Heat transfer and convection onset in a compressible fluid: 3He near the critical point

Thermal relaxation and critical instability of near-critical fluid microchannel flow

We present two-dimensional numerical investigations of the temperature and velocity evolution of a pure near-critical fluid confined in microchannels. The fluid is subjected to two sides heating after it reached isothermal steady state. We focus on the abnormal behaviors of the near-critical fluid in response to the sudden imposed heat flux. New thermal-mechanical effects dominated by fluid instability originating from the boundary and local equilibrium process are reported. Near the microchannel boundaries, the instability grows very quickly and an unexpected vortex formation mode is identified when near-critical thermal-mechanical effect is interacting with the microchannel shear flow. The mechanism of the new kind of Kelvin-Helmholtz instability induced by boundary expansion and density stratification processes is also discussed in detail. This mechanism may bring about innovations in the field of microengineering.

Thermoacoustic waves along the critical isochore

Near the liquid-gas critical point, thermal disturbances can generate sounds. We study the acoustic emission over four decades of reduced temperatures [defined as ɛ=(T-T(c))/T(c), with T(c) the critical temperature] along the critical isochore, under linear and nonlinear temperature perturbations, respectively. We identify various thermoacoustic behaviors by numerically solving the governing equations. It is shown that a homogeneous thermoacoustic-wave pattern dominates in the linear case, largely independent of ɛ; whereas under the nonlinear perturbation, variation in ɛ could lead to severe wavefront deformation. The strong nonlinear effect is found to be of a transient nature because, in due time, both cases tend to converge in terms of the energy yield of the adiabatic process.

Possibility of long-distance heat transport in weightlessness using supercritical fluids

Heat transport over large distances is classically performed with gravity or capillarity driven heat pipes. We investigate here whether the "piston effect," a thermalization process that is very efficient in weightlessness in compressible fluids, could also be used to perform long-distance heat transfer. Experiments are performed in a modeling heat pipe (16.5 mm long, 3 mm inner diameter closed cylinder), with nearly adiabatic polymethylmethacrylate walls and two copper base plates. The cell is filled with H2 near its gas-liquid critical point (critical temperature: 33 K). Weightlessness is achieved by submitting the fluid to a magnetic force that compensates gravity. Initially the fluid is isothermal. Then heat is sent to one of the bases with an electrical resistance. The instantaneous amount of heat transported by the fluid is measured at the other end. The data are analyzed and compared with a two-dimensional numerical simulation that allows an extrapolation to be made to other fluids (e.g., CO2, with critical temperature of 300 K). The major result is concerned with the existence of a very fast response at early times that is only limited by the thermal properties of the cell materials. The yield in terms of ratio, injected or transported heat power, does not exceed 10-30% and is limited by the heat capacity of the pipe. These results are valid in a large temperature domain around the critical temperature.

Adiabatic heating and convection in a porous medium filled with a near-critical fluid

Dynamics and heat transfer in a porous medium filled with a fluid phase at parameters near the gas-liquid critical point are studied. A two-dimensional numerical solver based on the hydrodynamic model for a porous medium with a high compressible fluid phase including the van der Waals equation of state is used. In weightlessness, adiabatic heating of fluid phase under the step-temperature heat supply is investigated analytically and numerically. In terrestrial conditions, gravity-driven convection in vertical rectangular cells generated by lateral heating in unsteady and steady-state regimes is simulated. The effects of high compressibility of near-critical fluid phase on convection are studied. Convective motions and heat transfer in horizontal rectangular cells consisting of two porous layers at different porosity and permeability heated from below are simulated as well. Adiabatic heating subjected to hydrostatic compressibility effects, the onset and development of convection, and convective structures in a steady-state regime are analyzed.

