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Behavior of the properties and critical exponents

The behavior of various thermo-physical properties can be analyzed if the relation between various thermodynamic variables is known, for example using an equation of state defining the relation between pressure, density and temperature. Intuitively, such an equation can also be used to analyze the phase transition and was first described using Van der Waals (vdW) equation of state near the critical point of a fluid. An elementary analysis using this equation of state shows an unusual behavior of various properties such as vanishing density difference, diverging isothermal compressibility etc. However, very close to the critical point, this has been found to be incapable to capture an accurate behavior of the thermo-physical properties [10]. This is primarily attributed to the behavior of fluctuations on approaching the critical point. It is known that locally, the various properties exhibit fluctuations from their mean values, which are more statistical in nature. With majority of the macroscopic thermo-physical properties being related to the statistical distribution of these fluctuations, it is evident that these fluctuations will have a significant effect on various properties. It has been established that on approaching the critical point, the behavior of macroscopic properties is no longer ascribed to the local behavior of molecules but due to the collective behavior arising from the long-range interactions. These long-range interactions, which exist along short-range interactions (defining the background contribution) thus govern the behavior of various properties. Consequently, the specificity of a system gets hidden leading to the universal behavior which has been termed as critical universality. This implies, that on approaching the critical point, the various properties will behave in a similar manner independently of the fluid being considered, and thus have been defined based on a single parameter with critical exponent of each property [10, 11] . These critical exponents have been derived by renormalizing the fluctuations (Renormalization Group Theory) and are described in Table 1-1 [10] . Thus, by virtue of the dominance of these long-range interactions, the various thermodynamic properties tend to show a singular behavior, . . some properties such as, isothermal compressibility( ), thermal expansion coefficient ( ), thermal conductivity ( ), specific heat at constant volume and constant pressure ( , ) diverge while thermal diffusivity ( ) and sound speed ( ) tend to zero as can be observed from the behavior of critical exponents in Table 1-1.
The general behavior of these thermodynamic properties is obtained by fitting experimental results to a law with a constant background term (corresponding to the short-range interactions or mean contribution) and a temperature dependent critical contribution term as a function of reduced proximity to the critical point, = − . These two contributions are generally separated from each other by assuming an additive relation, wherein the critical part is added to the regular part [10]. For any given quantity, , this can be expressed as, ( ) = ( ) + ( ), where ( ) and ( ) denote the critical and regular contribution. The background is generally expressed by a polynomial function, as [10], while the critical contribution is represented by non-analytical functions as a power law multiplied by crossover functions (to account for deviations away from the critical point) as [10], ( ) = 0 ± (1 + (1) + (2) 2 + ⋯ ) (1.2.2)
Here, 0 is the critical amplitude, while is a universal exponent and (1 + (1) + (2) 2 + ⋯ ) denotes the crossover function with (≈ 0.502) representing the universal exponent while ( ) represent coefficients which depend on the type of fluid. However, in most of the cases, the behavior can be well defined by neglecting the higher order terms in cross-over functions and the properties of a fluid in general can be represented by [10], ( ) = + 0 ± (1.2.3)
It can therefore be deduced that on approaching the critical point, the contribution from the critical term becomes predominant. When these distinctive properties of the fluid become distinguishable from mean contributions for certain values of pressure and temperature, the fluid is termed as near-critical fluid.

