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.
The origin of the power-law decay measured in the power spectra of low Prandtl number Rayleigh-Benard convection near the onset of chaos is addressed using long time numerical simulations of the three-dimensional Boussinesq equations in cylindrical domains. The power-law is found to arise from quasi-discontinuous changes in the slope of the time series of the heat transport associated with the nucleation of dislocation pairs and roll pinch-off events. For larger frequencies, the power spectra decay exponentially as expected for time continuous deterministic dynamics.
The Rayleigh-Benard theory by Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000)] is extended towards very large Prandtl numbers Pr. The Nusselt number Nu is found here to be independent of Pr. However, for fixed Rayleigh numbers Ra> 10^{10} a maximum around Pr= 2 in the Nu(Pr)-dependence is predicted which is absent for lower Ra. We moreover offer the full functional dependences of Nu(Ra,Pr) and Re(Ra,Pr) within this extended theory, rather than only giving the limiting power laws as done previously. This enables us to more realistically describe the transitions between the various scaling re…
In this paper we present various convective states of zero-Prandtl number Rayleigh-Bénard convection using direct numerical simulations (DNS) and a 27-mode low-dimensional 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 $(0,\pm i ω)$ generates chaotic attractors and associated periodic, quasiperiodi…
We study the scaling properties of heat transfer $Nu$ in turbulent thermal convection at large Prandtl number $Pr$ using a quasi-linear theory. We show that two regimes arise, depending on the Reynolds number $Re$. At low Reynolds number, $Nu Pr^{-1/2}$ and $Re$ are a function of $Ra Pr^{-3/2}$. At large Reynolds number $Nu Pr^{1/3}$ and $Re Pr$ are function only of $Ra Pr^{2/3}$ (within logarithmic corrections). In practice, since $Nu$ is always close to $Ra^{1/3}$, this corresponds to a much weaker dependence of the heat transfer in the Prandtl number at low Reynolds number than at large Rey…
We investigate instabilities and chaos near the onset of Rayleigh-Bénard convection (RBC) of electrically conducting fluids with free-slip, perfectly electrically and thermally conducting boundary conditions in presence of uniform rotation about vertical axis and horizontal external magnetic field by considering zero Prandtl-number limit ($\mathrm{Pr} \rightarrow 0$). Direct numerical simulations (DNS) and low dimensional modeling of the system are done for the investigation. Values of the Chandrasekhar number ($\mathrm{Q}$) and the Taylor number ($\mathrm{Ta}$) are varied in the range $0 < \m…
We investigate the onset of convective instability and subsequent transitions near it in Rayleigh-Bénard convection (RBC) of electrically conducting low Prandtl-number (Pr) fluids in the simultaneous presence of rotation about the vertical axis and external uniform horizontal magnetic field with free-slip boundary conditions. Both three dimensional direct numerical simulations (DNS) and low dimensional modeling of the system have been performed for the investigation. DNS results show a plethora of patterns near the onset of convection for weak magnetic field. Most interestingly, the Nusselt nu…
Turbulent Rayleigh-Bénard convection displays a large-scale order in the form of rolls and cells on lengths larger than the layer height once the fluctuations of temperature and velocity are removed. These turbulent superstructures are reminiscent of the patterns close to the onset of convection. They are analyzed by numerical simulations of turbulent convection in fluids at different Prandtl number ranging from 0.005 to 70 and for Rayleigh numbers up to $10^7$. For each case, we identify characteristic scales and times that separate the fast, small-scale turbulent fluctuations from the gradua…
We investigate numerically the statistical properties of the large scale flow in Rayleigh--Bénard convection. By using an external random perturbation on the temperature field, we were able to decrease the effective Prandtl number of the flow while keeping the Rayleigh number relatively small; this increases the Reynolds number thus making possible the numerical investigation of the long--term flow statistics. We also propose a simple and quantitative explanation for the experimental findings on the statistical distribution of flow reversals and reorientations.
