An overview of heat transfer near the liquid–gas critical point under the influence of the piston effect: Phenomena and theory

International Journal of Thermal Sciences 71 (2013) 1e19

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An overview of heat transfer near the liquidegas critical point under the influence of the piston effect: Phenomena and theory B. Shen, P. Zhang* Institute of Refrigeration and Cryogenics, MOE Key Laboratory for Power Machinery and Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 December 2012 Received in revised form 1 April 2013 Accepted 4 April 2013 Available online 18 May 2013

Recent advances in the knowledge of heat transport near the liquid-gas critical point under the influence of the piston effect are reviewed with an emphasis on the different physical mechanisms and timescales in regard to thermal and density relaxations. Near the critical point, thermophysical properties exhibit singular behaviors, such as the diverging compressibility and vanishing thermal diffusivity. The resulting fast thermalization leads to the unexpected discovery of the piston effect. We describe the previous theoretical, numerical, and experimental investigations of this unique critical phenomenon and related topics, including its thermoacoustic nature with various nonlinear features on the acoustic timescale. Hydrodynamic and thermovibrational instabilities on the diffusion timescale in near-critical fluids are addressed as well. The review ends with a brief discussion of the merits and limitations of selected research methods in common use. Ó 2013 Elsevier Masson SAS. All rights reserved.

Keywords: Critical point Piston effect Thermoacoustic waves Hydrodynamic instabilities Thermovibrational instabilities Heat transfer

1. Introduction The liquidegas critical point on the phase diagram represents a singular equilibrium state, beyond which the boundary between the liquid and gas phases no longer exists and the fluid becomes supercritical (as shown in Fig. 1). In general, a supercritical fluid can be regarded as an intermediate between a liquid and a gas, exhibiting unusual properties such as high density and solubility. In the absence of surface tension, a supercritical fluid can be readily turned more gas-like or more liquid-like without experiencing phase transition. Over the past several decades, there has been a growing demand for supercritical fluids in industrial applications as varied as alternative eco-friendly refrigerants [1], cold energy storage of Liquefied Natural Gas (LNG) [2], superconducting magnet cooling [3], chemical extraction/separation processes [4], supercritical water reactors [5], and high-performance rocket propellants [6]. Approaching the critical point (CP), more peculiar thermophysical properties emerge, including diverging compressibility and vanishing thermal diffusivity. These singular behaviors are the macroscopic manifestation of the critical fluctuations, which must be explained by statistical physics [7]. The dynamic relaxation

* Corresponding author. Tel.: þ86 21 34205505; fax: þ86 21 34206814. E-mail address: [email protected] (P. Zhang). 1290-0729/$ e see front matter Ó 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ijthermalsci.2013.04.010

processes should, as a result, be extremely slow at the CP, which is termed critical slowing down. In his wide-ranging and informative review of various experimentally observed critical phenomena in liquid-vapor, binary, magnetic and other systems, Heller [8] listed the exceedingly long transition time from one equilibrium state to another as one of the major difficulties facing experimenters in the critical region. Most perplexingly, however, in a microgravity experiment in the 1980s, instead of very slow thermalization of near-critical SF6, critical speeding-up of temperature homogenization was observed [9]. The physical interpretation of such an interesting yet puzzling phenomenon resulted in the unveiling of a unique mechanism of heat transfer, the so-called piston effect (PE). Over the past 30 years, an extensive volume of literature on the subject of the PE has been accumulated, which brought about an improved understanding of various thermodynamic and hydrodynamic processes near the CP. The aim of this article is to survey the latest progress made in research on the PE and related topics, and possibly to stimulate renewed interest in future work to investigate the interaction between the PE and other critical phenomena. In Section 2, a general description is given of the discovery of the PE, including the evidenced acceleration of thermal relaxation near the CP and the subsequent theoretical interpretations. In Section 3, we offer a summary of the findings concerning the thermoacoustic nature of the PE on a short timescale. The focus is placed on the nonlinear features of thermoacoustic waves, which have drawn the

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Nomenclature a A b CV CP DT e h H g kB L Ma n Nu P dP Pr q R Ra s t T dT T1 DT x y u v v vs V DV

parameter for the intermolecular forces in the van der Waals equation [Eq. (12)] surface area (m2) parameter for the finite molecular size in the van der Waals equation [Eq. (12)] isochoric specific heat (J kg1 K1) isobaric specific heat (J kg1 K1) thermal diffusivity (m2 s1) effusivity ratio boundary layer thickness (m) heating parameter gravitational acceleration (m s2) Boltzmann constant characteristic length (m) Mach number, Ma ¼ P/Lvs unit spatial vector normal to A Nusselt number, qL/lDT pressure (Pa) pressure variation (Pa) Prandtl number, Pr ¼ y/DT Heat flux (W/m2) perfect gas constant Rayleigh number, Ra ¼ aPgDTL3/yDT specific entropy (J kg1 K1) time (s) temperature (K) temperature variation (K) temperature change at the boundary (K) temperature drop across the fluid layer (K) space variable (m) space variable (m) x-velocity (m s1) velocity vector (m s1) y-velocity (m s1) speed of sound (m s1) volume of the fluid (m3) volume change of the one-dimensional thermal boundary layer (m)

Greek symbols a critical exponent, a y 0.11 aP isobaric thermal expansion coefficient (K1) g ratio of the specific heats, g ¼ CP/CV ε reduced temperature, ε ¼ (T  Tc)/Tc h coefficient of viscosity (Pa s) q potential temperature (K) i1 expansion parameter, i1 ¼ taPrIG/tD i2 expansion parameter, i2 ¼ (T  Tc)/Tc kT isothermal compressibility (Pa1) l thermal conductivity (W m1 K1) x correlation length (m) P characteristic velocity, P ¼ y/L r density (kg m3) dr density variation (kg m3) y kinematic viscosity (m2 s) F viscous dissipation function 4 critical exponent, 4 y 0.58 c critical exponent, c y 1.24 j energy efficiency of the piston effect Superscripts 0 nondimensional variables (0) zeroth-order terms (1) first-order terms Subscripts 0 initial state a acoustic ATG adiabatic temperature gradient b bulk viscosity c critical point D thermal diffusion IG ideal gas onset onset of convection p peak PE piston effect s shear viscosity v viscous regime W wall

attention of the latest numerical efforts [10,11]. Section 4 covers the presently available knowledge with respect to the complex interplay between the PE and mechanical instabilities such as gravity and small linear vibrations. In Section 5 we review the various research methods employed in past studies of theoretical, numerical, and experimental natures, with a discussion about their respective merits and limitations. Finally, the paper is concluded in Section 6. 2. The piston effect: phenomenon and theory 2.1. Asymptotic behavior near the CP

Fig. 1. Phase diagram of N2. Solid: liquid-gas coexistence curve; Dash-dot: pseudocritical line.

One of the most distinguished features of the liquid-gas CP is associated with the dramatic changes of thermophysical and transport properties (see Table 1): both the isothermal compressibility kT and the isobaric thermal expansion coefficient aP diverge; the isochoric specific heat CV, the thermal conductivity l, and the ratio of specific heats g all tend to infinity; whereas the sound velocity vs and the thermal diffusivity DT go to zero. Moreover, these

B. Shen, P. Zhang / International Journal of Thermal Sciences 71 (2013) 1e19

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Table 1 Thermophysical properties of supercritical N2 close to the CP along its critical isochorea [13]. εb (-) 3.96 3.96 3.96 3.96 1.58 a b

    

104 103 102 101 100

CV (kJ K1 kg1)

g (-)

l (W m1 K1)

108  DT (m2 s1)

aP (K1)

1.871 1.693 1.154 0.854 0.794

1001.610 101.780 11.960 2.524 1.660

0.198 0.087 0.048 0.037 0.048

0.034 0.161 1.104 5.541 11.648

2.826 2.556 1.799 1.356 3.097

    

kT (kPa1) 101 100 101 102 103

1.718 1.549 1.071 7.941 1.906

    

101 102 103 105 105

hs (mPa s)

vs (m s1)

18.773 18.800 19.061 21.571 28.761

136.411 144.814 188.794 318.538 527.156

Critical coordinate of N2: Tc ¼ 126.192 K, rc ¼ 313.3 kg/m3, Pc ¼ 3.396 MPa. Reduced temperature, ε ¼ (T  Tc)/Tc.

singularities can be modeled as universal power laws on the basis of a common parameter

ε ¼

T  Tc Tc

(1)

where Tc is the critical temperature. The reduced temperature ε measures the distance to the CP along the critical isochore. The collective asymptotic behavior of the thermodynamic properties close the CP thus can be expressed as

aP ; kT wεc ; CV wεa ; lwε4 ; DT wεc4 ; vs wεa=2

(2)

where the critical exponents are, regardless of the specific fluid, c y 1.24, a y 0.11, and 4 y 0.58, respectively [12]. The global critical exponents can be addressed by the concept of universality [14]. The differences between physical systems are determined more by the collective long-range interaction than by the individual molecular details of its components, which leads to a division of universality classes. Near-critical fluids belong to the same universality class, in which the correlation length x strongly diverges (as wε0.63) at the CP. In a liquid-gas equilibrium state, molecules in the liquid and gas phases have distinct properties, whose coexistence is ensure by the nonzero surface tension. As the CP is approached, the difference between the liquid and gas phases tends to become progressively small, which coincides with the disappearance of the phase boundary. The critically diverging x allows increasingly long-range microscopic interactions that are independent of local molecular behavior to be projected onto the macroscopic level. The resulting collective behavior of the system, which can be formulized through the renormalization group theory for the Ising model, gives rise to universal dynamic properties that have the same asymptotic critical exponents and amplitude ratios among different fluids. The statistical distributions of the growing density and energy fluctuations lead to the divergence of the isothermal compressibility and specific heats at the CP, respectively. For a more detailed description of the singular thermophysical properties near the CP, the interested reader is referred to an excellent review by Carlès and the references cited therein [7].

