Session EJ: Convection and Buoyancy Driven Flows III

How is the large-scale flow influenced by the aspect-ratio in turbulent cylindrical Rayleigh-B\'enard samples?

In turbulent Rayleigh-B\'enard convection the fluid develops a large-scale flow (LSF). The shadowgraph method is used to visualize plumes in cylindrical samples that are emitted from a top and bottom boundary layer. These plumes serve as tracers of the LSF. In aspect-ratio $\Gamma \equiv D/L$ = 1 ($D$ and $L$ are the diameter and height of the cell respectively), the LSF consists of a single roll circulating in a near-vertical plane and oscillating azimuthally with a well defined frequency\footnote{D. Funfschilling and G. Ahlers, Phys. Rev. Lett. {\bf 92}, 194502 (2004).}. For $\Gamma$ = 2 and 3, the LSF still consists of a single roll but has no detectable oscillations. For $\Gamma$ = 2 and 3 the ``density" of plumes above the bottom plate (the ratio of the area occupied by the plumes to the total area) does show clear oscillations at a well defined frequency. This later observation adds new interest to a model of plume emission by Villermaux\footnote{E. Villermaux, Phys. Rev. Lett. {\bf 75}, 4618 (1995)}. Our results provide evidence for a transition in the LSF dynamics between $\Gamma$ = 1 and $\Gamma$ = 2.   

Aspect-ratio effect on Natural Convection subject to horizontal temperature gradient

Direct numerical simulations of natural convection in low aspect-ratio (AR) cavities with heated and cooled side walls and adiabatic top and bottom are performed. Three cavities with $AR=0.5$, 1, and 2 are simulated in the range of Rayleigh number ($Ra$) from 1 to $10^8$ and Prandtl number $Pr=0.71$. Mean velocity magnitude, Nusselt number ($Nu$), and boundary layer thicknesses are computed as functions of $Ra$. In the laminar flow regime with $Ra$ approximately from $10^3$ to $10^7$, power-law scalings $\sim Ra^\beta$ of velocity magnitude, $Nu$, and boundary layer thicknesses are found in three cavities. Threshold Ra values are examined at the onset of the power-law growth in each cavity. Aspect-ratio effect on flow structure and heat transfer are demonstrated.   

Influence of Periodic Fins in a 2-D Rayeigh-B\'{e}nard Cavity

We examine the heat transfer augmentation of classic Rayleigh-B\'{e}nard convection resulting from the the addition of periodically-spaced tranverse fins attached to the heated, lower plate. The respective impacts of the fin size, the fin spacing and the thermal conductivity of the fin material are examined through numerical simulations. The present study has been primarily focused on laminar flow regimes and fin spacing-to-gap ratios of 5-to-1 and less. Weakly turbulent flow conditions have also been examined, as have instances of thermally-conducting and thermally-insulating fin materials. The results from numerical simulations have revealed that surprisingly rich fluid mechanical behavior is possible under certain parametric conditions, including flow bifurcations leading to dual-convection cells not found in the classic 2-D Rayleigh-B\'{e}nard problem. It is found that the impact of the fin is almost entirely due to its hydrodynamic role as a no-slip boundary condition, thus rendering the selection of fin material moot. For all but the shortest of fins, the heat transfer obtained for all spacing-to-gap ratios is less than that for the Rayleigh-B\'{e}nard scenario; in contrast, for very short fins, an enhancement of heat transfer is possible for the range of conditions examined. The existence of `optimal' conditions which maximize heat transfer is discussed.   

Heat flux and local temperature measurements in extremely small aspect ratio turbulent Rayleigh-B{\'e}nard convection

We report measurements of local temperature fluctuations and the heat flux in cylindrical convection cells of diameter $\sim 20$ cm and heights 1-m and 2-m respectively (aspect ratio $\Gamma =$ 0.2 and 0.1). It is found that local temperature fluctuations exhibit a Gaussian distribution even at cell center and it appears to be independent of positions along the central axis of the cell. It is also found that the mean temperature has a linear profile across the height of the cell. The Nusselt number (Nu) measured with both copper and aluminum plates show that the effect of the plates' finite thermal conductivity is very small in this system. The scaling of Nu with Ra exhibits a local exponent that varies continuously from less than $1/3$ to larger than $1/3$ over the range of Ra from $3\times 10^{10}$ to $1\times 10^{13}$.   