Thermovibrational instability in supercritical fluids under weightlessness

Low amplitude, high frequency vibrations can induce in fluids under weightlessness behaviors that resemble those induced by gravity. Supercritical fluids (above their gas-liquid critical point) are used in the space industry and also display universal behavior. They are particularly sensitive to gravity effects. When submitted to vibration (typically 0.1 to 0.5mm amplitude, 10 to 50Hz frequency), a Rayleigh-Bénard-like instability is observed in experiments with H2 and CO2 under weightlessness. The thermal boundary layer created during a temperature change displays periodic fingering perpendicular to the vibration direction. A systematic two-dimensional numerical study by the finite volume method is performed in CO2 that shows that the fingering pattern is due to a thermovibrational instability, characterized by a vibrational Rayleigh number. The simulation and a simplified dimensional analysis show that the fingering wavelength and the vibrational Rayleigh number decrease as a power law with the distance in temperature to the critical point. However, due to the oversimplification of the analysis, the exponent in the simulation is found to be somewhat different than in the theoretical approach, calling for a more complete investigation of the problem.

Modeling heat transfer in supercritical fluid using the lattice Boltzmann method

A lattice Boltzmann model has been developed to simulate heat transfer in supercritical fluids. A supercritical viscous fluid layer between two plates heated from the bottom has been studied. It is demonstrated that the model can be used to study heat transfer near the critical point where the so-called piston effect speeds up the transfer of heat and results in homogeneous heating in the bulk of the layer. We have also studied the onset of convection in a Rayleigh-Bénard configuration. It is shown that our model can well predict qualitatively the onset of convection near the critical point, where there is a crossover between the Rayleigh and Schwarzschild criteria.

Thermoacoustic effects in supercritical fluids near the critical point: Resonance, piston effect, and acoustic emission and reflection

We present a general theory of thermoacoustic phenomena in one phase states of one-component fluids. Singular behavior is predicted in supercritical fluids near the critical point. In a one-dimensional geometry we start with linearized hydrodynamic equations taking into account the effects of heat conduction in the boundary walls and the bulk viscosity. We introduce a coefficient Z(omega) characterizing reflection of sound with frequency omega at the boundary in a rigid cell. As applications, we examine acoustic eigenmodes, response to time-dependent perturbations, and sound emission and reflection. Resonance and rapid adiabatic changes are noteworthy. In these processes, the role of the thermal diffusion layers is enhanced near the critical point because of the strong critical divergence of the thermal expansion.

Hydrostatic compressibility phenomena: new opportunities for near-critical research in microgravity

Peculiarities of the isothermal and isentropic equilibrium in highly compressible media with nonperfect state equations are discussed. Formulation of the adiabatic temperature gradient using Schwarzschild criterion and equilibrium equations for nonperfect gas are considered. Criterion for the onset of convection in the van der Waals (VDW) gas is examined using direct numerical modeling on the basis of equation for convection in compressible viscous nonperfect gas. Results of three-dimensional modeling near-critical convection in the vicinity of the convection onset in space flight with quasi-steady microaccelerations are given. A proposal for the microgravity research of the near-critical convection is discussed.

Perturbation-controlled numerical simulations of the convection onset in a supercritical fluid layer

This paper reports 2D and 3D direct numerical simulations of the Rayleigh-Bénard convection in a fluid layer, 3He, close to its gas-liquid critical point. The main quantity of interest is the time when the convective motion becomes appreciable after the heat current has been started. A space periodic, time-independent temperature perturbation is applied to the top plate aiming to represent the various sources of noise in the experiments. A single amplitude of this additional perturbation reproduces the noise level of the experiments for all the used combinations of the critical point proximity and of the heating power. Over four decades of the Rayleigh number, this simple operation removes the systematic discrepancy reported in an earlier paper and brings the simulations into good agreement with the measurements. Scaling of the exponential convection growth rate is presented and discussed.

