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Papers by Supriyo Paul

International Journal of Non-linear Mechanics, 2011

Rayleigh–Bénard convection with large-Prandtl number (P) is studied using a low-dimensional model... more Rayleigh–Bénard convection with large-Prandtl number (P) is studied using a low-dimensional model constructed with the energetic modes of pseudospectral direct numerical simulations. A detailed bifurcation analysis of the non-linear response has been carried out for water at room temperature (P=6.8) as the working fluid. This analysis reveals a rich instability and chaos picture: steady rolls, time-periodicity, quasiperiodicity, phase locking, chaos, and crisis. Our low-dimensional model captures the reappearance of ordered states after chaos, as previously observed in experiments and simulations. We also observe multiple coexisting attractors consistent with previous experimental observations for a range of parameter values. The route to chaos in the model occurs through quasiperiodicity and phase locking, and attractor-merging crisis. Flow patterns spatially moving along the periodic direction have also been observed in our model.

In this paper we present various convective states of zero-Prandtl number Rayleigh-Bénard convect... more In this paper we present various convective states of zero-Prandtl number Rayleigh-Bénard convection using direct numerical simulations (DNS) and a 27-mode lowdimensional model containing the energetic modes of DNS. The origin of these convective states have been explained using bifurcation analysis. The system is chaotic at the onset itself with three coexisting chaotic attractors that are born at two codimension-2 bifurcation points. One of the bifurcation points with a single zero eigenvalue and a complex pair (0, ±iω) generates chaotic attractors and associated periodic, quasiperiodic, and phase-locked states that are related to the wavy rolls observed in experiments and simulations. The frequency of the wavy rolls are in general agreement with ω of the above eigenvalue of the stability matrix. The other bifurcation point with a double zero eigenvalue produces the other set of chaotic attractors and ordered states such as squares, asymmetric squares, oscillating asymmetric squares, relaxation oscillations with intermediate squares, some of which are common to the 13-mode model of Pal et al.

A low-dimensional model of large Prandtl-number ($P$) Rayleigh B\'{e}nard convection is construct... more A low-dimensional model of large Prandtl-number ($P$) Rayleigh B\'{e}nard 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=6.8P=6.8P=6.8 and aspect ratio of 2sqrt22\sqrt{2}2sqrt2 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.

Chaos, 2011

We present a detailed bifurcation scenario of zero-Prandtl number Rayleigh-Bénard convection usin... more We present a detailed bifurcation scenario of zero-Prandtl number Rayleigh-Bénard convection using direct numerical simulations (DNS) and a 27-mode low-dimensional model containing the most energetic modes of DNS. The bifurcation analysis reveals a rich variety of convective flow patterns and chaotic solutions, some of which are common to that of the 13-mode model of Pal et al. [EPL 87, 54003 (2009)]. We also observed a set of periodic and chaotic wavy rolls in DNS and in the model similar to those observed in experiments and numerical simulations. The time period of the wavy rolls is closely related to the eigenvalues of the stability matrix of the Hopf bifurcation points at the onset of convection. This time period is in good agreement with the experimental results for low-Prandtl number fluids. The chaotic attractor of the wavy roll solutions is born through a quasiperiodic and phase-locking route to chaos.

Using direct numerical simulations of Rayleigh-Bénard convection (RBC) under free-slip boundary c... more Using direct numerical simulations of Rayleigh-Bénard convection (RBC) under free-slip boundary condition, we show that the normalized correlation function between the vertical velocity field and the temperature field, as well as the normalized viscous dissipation rate, scales as Ra −0.22 for moderately large Rayleigh number Ra. This scaling accounts for the Nusselt number (N u) exponent to be around 0.3 observed in experiments. Numerical simulations also reveal that the above normalized correlation functions are constants for the convection simulation under periodic boundary conditions.

