scholarly journals Energy flux enhancement, intermittency and turbulence via Fourier triad phase dynamics in the 1-D Burgers equation

2018 ◽  
Vol 850 ◽  
pp. 624-645 ◽  
Author(s):  
Brendan P. Murray ◽  
Miguel D. Bustamante
Keyword(s):  
Burgers Equation ◽  
Numerical Study ◽  
Physical Space ◽  
Space Velocity ◽  
Inertial Range ◽  
Kuramoto Model ◽  
Fast Time ◽  
Phase Dynamics ◽  
Velocity Statistics ◽  
Phase Alignment

We present a theoretical and numerical study of Fourier-space triad phase dynamics in the one-dimensional stochastically forced Burgers equation at Reynolds number $Re\approx 2.7\times 10^{4}$. We demonstrate that Fourier triad phases over the inertial range display a collective behaviour characterised by intermittent periods of synchronisation and alignment, reminiscent of the Kuramoto model (Chemical Oscillations, Waves, and Turbulence, Springer, 1984) and directly related to collisions of shocks in physical space. These periods of synchronisation favour efficient energy fluxes across the inertial range towards small scales, resulting in strong bursts of dissipation and enhanced coherence of the Fourier energy spectrum. The fast time scale of the onset of synchronisation relegates energy dynamics to a passive role: this is further examined using a reduced system with the Fourier amplitudes fixed in time – a phase-only model. We show that intermittent triad phase dynamics persists without amplitude evolution and we broadly recover many of the characteristics of the full Burgers system. In addition, for both full Burgers and phase-only systems the physical-space velocity statistics reveals that triad phase alignment is directly related to the non-Gaussian statistics typically associated with structure-function intermittency in turbulent systems.

A numerical study of the Burgers’ equation

2008 ◽  
Vol 345 (4) ◽  
pp. 328-348 ◽  
Author(s):  
Bülent Saka ◽  
İdris Dağ
Keyword(s):  
Burgers Equation ◽  
Numerical Study

Scale-dependent alignment, tumbling and stretching of slender rods in isotropic turbulence

2018 ◽  
Vol 860 ◽  
pp. 465-486 ◽  
Author(s):  
Nimish Pujara ◽  
Greg A. Voth ◽  
Evan A. Variano
Keyword(s):  
Isotropic Turbulence ◽  
Fluid Velocity ◽  
Inertial Range ◽  
Small Scale ◽  
Strain Rate Tensor ◽  
Scaling Exponents ◽  
Slender Body Theory ◽  
Velocity Statistics ◽  
Rigid Rods ◽  
Dissipation Range

We examine the dynamics of slender, rigid rods in direct numerical simulation of isotropic turbulence. The focus is on the statistics of three quantities and how they vary as rod length increases from the dissipation range to the inertial range. These quantities are (i) the steady-state rod alignment with respect to the perceived velocity gradients in the surrounding flow, (ii) the rate of rod reorientation (tumbling) and (iii) the rate at which the rod end points move apart (stretching). Under the approximations of slender-body theory, the rod inertia is neglected and rods are modelled as passive particles in the flow that do not affect the fluid velocity field. We find that the average rod alignment changes qualitatively as rod length increases from the dissipation range to the inertial range. While rods in the dissipation range align most strongly with fluid vorticity, rods in the inertial range align most strongly with the most extensional eigenvector of the perceived strain-rate tensor. For rods in the inertial range, we find that the variance of rod stretching and the variance of rod tumbling both scale as $l^{-4/3}$, where $l$ is the rod length. However, when rod dynamics are compared to two-point fluid velocity statistics (structure functions), we see non-monotonic behaviour in the variance of rod tumbling due to the influence of small-scale fluid motions. Additionally, we find that the skewness of rod stretching does not show scale invariance in the inertial range, in contrast to the skewness of longitudinal fluid velocity increments as predicted by Kolmogorov’s $4/5$ law. Finally, we examine the power-law scaling exponents of higher-order moments of rod tumbling and rod stretching for rods with lengths in the inertial range and find that they show anomalous scaling. We compare these scaling exponents to predictions from Kolmogorov’s refined similarity hypotheses.