Onset of the convection in a supercritical fluid

A model is proposed that leads to the scaled relation tp/tau D=Ftp(Ra-Rac) for the development of convection in a pure fluid in a Rayleigh-Bénard cell after the start of the heat current at t=0. Here tp is the time of the first maximum of the temperature drop DeltaT(t) across the fluid layer, the signature of rapidly growing convection, tau D is the diffusion relaxation time, and Rac is the critical Rayleigh number. Such a relation was first obtained empirically from experimental data. Because of the unknown perturbations in the cell that lead to convection development beyond the point of the fluid instability, the model determines tp/tau D within a multiplicative factor Psi square root Rac(HBL), the only fit parameter product. Here Rac(HBL), of the order 10(3), is the critical Rayleigh number of the hot boundary layer and Psi is a fit parameter. There is then good agreement over more than four decades of Ra-Rac between the model and the experiments on supercritical 3He at various heat currents and temperatures. The value of the parameter Psi, which phenomenologically represents the effectiveness of the perturbations, is discussed in connection with predictions by El Khouri and Carlès of the fluid instability onset time.

Reverse transition to hydrodynamic stability through the Schwarzschild line in a supercritical fluid layer

We consider a fluid close to its gas-liquid critical point in a bottom-heated cavity. Due to strong density stratification, both the Schwarzschild and the Rayleigh stability criteria are relevant at the same space scale. Taking advantage of the competition between these two limits of the convection-onset criterion, we numerically exhibit striking non-Boussinesq effects: the reverse transition to stability through the Schwarzschild line of a heat diffusing layer subjected to convection, and the convection onset inside a few-millimeters-thick layer according to the Schwarzschild criterion.

Numerical simulation studies of the convective instability onset in a supercritical fluid

Numerical simulation studies in 2D with the addition of noise are reported for the convection of a supercritical fluid, 3He , in a Rayleigh-Bénard cell. The noise addition is to accelerate the instability growth after starting the heat flow across the fluid, so as to bring simulations into better agreement with published experimental observations. Homogeneous temperature noise and spatial longitudinal periodic temperature variations in either top or bottom plates were programmed into the simulations. The second method was the most effective in speeding up the instability onset. For a small amplitude of the longitudinal perturbations, a semiquantitative agreement with the observations was obtained. The results are discussed in relation to predictions by El Khouri and Carlès.

Scaling behavior for the onset of convection in a colloidal suspension

We investigate the early stages of mass convection in a colloidal suspension at high solutal Rayleigh number Ras. From the time evolution of shadowgraph images and by assuming a diffusive growth of the boundary layers we obtain an indirect measurement of the concentration boundary layer thickness delta* at the onset of convection. We show that the dimensionless boundary layer thickness delta=delta*/d scales as Ra-ps, where Ras=Rasdelta is a modified solutal Rayleigh number for convection which accounts for the actual density unbalance and d is the thickness of the sample layer. This scaling behavior is analogous to that reported at steady state for turbulent convection in simple fluids. We find p=0.35, a value compatible with the exponent 1/3, reported for turbulent heat convection in simple fluids at steady state.

Convection in a very compressible fluid: comparison of simulations with experiments

The time profile DeltaT(t) of the temperature difference, measured across a very compressible fluid layer of supercritical 3He after the start of a heat flow, shows a damped oscillatory behavior before steady-state convection is reached. The results for DeltaT(t) obtained from numerical simulations and from laboratory experiments are compared over a temperature range where the compressibility varies by a factor of approximately 40. First the steady-state convective heat current j(conv) as a function of the Rayleigh number Ra is presented, and the agreement is found to be good. Second, the shape of the time profile and two characteristic times in the transient part of DeltaT(t) from simulations and experiments are compared, namely (1) t(osc), the oscillatory period, and (2) t(p), the time of the first peak after starting the heat flow. These times, scaled by the diffusive time tau(D) versus Ra, are presented. The agreement is good for t(osc)/tau(D), where the results collapse on a single curve showing a power-law behavior. The simulation hence confirms the universal scaling behavior found experimentally. However for t(p)/tau(D), where the experimental data also collapse on a single curve, the simulation results show systematic departures from such a behavior. A possible reason for some of the disagreements, both in the time profile and in t(p), is discussed.