Pramana-journal of Physics, 2010

In this paper we investigate two-dimensional (2D) Rayleigh-Bénard convection using direct numeric... more In this paper we investigate two-dimensional (2D) Rayleigh-Bénard convection using direct numerical simulation in Boussinesq fluids with 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 a global flow reversal of the convective motion. The chaotic travelling rolls are observed in simulations with free-slip as well as no-slip boundary conditions on the velocity field. We show that the travelling rolls and the flow reversal are due to an interplay between the real and imaginary parts of the critical modes.

A detailed study of the Rayleigh-B\'enard convection in two-dimensions with free-slip boundaries ... more A detailed study of the Rayleigh-B\'enard 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 3.3times1073.3 \times 10^73.3times107. 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.

Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 2007

Effect of magnetic field on parametrically driven References ml#ref-list-1 Stability analysis of ... more Effect of magnetic field on parametrically driven References ml#ref-list-1 Stability analysis of parametrically driven surface waves in liquid metals in the presence of a uniform vertical magnetic field is presented. Floquet analysis gives various subharmonic and harmonic instability zones. The magnetic field stabilizes the onset of parametrically excited surface waves. The minima of all the instability zones are raised by a different amount as the Chandrasekhar number Q is raised. The increase in the magnetic field leads to a series of bicritical points at a primary instability in thin layers of a liquid metal. The bicritical points involve one subharmonic and another harmonic solution of different wavenumbers. A tricritical point may also be triggered as a primary instability by tuning the magnetic field.

Journal of Physics: Conference Series, 2011

Rotating helical turbulence: three-dimensionalization or self-similarity in the small scales?

A detailed study of the Rayleigh-Bénard convection in two-dimensions with free-slip boundaries is... more 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 × 10 7 . The system exhibits various convective states: stationary, oscillatory, chaotic and soft-turbulent. The 'travelling rolls' instability is observed in the chaotic regime.

We investigate the origin of various convective patterns using bifurcation diagrams that are cons... more 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 √ 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 ≃ 80 and r ≃ 500 respectively. The system becomes chaotic at r ≃ 750 through a quasiperiodic route to chaos. The size of the chaotic attractor increases at r ≃ 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 ≤ r ≤ 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. PACS numbers: 47.20.Bp, 47.27.ek, 47.52.+j

International Journal of Non-linear Mechanics, 2011

Rayleigh–Bénard convection with large-Prandtl number (P) is studied using a low-dimensional model... more Rayleigh–Bénard convection with large-Prandtl number (P) is studied using a low-dimensional model constructed with the energetic modes of pseudospectral direct numerical simulations. A detailed bifurcation analysis of the non-linear response has been carried out for water at room temperature (P=6.8) as the working fluid. This analysis reveals a rich instability and chaos picture: steady rolls, time-periodicity, quasiperiodicity, phase locking, chaos, and crisis. Our low-dimensional model captures the reappearance of ordered states after chaos, as previously observed in experiments and simulations. We also observe multiple coexisting attractors consistent with previous experimental observations for a range of parameter values. The route to chaos in the model occurs through quasiperiodicity and phase locking, and attractor-merging crisis. Flow patterns spatially moving along the periodic direction have also been observed in our model.

In this paper we present various convective states of zero-Prandtl number Rayleigh-Bénard convect... more In this paper we present various convective states of zero-Prandtl number Rayleigh-Bénard convection using direct numerical simulations (DNS) and a 27-mode lowdimensional model containing the energetic modes of DNS. The origin of these convective states have been explained using bifurcation analysis. The system is chaotic at the onset itself with three coexisting chaotic attractors that are born at two codimension-2 bifurcation points. One of the bifurcation points with a single zero eigenvalue and a complex pair (0, ±iω) generates chaotic attractors and associated periodic, quasiperiodic, and phase-locked states that are related to the wavy rolls observed in experiments and simulations. The frequency of the wavy rolls are in general agreement with ω of the above eigenvalue of the stability matrix. The other bifurcation point with a double zero eigenvalue produces the other set of chaotic attractors and ordered states such as squares, asymmetric squares, oscillating asymmetric squares, relaxation oscillations with intermediate squares, some of which are common to the 13-mode model of Pal et al.