This paper investigates the global structures of periodic orbits that appear in Rayleigh-Bénard convection, which is modeled by a two-dimensional perturbed Hamiltonian model, by focusing upon resonance, symmetry and bifurcation of the periodic orbits. First, we show the global structures of periodic orbits in the extended phase space by numerically detecting the associated periodic points on the Poincaré section. Then, we illustrate how resonant periodic orbits appear and specifically clarify that there exist some symmetric properties of such resonant periodic orbits which are projected on the…
We use nonlinear signal processing techniques, based on artificial neural networks, to construct an empirical mapping from experimental Rayleigh-Benard convection data in the quasiperiodic regime. The data, in the form of a one-parameter sequence of Poincare sections in the interior of a mode-locked region (resonance horn), are indicative of a complicated interplay of local and global bifurcations with respect to the experimentally varied Rayleigh number. The dynamic phenomena apparent in the data include period doublings, complex intermittent behavior, secondary Hopf bifurcations, and chaotic…
A global bifurcation of the blue sky catastrophe type has been found in a small Prandtl number binary mixture contained in a laterally heated cavity. The system has been studied numerically applying the tools of bifurcation theory. The catastrophe corresponds to the destruction of an orbit which, for a large range of Rayleigh numbers, is the only stable solution. This orbit is born in a global saddle-loop bifurcation and becomes chaotic in a period doubling cascade just before its disappearance at the blue sky catastrophe.
We study the dynamics of a system defined by the Navier-Stokes equations for a non-compressible fluid with Marangoni boundary conditions in the two dimensional case. We show that more complicated bifurcations can appear in this system for a certain nonlinear temperature profile as compared to bifurcations in the classical Rayleigh-Bénard and Bénard-Marangoni systems with simple linear vertical temperature profiles. In terms of the Bénard-Marangoni convection, the obtained mathematical results lead to our understanding of complex spatial patterns at a free liquid surface, which can be induced b…
We present the first large scale numerical simulation of three-dimensional Rayleigh-Bénard convection near onset, under free-free boundary conditions for a fluid of Prandtl number $σ=0.5$. We find that a spatiotemporally chaotic state emerges immediately above onset, which we investigate as a function of the reduced control parameter $ε$. We conclude that the transition from conduction to spatiotemporal chaos is second order and of ``mean field'' character. We also present a simple theory for the time-averaged convective current. Finally, we show that the time-averaged structure factor satisfi…
Direct measurements of the velocity in turbulent convection of $SF_6$ near its gas-liquid critical point by light scattering on the critical density fluctuations were conducted. The temperature, velocity and cross frequency power spectra in a wide range of the Rayleigh and Prandtl numbers show scaling behaviour with indices, which are rather close to the Bolgiano-Obukhov scaling in the wavenumber domain. The statistics of the velocity fluctuations remains Gaussian up to the Reynolds numbers $10^5$.
We investigate oscillatory instability and routes to chaos in Rayleigh-Bénard convection of electrically conducting fluids in presence of external horizontal magnetic field. Three dimensional direct numerical simulations (DNS) of the governing equations are performed for the investigation. DNS shows that oscillatory instability is inhibited by the magnetic field. The supercritical Rayleigh number for the onset of oscillation is found to scale with the Chandrasekhar number $\mathrm{Q}$ as $\mathrm{Q}^α$ in DNS with $α= 1.8$ for low Prandtl numbers ($\mathrm{Pr}$). Most interestingly, DNS shows …
$SF_6$ in the vicinity of its critical point was used to study turbulent convection up to exceptionally high Rayleigh numbers, $Ra$, (up to $5\cdot 10^{14}$) and to verify for the first time the generalized scaling laws for the heat transport and the large scale circulation velocity as a function of $Ra$ and the Prandtl number, $Pr$, in very wide range of these parameters. The both scaling laws obtained are consistent with theoretical predictions by B.Shraiman and E.Siggia, Phys. Rev. {\bf A 42}, 3650 (1990).
A solvable turbulent model is used to predict both the structure of the boundary layer and the scaling laws in thermal convection. The transport of heat depends on the interplay between the thermal, viscous and integral scales of turbulence, and thus, on both the Prandtl number and the Reynolds numbers. Depending on their values, a wide variety of possible regimes is found, including the classical 2/7 and 1/3 law, and a new $4/13=0.308$ law for the Nusselt power law variation with the Rayleigh number.