Piston Effect: Heat transfer mechanism in near-critical fluids

The heat transfer in any substance or system is primarily governed by its thermal diffusivity, which reflects how fast the thermal perturbation will homogenize the system. On approaching the critical point, the thermal diffusivity tends to vanish and thus, intuitively it was believed that this will have as an impeding effect on the thermal homogenization of a supercritical (near-critical) fluid when subjected to thermal heating. However, Nitsche and Straub [12] in their experimental studies with supercritical SF6 observed that the bulk temperature followed the wall temperature very fast thereby causing thermal relaxation in seconds as opposed to the predicted scale of days based on thermal diffusion. This phenomenon was explained in 1990s by three independent teams (Onuki et al. [13] , Boukari et al. [14], and Zappoli et al. [15]) and was termed as piston effect and can be explained as follows.
The phenomenon can be understood by considering a cell filled entirely with a supercritical fluid subjected to a heat flux at one of its boundary. A very thin thermal boundary (TBL) layer is formed due to the vanishing thermal diffusivity. The fluid in the TBL expands due to its high thermal expansion at constant pressure ( ) causing a propagation of an acoustic wave in the bulk which compresses the bulk due to its very high isothermal compressibility ( ). With each passing compression wave, the fluid converts some of the kinetic energy into thermal energy, heating the bulk adiabatically. Further, the compression wave is associated with change in pressure and thus of the same nature as acoustic wave, which results in thermal homogenization to occur on very short time scales termed as piston effect time scale. Similar argument can be applied when the boundary wall is cooled wherein the fluid in the TBL contracts and adiabatic cooling thermalizes the bulk fluid. Since this resembles to the action of a piston which compresses a gas in a closed cylinder, this phenomenon has been termed as the piston effect and forms the basis of understanding of most of the phenomena in supercritical (near-critical) fluids.
It was shown by Onuki et al. [16] that the piston effect can be present in any compressible fluid which can be summarized as follows. The temperature evolution of a fluid, in the absence of any convection can be described by the classical heat diffusion equation. However, for a compressible fluid, it is necessary to add the effect of thermal expansion in the form of work done by the pressure force. This work done varies with the average temperature of the fluid and the ratio of heat capacities, ( = ). What makes this effect dominant and observable in supercritical (near-critical) fluids is the diverging behavior of ( diverges with a critical exponent much higher than , see Table 1-1). Thus, even though the piston effect can be considered ubiquitous in any compressible medium, it becomes significant only near the critical point where the effects of diffusion are subdued while those of compressibility increase rapidly. The time scale corresponding to the homogenization caused by the piston effect is defined as a piston effect time scale and is defined by [16], ≈ (1.3.1) ( −1)2 where represents the diffusion time scale ( 2 ), being the characteristic length scale.
Table 1-2 compares the various time scales for hydrogen with varying proximities to the critical point.

Applications of near-critical fluids

The need of scientific research has been largely driven by challenging applications in various fields, varying from space to micro technology. Over the past several decades, there has been a growing demand for supercritical fluids in various industrial and scientific applications such as varied as alternative eco-friendly refrigerants [17, 18], chemical extraction/separation processes, supercritical chromatography, drying and catalysis [19]. In addition, these also find their applications in particle formation ranging from nano to macro dimensions, supercritical water oxidation for destruction of aqueous based organic waste [20] , electronic chip manufacturing [21], drilling technologies [22, 23] and rocket fuels [24, 25]
In space applications, the storage of cryogenic fluids (Oxygen, Hydrogen) is done under supercritical conditions as the management of these fluids is a real problem due to the uncertain localization of gas and liquid phases. Among these several applications, let us look at some applications pertinent to supercritical (near-critical) fluid and see what additional attributes lie in these.
➢ In rocket engines, methane is being considered as a denser and cheaper replacement of hydrogen in launch vehicles. The interest in such flows for liquid rocket engines is driven by the fact that the thrust chamber is cooled by one of the available propellants, which flows in a suitable narrow channel. A peculiar behavior is observed if methane is used as coolant: it enters channels at supercritical pressure and subcritical temperature and then, under heating from the hot gas, its temperature increases and can reach the supercritical regime. A further interesting behavior can be ascribed when just at the start of rocket engine, the pressure will not be high and thus the methane can transition from supercritical to sub-critical regime.
➢ In CO2 air to-water heat pump or in supercritical CO2 based refrigeration systems, the evaporation is performed at low temperatures in the subcritical regime whereas the condensation occurs at the supercritical state. The overall efficiency is increased which is beneficiary in a certain number of applications such as the production of domestic hot water or industrial drying processes. This is attributed to the fact that heat rejection in supercritical state is primarily a single-phase phenomenon unlike conventional system wherein phase transition reduces the thermodynamic efficiency. An important path that completes the refrigeration/heating cycle can be identified with transition from supercritical to sub-critical regime.
➢ In super-critical water reactors, supercritical water oxidation (SCWO) (a high-pressure and a high-temperature process for the destruction of toxic, hazardous or non-bio-degradable aqueous organic waste) is carried out above the critical point of water. Corrosion of reaction vessels and plugging due to salt precipitation are one of the major challenges of the process. In order to prevent these issues, it is desirable to have the inlet flow entering the reactor under subcritical conditions and thus the analysis of correct physics involves transition from subcritical to supercritical regime.
➢ Recently, hydrothermal spallation drilling has been considered as an alternative for deep drilling, depths > 10 , for geothermal applications. In order to induce thermal spallation, the necessary high temperatures and heat fluxes can be achieved using hydrothermal flame technologies. The hydrothermal flames in such conditions are usually formed using water which at such depth exceeds its critical conditions and unlike in its liquid state, supercritical water is a good solvent for compounds such as oxygen, nitrogen or carbon dioxide. Further, in the absence of any mass transfer limiting phase boundaries, combustion reactions between organic fuels (e.g. methane) and oxidizing agents (e.g. oxygen, air) are easily facilitated in supercritical water. Therefore, while initial conditions on the ground pertain to the sub-critical state, below the ground water is in supercritical state and hence a transition from one state to other.
It can be seen that while in some of the applications, only the supercritical pressures may be attained with sub-critical temperature, in others there is a continuous transition from single phase to two phase regions.

