

PROJECTIONS OFF FRACTAL FUNCTIONS: A NEW VISION OF NATURE'S COMPLEXITY

Fractals ◽  
1999 ◽  
Vol 07 (04) ◽  
pp. 387-401 ◽  
Author(s):  
CARLOS E. PUENTE ◽  
NELSON OBREGON ◽  
OSCAR ROBAYO ◽  
MARTA G. PUENTE ◽  
DEMIRAY SIMSEK
Keyword(s):  
Two Dimensions ◽  
Rainfall Time Series ◽  
Two Dimensional ◽  
Infinite Class ◽  
Gaussian Distributions ◽  
Contaminant Plumes ◽  
Developed Turbulence ◽  
Fractal Functions ◽  
Bivariate Gaussian Distribution ◽  
New Vision

The construction of a vast class of deterministic derived measures, defined as transformations of simple multinomial multifractals via fractal interpolating functions, is reviewed.1,2 It is illustrated that these objects, which are projections of unique measures defined over the graphs of fractal interpolating functions, provide a new vision to address the complexity of some of nature's tangled patterns over one and two dimensions, as they may be used to model: (a) Rainfall time series, (b) energy dissipation in fully developed turbulence, (c) two-dimensional contaminant plumes within a porous medium, and (d)"chaotic" and "stochastic" signals, as encountered in applications. A recently discovered universal connection between arbitrary measures with continuous cumulative distributions and the univariate and bivariate Gaussian distributions, via plane- and space-filling fractal interpolating functions respectively, is also reviewed.3,4 It is explained how this relationship yields (via projections) an infinite class of two-dimensional symmetric crystalline sets making up exotic kaleidoscopes of arbitrary symmetries, which decompose the bivariate Gaussian distribution.5 It is shown that these ideas enhance the vision that projections may be useful in approaching natural complexity, as some of these sets closely resemble key biochemical structures which even include life's own DNA rosette.

Reynolds-number dependency in homogeneous, stationary two-dimensional turbulence

2010 ◽  
Vol 646 ◽  
pp. 517-526 ◽  
Author(s):  
ANNALISA BRACCO ◽  
JAMES C. MCWILLIAMS
Keyword(s):  
Reynolds Number ◽  
Structure Function ◽  
Large Scale ◽  
Velocity Structure ◽  
Stokes Equations ◽  
Turbulent Transport ◽  
Distribution Functions ◽  
Two Dimensional ◽  
Random Forcing ◽  
Statistical Measures

Turbulent solutions of the two-dimensional Navier–Stokes equations are a paradigm for the chaotic space–time patterns and equilibrium distributions of turbulent geophysical and astrophysical ‘thin’ flows on large horizontal scales. Here we investigate how homogeneous, stationary two-dimensional turbulence varies with the Reynolds number (Re) in stationary solutions with large-scale, random forcing and viscous diffusion, also including hypoviscous diffusion to limit the inverse energy cascade. This survey is made over the computationally feasible range in Re ≫ 1, approximately between 1.5 × 103 and 5.6 × 106. For increasing Re, we witness the emergence of vorticity fine structure within the filaments and vortex cores. The energy spectrum shape approaches the forward-enstrophy inertial-range form k−3 at large Re, and the velocity structure function is independent of Re. All other statistical measures investigated in this study exhibit power-law scaling with Re, including energy, enstrophy, dissipation rates and the vorticity structure function. The scaling exponents depend on the forcing properties through their influences on large-scale coherent structures, whose particular distributions are non-universal. A striking result is the Re independence of the intermittency measures of the flow, in contrast with the known behaviour for three-dimensional homogeneous turbulence of asymptotically increasing intermittency. This is a consequence of the control of the tails of the distribution functions by large-scale coherent vortices. Our analysis allows extrapolation towards the asymptotic limit of Re → ∞, fundamental to geophysical and astrophysical regimes and their large-scale simulation models where turbulent transport and dissipation must be parameterized.

On the scaling of air entrainment from a ventilated partial cavity

2013 ◽  
Vol 732 ◽  
pp. 47-76 ◽  
Author(s):  
Simo A. Mäkiharju ◽  
Brian R. Elbing ◽  
Andrew Wiggins ◽  
Sarah Schinasi ◽  
Jean-Marc Vanden-Broeck ◽  
...  
Keyword(s):  
Reynolds Number ◽  
Weber Number ◽  
Large Scale ◽  
Air Entrainment ◽  
Small Scale ◽  
Length Scale ◽  
Entrainment Rate ◽  
Two Dimensional ◽  
Wide Range ◽  
Stable Cavity

AbstractThe behaviour of a nominally two-dimensional ventilated partial cavity was examined over a wide range of size scales and flow speeds to determine the influence of Froude, Reynolds, and Weber number on the cavity shape, dynamics, and gas entrainment rate. Two geometrically similar experiments were conducted with a 14:1 length scale ratio. The results were compared to a two-dimensional semi-analytical model of the cavity flow, and Froude scaling was found to be sufficient to match basic cavity shapes. However, the air flux required to maintain a stable cavity did not scale with Froude number alone, as the dynamics of the cavity closure changed with increasing Reynolds number. The required air flux differed over one order of magnitude between the lowest and highest Reynolds number flows. But, for sufficiently high Reynolds numbers, the rate of scaled entrainment appeared to approach Reynolds number independence. Modest changes in surface tension of the small-scale experiment suggested that the Weber number was important only at the lowest speeds and smaller length scale. Otherwise, the Weber numbers of the flows were sufficiently high to make the effects of interfacial tension negligible. We also observed that modest unsteadiness in the inflow to the large-scale cavity led to a significant increase in the required air flux needed to maintain a stable cavity, with the required excess gas flux nominally proportional to the flow’s perturbation amplitude. Finally, discussion is provided on how these results relate to model testing of partial cavity drag reduction (PCDR) systems for surface ships.

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