Convective instabilities in thermogravitational columns

Convective instabilities in side heated infinite vertical slots containing a single fluid are stationary, shear driven when the Prandtl number $Pr\tilde {\le }12.5$ while they are oscillatory, buoyancy dominated with $Pr\tilde {\ge }12.5$ due to the diminished influence of the thermal diffusivity with increasing $Pr$. Here we examine the influence of the concentration field generated by thermodiffusion in a binary mixture of otherwise uniform concentration on this phenomenon. While positive/negative separation corresponds to enhanced/diminished buoyancy and should promote instability/stability, the induced positive/negative vertical concentration gradient of the light component, i.e., stable/unstable stratification, and the nonmonotic horizontal species separation demanded by the vanishing vertical mass flux, combine to result in the opposite effect. Thus increasing/decreasing the separation ratio $\varepsilon $ is found to stabilize/destabilize both instability branches so that the cutoff $Pr$ where there is a switch from preferred stationary to oscillatory states is a monotonic increasing function of $\varepsilon $.   

Route to Turbulence in Sheared Annular Electroconvection

We studied the route to turbulence of a 2D, electrically-driven annular film, using direct numerical simulation. The film can simultaneously be sheared by rotating the inner edge of the annulus. The simulation models a laboratory experiment which consists of a weakly conducting liquid crystal film suspended between concentric electrodes. The film convects when a sufficiently large voltage $V$ is applied. The flow is driven by a surface charge density inversion unstable to the applied potential. The important dimensionless parameters are a Rayleigh-like number $Ra$, proportional to $V^2$, a Prandtl-like number $Pr$ and the radius ratio $\alpha$, characterizing the annular geometry. The applied shear has Reynolds number $Re$. The simulation uses a pseudo-spectral method with radial Chebyshev polynomials and azimuthal Fourier modes. The numerical results show a suppression of the onset of convection under the influence of shear that quantitatively agrees with previous theoretical and experimental results. Just above onset under shear, the numerical results reveal a Ruelle-Takens- Newhouse scenario in which there are bifurcations between various periodic and quasi-periodic flows. With increasing $Ra$, at constant applied $Re$, we observe subcritical bifurcations, indicated by sudden increases in convective charge transport. These jumps are also seen experimentally, and correspond to bifurcations between different azimuthal mode numbers.   

Globally Coupled Ginzburg Landau Equations for Electroconvection in Nematic Liquid Crystals

For certain materials, the electrohydrodynamic instability leading to convection in nematic liquid crystals is a Hopf bifurcation with four critical wave numbers. As a consequence, the linearized problem admits solutions in the form of two pairs of oblique counterpropagating travelling rolls. To describe this instability in a weakly nonlinear analysis, a system of four globally coupled Ginzburg Landau equations is introduced, whose coefficients can be computed from the weak electrolyte model of Kramer and Treiber. Some aspects of the solution behaviour of this system are discussed and related to recent experiments conducted at Kent State University for the nematic I52.   

Relaxation-Oscillation and Eckhaus -Unstable Roll Dynamics Observed Within Laser-Localized Electroconvection Spots.

We use modest laser powers to create slightly warmed regions within a nematic liquid crystal to generate localized electrconvection patterns [1]. Using a laser profile with an intensity that varies linearly with position, we observe counterpropagating waves with nonzero group and phase velocity; the latter is the largest and increases significantly if the applied frequency and voltage are increased in a manner that holds the control parameter fixed. The temporal dependence of the amplitude at a point is periodic but not sinusoidal: its temporal profile is that of a relaxation-oscillator. Measurement of the wavevector profiles shows the wavevector range for the traveling waves extends well beyond the Eckhaus-stable band. This type of measurement can in principle be used to provide an experimental system-size control parameter for spatial-temporal chaos; results of preliminary experiments on this problem will be presented.