Search Results (151 total)

As shown in earlier work [Ahlers , J. Fluid Mech. 569, 409 (2006)], non-Oberbeck-Boussinesq (NOB) corrections to the center temperature in turbulent Rayleigh-Bénard convection in water and also in glycerol are governed by the temperature dependences of the kinematic viscosity and the thermal diffusion coefficient. If the working fluid is ethane close to the critical point, the origin of non-Oberbeck-Boussinesq corrections is very different, as will be shown in the present paper. Namely, the main origin of NOB corrections then lies in the strong temperature dependence of the isobaric thermal expansion coefficient β(T). More precisely, it is the nonlinear T dependence of the density ρ(T) in the buoyancy force that causes another type of NOB effect. We demonstrate this through a combination of experimental, numerical, and theoretical work, the last in the framework of the extended Prandtl-Blasius boundary-layer theory developed by Ahlers as cited above. The theory comes to its limits if the temperature dependence of the thermal expension coefficient β(T) is significant. The measurements reported here cover the ranges 2.1≲Pr≲3.9 and 5×10⁹≲Ra≲2×10¹² and are for cylindrical samples of aspect ratios 1.0 and 0.5.

A model for the large-scale-circulation (LSC) of turbulent Rayleigh-Bénard convection in cylindrical samples is presented. It consists of two physically motivated stochastic ordinary differential equations, one each for the strength and the azimuthal orientation of the LSC. Stochastic forces represent phenomenologically the influence of turbulent fluctuations. Consistent with measurements, the model yields an azimuthally meandering LSC with occasional rotations, and with more rare cessations. As in experiment, cessations have a uniform distribution of LSC orientation changes.

Non-Oberbeck-Boussinesq (NOB) effects are measured experimentally and calculated theoretically for strongly turbulent Rayleigh-Bénard convection of ethane gas under pressure where the material properties strongly depend on the temperature. Relative to the Oberbeck-Boussinesq case we find a decrease of the central temperature as compared to the arithmetic mean of the top- and bottom-plate temperature and an increase of the Nusselt number. Both effects are of opposite sign and greater magnitude than those for NOB convection in liquids like water.

We present patterns of electroconvection (EC) for the homeotropically aligned nematic liquid crystal MBBA. A voltage V=sqrt[2]V₀  sin(2πft) was applied. With increasing V₀, the bend Freedericksz transition at V_F was followed by the onset of EC at V_c>V_F. We found four distinct pattern types. First, a primary supercritical Hopf bifurcation to traveling waves (TW’s) of convection rolls occurred. The structure factor S(k) of this state reflected the azimuthal anisotropy of the underlying Freedericksz state. For f<f_L≃75  Hz there was a superposition of two oblique-roll modes (pattern I). These patterns were chaotic in space and time. For larger f the patterns consisted of chaotic TW normal rolls (pattern II). Here the chaos was attributable to the motion of dislocations and domain walls between left- and right-traveling waves. A secondary bifurcation yielded pattern III; it had no dominant TW frequency but had broadband chaotic dynamics dominated by the motion of dislocations. This pattern type had been referred to by others as a “chevron pattern;” its structure factor still revealed azimuthal anisotropy. Finally, at somewhat larger ϵ≡V²∕V_c²−1 a highly disordered pattern IV with defect dynamics was found. This state had been studied before by Kai and co-workers and was referred to by them as “phase turbulence.” It had a structure factor that was (within our resolution) invariant under rotation. For patterns I, II, and III, S(k) contained crescent-shaped peaks. The peak shape was qualitatively different from the case of planar EC where the structure factor has an elliptical cross section. We present measurements of the widths 1∕ξ_k and 1∕ξ_θ in the radial (k) and the azimuthal (θ) directions. For small ϵ (patterns I and II) we found that ξ_k was consistent with the usual Ginzburg-Landau scaling ξ_k∼ϵ^−ν_k with ν_k≃1∕2. However, for ξ_θ we found ξ_θ∼ϵ^−ν_θ with ν_θ≃3∕4. Presumably this anomalous scaling of ξ_θ is associated with the Goldstone mode of homeotropic EC. We also show data for the height S₀ of the structure factor that are consistent with S₀∼ϵ^β with β≃−0.5, implying that S₀ diverges at onset. This differs from the case of domain chaos in rotating Rayleigh-Bénard convection where experiment is consistent with β=1∕2 and thus with a vanishing S₀. The difference between the shape of the structure-factor cross section and between the exponents, for the present case, for planar EC, and for domain chaos suggests that there are different universality classes for spatiotemporal chaos.