scholarly journals A numerical study of a semi-Lagrangian Parareal method applied to the viscous Burgers equation

2018 ◽  
Vol 19 (1-2) ◽  
pp. 45-57 ◽  
Author(s):  
A. Schmitt ◽  
M. Schreiber ◽  
P. Peixoto ◽  
M. Schäfer
Keyword(s):  
Burgers Equation ◽  
Numerical Study ◽  
Parareal Method

scholarly journals Numerical study of pressure and velocity fluctuations in nearly isotropic turbulence

1978 ◽  
Vol 88 (4) ◽  
pp. 685-709 ◽  
Author(s):  
U. Schumann ◽  
G. S. Patterson
Keyword(s):  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Numerical Study ◽  
Pressure Fluctuations ◽  
Reynolds Numbers ◽  
Companion Paper ◽  
Real Space ◽  
Pressure Gradients ◽  
Velocity Statistics ◽  
Decaying Turbulence

The spectral method of Orszag & Patterson has been extended to calculate the static pressure fluctuations in incompressible homogeneous decaying turbulence at Reynolds numbers Reλ [lsim ] 35. In real space 323 points are treated. Several cases starting from different isotropic initial conditions have been studied. Some departure from isotropy exists owing to the small number of modes at small wavenumbers. Root-mean-square pressure fluctuations, pressure gradients and integral length scales have been evaluated. The results agree rather well with predictions based on velocity statistics and on the assumption of normality. The normality assumption has been tested extensively for the simulated fields and found to be approximately valid as far as fourth-order velocity correlations are concerned. In addition, a model for the dissipation tensor has been proposed. The application of the present method to the study of the return of axisymmetric turbulence to isotropy is described in the companion paper.

scholarly journals Fractional hyperviscosity induced growth of bottlenecks in energy spectrum of Burgers equation solutions

2019 ◽  
Vol 92 (9) ◽  
Author(s):  
Debarghya Banerjee
Keyword(s):  
Energy Spectrum ◽  
Velocity Field ◽  
Burgers Equation ◽  
Space Velocity ◽  
Real Space ◽  
Analytical Results ◽  
Turbulent Fluids ◽  
Dissipation Range

Abstract Energy spectrum of turbulent fluids exhibit a bump at an intermediate wavenumber, between the inertial and the dissipation range. This bump is called bottleneck. Such bottlenecks are also seen in the energy spectrum of the solutions of hyperviscous Burgers equation. Previous work have shown that this bump corresponds to oscillations in real space velocity field. In this paper, we present numerical and analytical results of how the bottleneck and its real space signature, the oscillations, grow as we tune the order of hyperviscosity. We look at a parameter regime α ∈ [1, 2] where α = 1 corresponds to normal viscosity and α = 2 corresponds to hyperviscosity of order 2. We show that even for the slightest fractional increment in the order of hyperviscosity (α) bottlenecks show up in the energy spectrum. Graphical abstract

scholarly journals Equilibrium and travelling-wave solutions of plane Couette flow

2009 ◽  
Vol 638 ◽  
pp. 243-266 ◽  
Author(s):  
J. F. GIBSON ◽  
J. HALCROW ◽  
P. CVITANOVIĆ
Keyword(s):  
Couette Flow ◽  
Three Dimensional ◽  
Physical Space ◽  
Space Velocity ◽  
Travelling Wave ◽  
Isotropy Group ◽  
Travelling Wave Solutions ◽  
Plane Couette Flow ◽  
Equilibrium Solutions ◽  
Wave Solutions