example, the Raleigh number for supercritical CO2 could reach as high as 1013 even at small space scales [16]. It was widely considered that, in microgravity experiments (with natural convection being mostly suppressed), thermal equilibration must be realized by diffusion alone, and consequently should be increasingly retarded approaching the CP. Due to the practical limitations usually associated with space experiments, not until 1985 was the first successful microgravity experiment of supercritical fluids performed. Nitsche and Struab carried out a measurement of the heat capacity at fixed volume of supercritical SF6 under reduced gravity aboard the ballistic TEXUS rocket as part of the German Spacelab mission D1 [9]. The results led to a very puzzling discovery that directly contradicted the assumption of critical slowing down (see Fig. 2): the temperature at the center of the fluid cell containing supercritical SF6 followed the rising wall temperature closely even in the absence of apparent natural convection. In comparison, for the same period of microgravity, the diffusion theory predicted hardly any temperature change (dash line). The thermal relaxation was completed within seconds as opposed to days based on pure thermal diffusion. Instead of critical slowing down, a prominent critical speeding up was observed nearing the critical point. 2.3. Thermal relaxation accelerated by the PE No tenable theoretical explanation had been proposed until 1990 when three teams independently uncovered the physical mechanism behind the unexpected acceleration of thermal homogenization near the CP, based on different approaches. From a purely thermodynamic perspective, Onuki et al. [17,18] have derived a heuristic model to explain the remarkably rapid equilibration. Imagine a supercritical fluid of length L, whose temperature at one of the bounding walls is raised by T1. Because of

2.2. Observation of critical speeding up Due to the vanishing thermal diffusivity close to the CP (as shown in Table 1), the thermal relaxation of a near-critical fluid was long thought to experience a critical slowing down nearing the CP. However, very fast heat transport was consistently reported in ground-based calorimetric experiments. The thermal relaxation times of supercritical 3He measured by Dahl and Moldover in 1972 were appreciably shorter than the thermal diffusion model predicted [15]. The authors ascribed the failure of the model to the interference of turbulent buoyancy-induced convection. Because of the unusual combination of exceptionally high compressibility and low thermal diffusivity of supercritical fluids, temperature perturbations are indeed prone to cause convective instabilities. For

Fig. 2. Experimental results of heating near-critical SF6 from T-Tc ¼ 0.4 K to 0.4 K along the critical isochore [9]. The thermal response Tcenter to the ramping of Twall under microgravity shows little difference from that under terrestrial conditions. With convection suppressed, the fast thermalization was largely driven by the PE.

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the diverging compressibility near the CP, the thermal boundary layer, which grows as h ¼ (DTt)1/2, expands by

DV ¼ r

! vr1 T1 h vT



r

Assuming no entropy transfer between the boundary layer and the bulk, one can obtain the space-averaged temperature change in the interior of the fluid,



vT vr



DV

s

L

¼

T1 ðg  1Þh L

(4)

By letting dT ¼ T1/2 one arrives at the typical equilibrium time (viz., the PE timescale)

tPE ¼

L2 4ðg  1Þ2 DT

wε1:6

vr þ V$ðrvÞ ¼ 0 vt

(6)

Table 2 Typical physical timescales of dynamic processes in near-critical N2 along its critical isochore confined in a one-dimensional slab of 10 mm.

3.96 3.96 3.96 3.96 1.58

    

104 103 102 101 100

¼ gradP þ hs Dv þ



 1 hs þ hb graddivv 3

  vs þ v$grads ¼ divðlgradTÞ þ F vt

ta ¼ L/vs (ms)

tPE ¼ L2/DT(g  1)2 (s)

tD ¼ L2/DT (s)

73.5 69.1 52.9 31.4 19.0

0.3 6.1 75.4 777.0 1970.8

29,4117.6 62,111.8 9057.8 1804.7 858.5

(8)

where hs is the shear viscosity, hb is the bulk viscosity, and F is the viscous dissipation function. Equations (6) through (8) represent the conservation equations for mass, momentum, and entropy, respectively. The left-hand side of Eq. (8) can be written as

rT

Ds ¼ rT Dt

(5)

which vanishes near the CP. Obviously tPE is significantly less than the characteristic time of thermal diffusion, tD ¼ L2/DT w ε0.66 (see Table 2), as the ratio of the specific heats g w ε1.13 goes to infinity at the CP. The theoretical analysis shows that the strong expansion of the thermal boundary layer, which is caused by the imposed thermal perturbations, squeezes the rest of the fluid like a moving piston. The compression constitutes adiabatic heating of the bulk, through which local temperature change is spread at a much faster rate than by diffusion alone. Consequently the thermal equilibration is considerably facilitated. The heuristic model has revealed some basic features of the PE as well [17]. For example, the effect of gravity should be of little significance to thermal relaxation in the initial stage of the process, which implies that the PE might be active even on earth. Also, in a two-phase configuration, the adiabatic process exhibits a distinct non-uniform behavior. The gas phase and the liquid phase undergo different temperature changes, which could lead to slow motion of the liquid-gas interface and markedly longer relaxation time than is with the single-phase (supercritical) case. Zappoli et al. [19] developed a numerical solution to the nonlinear hydrodynamic model near the CP, which captured the essence of the PE. It has been shown that the expansion of the thermal boundary layer comprises propagating waves with a velocity slightly lower than vs on the acoustic timescale (denoted by ta ¼ L/vs). The evolution of the temperature field, driven by the induced convection, reaches a nearly equilibrium state in a much more efficient way than by pure thermal diffusion. Under the continuum approximation, the complete hydrodynamic description for an isotropic, Newtonian, compressible, and dissipative (viscous and heat-conducting) fluid consists of [20]

ε ¼ (T  Tc)/Tc (-)



(7) (3)

rT

dT ¼

vv þ ðv$gradÞv vt



vs vT



   DT vs DP þ vP T Dt P Dt

¼ rCP

k DP DT  rðCP  CV Þ T aP Dt Dt

¼ rCV

DT rðCP  CV Þ divv þ aP Dt

(9)

where D=Dthv=vt þ v$grad denotes the material derivative. Substituting Eq. (9) into Eq. (8) leads to

rCP

k DP DT ¼ divðlgradTÞ þ F þ rðCP  CV Þ T aP Dt Dt

(10)

or, alternatively,

rCV

rðCP  CV Þ DT divv ¼ divðlgradTÞ þ F  aP Dt

(11)

The above equations are the CV- and CP-formulations of the energy equation, respectively. Nikolayev et al. [21] noted that the latter better accommodates numerical simulation because of the much weaker divergence of CV close to the CP. The extra terms on the right-hand side of Eqs. (10) and (11) represent the adiabatic temperature change (i.e., the PE) caused by the propagating pressure perturbations and consequent fluid motion, which is incidentally zero in incompressible fluids (CV ¼ CP). As a result, Eqs. (10) and (11) are reduced to the classical conductive-convective energy equation. This important observation has broad implications for the critical speeding-up phenomenon. The PE, as it turns out, is ubiquitous in any compressible fluids, whose contribution to temperature relaxation however would be negligibly small far from the CP when thermal diffusion is not particularly subdued. To complete the physical description, an equation of state (EOS) is needed. For its simple form, the van der Waals (VdW) equation has long been regarded as an acceptable approximation of the singular thermodynamic behavior near the CP [19,22,23], which reads

P ¼

rRT  ar2 1  br

(12)

In Eq. (12), R is the perfect gas constant, and the parameters a and b can be calculated from Tc ¼ 8a/(27Rb) and rc ¼ 1/3b. Quantitatively though, the VdW equation is unable to model the critical singularities correctly. In practice, the critical divergence of l needs to be taken into account separately, and CV and hs are usually regarded as constants and equal to the values of a perfect gas. To obtain a more accurate picture close to the CP, one can resort to a real-fluid EOS that is based on the thermodynamic identity [2,10,11],

dP ¼

1

rkT

dr þ

aP dT kT

(13)

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Note that the above relation is deemed appropriate only in situations involving not too large a departure from the equilibrium state. A few comments are in order about the general applicability of the hydrodynamic model. As the correlation length becomes as large as the physical scale of the process, the macroscopic description might eventually fail, as duly noted by Zappoli [16]. For CO2 in a 10 mm container, for example, the model ceases to be reliable at a few milikelvins above the CP [24]. Most inquiries, however, focus on conditions that are not in the immediate vicinity of the CP and well beyond such thresholds. The physics involved ought, as a result, to fall safely in the realm of continuum mechanics. By means of the matched asymptotic method, Zappoli and Carlès [25] discovered that underneath the PE lies a prominent thermoacoustic nature on the acoustic timescale. The bulk homogeneous heating is in fact realized by a system of linear thermoacoustic compression waves of vanishingly small amplitudes. One typical example of such wave propagation [26] is shown in Fig. 3(a). Here one-dimensional supercritical N2 (T0 ¼ 130 K, P0 ¼ 3.815 MPa, L ¼ 10 mm) is subjected to a step temperature rise of 1 K at the left boundary. The time sequence of numerically obtained short-time temperature distributions (by an explicit finite-difference scheme) clearly captures the passage of a temperature perturbation inside the bulk. The fluid velocity produced by the hot boundary layer expansion reaches its maximum value at the edge of the boundary layer, which is considered to be the true driving

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force for the PE. Thus, the boundary layer acts as a converter transforming thermal energy into kinetic energy through sound emission, which has surprisingly low energy transfer efficiency (i.e., the ratio of the energy added to the bulk to the total heat input) close to the CP [11,17]. The generated pressure perturbation (sound) travels back and forth within the fluid. Owing to the strong thermoacoustic coupling near the CP, the propagation of sound could have a significant impact on the temperature field as well, as is suggested by the dominating pressure force term in Eq. (10). It should be noted that, notwithstanding the apparently efficient temperature equilibration, the thermoacoustic process is mostly isentropic, as has been numerically confirmed in [11]. The adiabatic nature of thermoacoustic waves is also consistent with the supposed negligible entropic exchange between the expanding boundary layer and the bulk when viewed on a much longer timescale [17,18]. Hence, it is somewhat an overstatement to categorize the piston effect as the fourth mode of heat transfer in addition to heat conduction, convection, and radiation. Following many successive traversals of the thermoacoustic wave, the temperature in the bulk rises homogeneously. The signature impact of the PE on the temperature distribution, which amounts to the long-term accumulation of the transient thermoacoustic effect, is clearly visible in the evolution of the temperature distribution over an extended period of time [26] in Fig. 3(b). Thermal boundary layers are shown to form near both the hot and cold ends of the fluid. Bourkari et al. [27] have proposed a semi-hydrodynamic model for temperature relaxation in a pure fluid near its critical point, which describes the adiabatic process in response to local temperature inhomogeneities on a much longer timescale than ta. In developing the physical model, zero gravity and no convection is assumed of the fluid. The energy Eq. (10) thus becomes

  vT 1 1 kT vP divðlgradTÞ þ 1  ¼ rCP g aP vt vt

(14)

The pressure derivative represents the space-averaged pressure variation caused by the adiabatic compression of the bulk, and can be obtained through integrating the EOS (13) over the entire volume of the fluid, along with the argument of mass conservation. The result reads

Z

raP ðvT=vtÞdV vP ¼ vt

V

Z

(15)

rkT dV V

Eqs. (14) and (15), in spite of being limited to times much longer than the acoustic timescale, managed to accurately predict the critical speeding up in near-critical xenon [27]. The iterative numerical solution of the model has showed that the boundary temperature and the midpoint temperature in the xenon sample vary at the same rate after a sudden temperature quench, similar to the experimental findings of Nitsche and Straub [9]. 2.4. Decoupling between temperature and density relaxations

Fig. 3. Numerical simulation of (a) the short-time and (b) long-time temperature distributions of supercritical N2 (T0 ¼ 130 K, P0 ¼ 3.815 MPa) confined in a onedimensional slab (L ¼ 10 mm) with the temperature at the left end increased by 1 K and the right side kept isothermal [26]. The arrows in (a) indicate the direction of the thermoacoustic wave propagation.