We present experimental measurements near the onset of electroconvection (EC) of homeotropically aligned nematic liquid crystals Phase 5A and MBBA. A voltage of amplitude sqrt[2]V₀ and frequency f was applied. With increasing V₀, EC occurred after the bend Freedericksz transition. We found supercritical bifurcations to EC that were either stationary bifurcations or Hopf bifurcations to traveling convection rolls, depending on the sample conductances. Results for the onset voltages V_c, the critical wave numbers k_c, the obliqueness angles θ_c, and the traveling-wave (Hopf) frequencies at onset ω_c over a range of sample conductances and driving frequencies are presented and compared, to the extent possible, with theoretical predictions. For the most part good agreement was found. However, the experiment revealed some unusual results for the orientations of the convection rolls relative to the direction selected by the Freedericksz domain.

Experiments and simulations from a variety of sample sizes indicated that the centrifugal force significantly affects the domain-chaos state observed in rotating Rayleigh-Bénard convection-patterns. In a large-aspect-ratio sample, we observed a hybrid state consisting of domain chaos close to the sample center, surrounded by an annulus of nearly stationary nearly radial rolls populated by occasional defects reminiscent of undulation chaos. Although the Coriolis force is responsible for domain chaos, by comparing experiment and simulation we show that the centrifugal force is responsible for the radial rolls. Furthermore, simulations of the Boussinesq equations for smaller aspect ratios neglecting the centrifugal force yielded a domain precession-frequency f∼ε^μ with μ≃1 as predicted by the amplitude-equation model for domain chaos, but contradicted by previous experiment. Additionally the simulations gave a domain size that was larger than in the experiment. When the centrifugal force was included in the simulation, μ and the domain size were consistent with experiment.

The disagreement of the scaling of the correlation length ξ between experiment and the Ginzburg-Landau (GL) model for domain chaos was resolved. The Swift-Hohenberg (SH) domain chaos model was integrated numerically to acquire test images to study the effect of a finite image size on the extraction of ξ from the structure factor (SF). The finite image size had a significant effect on the SF determined with the Fourier-transform (FT) method. The maximum entropy method (MEM) was able to overcome this finite image-size problem and produced fairly accurate SFs for the relatively small image sizes provided by experiments. Correlation lengths often have been determined from the second moment of the SF of chaotic patterns because the functional form of the SF is not known. Integration of several test functions provided analytic results indicating that this may not be a reliable method of extracting ξ. For both a Gaussian and a squared SH form, the correlation length ξ̅ ≡1∕σ, determined from the variance σ² of the SF, has the same dependence on the control parameter ε as the length ξ contained explicitly in the functional forms. However, for the SH and the Lorentzian forms we find ξ̅ ∼ξ^1∕2. Results for ξ determined from new experimental data by fitting the functional forms directly to the experimental SF yielded ξ∼ε^−ν with ν≃1 / 4 for all four functions in the case of the FT method, but ν≃1 / 2, in agreement with the GL prediction, in the case of the MEM. Over a wide range of ε and wave number k, the experimental SFs collapsed onto a unique curve when appropriately scaled by ξ.