Piston-effect-induced thermal oscillations at the Rayleigh-Bénard threshold in supercritical 3He

We perform a Navier-Stokes numerical simulation of the transient Rayleigh-Bénard convection onset in nearly supercritical 3He in the exact conditions in experiments performed by Kogan, Murphy, and Meyer [Phys. Rev. Lett. 82, 4635 (1999)]Phys. Rev. E 63, 056310 (2001)]]. We find an interpretation of the observed unexpected temperature oscillations at the convection onset in terms of the piston effect. This is our first result towards the exploration of the whole instability diagram as recently mapped in those experiments.

Scenarios for the onset of convection close to the critical point

We perform a theoretical analysis of the onset of convection in a layer of near-critical 3He submitted to an unsteady bottom heating. A theoretical model previously presented [P. Carlès, Physica D 147, 36 (2000)] is adapted to the corresponding physical conditions, and a method is proposed to solve the associated equations. We predict, for different intensities of heating and different initial temperatures, when convection will start and what will be the shape of the dominant growing perturbations. A systematic parametric analysis shows that the onset of convection in a supercritical fluid can take place following four distinct scenarios, depending on the initial temperature and the intensity of the heating. Two of these scenarios are entirely specific to near-critical fluids, being impossible to observe in classical Boussinesq fluids.

Onset of convection in a very compressible fluid: the transient toward steady state

We analyze the time profile deltaT(t) of the temperature difference, measured across a very compressible supercritical 3He fluid layer in its convective state. The experiments were done along the critical isochore in a Rayleigh-Bénard cell after starting the vertical constant heat flow q. For q sufficiently well above that needed for the convection onset, the transient deltaT(t) for a given epsilon identical with (T-T(c))/T(c), with T(c)=3.318 K, shows a damped oscillatory profile with period t(osc) modulating a smooth base profile. The smooth profile forms the exponential tail of the transient which tends to the steady-state deltaT( infinity ) with a time constant tau(tail). The scaled times t(osc)/t(D) and tau(tail)/t(D) from all the data could be collapsed onto two curves as a function of the Rayleigh number over approximately 3.5 decades. Here t(D) is the characteristic thermal diffusion time. Furthermore, comparisons are made between measurements of a third characteristic time t(m) between the first peak and the first minimum in the deltaT(t) profile and its estimation by Onuki et al. Also, comparisons are made between the observed oscillations and the two-dimensional simulations by Onuki et al. and by Amiroudine and Zappoli. For epsilon<9 x 10(-3), the experiments show a crossover to a different transient regime. This regime, which we briefly describe, is not understood at present.

Convective heat transport in compressible fluids

We present hydrodynamic equations of compressible fluids in gravity as a generalization of those in the Boussinesq approximation used for nearly incompressible fluids. They account for adiabatic processes taking place throughout the cell (the piston effect) and those taking place within plumes (the adiabatic temperature gradient effect). Performing two-dimensional numerical analysis, we reveal some unique features of plume generation and convection in transient and steady states of compressible fluids. As the critical point is approached, the overall temperature changes induced by plume arrivals at the boundary walls are amplified, giving rise to overshoot behavior in transient states and significant noise in the temperature in steady states. The velocity field is suggested to assume a logarithmic profile within boundary layers. Random reversal of macroscopic shear flow is examined in a cell with unit aspect ratio. We also present a simple scaling theory for moderate Rayleigh numbers.

Thermal plumes and convection in highly compressible fluids

We present simple hydrodynamic equations of supercritical fluids close to the gas-liquid critical point. We numerically solve them to examine plume generation and convection under gravity. These results are in good agreement with the experiment [A. B. Kogan and H. Meyer, Phys. Rev. E 63, 056310 (2001)]. This Letter is a first study of transient behavior of convection, which is unique in compressible fluids due to the piston effect.