One of the consequences of the PE is that the thermoacoustic convection caused by the propagating sound pushes warm fluid near the heated end to the opposing cold end. Strong density gradients emerge due to the continuous expansion of the thermal boundary layer and ensuing compression of the bulk. Based on the matched asymptotic analysis, Bailly and Zappoli [28] have

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Fig. 4. Evolution of the normalized density and temperature perturbations within a near-critical VdW fluid [28]. Here the parameters are given as, pffiffiffiffiffi m ¼ ðT0  Tc Þ=Tc ; q ¼ w mt , with w ¼ ta PrIG =tD .

developed a hydrodynamic theory of density relaxation in a nearcritical VdW fluid. The model indicates that the process of density equilibration involves two timescales, the PE timescale and the diffusion timescale. The density relaxation, as it turns out, is governed by two fundamentally different mechanisms. On the PE timescale, the initial temperature perturbations are almost homogenized by the PE except for hardly measurable temperature differences in the boundary layers, while significant density inhomogeneities still prevail in the fluid. The density field consists of two distinct regions: the deep depletion of fluid in the vicinity of the heated wall, and the adiabatically compressed bulk generated by the mass transfer from the thermally expanding boundary layer. Then, on the diffusion timescale when heat diffusion replaces the PE as the dominant regime in thermal equilibration, the remaining density inhomogeneities are slowly evened out by a diffusive mechanism. The analysis suggests that the process is driven by the slow movement of a damped expansion-compression zone in the bulk at the speed of diffusion. Fig. 4 plots the different paths of the temperature and density relaxation, which are increasingly decoupled closer to the CP. The diffusive nature of the density relaxation was confirmed in Zhong and Meyer’s experiment in near-critical 3He [29] and Guenoun et al.’s experiment in CO2 [30]. 2.5. Experimental verification of the PE To confirm the fast homogeneous thermalization of near-critical fluids and, for that matter, the existence of the PE, a series of microgravity experiments were conducted in the 1990s. Klein et al. [31] conducted an experiment aboard a sounding rocket to study the propagation of a temperature change through near-critical SF6 undergoing phase transition. The temperature in the interior region of the fluid exhibited a rapid response to the imposed temperature drop, which corroborated the strong adiabatic heating near the CP. Also noted was an interesting observation that the temperature equilibration slightly slowed down during phase separation. Guenoun et al. [30] designed and carried out a thermal cycle experiment targeting the heat transport in CO2 near its critical point under reduced gravity. The thermal cycle consisted of a temperature quench from Tc þ 2.3 mK to Tc þ 1.3 mK, then to Tc  0.8 mK, followed by a temperature ramp back to Tc þ 2.0 mK. The temperature measurements showed that the PE, even with the onset of phase separation, significantly accelerated the thermal relaxation inside the experimental cell, which took only a fraction of the typical diffusion time. It was observed that the density

inhomogeneity, on the other hand, relaxed largely through diffusion after the PE had adiabatically homogenized the temperature field. Bonetti et al. [32] reported an experimental study of the timeevolution of the thermal boundary layer and the corresponding bulk temperature variation in near-critical CO2 in the absence of gravity. The quantitative interferometric results helped shed light on some fundamental characteristics of the PE, including the fast homogenous density change in the bulk fluid. The observation of decreasing density in the boundary layer and increasing density in the bulk was consistent with the analogy of a moving piston compressing the rest of the fluid. A simple scaled function was derived to describe the diffusive growth of the thermal boundary layer. During the German Spacelab Mission D-2, the calorimetric experiment originally designed for the measurement of CV was used to investigate the dynamic temperature propagation near the CP [33]. The experimental setup had a heat pulse lasting 10 s applied to a spherical cell filled with pure SF6 at the critical density. A total of 39 different initial temperatures ranging 0.03 K < jT  Tcj < 5.25 K in both the one-phase and two-phase regions were studied. The results showed that, within a short time of the heating, the isentropic PE played the leading role in heat transfer, overshadowing diffusive effects. The measured temperature outside the boundary layer followed a fast homogeneous rise (see Fig. 5), which compared well with the theorized temperature distributions based on Boukari et al.’s simplified model [27]. The notably different behavior of the two-phase system during heating and cooling was explained with the help of the T  s phase diagram, which matched the observed hysteresis during previous CV measurements. By use of a TwymanneGreen interferometer, Fröhlich et al. [34] provided unambiguous experimental evidence that the homogeneous temperature rise in the bulk of supercritical SF6dstill untouched by thermal diffusiondwas in fact induced by the adiabatic heating (namely, the PE). The experimental sample was a cylindrical cell (of 11.9 mm in diameter) filled with slightly off-critical (r0 ¼ 1.27rc) SF6. Three thermistors (TH1, TH2, and TH3) were used to monitor the temperature variations at different locations in the cell. After 15 s of heating through TH2, a hot boundary layer [denoted by the gray zone in Fig. 6(b)] emerged from the initial homogeneous interference fringes [see Fig. 6(a)] and spread form the center of the fluid. It can be clearly seen that the diffusive temperature front had not reached TH2 yet. The rapid temperature response of TH2 shown in Fig. 6(c) was confirmed to be caused by the adiabatic compression of the bulk. The density distributions obtained via the fringe shifts were also found to be in very good agreement with the calculations, which was corrected for the particular thermodynamic state of the fluid. As was mentioned above, the efficient thermalization of supercritical fluids is realized by thermoacoustic waves. In what follow, we will examine the recent findings with regard to the transient thermoacoustic process in supercritical fluids. 3. Thermoacoustic nature of the PE 3.1. Thermoacoustic interaction In compressible fluids at fixed volume, by virtue of Eq. (10), there exist varying degrees of inescapable thermomechanical coupling. Local thermal disturbances could cause sound emission, and on the other hand, sound propagation is expected to have an impact on the thermomechanical equilibrium in the fluid. The subtle interaction between thermal and acoustic processes was first approached by Lord Rayleigh [35]. By use of Laplace transform,

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Fig. 6. Interferometric measurement of the temperature responses in supercritical SF6 to an internal heat pulse of 15 s [34]. In (a) is shown the initial homogeneous state. At the end of the heat pulse released through the thermistor TH1, both the hot boundary layer (HBL, denoted by the diffusive gray zone) and the cold boundary layer (CBL) near the wall had apparently not reached TH2 yet in (b). The fast temperature change at TH2 in (c) was believed to be caused by the PE instead. Fig. 5. Comparison between the measured and calculated temperature responses of supercritical SF6 to the heat pulse at T0-Tc ¼ 1 K [33]. Specifically, (a) temporal variations of temperature at different locations inside the cell (denoted by the corresponding symbols in the inset), and (b) spatial distribution of temperature in the cell at the end of the heat pulse (t ¼ 10 s). The numerical results are represented by the dashed line in (a) and solid line in (b), respectively.

Trilling [36] has postulated that by imposing a small temperature perturbation one can create a mechanical reaction in the form of an acoustic wave in the rest of the fluid. The resulting fluid motion from rapid heating, termed thermoacoustic convection, was later numerically investigated by Ozoe et al. [37]. Kassoy [38] and Radhwan et al.’s [39] theoretical study of the physical characteristics of the thermoacoustic interaction has revealed two different regimes that dominate heat transport in a confined perfect gas, which are thermomechanical convection on the acoustic timescale, and strong conductiveeconvective interaction on the long diffusive timescale. It has been shown that, if the external perturbation is particularly strong, the fast expanding hot thermal boundary could even generate shock waves. Huang and Bau’s [40,41] numerical modeling of the thermoacoustic effect in gases conjectured that thermoacoustic waveforms, instead of having a rounded symmetric shape as the then prevailing theory indicated, comprise a sharp front and a long tail. This description of thermoacoustic waves was subsequently confirmed by pressure measurements in various gases by Brown and Churchill [42], Hwang and Kim [43], and Lin and Farouk [44]. The incorrect prediction of symmetric thermoacoustic waves was, according to Brown and Churchill [45], caused by inadequate discretization and improbable convergence criteria in previous numerical simulations. In the past decade or so, numerical simulations of the thermoacoustic process have been extended to two-dimensional

configurations [46], the interaction between thermoacoustic waves and gravity [47], and even conventionally incompressible media such as water [48]. Due to the presence of viable thermal diffusion, the contribution of the thermoacoustic coupling to global thermalization is negligible. Near the CP, the thermoacoustic behavior becomes physically

Fig. 7. Comparison between the experimental observation and numerical simulation of propagating thermoacoustic waves in a container (L ¼ 10.3 mm) filled with supercritical CO2 at T0  Tc ¼ 150 mK and 30 mK along the critical isochore [51], which was produced by continuous heating through an immersed film heater. A total of 367 erg of heat was released into the fluid over 0.2 ms.