We present measurements of the orientation θ₀(t) of the large-scale circulation (LSC) of turbulent Rayleigh-Bénard convection in cylindrical cells of aspect ratio 1. θ₀(t) undergoes irregular reorientations. It contains reorientation events by rotation through angles Δθ with a monotonically decreasing probability distribution p(Δθ), and by cessations (where the LSC stops temporarily) with a uniform p(Δθ). Reorientations have Poissonian statistics in time. The amplitude of the LSC and the magnitude of the azimuthal rotation rate have a negative correlation.

We fitted C(k,τ,ϵ)∝exp⁡[-σ(k,ϵ)τ] to time-correlation functions C(k,τ,ϵ) of structure factors S(k,t,ϵ) of shadowgraph images of fluctuations below a supercritical bifurcation at V₀=V_c to electroconvection of a planar nematic liquid crystal in the presence of a voltage V=sqrt[2]V₀cos⁡(2πft) [k=(p,q) is the wave vector and ϵ≡V₀²/V_c²-1]. There were stationary oblique (normal) rolls at small (large) f. Fits of a modified Swift-Hohenberg form to σ(k,ϵ) gave f-dependent critical behavior for the minimum decay rates σ₀(ϵ) and the correlation lengths ξ_p,q(ϵ).

We present experimental results for patterns of Rayleigh-Bénard convection in a cylindrical container with static sidewall forcing. The fluid used was methanol, with a Prandlt number σ=7.17, and the aspect ratio was Γ≡R∕d≃19 (R is the radius and d the thickness of the fluid layer). In the presence of a small heat input along the sidewall, a sudden jump of the temperature difference ΔT from below to slightly above a critical value ΔT_c produced a stable pattern of concentric rolls (a target pattern) with the central roll (the umbilicus) at the center of the cell. A quasistatic increase of ε≡ΔT∕ΔT_c−1 beyond ε_1,c≃0.8 caused the umbilicus of the pattern to move off center. As observed by others, a further quasistatic increase of ε up to ε=15.6 caused a sequence of transitions at ε_i,b,i=1,...,8, each associated with the loss of one convection roll at the umbilicus. Each loss of a roll was preceded by the displacement of the umbilicus away from the center of the cell. After each transition the umbilicus moved back toward but never quite reached the center. With decreasing ε new rolls formed at the umbilicus when ε was reduced below ε_i,a<ε_i,b. When decreasing ε, large umbilicus displacements did not occur. In addition to quantitative measurements of the umbilicus displacement, we determined and analyzed the entire wave-director field of each image. The wave numbers varied in the axial direction, with minima at the umbilicus and at the cell wall and a maximum at a radial position close to 2Γ∕3. The wave numbers at the maximum showed hysteretic jumps at ε_i,b and ε_i,a, but on average agreed well with the theoretical predictions for the wave numbers selected in the far field of an infinitely extended target pattern. To our knowledge there is as yet no prediction for the wave number selected by the umbilicus itself, or by the cell wall of the finite experimental system.

We used the time correlation of shadowgraph images to determine the angle Θ of the horizontal component of the plume velocity above (below) the center of the bottom (top) plate of a cylindrical Rayleigh-Bénard cell of aspect ratio Γ≡D/L=1 (D is the diameter and L≃87  mm is the height) in the Rayleigh-number range 7×10⁷≤R≤3×10⁹ for a Prandtl number σ=6. We expect that Θ gives the direction of the large-scale circulation. It oscillates time periodically. Near the top and bottom plates Θ(t) has the same frequency but is anticorrelated.

We present experimental data and their theoretical interpretation for the decay rates of temperature fluctuations in a thin layer of a fluid heated from below and confined between parallel horizontal plates. The measurements were made with the mean temperature of the layer corresponding to the critical isochore of sulfur hexafluoride above but near the critical point where fluctuations are exceptionally strong. They cover a wide range of temperature gradients below the onset of Rayleigh-Bénard convection, and span wave numbers on both sides of the critical value for this onset. The decay rates were determined from experimental shadowgraph images of the fluctuations at several camera exposure times. We present a theoretical expression for an exposure-time-dependent structure factor which is needed for the data analysis. As the onset of convection is approached, the data reveal the critical slowing down associated with the bifurcation. Theoretical predictions for the decay rates as a function of the wave number and temperature gradient are presented and compared with the experimental data. Quantitative agreement is obtained if allowance is made for some uncertainty in the small spacing between the plates, and when an empirical estimate is employed for the influence of symmetric deviations from the Oberbeck-Boussinesq approximation which are to be expected in a fluid with its density at the mean temperature located on the critical isochore.

