arXiv subjects

Mahendra K. Verma

Publications and source records attributed to Mahendra K. Verma.

At least 109 records · Page 6Linked to original sources

arXiv2010

Chaotic dynamics in two-dimensional Rayleigh-Bénard convection

We investigate the origin of various convective patterns using bifurcation diagrams that are constructed using direct numerical simulations. We perform two-dimensional pseudospectral simulations for a Prandtl number 6.8 fluid that is confined in a box with aspect ratio $Γ= 2\sqrt{2}$. Steady convective rolls are born from the conduction state through a pitchfork bifurcation at $r=1$, where $r$ is the reduced Rayleigh number. These fixed points bifurcate successively to time-periodic and quasiperiodic rolls through Hopf and Neimark-Sacker bifurcations at $r \simeq 80$ and $r \simeq 500 $ respectively. The system becomes chaotic at $r \simeq 750$ through a quasiperiodic route to chaos. The size of the chaotic attractor increases at $r \simeq 840$ through an "attractor-merging crisis" which also results in travelling chaotic rolls. We also observe coexistence of stable fixed points and a chaotic attractor for $ 846 \le r \le 849$ as a result of a subcritical Hopf bifurcation. Subsequently the chaotic attractor disappears through a "boundary crisis" and only stable fixed points remain. Later these fixed points become periodic and chaotic through another set of bifurcations which ultimately leads to turbulence.

physics.flu-dyn↗

arXiv2010

Energy spectra and fluxes for Rayleigh-Benard convection

We compute the spectra and fluxes of the velocity and temperature fields in Rayleigh-Bénard convection in turbulent regime for a wide range of Prandtl numbers using pseudo-spectral simulations on $512^3$ grids. Our spectral and flux results support the Kolmogorov-Obukhov (KO) scaling for zero Prandtl number and low Prandtl number ($P=0.02$) convection. The KO scaling for the velocity field in zero-Prandtl number and low-Prandtl number convection is because of the weak buoyancy in the inertial range (buoyancy is active only at the very low wavenumbers). We also observe that for intermediate Prandtl numbers ($P=0.2$) the KO scaling fits better with the numerical results than the Bolgiano-Obukhov (BO) scaling. For large Prandtl number ($P=6.8$), the spectra and flux results are somewhat inconclusive on the validity of the KO or BO scaling, yet the BO scaling is preferred over the KO scaling for these cases. The numerical results for P=1 is rather inconclusive.

physics.flu-dyn↗

arXiv2009

Patterns and bifurcations in low-Prandtl number Rayleigh-Benard convection

We present a detailed bifurcation structure and associated flow patterns for low-Prandtl number ($P=0.0002, 0.002, 0.005, 0.02$) Rayleigh-Bénard convection near its onset. We use both direct numerical simulations and a 30-mode low-dimensional model for this study. We observe that low-Prandtl number (low-P) convection exhibits similar patterns and chaos as zero-P convection \cite{pal:2009}, namely squares, asymmetric squares, oscillating asymmetric squares, relaxation oscillations, and chaos. At the onset of convection, low-P convective flows have stationary 2D rolls and associated stationary and oscillatory asymmetric squares in contrast to zero-P convection where chaos appears at the onset itself. The range of Rayleigh number for which stationary 2D rolls exist decreases rapidly with decreasing Prandtl number. Our results are in qualitative agreement with results reported earlier.

physics.flu-dyn↗

arXiv2009

Bifurcations and chaos in large Prandtl-number Rayleigh-Bénard Convection

A low-dimensional model of large Prandtl-number ($P$) Rayleigh Bénard convection is constructed using some of the important modes of pseudospectral direct numerical simulations. A detailed bifurcation analysis of the low-dimensional model for $P=6.8$ and aspect ratio of $2\sqrt{2}$ reveals a rich instability and chaos picture: steady rolls, time-periodicity, quasiperiodicity, phase locking, chaos, and crisis. Bifurcation analysis also reveals multiple co-existing attractors, and a window with time-periodicity after chaos. The results of the low-dimensional model matches quite closely with some of the past simulations and experimental results where they observe chaos in RBC through quasiperiodicity and phase locking.

physics.flu-dyn↗

arXiv2009

Bifurcation and chaos in zero Prandtl number convection

We present the detailed bifurcation structure and associated flow patterns near the onset of zero Prandtl number Rayleigh Bénard convection. We employ both direct numerical simulation and a low-dimensional model ensuring qualitative agreement between the two. Various flow patterns originate from a stationary square observed at a higher Rayleigh number through a series of bifurcations starting from a pitchfork followed by a Hopf and finally a homoclinic bifurcation as the Rayleigh number is reduced to the critical value. Global chaos, intermittency, and crises are observed near the onset.