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more relevant because of the combination of extremely low thermal diffusivity and high compressibility. Although the PE has long been assumed to be of thermoacoustic origin [25], only recently did Miura et al. [49] make the first experimental observation of the thermally-induced acoustic emission near the PE. Part of the experimental dada was later verified analytically by Carlès [50], and numerically by Zhang and Shen [51]. It should be noted that in Carlès’s model, essentially no viscous dissipation was allowed. Fig. 7 shows the comparison between the results of the high-speed interferometric measurement of the thermoacoustic waveforms [49] and the numerical reproductions [51]. The propagating sound (represented by the density perturbation) was caused by continuous heating for 200 ms via a film heater immersed in a cell filled with near-critical CO2 at T0  Tc ¼ 150 mK and 30 mK, respectively, along the critical isochore. Generally good agreement is obtained, as shown in the figure. 3.2. Influences of the bulk viscosity The definition of the PE timescale tPE implies a nonphysical notion of instantaneous relaxation infinitely close to the CP. Zappoli and Carlès [52] have proposed an ultimate acoustic saturation of critical speeding up. As tPE collides with the acoustic timescale ta, their model suggests that energy should be completely transported by acoustic processes. In the region of acoustic saturation, as was shown elsewhere [53], an inversion of the reflection rules of acoustic waves on thermostated boundaries could materialize. Specifically, the apparent non-zero velocity at the edge of the thermal adaption layer could become the new boundary condition for the governing equations, replacing the imposed no-slip condition. This might in turn alter the wave reflection mechanism. However, these phenomena could be masked by the critical divergence of the bulk viscosity hb, which is formulated, in the vanishing-frequency limit, as [49,54,55]

0:18rv2s phs x kB T

3

hb y

(16)

where kB is the Boltzmann constant. Based on the above relation, the critical exponent of hb could be as low as 1.94, which is the strongest divergence among all relevant thermophysical properties. The inclusion of the bulk viscosity has been shown to lead to more exciting features about the PE. On a long timescale, pioneering theoretical work by Carlès [24] has clarified the role of the bulk viscosity in temperature homogenization of near-critical fluids. The asymptotic analysis shows that, the enlarged viscous drag tends to create a strong pressure gradient in the boundary layer, which weakens the interaction between the temperature and density perturbations. The engendered pressure gradient, on the other hand, has an impeding effect on the expansion of the boundary layer. As a result, the PE enters a new physical regime, named viscous regime, whereby critical speeding up turns into actual critical slowing down. Under the viscous regime, the theorized time-evolution of the boundary layer indicates that the size of the boundary layer should finally stabilize to a fixed thickness, defying predictions of continuous growth by the nonviscous model. As an extension of the previous work, Carlès and Dadzie [56] further quantified the typical timescale of the newly minted viscous regime, which was shown to grow as tv w ε0.74 approaching the CP. As is indicated in Fig. 8, thermal relaxation decelerates significantly after the crossover from the PE regime to the viscous regime. The transition was predicted to occur at ε y 2  104 for a 10-mm cell filled with CO2 [24], and at

Fig. 8. Typical timescales of the viscous and PE regimes of temperature relaxation for 1-mm cell filled with supercritical 3He [56]. For comparison, the diffusion timescale tD and the acoustic timescale ta are also included.

ε y 5  104 for a 1-mm cell filled with 3He [56]. The model also suggests that the threshold is quite sensitive to the characteristic length of the fluid, showing a dependence of L2/5 [24]. This feature has yet to be verified experimentally. On a short timescale, Zhang and Shen [51] have conducted a numerical study of the impact of the diverging bulk viscosity on the thermoacoustic waves. The fluid is supercritical CO2 confined in a one-dimensional enclosure (L ¼ 10.3 mm), into which a short heat pulse with a width of 10 ms is added through an internal heat source. In Fig. 9, we plot the snapshots of the temperature, pressure, density, and velocity profiles close to the left boundary after the pulse reflection. Dramatically different boundary behaviors can be seen under the viscous (denoted by the solid lines, with T0  Tc ¼ 60 mK) and PE regimes (denoted by the dash-dot lines, with T0  Tc ¼ 500 mK). Due to the enlarged viscous stress near the boundary, large-scale fluid accumulation is visible within the boundary layer under the viscous regime, which then translates into a significant pressure surge. The resulting pressure difference across the boundary layer causes its expansion, generating a thermoacoustic flow. As a result, the singular cold ‘pocket’ in the temperature distribution resulted from the reflection of the pulse at the thermostated boundary almost disappears. The accelerated

Fig. 9. Plots of the numerically obtained spatial profiles for (a) temperature, (b) pressure, (c) density, and (d) velocity close to the left boundary of a near-critical (r0 ¼ rc) CO2-filled enclosure after the reflection of a thermoacoustic pulse [51]. Dash dot: T0  Tc ¼ 500 mK (under the PE regime) at t ¼ 400 ms; solid: T0  Tc ¼ 60 mK (under the viscous regime) at t ¼ 440 ms.

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homogenization of the local temperature imbalance is driven by the hb-induced boundary layer expansion, which constitutes a secondary PE. 3.3. Nonlinear thermoacoustic features Thermoacoustic waves are temperature/pressure disturbances that travel within the fluid from where local thermal inhomogeneities arise, whose characters depend in large part on the thermomechanical interaction inside the thermal boundary layer. Through modifications to the initial perturbation, nonlinear features can be introduced to the thermoacoustic process. One important question emerges: could these nonlinear effects have a lasting impact on the long-term dynamics of the PE? Shen and Zhang [10,11] have conducted a series of numerical investigations of the nonlinear thermoacoustic wave generation near the CP to identify various intrinsic and extrinsic factors. Based on those results, the answer seems to be negative. It has been shown that, by adjusting the heating rate, extreme thermoacoustic wave transformation might occur, including shockwave-like features [10]. Consider thermally quiescent and motion-free supercritical N2 (T0 ¼ 126.5 K and P0 ¼ 3.5 MPa) confined between two solid walls (L ¼ 10 mm). An exponential temperature variation (denoted by T  T0 ¼ T1[1  exp(t/Hta)], where T1 ¼ 10 mK) is imposed at the left boundary, while the right boundary is kept thermostated. The rate of the temperature rise is controlled by the parameter H, with the impulsive heating (viz., a stepwise temperature increase) corresponding to H ¼ 0 and more gentle heating corresponding to larger H. With varying H, the boundary temperature increase gives rise to markedly different pressure responses. For H ¼ 0, a single strong pressure oscillation results from the sudden temperature change. Apparently, as the temperature eventually stabilizes, the local pressure cannot sustain the gradient across the boundary layer, and is forced to return to its initial state. As H grows, the dampened temperature rise causes more and more stable boundary pressure buildups, which help maintain the expansion of the boundary layer. These local pressure perturbations then propagate as thermoacoustic waves into the bulk. In Fig. 10 are depicted the spatial profiles of the traveling temperature waves for H ¼ 0 and 100, respectively. Features of the boundary pressure variations can be found in their corresponding thermoacoustic waveforms. Under the impulsive heating, the

Fig. 10. Calculated temperature variations at the center of a one-dimensional cell (L ¼ 10 mm) filled with supercritical N2, following prescribed temperature variations at the boundary [10]. The initial state of the fluid is T0 ¼ 126.5 K and P0 ¼ 3.5 MPa. The boundary temperature change is in an exponential form, T  T0 ¼ T1[1  exp(t/Hta)], where T1 ¼ 10 mK. Here H denotes the rate of the boundary temperature rise, with smaller H corresponding to faster heating.

9

induced pressure fluctuation practically emits an acoustic pulse; on the other hand, the steady pressure surge under the gradual heating is consistent with the long plateau in the waveform. However, such stark nonlinear differences have a very limited influence on the thermal relaxation on a longer timescale. Except for a few quantitative differences, the simulated evolution of the temperature distribution features a homogeneously heated bulk inserted between two thermal boundary layers for both the rapid and gradual heating, much similar to what is depicted in Fig. 3(b). With changing initial distances to the CP, thermoacoustic waves also undergo dramatic changes. Shen and Zhang [11] have examined the influence of the degree of criticality on the acoustic emission by performing numerical simulations over four decades of ε (ranging from 104 to 100) along the critical isochore of N2. The fluid is subjected to a constant 100 W/m2 heat flux at one end of a one-dimensional cell (L ¼ 10 mm) when the opposite end is thermally insulated. Since thermal diffusion is mostly confined within the thermal boundary layer, the temperature at the heated end increases as wt1/2. Under the nonlinear temperature disturbance, the initial sharp pressure surge becomes increasingly damped as ε diminishes, which is then followed by a commensurate decline in response to the decelerating temperature rise as time passes. Eventually the pressure stabilizes at an increasingly lower level the further from the CP. The nonlinear distortions in the generated thermoacoustic waveform (see Fig. 11) are consistent with the boundary pressure variations associated with varying ε, which are of a transient nature. The long-term energy yields j of the thermoacoustic process (i.e., the PE energy efficiency) at different ε, as shown in Fig. 12, suggest that the nonlinearity is slowly smeared out. The solid line in the figure represents the theoretical j based on the linearized analysis [17,57],



jy 1 

     1 P vT 1 P kT ¼ 1 g T vP r g T aP

(17)

which tends to a constant near the CP. As temperature inhomogeneities quickly equilibrate under the PE, fluid is redistributed from the high-temperature region toward the low-temperature region. The growing density imbalance is capable of triggering convection, which is the topic of the following section.

Fig. 11. Spatial temperature profiles at t ¼ ta/2 for different initial ε along the critical isochore of N2 following the application of a constant heat flux of 100 W/m2 at the left end of a one-dimensional enclosure of L ¼ 10 mm [11]. The resulting temperature perturbations travel as thermoacoustic waves, assuming different waveforms consistent with the local pressure responses within the boundary layer.

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Fig. 12. Calculated j versus ε at various times for supercritical N2 at critical density [11]. The solid line represents the theoretical values according to Eq. (17). As time increases, the initial nonlinearity slowly disappears.

