Convective heat transport in compressible fluids

Convection in an ideal gas at high Rayleigh numbers

Numerical simulations of convection in a layer filled with ideal gas are presented. The control parameters are chosen such that there is a significant variation of density of the gas in going from the bottom to the top of the layer. The relations between the Rayleigh, Peclet, and Nusselt numbers depend on the density stratification. It is proposed to use a data reduction which accounts for the variable density by introducing into the scaling laws an effective density. The relevant density is the geometric mean of the maximum and minimum densities in the layer. A good fit to the data is then obtained with power laws with the same exponent as for fluids in the Boussinesq limit. Two relations connect the top and bottom boundary layers: The kinetic energy densities computed from free fall velocities are equal at the top and bottom, and the products of free fall velocities and maximum horizontal velocities are equal for both boundaries.

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.

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.

Dynamic van der Waals theory

We present a dynamic van der Waals theory starting with entropy and energy functional with gradient contributions. The resultant hydrodynamic equations contain the stress arising from the density gradient. It provides a general scheme of two-phase hydrodynamics involving the gas-liquid transition in nonuniform temperature. Some complex hydrodynamic processes with evaporation and condensation are examined numerically. They are (i) adiabatically induced spinodal decomposition, (ii) piston effect with a bubble in liquid, (iii) temperature and velocity profiles around a droplet in heat flow, (iv) efficient latent heat transport at small liquid densities (the mechanism of heat pipes), (v) boiling in gravity with continuous bubble formation and rising, and (vi) spreading and evaporation of liquid on a heated boundary wall.

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.

Droplet motion with phase change in a temperature gradient

We examine the droplet motion in one-component fluids in a small temperature gradient by solving linearized hydrodynamic equations supplemented with appropriate surface boundary conditions. We show that the velocity field and the temperature around the droplet are strongly influenced by first-order phase transition taking place at the interface. Latent heat released or absorbed at the interface drastically changes the hydrodynamic flow around the droplet. As a result, the temperature becomes almost homogeneous inside the droplet and the Marangoni effect arising from the surface tension gradient is much suppressed. The droplet velocity is also much decelerated.

Critical fluctuation of wind reversals in convective turbulence

The irregular reversals of wind direction in convective turbulence are found to have fluctuating intervals that can be related, under certain circumstances, to critical behavior. In particular, by focusing on its temporal evolution, the net magnetization of a two-dimensional Ising lattice of finite size is observed to fluctuate in the same way. Detrended fluctuation analysis of the wind reversal time series results in a scaling behavior that agrees remarkably well with that of the Ising problem. The specific properties found here, as well as the lack of an external tuning parameter, also suggest that the wind reversal phenomenon exhibits signs of self-organized criticality.

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.

Wind reversals in turbulent Rayleigh-Bénard convection

The phenomenon of irregular cessation and subsequent reversal of the large-scale circulation in turbulent Rayleigh-Bénard convection is theoretically analyzed. The force and thermal balance on a single plume detached from the thermal boundary layer yields a set of coupled nonlinear equations, whose dynamics is related to the Lorenz equations. For Prandtl and Rayleigh numbers in the range 10(-2) < or = Pr < or = 10(3) and 10(7) < or = Ra < or = 10(12), the model has the following features: (i) chaotic reversals may be exhibited at Ra > or = 10(7); (ii) the Reynolds number based on the root mean square velocity scales as Re(rms) approximately Ra([0.41...0.47]) (depending on Pr), and as Re(rms) approximately Pr(-[0.66...0.76]) (depending on Ra); and (iii) the mean reversal frequency follows an effective scaling law omega/(nu L(-2)) approximately Pr(-(0.64 +/- 0.01))Ra(0.44 +/- 0.01). The phase diagram of the model is sketched, and the observed transitions are discussed.

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.

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.

Adiabatic heating and convection caused by a fixed-heat-flux source in a near-critical fluid

Dynamics and heat transfer in a near-critical fluid in a square cavity with a finite heat source located at the bottom are studied numerically. A thermally insulated enclosure and a fixed-heat-flux source are considered. The two-dimensional simulation is based on the full Navier-Stokes equations with two-scale splitting of the pressure and the van der Waals equation of state. It is shown that the piston effect is independent of convection. Near the critical point, this effect becomes independent of criticality and convective motions are damped.

Fast heat transfer calculations in supercritical fluids versus hydrodynamic approach

This study investigates the heat transfer in a simple pure fluid whose temperature is slightly above its critical temperature. We propose an efficient numerical method to predict the heat transfer in such fluids when the gravity can be neglected. The method, based on a simplified thermodynamic approach, is compared with direct numerical simulations of the Navier-Stokes and energy equations performed for CO2 and SF6. A realistic equation of state is used to describe both fluids. The proposed method agrees with the full hydrodynamic solution and provides a huge gain in computation time. The connection between the purely thermodynamic and hydrodynamic descriptions is also discussed.

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.