nlin.CD↗

arXiv2009

Order and chaos in two-dimensional Rayleigh-Bénard convection

A detailed study of the Rayleigh-Bénard convection in two-dimensions with free-slip boundaries is presented. Pseudo-spectral method has been used to numerically solve the system for Rayleigh number up to $3.3 \times 10^7$. The system exhibits various convective states: stationary, oscillatory, chaotic and soft-turbulent. The `travelling rolls' instability is observed in the chaotic regime. Scaling of Nusselt number shows an exponent close to 0.33. Studies on energy spectrum and flux show an inverse cascade of kinetic energy and a forward cascade of entropy. This is consistent with the shell-to-shell energy transfer in wave number space. The shell-to-shell energy transfer study also indicates a local energy transfer from one shell to the other.

physics.flu-dyn↗

arXiv2008

Chaotic traveling rolls in Rayleigh-Bénard convection

In this paper we investigate two-dimensional (2D) Rayleigh-Bénard convection using direct numerical simulation in Boussinesq fluids of Prandtl number $P = 6.8$ confined between thermally conducting plates. We show through the simulation that in a small range of reduced Rayleigh number $r$ ($770 < r < 890$) the 2D rolls move chaotically in a direction normal to the roll axis. The lateral shift of the rolls may lead to global flow reversal of the convective motion. The chaotic traveling rolls are observed in simulation with {\it free-slip} as well as {\it no-slip} boundary conditions on the velocity field. We show that the traveling rolls and the flow reversal are due to an interplay between the real and imaginary parts of the critical modes.

nlin.CD↗

arXiv2008

Analysis of music: controlled random music and probability distribution function of recurrence time of amplitude peaks

Correlations in music that exist within its waveform are studied. Monophonic wave files of random music are generated and the probability distribution function of time interval between large signal values is analyzed. A power law behavior for the distribution function in the range from 0.1 millisecond to 20 milliseconds is observed. An attempt is made to investigate the origin of these correlations by randomizing each of the factors (frequencies, intensities and durations of notes of the random music files) separately.

nlin.AO↗

arXiv2008

Dynamo Transition in Low-dimensional Models

Two low-dimensional magnetohydrodynamic models containing three velocity and three magnetic modes are described. One of them (nonhelical model) has zero kinetic and current helicity, while the other model (helical) has nonzero kinetic and current helicity. The velocity modes are forced in both these models. These low-dimensional models exhibit a dynamo transition at a critical forcing amplitude that depends on the Prandtl number. In the nonhelical model, dynamo exists only for magnetic Prandtl number beyond 1, while the helical model exhibits dynamo for all magnetic Prandtl number. Although the model is far from reproducing all the possible features of dynamo mechanisms, its simplicity allows a very detailed study and the observed dynamo transition is shown to bear similarities with recent numerical and experimental results.

nlin.CD↗

arXiv2007

Energy Flux and Bottleneck Effect in Turbulence

Past numerical simulations and experiments of turbulence exhibit a hump in the inertial range, called the bottleneck effect. In this paper we show that sufficiently large inertial range (four decades) is required for an effective energy cascade. We propose that the bottleneck effect is due to the insufficient inertial range available in the reported simulations and experiments. To facilitate the turbulent energy transfer, the spectrum near Kolmogorov's dissipation wavenumber has a hump.

nlin.CD↗

arXiv2005

Incompressible Turbulence as Nonlocal Field Theory

It is well known that incompressible turbulence is nonlocal in real space because sound speed is infinite in incompressible fluids. The equation in Fourier space indicates that it is nonlocal in Fourier space as well. Contrast this with Burgers equation which is local in real space. Note that the sound speed in Burgers equation is zero. In our presentation we will contrast these two equations using nonlocal field theory. Energy spectrum and renormalized parameters will be discussed.

nlin.CD↗

arXiv2005

Local shell-to-shell energy transfer via nonlocal Interactions in fluid turbulence

In this paper we analytically compute the strength of nonlinear interactions in a triad, and the energy exchanges between wavenumber shells in incompressible fluid turbulence. The computation has been done using first-order perturbative field theory. In three dimension, magnitude of triad interactions is large for nonlocal triads, and small for local triads. However, the shell-to-shell energy transfer rate is found to be local and forward. This result is due to the fact that the nonlocal triads occupy much less Fourier space volume than the local ones. The analytical results on three-dimensional shell-to-shell energy transfer match with their numerical counterparts. In two-dimensional turbulence, the energy transfer rates to the near-by shells are forward, but to the distant shells are backward; the cumulative effect is an inverse cascade of energy.