We report measurements of fluctuation and roll patterns near the transition to Rayleigh-Bénard convection which are consistent with a fluctuation-induced first-order transition, as predicted by Swift and Hohenberg. Above onset, we find convection rolls with noise-induced fluctuations, time-dependent amplitude modulation and roll undulation, and homogeneous dislocation nucleation.

We present high-precision measurements of the Nusselt number N as a function of the Rayleigh number R for a cylindrical sample of water (Prandtl number σ=4.4) of height L≃50   cm and aspect ratio Γ≡D/L≃1 (D is the diameter) for 3×10⁹≤R≤6×10¹⁰. For R≃3×10⁹ the data are consistent with existing results for acetone (σ=4.0, R≤3×10⁹). There the measurements are also consistent with a model by Grossmann and Lohse (GL). As R increases, the measurements fall below the GL prediction. Near R=6×10¹⁰ the prediction is 8% above the data.

We present measurements of the thermal resistivity ρ(t,P,L) near the superfluid transition of ⁴He at saturated vapor pressure and confined in cylindrical geometries with radii L=0.5 and 1.0   μm [t≡T/T_λ(P)-1]. For L=1.0   μm measurements at six pressures P are presented. At and above T_λ the data are consistent with a universal scaling function F(X)=(L/ξ₀)^x/ν(ρ/ρ₀), X=(L/ξ₀)^1/νt valid for all P (ρ₀ and x are the pressure-dependent amplitude and effective exponent of the bulk resistivity ρ, and ξ=ξ₀t^-ν is the correlation length). Indications of breakdown of scaling and universality are observed below T_λ.

We report on defect formation in convection patterns of stadium-shaped and elliptical horizontal layers of fluid heated from below (Rayleigh-Bénard convection). The fluid was ethanol with a Prandtl number σ=14.2. The outermost convection roll was forced to be parallel to the sidewall by a supplementary wall heater. The major- and minor-axis aspect ratios Γ_i=D_i/2d, i=1, 2 (D_i are the major and minor diameter and d the thickness) were 19.4 and 13.0, respectively. For the stadium shape, we found a stable pattern that was reflection-symmetric about the major diameter and had a downflow roll of length L_s along a large part of this diameter. This roll terminated in two convex disclinations, as expected from theory. No other patterns with the outermost roll parallel to the sidewall were found. The wave numbers of the rolls in the curved sections and L_s decreased with increasing ε≡ΔT/ΔT_c-1, consistent with a prediction for wave-number selection by curved rolls in an infinite system. At large ε, the roll adjacent to the sidewall became unstable due to the cross-roll instability. For the elliptical shape, wave-director frustration yielded a new defect structure predicted by Ercolani et al. Depending on the sample history, three different patterns with the outermost roll parallel to the wall were found. For one, the central downflow roll seen in the stadium was shortened to the point where it resembled a single convection cell. Along much of the major diameter there existed an upflow roll. The new defect structure occurred where the two downflow rolls surrounding the central upflow roll joined. This joint, instead of being smooth as in the stadium case, was angular and created a protuberance pointing outward along the major diameter. We also found a pattern with an upflow roll along the major diameter without the central downflow cell. A third pattern contained a downflow cell, but this cell was displaced by a roll width from the center along a minor diameter. As ε increased, the length L_e between the two protuberances and the wave numbers along the outer parts of the major diameter decreased for all three patterns, analogous to what was found for the stadium. The upper stability limit of these patterns was also set by the cross-roll instability.