4. Thermal convection near the CP 4.1. PE under gravity For a long time after its discovery in microgravity experiments, one issue remained of significant controversy: can the PE survive buoyancy-induced convection? Boukari et al. [58] made the first attempt to prove the existence of critical speeding up under terrestrial conditions. Although the experimental results indeed indicated the time for supercritical xenon to reach equilibrium evidently declined closer to the CP, the possible contribution of natural convection could not be completely eliminated. By use of laser holographic interferometry, Nakano et al. [59] studied the PE accelerated-heat transport in supercritical N2 on a ground-based facility. They avoided buoyant convection by placing the heat source near the top of the experimental chamber. The strengthened density stratification due to the PE heating managed to suppress hydrodynamic instabilities. Dynamic relaxation processes in near-critical fluids involve multiple timescales, and are accordingly driven by different mechanisms. The PE and thermogravitational convection are largely decoupled as the PE timescale precedes that of natural convection close to the CP. Boukari et al.’s [60] numerical study of the dynamic behavior of a fluid (xenon) near its liquidevapor critical point in earth’s gravity shows that the adiabatic heating remains strong even when gravity is present. By means of numerical simulation, Zappoli [61] has confirmed the non-negligible role that the PE plays near the CP, even in the presence of natural convection. The two-dimensional simulation concerning side-heated supercritical CO2 shows that the global thermalization is still dominated by the PE regardless of gravity. In addition, the large density gradients in the boundary layer caused by strong compressibility seem to need a long time to be fully resolved after the temperature is nearly completely homogenized, which results in the remarkable appearance of quasi-isothermal convection. In Ref. [23], Zappoli et al. have numerically studied the coupling effect of buoyancy-induced convection and thermoacoustic heat transfer in a near-critical VdW fluid. The convection is initiated in the form of a single thermal plume rising from a local heat source at the center of a two-dimensional square cavity, which has apparently a twofold effect on the fluid thermalization. First, the thermal

plume strongly enhances the cooling PE by impinging on the cold boundary, accelerating a quasi-thermal equilibrium between the heating PE and the cooling PE in the bulk of the fluid. Second, diffusion is mostly confined in the thermal boundary as the convection develops, whose long-term influence on heat transport hence would be rather insignificant. Soboleva’s [62] modeling of near-critical CO2 (based on the VdW equation) heated from below through a finite fixed-heat-flux source in an otherwise thermally insulated square cavity focused on the interaction between the PE and convective motion. The comparison between the simulations in terrestrial and zero-gravity conditions has led to the conclusion that the PE is independent of convection. The limited heat fluxes from the heat source could enhance the local convection near the heat source as the CP is approached, while dampening the bulk convection. Consequently the PE in global thermalization becomes distinguishable even in the presence of gravity. Recent numerical results from Shen and Zhang [63] of the evolution of buoyancy-induced convection in supercritical N2 clearly show that, since the timescale tPE is much shorter than that of natural convection, the PE continues to dominate the temperature equilibration very close to the CP. Fig. 13 illustrates the vertical temperature variations at different times within a rectangular cell with an aspect ratio of 4, which is heated from below with a constant heat flux of 1 W/m2. As convection develops, the temperature field becomes affected by modest local fluctuations. All in all the temperature distributions shown in the figure are still strongly reminiscent of the PE in that the temperature varies more or less uniformly in the bulk [cf. Fig. 3(b)]. It is worth mentioning Ishii et al.’s [64] interesting work on the temperature wave propagation in near-critical CO2 under the influence of gravitational acceleration. The linear analysis has led to a striking observation that the amplitude of the thermoacoustic wave increases with time in the anti-gravitational direction and decreases along the gravitational direction, which, needless to say, needs further experimental verification. Notwithstanding, the results show that under the effect of gravity, perturbations of small wave numbers do not dissolve immediately, which implies that the PE should remain observable on earth.

Fig. 13. Snapshots of the vertical temperature distributions at different times, in a rectangular cavity (with an aspect ratio of 4 and a height of 2.5 mm) filled with supercritical N2 (ε ¼ 5.55  104, r0 ¼ rc) heated from below with a constant heat flux of 1 W/m2 [63]. The local fluctuations in the bulk are caused by the growing convective motion of the rising and falling thermal plumes.

B. Shen, P. Zhang / International Journal of Thermal Sciences 71 (2013) 1e19

4.2. Hydrodynamic instabilities As a classical dynamic system problem, RayleigheBénard (RB) convection (namely, heated from below) remains a subject of fundamental interest [65]. For a more thorough summary of the nonlinear fluid dynamics, including classification of instabilities, convection pattern formation, transition to turbulence, in-depth reviews by Busse [66], Behringer [67], and Ahlers et al. [68] are highly recommended. In a RB configuration (viz., a vertically confined compressible fluid subjected to an adverse temperature gradient that is parallel to gravity), warmer fluid near the bottom tends to flow upward by buoyancy force, and cooler fluid near the top in turn falls down to take its place. Once the critical temperature gradient across the fluid layer is exceeded, sustained convection arises carrying energy in a much more efficient fashion than pure diffusion. There are two sources of stability in a compressible fluid, namely, dissipation mechanisms and compressibility. The former is described by the Rayleigh criterion, which leads to the classical definition of the dimensionless Rayleigh number as Ra ¼ aPgDTL3/yDT. The latter permits strong density stratification, which results in the Schwarzschild criterion. High compressibility allows a larger temperature gradient to be imposed across the fluid layer without triggering sustained motion. As the Schwarzschild criterion states, convective instabilities cannot grow if an adiabatically rising heated fluid element remains heavier than the surrounding fluid, whose distribution depends on the density stratification under the influence of the hydrostatic pressure. In other words, the ascending fluid needs to overcome the density gradient imposed by the compression of the fluid under its own weight [20]. For an ordinary compressible fluid, the Rayleigh criterion is adequate when the fluid height is small. Usually only at meteorological scales (involving large-scale geophysical systems) does the compressibility become an essential factor in determining fluid stability. Gitterman [69] recognized that hydrodynamic processes near the CP could be explained by neither ordinary hydrodynamics (due to the diverging compressibility) nor gasdynamics (due to the nonideal EOS). There could be a crossover between the Rayleigh and Schwarzschild criteria in a supercritical fluid within the same space scale. The team led by Carlès [70,71] performed a rigorous stability analysis of a near-critical fluid based on a more accurate description of the asymptotic thermophysical properties at the CP. The results show that the threshold of free convection near the CP depends on both contributions from compressibility and dissipation. Convection is not initiated until the temperature difference across a layer of height L reaches



DT L





11

impact on the onset of convection. In the case of unsteady bottom heating, to a first-order approximation, the velocity gradient caused by the PE appears to have little influence on the momentum equation, which was also noted in a similar linear analysis later by Gitterman [73]. On the other hand, the particular temperature profile under the PE allows simultaneous generation of instabilities from both the top and bottom ends of the cell, which differs considerably from an ordinary compressible fluid. It should be noted that the dispersion equations derived for supercritical fluids were found to be almost identical to those common to incompressible fluids. As a result, the size of the most unstable perturbations (thermal plumes) turns out to be, after rescaled on the basis of the boundary layer thickness (rather than the distance between the plates), comparable to that under a more convectional RB configuration. The contribution of the ATG to the hydrodynamic stability near the CP was experimentally confirmed by Kogan and coworkers [7476]. The transient and steady-state temperature differences DT across a flat RB cell (of 1 mm in height and 57 mm in diameter) filled with near-critical 3He on its critical isochore (5  104  ε  2  101) under different heat currents q were measured in a systematic study. The variation of the critical temperature difference at the onset of the convection (DTonset) with ε was found to be in excellent quantitative agreement with the theorized transition from the Rayleigh criterion to the Schwarzschild criterion, as is shown in Fig. 14. In the experiment, DTonset was determined as the inflection point in the steadystate DT(q) relation suggesting the initiation of convection. On the other hand, according to Eq. (18), DTonset can be expressed as the sum of two different mechanisms, DTonset ¼ RacyDT/aPgL3þgaPTL/Cp. Both the experimental observation and theory perfectly capture the crossover between the Rayleigh regime relatively far from the CP (dash dot) and the Schwarzschild regime near the CP (dash). Moreover, in the convective regime, after being corrected for the ATG with the use of the potential temperature (defined as, q ¼ T  TATG), the plots of the convective current versus the reduced Rayleigh number seemed to collapse onto a single curve, whose initial slope was typical of the flow pattern of straight parallel rolls. It was a surprising observation that the NueRa relation deviated substantially from the



yDT dT þ dy aP gL4 yDT g aP T ¼ Rac þ CP aP gL4

¼ Rac onset

ATG

(18)

where Rac is the critical Rayleigh number. For stress-free and thermostated solid boundaries, Rac y 1708 [65]. The first term on the right-hand side of Eq. (18) corresponds to the contribution of the Rayleigh criterion, whereas the second term corresponds to the contribution of the Schwarzschild criterion, also known as the adiabatic temperature gradient (ATG). The magnitude of the ATG, which actually tends to a constant near the CP, becomes physically more relevant as thermal dissipation diminishes. (We note that effects of the ATG was excluded in the experiment of the RB convection in near-critical SF6 by Assenheimer and Steinberg [72] on account of the fluid layer being extremely thin.) The matched asymptotic analysis [71] shows that the PE only has an indirect

Fig. 14. Comparison between the experimentally determined steady-state DTonsest versus ε (symbols) along the isochore of 3He and theoretical values (lines) derived from Eq. (18) [76]. The experimental data agree well with the modified model (solid) [70,71], which is a superposition of the Rayleigh criterion (dash dot) and the Schwarzschild criterion (dash).