nlin.CD↗

arXiv2005

Energy transfer and locality in magnetohydrodynamic turbulence

The shell-to-shell energy transfer rates for magnetohydrodynamic (MHD) turbulence are computed analytically, which shows local energy transfer rates from velocity to velocity, velocity to magnetic, magnetic to velocity, and magnetic to magnetic fields for nonhelical MHD in the inertial range. It is also found that for kinetic-energy dominated MHD fluid, there is a preferential shell-to-shell energy transfer from kinetic to magnetic energy; the transfer is reversed for magnetic-energy dominated MHD fluid. This property is the reason for the asymptotic value of Alfven ratio to be close to 0.5. The analytical results are in close agreement with recent numerical results. When magnetic and kinetic helicities are turned on, the helical contributions are opposite to the corresponding nonhelical contributions. The helical energy transfers have significant nonlocal components.

nlin.CD↗

arXiv2004

Statistical Theory of Magnetohydrodynamic Turbulence: Recent Results

In this review article we will describe recent developments in statistical theory of magnetohydrodynamic (MHD) turbulence. Kraichnan and Iroshnikov first proposed a phenomenology of MHD turbulence where Alfven time-scale dominates the dynamics, and the energy spectrum E(k) is proportional to k^{-3/2}. In the last decade, many numerical simulations show that spectral index is closer to 5/3, which is Kolmogorov's index for fluid turbulence. We review recent theoretical results based on anisotropy and Renormalization Groups which support Kolmogorov's scaling for MHD turbulence. Energy transfer among Fourier modes, energy flux, and shell-to-shell energy transfers are important quantities in MHD turbulence. We report recent numerical and field-theoretic results in this area. Role of these quantities in magnetic field amplification (dynamo) are also discussed. There are new insights into the role of magnetic helicity in turbulence evolution. Recent interesting results in intermittency, large-eddy simulations, and shell models of magnetohydrodynamics are also covered.

nlin.CD↗

arXiv2004

Field theoretic calculation of energy cascade rates in nonhelical magnetohydrodynamic turbulence

Energy cascade rates and Kolmogorov's constant for nonhelical steady magnetohydrodynamic turbulence have been calculated by solving the flux equations to the first order in perturbation. For zero cross helicity and space dimension $d=3$, magnetic energy cascades from large length-scales to small length-scales (forward cascade). In addition, there are energy fluxes from large-scale magnetic field to small-scale velocity field, large-scale velocity field to small-scale magnetic field, and large-scale velocity field to large-scale magnetic field. Kolmogorov's constant for magnetohydrodynamics is approximately equal to that for fluid turbulence ($\approx 1.6$) for Alfvén ratio $0.5 \le r_A \le \infty$. For higher space-dimensions, the energy fluxes are qualitatively similar, and Kolmogorov's constant varies as $d^{1/3}$. For the normalized cross helicity $σ_c \to 1$, the cascade rates are proportional to $(1-σ_c)/ (1+σ_c)$, and the Kolmogorov's constants vary significantly with $σ_c$.

nlin.CD↗

arXiv2004

Energy fluxes in helical magnetohydrodynamics and dynamo action

Renormalized viscosity, renormalized resistivity, and various energy fluxes are calculated for helical magnetohydrodynamics using perturbative field theory. The calculation is to first-order in perturbation. Kinetic and magnetic helicities do not affect the renormalized parameters, but they induce an inverse cascade of magnetic energy. The sources for the the large-scale magnetic field have been shown to be (1) energy flux from large-scale velocity field to large-scale magnetic field arising due to nonhelical interactions, and (2) inverse energy flux of magnetic energy caused by helical interactions. Based on our flux results, a premitive model for galactic dynamo has been constructed. Our calculations yields dynamo time-scale for a typical galaxy to be of the order of $10^8$ years. Our field-theoretic calculations also reveal that the flux of magnetic helicity is backward, consistent with the earlier observations based on absolute equilibrium theory.

nlin.CD↗

arXiv2003

Large-Eddy Simulations of Fluid and Magnetohydrodynamic Turbulence Using Renormalized Parameters

In this paper a procedure for large-eddy simulation (LES) has been devised for fluid and magnetohydrodynamic turbulence in Fourier space using the renormalized parameters. The parameters calculated using field theory have been taken from recent papers by Verma [Phys. Rev. E, 2001; Phys. Plasmas, 2001]. We have carried out LES on $64^3$ grid. These results match quite well with direct numerical simulations of $128^3$. We show that proper choice of parameter is necessary in LES.

nlin.CD↗

arXiv2002

Field theoretic calculation of scalar turbulence

The cascade rate of passive scalar and Bachelor's constant in scalar turbulence are calculated using the flux formula. This calculation is done to first order in perturbation series. Batchelor's constant in three dimension is found to be approximately 1.25. In higher dimension, the constant increases as $d^{1/3}$.

nlin.CD↗