We present experimental and theoretical results near the onset of the Rayleigh-Bénard convection with rotation about a vertical axis in a fluid with a Prandtl number σ close to 0.18. In the experiment we used a H₂-Xe gas mixture with a separation ratio Ψ=0.22 and a Lewis number L=1.22 at various pressures and dimensionless rotation rates Ω up to 400. On the basis of a standard weakly nonlinear stability analysis, we found a supercritical, stationary bifurcation for Ω≲13, which became subcritical over the range 13≲Ω≲160. For Ω≳160 a supercritical Hopf bifurcation precedes the stationary instability of the uniform state. Following the unstable straight-roll fixed point in the subcritical regime by Galerkin methods we determined the location of the saddlenode and the stability of the nonlinear two-dimensional straight-roll state. The rolls were found to be unstable to three-dimensional Küppers-Lortz perturbations for 3.8≲Ω≲160. Theoretical results for a pure fluid with the same σ were qualitatively similar. Measurements using shadowgraph flow visualization yielded a bifurcation line and an Ω range of subcriticality, which agreed with the stability analysis. In the subcritical range the experiment revealed a discontinuity of the pattern amplitude at onset, but was unable to find any hysteresis. Patterns at onset fluctuated irregularly between the ground state and the finite-amplitude state. In this parameter range the convection pattern further above onset was chaotically time dependent. Investigation of the Hopf bifurcation line was difficult because of a wall mode that, for large Ω, preceded the bulk instability. For Ω≃400, patterns were found in the sample interior only when the expected Hopf bifurcation was exceeded by about 10%. This is consistent with the convective nature of the bifurcation. However, the observed structure, although time periodic, was spatially disordered and had a frequency that was considerably larger than the expected Hopf frequency. In a separate sample cell with a radial ramp in the spacing no structure was observed at all in the cell interior until the expected stationary instability was reached.

We present measurements of thermally-induced oblique-roll traveling-wave (TW) fluctuations below the supercritical primary bifurcation to electroconvection (EC) in the nematic liquid crystal 4-ethyl-2-fluoro-4^′-[2-(trans-4-pentylcyclohexyl)ethyl]-biphenyl (I52). First we analyze time sequences of one-dimensional shadowgraph images taken parallel to the director to obtain the TW frequency ω and the fluctuation lifetime τ. Within our resolution we find that ω is independent of ε≡V/V_c-1 (V is the applied voltage amplitude and V_c its value at the onset of convection). Contrary to linear theory, the relaxation rate 1/τ remains finite at the bifurcation. Next we present the analysis of temporally uncorrelated two-dimensional shadowgraph images of the fluctuations for several values of the electrical conductivity σ. We fitted an anisotropic two-dimensional Lorentzian function, corresponding to oblique-roll EC, to the time-averaged structure factors S(k) derived from the images. This yielded information about the components of the mean wave vector k₀ and about the correlation length ξ as a function of σ and ε. The angle of obliqueness ϑ of the roll patterns was independent of σ but decreased anomalously as ε approached zero. The modulus k₀ of k₀ depended on σ. It also showed an anomalous reduction close to onset. The anomalous ε dependence of k₀ and ϑ disagrees with linear theory, which predicts a smooth, essentially linear dependence on ε, and presumably is caused by nonlinear interactions between the fluctuations.

We present experimental results for pattern formation in Rayleigh-Bénard convection with modulated rotation about a vertical axis. The dimensionless rotation rate Ω was varied as Ω_m=Ω[1+δ cos(φΩt)] (time is scaled by the vertical viscous diffusion time of the cell). We used a cylindrical cell of aspect ratio (radius/height) Γ=11.8 and varied Ω, δ, φ, and ε≡R/R_c(Ω)-1 (R is the Rayleigh number). The fluid was water with a Prandtl number of 4.5. Sufficiently far above onset even a small δ≳0.02 stabilized a concentric-roll (target) pattern. Multiarmed spirals were observed close to onset. The rolls of the target patterns traveled radially inward independent of the sense of rotation. The radial speed v was nearly independent of ε for fixed Ω, δ, and φ. However, v increased with any one of Ω, δ, and φ when all the other parameters were held fixed.