We present 10 new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds number Re and two new travelling-wave solutions. The solutions are continued under changes of Re and spanwise period. We provide a partial classification of the isotropy groups of plane Couette flow and show which kinds of solutions are allowed by each isotropy group. We find two complementary visualizations particularly revealing. Suitably chosen sections of their three-dimensional physical space velocity fields are helpful in developing physical intuition about coherent structures observed in low-Re turbulence. Projections of these solutions and their unstable manifolds from their ∞-dimensional state space on to suitably chosen two- or three-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows.

scholarly journals THE KURAMOTO MODEL SUBJECT TO A FLUCTUATING ENVIRONMENT: APPLICATION TO BRAINWAVE DYNAMICS

2012 ◽  
Vol 11 (01) ◽  
pp. 1240011 ◽  
Author(s):  
A. C. HALE ◽  
T. HANSARD ◽  
L. W. SHEPPARD ◽  
P. V. E. McCLINTOCK ◽  
A. STEFANOVSKA
Keyword(s):  
Quantitative Measure ◽  
Kuramoto Model ◽  
Living Systems ◽  
Phase Dynamics ◽  
Fluctuating Environment ◽  
Kuramoto Oscillators ◽  
The Kuramoto Model ◽  
Model Subject

We consider the phase dynamics of an ensemble of Kuramoto oscillators whose eigenfrequencies are perturbed to model the openness of living systems, and we show that it exhibits time-localized epochs of synchrony. A new quantitative measure is used to show that the model compares well with electroencephalography data recorded from a healthy awake human.

scholarly journals Large deviation for two-time-scale stochastic burgers equation

2020 ◽  
pp. 2150023
Author(s):  
Xiaobin Sun ◽  
Ran Wang ◽  
Lihu Xu ◽  
Xue Yang
Keyword(s):  
Burgers Equation ◽  
Large Deviation ◽  
Slow Component ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Convergence Criterion ◽  
Fast Time ◽  
Infinite Dimensional ◽  
Stochastic Burgers Equation ◽  
Stochastic Reaction

A Freidlin–Wentzell type large deviation principle is established for stochastic partial differential equations with slow and fast time-scales, where the slow component is a one-dimensional stochastic Burgers equation with small noise and the fast component is a stochastic reaction-diffusion equation. Our approach is via the weak convergence criterion developed in [A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist. 20 (2000) 39–61].

Vorticity and passive-scalar dynamics in two-dimensional turbulence

1987 ◽  
Vol 183 ◽  
pp. 379-397 ◽  
Author(s):  
Armando Babiano ◽  
Claude Basdevant ◽  
Bernard Legras ◽  
Robert Sadourny
Keyword(s):  
Similarity Theory ◽  
Physical Space ◽  
Passive Scalar ◽  
Inertial Range ◽  
Flow Dynamics ◽  
Two Dimensional ◽  
Closure Model ◽  
Coherent Vortices ◽  
Vorticity Transport ◽  
Kolmogorov Theory

The dynamics of vorticity in two-dimensional turbulence is studied by means of semi-direct numerical simulations, in parallel with passive-scalar dynamics. It is shown that a passive scalar forced and dissipated in the same conditions as vorticity, has a quite different behaviour. The passive scalar obeys the similarity theory à la Kolmogorov, while the enstrophy spectrum is much steeper, owing to a hierarchy of strong coherent vortices. The condensation of vorticity into such vortices depends critically both on the existence of an energy invariant (intimately related to the feedback of vorticity transport on velocity, absent in passive-scalar dynamics, and neglected in the Kolmogorov theory of the enstrophy inertial range); and on the localness of flow dynamics in physical space (again not considered by the Kolmogorov theory, and not accessible to closure model simulations). When space localness is artificially destroyed, the enstrophy spectrum again obeys a k−1 law like a passive scalar. In the wavenumber range accessible to our experiments, two-dimensional turbulence can be described as a hierarchy of strong coherent vortices superimposed on a weak vorticity continuum which behaves like a passive scalar.

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