A low-dimensional model of large Prandtl-number ($P$) Rayleigh B\'{e}nard convection is construct... more A low-dimensional model of large Prandtl-number ($P$) Rayleigh B\'{e}nard 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=6.8P=6.8P=6.8 and aspect ratio of 2sqrt22\sqrt{2}2sqrt2 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.

Chaos, 2011

We present a detailed bifurcation scenario of zero-Prandtl number Rayleigh-Bénard convection usin... more We present a detailed bifurcation scenario of zero-Prandtl number Rayleigh-Bénard convection using direct numerical simulations (DNS) and a 27-mode low-dimensional model containing the most energetic modes of DNS. The bifurcation analysis reveals a rich variety of convective flow patterns and chaotic solutions, some of which are common to that of the 13-mode model of Pal et al. [EPL 87, 54003 (2009)]. We also observed a set of periodic and chaotic wavy rolls in DNS and in the model similar to those observed in experiments and numerical simulations. The time period of the wavy rolls is closely related to the eigenvalues of the stability matrix of the Hopf bifurcation points at the onset of convection. This time period is in good agreement with the experimental results for low-Prandtl number fluids. The chaotic attractor of the wavy roll solutions is born through a quasiperiodic and phase-locking route to chaos.

Using direct numerical simulations of Rayleigh-Bénard convection (RBC) under free-slip boundary c... more Using direct numerical simulations of Rayleigh-Bénard convection (RBC) under free-slip boundary condition, we show that the normalized correlation function between the vertical velocity field and the temperature field, as well as the normalized viscous dissipation rate, scales as Ra −0.22 for moderately large Rayleigh number Ra. This scaling accounts for the Nusselt number (N u) exponent to be around 0.3 observed in experiments. Numerical simulations also reveal that the above normalized correlation functions are constants for the convection simulation under periodic boundary conditions.

Pramana-journal of Physics, 2010

In this paper we investigate two-dimensional (2D) Rayleigh-Bénard convection using direct numeric... more In this paper we investigate two-dimensional (2D) Rayleigh-Bénard convection using direct numerical simulation in Boussinesq fluids with 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 a global flow reversal of the convective motion. The chaotic travelling rolls are observed in simulations with free-slip as well as no-slip boundary conditions on the velocity field. We show that the travelling rolls and the flow reversal are due to an interplay between the real and imaginary parts of the critical modes.

A detailed study of the Rayleigh-B\'enard convection in two-dimensions with free-slip boundaries ... more A detailed study of the Rayleigh-B\'enard 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 3.3times1073.3 \times 10^73.3times107. 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.

Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 2007

Effect of magnetic field on parametrically driven References ml#ref-list-1 Stability analysis of ... more Effect of magnetic field on parametrically driven References ml#ref-list-1 Stability analysis of parametrically driven surface waves in liquid metals in the presence of a uniform vertical magnetic field is presented. Floquet analysis gives various subharmonic and harmonic instability zones. The magnetic field stabilizes the onset of parametrically excited surface waves. The minima of all the instability zones are raised by a different amount as the Chandrasekhar number Q is raised. The increase in the magnetic field leads to a series of bicritical points at a primary instability in thin layers of a liquid metal. The bicritical points involve one subharmonic and another harmonic solution of different wavenumbers. A tricritical point may also be triggered as a primary instability by tuning the magnetic field.

Journal of Physics: Conference Series, 2011

Rotating helical turbulence: three-dimensionalization or self-similarity in the small scales?

A detailed study of the Rayleigh-Bénard convection in two-dimensions with free-slip boundaries is... more 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 × 10 7 . The system exhibits various convective states: stationary, oscillatory, chaotic and soft-turbulent. The 'travelling rolls' instability is observed in the chaotic regime.

We investigate the origin of various convective patterns using bifurcation diagrams that are cons... more 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 √ 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 ≃ 80 and r ≃ 500 respectively. The system becomes chaotic at r ≃ 750 through a quasiperiodic route to chaos. The size of the chaotic attractor increases at r ≃ 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 ≤ r ≤ 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. PACS numbers: 47.20.Bp, 47.27.ek, 47.52.+j