We report new measurements in four cells of the thermal boundary resistance R between copper and ⁴He below but near the superfluid-transition temperature T_λ. For 10^-7≤t≡1-T/T_λ≤10^-4 fits of R  =  R₀t^-x_b+R_B to the data yielded x_b≃0.18, whereas a fit to theoretical values based on the renormalization-group theory yielded x_b  =  0.23. Alternatively, a good fit of the theory to the data could be obtained if the amplitude of the prediction was reduced by a factor close to 2. The results raise the question whether the boundary conditions used in the theory should be modified.

We present measurements of the Nusselt number N as a function of the Rayleigh number R and the Prandtl number σ in cylindrical cells with aspect ratios Γ  =  0.5 and 1.0. We used acetone, methanol, ethanol, and 2-propanol with Prandtl numbers σ  =  4.0, 6.5, 14.2, and 34.1, respectively, in the range 3×10⁷≲R≲10¹¹. At constant R, N(R,σ) varies with σ by only about 2%. This result disagrees with the extrapolation of the Grossmann and Lohse theory beyond its range of validity, which implies a decrease by 20% over our σ range, but agrees with their recent extension of the theory to small Reynolds numbers.

For measurements of turbulent heat transport in Rayleigh-Bénard convection the correction for the sidewall conductance is usually neglected or based on measurements or estimates for the empty cell. It is argued that the lateral thermal coupling between the fluid and the wall can invalidate these approaches, and that corrections based on calculations of the two-dimensional temperature fields are required in some cases. These corrections can increase γ obtained from fits of N=N₀R^γ (R is the Rayleigh number) to the Nusselt number N(R) by 0.02 or more, yielding values in the range 0.30 to 0.33, which are larger than most theoretical predictions.

Over two decades ago it was predicted that nonlinear interactions between thermally driven fluctuations in dissipative nonlinear nonequilibrium systems lead to deviations from mean-field theory. Here we report experimental observations of such deviations as a supercritical primary bifurcation is approached. We measured the mean-square director-angle fluctuations 〈θ²〉 below the bifurcation to electroconvection of two different nematic liquid crystals. For ε_mf≡V²/V_c,mf²-1≲-0.1 ( V is the applied voltage) we find 〈θ²〉∝|ε_mf|^-γ with γ given by linear theory (LT). Closer to the bifurcation there are deviations from LT with a smaller γ and with V_c²>V_c,mf².

We present measurements of the Nusselt number N as a function of the Rayleigh number R in cylindrical cells with aspect ratios 0.5≤Γ≡D/d≤12.8 ( D is the diameter and d is the height). We used acetone with a Prandtl number σ  =  4.0 for 10⁵≲R≲4×10¹⁰. A fit of a power law N  =  N₀R^γ_eff over limited ranges of R yielded values of γ_eff from 0.275 near R  =  10⁷ to 0.300 near R  =  10¹⁰. The data are inconsistent with a single power law for N(R). For R>10⁷ they are consistent with N  =  aσ^-1/12R^1/4+bσ^-1/7R^3/7 as proposed by Grossmann and Lohse for σ≳2.

We present experimental results for wave numbers q_s selected in a thin horizontal fluid layer heated from below. The cylindrical sample had an interior section of uniform spacing d  =  d₀ for radii r<r₀ ( Γ₀≡r₀/d₀  =  43) and a ramp d(r) for r>r₀. For Rayleigh numbers R₀>R_c  =  1708 in the interior, straight or slightly curved rolls with an average 〈q_s〉  =  q̃_c+αε₀(ε₀≡R₀/R_c-1) and q̃_c<q_c  =  3.117 were selected, and q_s varied on two length scales approximately equal to Γ₀ and to four roll wavelengths. For ε≲0.03 and ε≳0.18 the pattern repeatedly formed defects.