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‘2/7’ power law in the region of very high Pr (incidentally, extremely close to the CP). The experiment of turbulent convection at extreme Rayleigh numbers by Ashkenazi and Steinberg [77] provided evidence supporting the dominant role of the Schwarzschild criterion as well. The velocity measurement by laser Doppler velocimetry (LDV) indicated that convection in near-critical SF6 became visible only after the temperature difference exceeded that according to the ATG. Based on the configuration in Kogan et al.’s experiment [74e76], the ensuing theoretical [78,79] and numerical efforts [80,81] have successfully reproduced most of the experimental results. The apparent ‘overshoot’ in the temperature trace, which was then followed by damped oscillations, was found to be the result of the competition between the heating and cooling PE [80]. At the initial stage of the bottom heating, the bulk temperature evolved homogeneously while two heat diffusion layers grew near both the bottom and the top plates. As the temperature gradient inside the boundary layers surpassed the convection threshold, hydrodynamic instabilities emerged as thermal plumes. Once the rising thermal plumes reached the top wall, the impact provoked a cooling PE and caused a homogeneous decrease of the bulk temperature. At the same time, the temperature drop at the edge of the upper thermal boundary layer started to weaken the cooling PE. The process repeated itself with the regeneration of thermal plumes, which produced continual oscillations. It was also found that the departure of the experimental data of the first peak time tp of DT(t) versus the reduced Rayleigh number from a universal scaling representation was attributable to small parasitic thermal noises [82]. The hydrodynamic model was accordingly modified to include time-independent thermal fluctuations so as to reflect the various imperfections of the experimental apparatus, which resulted in greatly improved agreement with the experimental data [83,84]. A semi-empirical scaling relation was later derived to describe the development of convection under these externally imposed perturbations [85]. Maekawa et al. [86] has conducted a linear stability analysis of the onset of buoyant convection under different perturbations, including constant temperature rises and heat fluxes applied at the bottom surface. The critical Rayleigh number and the corresponding nondimensional wave number were obtained for near-critical CO2. The delay of the convection onset and the wave number could be modeled as power laws based on the critical exponents of the relevant thermophysical properties. The coexistence of two stability criteria in the same fluid system could lead to some interesting complications. The numerical study by Accary et al. [87] shows that the stabilizing effect of the stratification (i.e., the Schwarzschild criterion) permits regaining stability in a near-critical VdW fluid without any external intervention. Consider a cavity with a height large enough to allow the growth of the thermal boundary layers, whose stability is vital to the initiation of convection. As heat is continuously added to the fluid from below, a hot boundary layer grows with time at a diffusive rate and expands upwards compressing the remaining fluid. Based on the thickness of the boundary layer h, the convection-onset criterion for the boundary layer can be written as

Ra ¼

  gh4 aP dT gT aP > Rac  yDT h CP

(19)

As Fig. 15 shows, the neutral stability curve derived from Eq. (19) comprises two branches dependent on the size of the boundary layer. For small h, the ATG effect can be omitted. The leading Rayleigh criterion has dTh3 > RacyDT/gaP when the boundary layer becomes unstable. On the other hand, for larger h, the Schwarzschild criterion dominates in the generation of hydrodynamic instabilities meaning no convection until dT/h > gTaP/CP. A reverse

Fig. 15. Neutral stability curve of RB convection based on the evolution of the temperature difference dT across the hot boundary layer versus the size of the boundary layer h, as derived from Eq. (19) [87]. The ‘circle’ and ‘triangle’ denote the onset of convection and the collapse of the boundary layer, respectively. A reverse transition to stability (RTS) is possible by crossing the Schwarzschild line if the boundary layer is allowed to grow continuously.

transition to stability (RTS) is possible if the hot boundary layer is allowed to grow uninterrupted and cross the Schwarzschild line of the stability curve. The ‘circles’ in the figure indicate the beginning of convection when the intensity of vorticity starts to grow exponentially based on the numerical simulation; while the ‘triangles’ represent the complete collapse of the boundary layer that prevents the evolution of dT from reaching the Schwarzschild line. In order to avoid the collapse of the boundary layer, particularly weak heating must be applied. One case of RTS was obtained numerically for DT ¼ 0.24 mK and cavity height L ¼ 15 mm, as shown in the figure. The resulting convection only causes minor distortion to the isotherms without destroying the boundary layer (point B), which returns to its undisturbed state once the convection is damped out in the stable zone (point C). At different distances to the CP, RB convection patterns vary considerably due to the wide range of thermophysical properties (see Table 1). Shen and Zhang [63] have numerically obtained the evolution of buoyancy-induced convection in supercritical N2 along its critical isochore in a two-dimensional rectangular cell (with an aspect ratio of 4 and a height of 2.5 mm). In Fig. 16 are plotted the temperature fields for ε ¼ 5.55  104 (Ra ¼ 8.35  107) and ε ¼ 7.92  101 (Ra ¼ 4.88  103), respectively. Close to the CP, the thermal boundary layers formed near both the hot and cold ends under the PE become unstable almost simultaneously, as many mushroom-shaped thermal plumes rise from the bottom and fall from the top [Fig. 16(a)]. It was argued elsewhere [88,89] that the dynamics of the hot and cold plume generation are nearly identical except for some nonlinear effects under infinitesimal perturbations. As the convection enters the quasi-steady stage, the largescale plumes directly shuttle fluid form the hot and cold regions across an otherwise homogenous bulk, leading to greatly enhanced heat transport [Fig. 16(b)]. As ε increases, a periodic self-organized convection pattern emerges. The hydrodynamic instabilities are originated in the bulk due to the weakened PE, as evidenced in the wavy isotherms [Fig. 16(c)]. Then three well-defined

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taken as a prelude to the ultimate regime for high Rayleigh numbers and modest Prandtl numbers predicted by Kraichnan [93] in 1962, under which convection is dominated by almost ballistic motions of hot fluid (represented by NuwRa1/2) [94]. However, this observation was challenged by other experimenters. In a study by Urban et al. [95], the measured Nu in a cell designed to minimize unwanted interference with convective motion was found to follow NuwRa1/3 more closely, and therefore the crossover was believed not to be the transition to the ultimate regime after all. Much still remains unknown on this subject. 4.3. Thermovibrational instabilities

Fig. 16. Numerically obtained temperature contours of a RB cell (with an aspect ratio of 4 and a height of 2.5 mm) filled with supercritical N2 along its isochore for ε ¼ 5.55  104 (Ra ¼ 8.35  107) at (a) t ¼ 10.3 s and (b) t ¼ 42.3 s, and for ε ¼ 7.92  101 (Ra ¼ 4.88  103) at (c) t ¼ 43.3 s and (d) t ¼ 100.0 s [63].

fully-developed hot ‘fingers’ rise from the bottom, as a result of the steady heat currents across the layer [Fig. 16(d)]. The pattern evolution is also visible in the scaling relation of the Nusselt number, represented by changes in the asymptotic slope. Accary et al. [90] has carried out a state-of-the-art threedimensional simulation of RB convection in a supercritical fluid. A detailed description of the convection was presented from the onset of instability to the steady state with Ra ranging from 106 to 108, including flow patterns, spatial distributions of the local Nusselt number, temperature contours, and global thermal balance. The results show that the hydrodynamic stability criterion for the cold top boundary layer is the same as that for the hot bottom boundary layer. Compared with two-dimensional simulations, the presence of lateral walls in the three-dimensional configuration notably accelerates the development of the convective regime. As the convection grows, the spectral distribution of velocity captures an increasing contribution of large wavelengths, which is consistent with the observed turbulent flow of large-scale vertical structures. The steady-state NueRa correlation (after being corrected for the ATG) was found to follow the ‘2/7’ power law. Owing to the peculiar properties of supercritical fluids (rising compressibility and falling thermal diffusivity), experimental explorations around the CP provide a unique opportunity to reach extremely high Rayleigh numbers. By using low-temperature nearcritical 4He as the working fluid, Chavanne et al.’s [91,92] efforts in investigating the ever-elusive ultimate regime of RB convection found that the ‘2/7’ regime held up to Ra y 1011, beyond which the Nu variation approached asymptotically Nu w Ra0.38 for 1011 < Ra < 1014. The authors claimed such a transition could be

As the CP is approached, the sound velocity decreases significantly, which makes it possible for vibrational convection and thermoacoustic wave processes to occur on the same timescale. Their interaction still lacks a thorough investigation. On the other hand, typical g-jitters frequently encountered aboard space vehicles entail vibrations with different amplitudes and frequencies, which are often referred to as ‘artificial gravity’. The effect of vibrations tends to affect microgravity experimental fluid systems. The resulting convective flows exhibit unique features. For these reasons, hydrodynamic behaviors caused by low amplitude (w104 m), low frequency (w101 Hz) vibrations under weightless conditions have attracted growing attention over the past two decades. Carlès and Zappoli [96] first studied the different responses of a one-dimensional near-critical fluid to mechanical vibrations over a wide range of frequencies based on a matched asymptotic analysis of the hydrodynamic model. The results show that there exist three distinct characteristic regimes unique to near-critical fluids (see Fig. 17), including the acoustic regime (where the period of vibrations matches the acoustic timescale), the medium regime (where the period of vibrations is longer than the typical acoustic timescale but shorter than the PE timescale), and the quasi-solid regime (where the period of vibrations nears the characteristic PE timescale). The main feature of the acoustic regime lies in its continuous generation of compressive and expansive waves by the motion of the wall, which seems independent of the initial proximity to the CP. In addition, a strong case of resonance is expected for certain frequencies when the time for a wave to cross the fluid bulk matches the time necessary for the production of another wave at the wall by vibrations. In the medium regime, the fluid bulk reacts in the same manner as a perfect gas since the particular thermal process is too slow to take effect. Finally, the quasi-solid regime displays unique characteristics that are closely related to the thermomechanical effect near the CP. The bulk of the fluid behaves

Fig. 17. Maximum bulk velocity of near-critical CO2 (based on the VdW description) of length L ¼ 10 mm versus frequency of the imposed mechanical vibrations [96]. Three regimes are visible dependent on the fluid behaviors. The dotted line represents the asymptotic response of a perfect gas.

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nearly like a solid in response to low-frequency vibrations, bouncing back and forth between two highly compressible boundary layer at the isothermal walls. Based on the same model, Jounet et al. [97] have numerically investigated the near-critical fluid behavior under low-frequency vibrations in a two-dimensional cavity. The simulation revealed the difference between adiabatic and thermostated longitudinal boundaries. The generated thermovibrational convection has been shown to have a pronounced two-dimensional pattern in areas where the thermomechanical coupling prompts bending of the stream lines near the thermostated boundaries. On the other hands, the flow remains unaffected at the adiabatic boundaries as the dynamic boundary layers forming along the wall turn out to be extremely thin due to the small kinematic viscosity. Recent experimental studies have revealed more intriguing features. Amiroudine et al. [98] imposed on a sample of H2 (at critical density and confined in a cylindrical cell of 3 mm in diameter) a linear harmonic translational vibration with 0.4 mm amplitude and 50 Hz frequency while the boundary temperature was being lower from yTc þ 3 mK to yTc þ 2 mK. Gravity was compensated by a strong magnetic field gradient. The temperature quench caused destabilization of the boundary layer as periodic ‘fingers’ formed perpendicular to the direction of the vibration, as shown in Fig.18(a). The observation of the fingering pattern has been qualitatively confirmed through a two-dimensional numerical study using nearcritical CO2 [see Fig. 18(b)]. The initial temperature varies form Tc þ 30 K to Tc þ 0.3 K along the critical isochore. The fluid is subjected to a step boundary temperature quench of 3 K, 0.2 K, and 0.03 K, respectively, and a sinusoidal vibration with an amplitude of 0.5 mm and a frequency of 20 Hz. As the figure shows, similar instabilities of ‘fingers’ appear within the boundary layer as well, whose horizontal wavelength apparently varies with the distance to the CP. The thermovibrational instability can be explained by analogy with RB convection, in which the driving buoyancy is replaced

by a Bernoulli-like pressure difference induced by the velocity difference between the fluid and its surroundings. It has been determined that the vibrational Rayleigh number strongly diverges and the most unstable wavelength goes to zero near the CP. Garrabos et al. [99] carried out a microgravity experiment of heat transport in near-critical SF6 under linear harmonic vibrations. The interferometric images of the fluid subjected to pulse heating clearly captured the effect of the vibrations on the shape of the thermal boundary layer forming near the thermistor (as a pointlike heat source). It was observed that the initial convection within the boundary layer resulted in two symmetrical hot plumes. The continuous spreading of the extremities of the plumes evolved into what the authors described as ‘pancakes’ perpendicular to the direction of the vibrations, which was understood to be the result of vibrational RB instabilities. The two-dimensional numerical simulation confirmed that the deformation of the thermal boundary layer was caused by the emergence of two counter-rotating rolls, which seemed to grow thinner and smaller nearing the CP. Beysens et al. [100] found that high-frequency vibrations could even speed up the phase transition in H2 near its liquid-gas CP by greatly modifying the relevant phase transient kinetics. Fluid motion in near-critical fluids under space flight conditions exhibits great diversity and variability. Owing to the sensitive nature in the vicinity of the CP, nongravitational effects in thermoand hydro-dynamic processes have given rise to new lines of inquiry that are of promising scientific and engineering potentials. Polezhaev’s reviews [101,102] on this vibrant research subject have provided valuable insights into a wide growing range of topics, such as the interaction between thermo-convection and thermoacoustics in supercritical media, weightless hydromechanics, and nearcritical vibration phenomena. In particular, the past 15 or so years of the joint Franco-Russian experimentation program using the ALICE-series facilities aboard the Russian Mir station, which results from [32,57,98] were part of, were heavily featured. 5. Methods and methodology 5.1. Analytical approaches

Fig. 18. (a) Experimentally observed fingering pattern caused by the deformation of the thermal boundary layer in near-critical H2 at critical density under a linear harmonic mechanical vibration of 0.4 mm amplitude and 50 Hz frequency (with the white arrow denoting the direction of the vibration), while the boundary temperature of the cylindrical container was lowered from T0 y Tc þ 3 mK to T0 y Tc þ 2 mK. (b) Simulated evolution of the temperature fields of supercritical CO2 in a square cavity of 10 mm in length under a uniform harmonic vibration (of 0.5 mm amplitude and 20 Hz frequency) at different initial temperatures of Tc þ 30 K, Tc þ 2 K, and Tc þ 0.3 K, with respective boundary temperature quenches of 3 K, 0.2 K, and 0.03 K [98].

Despite its questionable applicability extremely close to the CP, the complete hydrodynamic description offers the most accurate macroscopic account to date of heat transfer in supercritical fluids under the influence of the PE. Taking advantage of the asymptotic behavior of the thermophysical properties near the CP, the matched asymptotic expansions method can be relied on to provide analytical solutions across different timescales. A wide variety of critical phenomena have been revealed by performing perturbation analysis of the fluid dynamic equations, including strong thermoacoustic interaction on the acoustic timescale [25], possible acoustic saturation [52] or instead the dominance of the viscous regime in the close vicinity of the CP [24,56], the inherently different relaxation mechanisms of temperature and density inhomogeneities [27,57], the crossover between the Rayleigh and Schwarzschild criteria regarding the onset of RB convection [70,71], and the interesting responses of near-critical fluids to vibrational disturbances [96]. At the core of the analytical method are asymptotic expansions of the governing equations based on small perturbation parameters, say, i1 and i2. In practice, i1 and i2 are usually chosen as

l1 ¼

ta T  Tc Pr ; l ¼ ε ¼ tD IG 2 Tc

(20)

where PrIG is the Prandtl number written for an ideal gas. Since these two quantities go to the limits l1 /0 and l2 /0 at different

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rates, the expanded system suffers from asymptotic singularity. To ensure the proper global hierarchy of the expansions, the relative magnitudes of i1 and i2 need to be taken into consideration, which leads to the so-called inner description corresponding to i2¼O(i2/3 1 ), and the first and second outer descriptions correspond to i2>>i1 and i1 [ i2, respectively. The asymptotic matching between solutions under various descriptions contributes to a complete picture of dynamic processes near the CP. The resulting linearized equations, from which negligible terms are discarded on the basis of dimensional analysis, can be solved using Laplace transform method. Boukari et al.’s semi-hydrodynamic approach [denoted by Eqs. (14) and (15)] was, for being simple and amenable to calculation, preferred by many researchers as an expedient analytical tool to interpret experimental data [30,32]. The solution involves iterative steps in computing the volume integrals in Eq. (15), which requires significant computing capacity, especially when extended to higher dimensions. Nikolayev et al. [21] has proposed a Boundary Element Method (BEM) to remedy the problem. By assuming spatially homogeneous thermophysical properties, one can rewrite Eq. (14) as





vT 1 1 ¼ DT V2 T þ 1  g hriVCV vt

Z A

l

vT dA vn

(21)

where <.> indicates the space average, V is the volume of the fluid, and n represents the unit vector normal to the boundary area A. The approximate energy equation, with the volume integrals replaced by the surface integral, results in a great reduction in computational time. However, the modified model is still at risk of over-simplification as no momentum transfer is considered. Exclusion of fluid motion limits its application to temperature relaxation in reduced gravity on a particularly long timescale. Rigorous analyses of the hydrodynamic model are often coupled with the VdW equation as the complementary EOS. As mentioned in Section 2.3, the VdW description does not lead to correct critical exponents and hence is only phenomenologically correct. The accuracy of the VdW equation near the CP of O2 has been discussed by Wagner et al. [103]. Compared with a more accurate EOS based on real-fluid properties, the VdW equation might result in approximately 38% underestimation of the pressure rise and roughly 13% overestimation of the acoustic heating. By use of a linearized EOS similar to Eq. (13), Carlès’s [50] asymptotic modeling of thermoacoustic waves compared fairly well quantitatively with the experimental observation [49] without recourse of any fitting techniques. Wagner and Bayazitoglu [104] have examined the fine differences among the thermoacoustic responses of various critical fluids with actual thermophysical properties data. The systematic comparison was carried by means of numerical simulation using the same one-dimensional configuration with small boundary thermal disturbances. The results show that, on the acoustic timescale, the pressure buildup in H2 is only roughly 17% of that in O2, and the temperature increase in H2 is no more than 30% of that in O2. On the diffusion timescale, the lead of the pressure rise in O2 over that in H2 could grow to an order of magnitude, with the temperature rise in O2 stretching to no less than four times greater than that in H2. 5.2. Numerical approaches With the ever-growing computing power of modern computers, direct numerical simulation (DNS) of heat and mass transport near the CP has been pursued extensively since the mid-1990s. Given the strong nonlinearity of dynamic processes near the CP, a full-spectral universal numerical approach still remains elusive at the present time. Based on the shift in the dominant mechanism across various

15

timescales, two types of numerical approaches are presently available for short-time and long-time thermal behaviors of nearcritical fluids, respectively. On the acoustic timescale, in order to resolve the thermoacoustic nature of the PE, the system of Navier-Stokes equations [Eqs. (6) to (8)] needs to be solved in its full form, which is mathematically rather demanding. The explicit finite-difference MacCormack method, which has been widely used to predict shockwave phenomena in highly compressible fluids [105], proves to be up to the task to generate satisfactory results [2,10,11,26,51]. On the basis of the high-order accurate scheme, an extra fluxcorrected transport (FCT) algorithm can be incorporated to reduce spurious numerical oscillations and avoid possible reduction in the predicted magnitude of physical discontinuities. Furthermore, in the formulation of boundary conditions, density variations at solid walls can be updated using the characteristicsbased relation proposed by Poinsot and Lele [106],



vr vt

 ¼ W

1 vs

     vP vðv$nÞ  rvs vn W vn W

(22)

The subscript W denotes the wall. The above relation is free from common numerical nuisances, such as over-specification that simple extrapolation often engenders. The extension of the method to a longer timescale could, however, be somewhat problematic due to the accumulating errors and onerous computational burdens. Instead of using explicit method, Hasan and Farouk [107] reported in a recent study the simulation of thermoacoustic waves in supercritical CO2 by a finite-volume method that was based on a central-difference scheme for the spatial discretization and a Crank-Nicolson scheme for the temporal discretization of the hydrodynamic equations. The velocity and pressure fields were successfully decoupled by the SIMPLEC algorithm. On the diffusion timescale, a low-Mach-number (LMN) approximation is often used to remove repeated acoustic motion from the system so as to effectively reduce computational time [108]. For a two-dimensional flow, assuming the fluid velocity much slower than the speed of sound, one can expand the nondimensional fluid variables (denoted by primes) in such a way [63],

  T 0 ¼ T ð0Þ þ Ma2 T ð1Þ þ O Ma2 

r0 ¼ rð0Þ þ Ma2 rð1Þ þ O Ma2



(23)

(24)

  P 0 ¼ P ð0Þ þ Ma2 P ð1Þ þ O Ma2

(25)

i h  u0 ¼ Ma uð0Þ þ Ma2 uð1Þ þ O Ma2

(26)

i h  v0 ¼ Ma vð0Þ þ Ma2 vð1Þ þ O Ma2

(27)

Here Ma¼P/vs is the Mach number, with P¼y/L. The governing equations can, with the introduction of Eqs. (23)-(27), be significantly simplified. Through dropping all O(1) terms (except for P(1) in the momentum equations), frequent acoustic behavior is completely eliminated from the physical description as only longterm space-averaged impacts remain, which include the adiabatic homogeneous heating of the bulk by the PE. The strong pressurevelocity coupling in the simplified equations can be adequately treated by the SIMPLE [61,63], SIMPLER [23,80], and PISO [22,90] algorithms. Amiroudine et al. [109] compared the numerical solutions to the LMN hypercompressible Navier-Stokes equations using

16

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Fig. 19. Images of the interferograms of supercritical N2 (with initial temperature T0 y 126 K and pressure P0 y 3.6 MPa) under a constant heat flux q ¼ 3334 W/m2 imposed at the top plate, at (a) t ¼ 10 s, (b) t ¼ 20 s, and (c) t ¼ 30 s. The experimental cell was 16 mm wide, 65 mm deep, and 38 mm high. In (d) and (e) are shown the enlarged images of the portion near the bottom of (b) and (c), respectively [59].

these different methods, which showed that SIMPLER seemed to have the best performance in terms of computational accuracy and efficiency. Regarding the physical appropriateness of the approximate model, a short comment can be made. Shen and Zhang [63] have found that the removal of sound might cause a significant departure from physical reality if particularly strong nonlinearity is involved. The comparison between the predicted temperature distributions with and without acoustic filtration shows that, because the LMN model is largely unable to resolve the possibly pulselike thermoacoustic behavior from a sudden local thermal disturbance, it could result in a gross overestimation of the bulk temperature. Nevertheless, as time elapses, the agreement appears to improve gradually as the acoustic oscillations are eventually damped out. The most egregious theoretical flaw of the LMN approximation probably lies in its omission of gravity-induced density stratification, which amounts to excluding the stabilizing effect of the ATG altogether with respect to the hydrodynamic stability of the fluid. The dynamics of RB convection based on the LMN model exhibits similar behaviors to a Boussinesq fluid, which is also consistent with the analytical [70,71] and experimental results [82,84] derived with the contribution of the compressibility (namely, the Schwarzschild criterion) removed subsequently. In cases very close to the CP [where the effect of the ATG dominates in Eq. (18)], the approximate model could lead to glaring overestimation of hydrodynamic instabilities in the fluid and erroneous predictions of the convection onset. Accary et al. [110] has proposed an adapted low-Mach-number (ALMN)

approximation for supercritical fluids. By retaining the high-order buoyancy term with r(1) in the momentum equation, one can add a background density variation into the fluid system that is comparable to that induced by weak bottom heating. The comparison between the LMN and ALMN approximations shows little if any difference under strong heating; whereas the additional stabilizing factor under the ALMN model starts to emerge under weak heating as the free growth of rising thermal plumes is shown to be increasingly hindered by the initial stratification. With the application of the ALMN approximation, the authors qualitatively reproduced elsewhere [111] large-scale geophysical phenomena in a 10-mm high cavity filled with supercritical CO2 thanks to the diverging compressibility and severe stratification near the CP. 5.3. Experimental techniques Since the discovery that the PE is the main driving force behind the fast thermalization near the CP, many experimental efforts have been devoted to exploring such a unique critical phenomenon. Due to the unstable nature of near-critical fluids, experiments face many technical constraints. The extreme sensitivity to convective instabilities makes stable equilibrium states very difficult to attain on earth. Even small thermal disturbances are likely cause convective motion strong enough to destroy delicate structures such as thermal boundary layers, or disrupt precise calorimetric measurements [7]. Note that top heating of a supercritical fluid sample, as suggested by Nakano et al. [59], seems to alleviate the problem to a certain extent. As for microgravity experiments, limited

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experimental time (for instance, only 6 minutes in total of reduced gravity were realized for supercritical SF6 experiment [9], as shown in Fig. 2) is one of the major practical challenges [16]. As discussed in Section 2.3, supercritical fluids could still suffer from strong density inhomogeneities long after temperature perturbations are nearly homogenized since density equilibration occurs largely through slow diffusion. Also the problem is getting worse approaching the CP because of the diverging diffusion time. To reduce the influence of residual inhomogeneities, most experiments have been carried out using milimetric fluid cells [30,32,34,57]. Moreover, the mismatch between the heat capacities of the bounding wall and the fluid is an important factor that needs to be considered during design of experiments. The effusivity ratio defined as e ¼ (rWlWCW/rlCP)1/2 is expected to decrease considerably near the CP, which would make ideal isothermal boundary conditions harder to achieve [49]. It is quite common to impose thermal disturbances in near-critical fluids through immersed thermistors [16,32,34,99] or thin film heaters [49], which are claimed to provide better heating control. Given the difficulties facing experimental observation of dynamic processes close to the CP, optical diagnostics are ideally suited to measuring transient density variations (or temperature variations, for that matter) at minimum interference. Being noninvasive, optical techniques have been increasingly used in applications with hypercompressible near-critical fluids. Boukari et al. [58] measured the laser beam transmission through a critical xenon column under short pulse heating. The experimental data was then converted into temperature transients of the heated fluid based on the turbidity equation (assuming fixed density). The results were claimed to prove the existence of efficient PE-induced thermal relaxation near the CP even on earth. Guenoun et al. [30] used the comb-shadow technique to measure the density profile in nearcritical CO2 during a thermal cycle under weightlessness conditions, which confirmed the homogeneity of the bulk heating by the PE and the slow diffusive density relaxation. According to the Lorentz-Lorenz equation, density variations in compressible fluids are strongly coupled to variations of the refractive index, which can be measured by interferometry. A Twyman-Green interferometer was used to observe the evolution of the boundary layers (represented by the interferometric fringe shift) in supercritical SF6 around a heating thermistor in the absence of gravity [32,34] and under linear harmonic vibrations [99], and to detect the minute density change caused by the propagating thermoacoustic waves in critical CO2 with microsecond temporal resolution [49]. By means of laser holographic interferometry, Nakano et al. [59] investigated the heat transfer in N2 near its CP. As shown in Fig. 19, the interference pattern following the top heating clearly had similar features to the typical temperature profile under the PE [see Fig. 3(b)], with a growing number of fringes near the top and bottom walls representing the emergence of the thermal boundary layers. However, the numerical reproduction of the experimental results [2] was not successful as the fringe interpretation assumed that the temperature variation was solely the product of the density variation, neglecting the pressure contribution. It should be noted that in the LMN approximation of long-time heat transport in supercritical fluids [63], the temporal change of space-averaged P(0) reflects the dominant cumulative effect of the acoustic process, in other words, the PE itself. In addition, the infinite-fringe method was employed in the experiment, which suffers from well-documented fringe (sign) ambiguity. Finally, since the experimental setup lacked temperature control, the top plate was conveniently considered isothermal and chosen as the reference, which could contribute to the difference between the simulation and experimental observation.

17

6. Conclusions In this paper, we have presented an overview of the recent development in the theoretical, numerical, and experimental research of heat and mass transport in near-critical fluids. Due to the asymptotic behavior of thermophysical properties, the puzzling observation of critical speeding up is mainly driven by the PE, which has its origin in the particularly strong thermoacoustic coupling at the CP. On the other hand, density inhomogeneities relax by slow diffusion, which could trigger hydrodynamic instabilities under mechanical perturbations such as gravity or vibrations. Largely decoupled temperature and density relaxations lead to different mechanisms dominating dynamic processes on different timescales. On the acoustic timescale, numerical and experimental studies show homogenous adiabatic heating of the bulk is realized through thermoacoustic waves. The nonlinear wave features under various intrinsic and extrinsic circumstances seem to gradually diminish as time passes. Under the viscous regime (i.e., extremely close to the CP), the possibly diverging bulk viscosity could cause secondary hydrodynamic effects including the expansion of density boundary layers. On the diffusion timescale, interaction between the PE and gravity results in complex RB flows, whose growth in the fluid must, according to experimental measurements and theoretical analyses, be described by a crossover between the Rayleigh criterion and the Schwarzschild criterion. By means of microgravity experiments and numerical simulation, it has been shown that linear vibrations are capable of creating vibrational RB instabilities in near-critical fluids that exhibit intricate convective patterns. For the sake of brevity, the following topics are not included herein: A local overheating (viz., the bulk temperature rising above the imposed boundary temperature) phenomenon has been experimentally confirmed in a two-phase system [112], which seemingly violates the second law of thermodynamics. The numerical and experimental work by Nakano et al. [113,114] reported a very interesting interaction between the PE and the soret effect in a binary mixture supercritical fluid. Heterogeneous reactions were shown be possible at solid surfaces by heating a near-critical dilute mixture thanks to the critical solubility [115]. Extreme jet flows were observed to arise from a locally damaged thermistor in CO2 and SF6, which generally agrees with the hydrodynamic theory of the PE [116]. Supercritical fluids, as was pointed by Carlès [7], have the potential of being used as scaled-down models to study geophysical phenomena [111]. Further research in these areas could lead to new exciting physics and a better understanding of critical behaviors. Acknowledgments This research is supported by the National Natural Science Foundation of China under Contract No. 50976068. References [1] G. Lorentzen, J. Pettersen, A new, efficient and environmentally benign system for car air conditioning, International Journal of Refrigeration 16 (1) (1993) 4e12. [2] A. Nakano, M. Shiraishi, Numerical simulation for the piston effect and thermal diffusion observed in supercritical nitrogen, Cryogenics 44 (12) (2004) 867e873. [3] K. Dobashi, A. Kimura, Y. Oka, S. Koshizuka, Conceptual design of a high temperature power reactor cooled and moderated by supercritical light water, Annals of Nuclear Energy 25 (8) (1998) 487e505. [4] M. Hoenig, D. Montgomery, Dense supercritical-helium cooled superconductors for large high field stabilized magnets, IEEE Transactions on Magnetics 11 (2) (1975) 569e572. [5] P.J. Lu, D.A. Weitz, M.C. Foale, et al. Microgravity phase separation near the critical point in attractive colloids, 45th AIAA Aerospace Sciences Meeting and Exhibit, AIAA paper (2007) 1152.

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