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Motivation and Scope of the current work

With the description of piston effect, explaining the speeding up of thermal homogenization, it can be well ascertained that the singular behavior of the thermo-physical properties can cause various intriguing phenomena. While experimental constraints can limit an in-depth insight into the flow behavior due to complex conditions near the critical point, numerical modelling can thus prove to be a practical and flexible tool in understanding the fundamentals of flow and transport characteristics in this region, especially a continuous transition from supercritical to sub-critical state and vice-versa in the realm of continuum mechanics. The current thesis thus marks a first step towards this goal and the important objectives of the thesis are three folds,
• First, to develop a mathematical and numerical model which can capture the physics of highly compressible supercritical fluid. The importance of such an objective is motivated by the use of a linear state equation in the existing models which circumvents the analysis to a higher thermal heating/cooling and the closest approach to the critical point.
• Secondly, to perform numerical investigations using the developed model in order to gain insights into the various fundamental mechanisms that arise in supercritical fluids in weightlessness conditions when subjected to simultaneous thermal quench and longitudinal vibrations. While this has been primarily motivated by applications in space technology, the absence of gravity evades the effect of convection which may otherwise subdue the effect arising solely from the thermo-mechanical coupling in supercritical fluid.
• Lastly, to explore the possibility to extend this model to analyze the flow behavior in sub-critical state which may then be extended to study a continuous transition. One of the primary challenge attributed to this, unlike usual two-phase liquid-vapor system wherein the liquid phase is considered to be incompressible, both the phases, viz. liquid and gas, are highly compressible.
In order to reach these objectives, the current work is primarily oriented towards the development of mathematical and numerical models to analyze the flow behavior in near-critical fluids, primarily in supercritical state. The thesis is organized as follows, Chapter 2 describes the mathematical model developed to analyze highly compressible supercritical fluid. The current model directly incorporates the dependence of pressure on density and temperature into the momentum equation thereby circumventing the need of any pressure velocity coupling algorithm. In addition, the density is calculated directly from mass-conservation without the need of any explicit equation of state for the calculation of density. In order to analyze the two-phase flow in sub-critical regime near the critical point, phase-field model is developed to analyze flow in sub-critical state under isothermal conditions. One of the primary advantage of phase-field modelling is ascribed to its capability to model the appearance of interface.
Chapter 3 describes the numerical algorithm to solve the mathematical model developed in Chapter 2 for a single and two-phase flow which is succeeded by validation studies of these models. In addition, an in-depth error analysis is presented which highlights unusual behavior of Courant number on the accuracy of solution.
Chapter 4 presents the results pertaining to the investigation of thermo-vibrational instabilities, mainly Rayleigh-vibrational and parametric. The results are validated with experimental observations followed by the description of the physical mechanism causing these instabilities. In addition, the effect of various parameters on the critical amplitude for onset of these instabilities, effect on wavelength and a stability plot is described.
Chapter 5 highlights two intriguing phenomena which were observed in conjunction with Rayleigh-vibrational instabilities. The first one being the drop of fluid temperature below the imposed temperature at the boundary whilst the second one explains the observed see-saw motion of the thermal boundary layer. Both these have been ascribed to the high compressibility leading to strong thermo-mechanical coupling in supercritical fluids.
Chapter 6 presents elementary results in sub-critical state investigated using phase-field modelling approach under isothermal conditions, such as stability of stagnant bubble, coalescence of two drops, phase-separation etc. The primary difference with regard to other models is the use of mass-fraction as the phase-field parameter.
Chapter 7 summarizes the work presenting the concluding remarks, highlighting the challenges and future perspectives derived from the current work.


The mathematical analysis of a fluid flow comprises of translating the physical laws of conservation, namely, mass, momentum and energy into the mathematical relations. These relations which are generally in the form of partial differential equations are termed as mathematical model of the fluid system. In case of a compressible fluid, such as near-critical fluids in the context of present work, an additional relation between pressure ( ), temperature ( ) and density ( ), known as the equation of state, is required to close the set of conservation equations. In general, it is inevitable to use a pressure-velocity coupling algorithm when solving the compressible model which renders the solution more intricate in addition to limiting the accuracy of density not reaching the machine precision. As in near-critical fluids, even a small variation in density can significantly affect the flow characteristics, it is thus highly desirable to calculate density with high precision.
In order to address these issues, a comprehensive description of a mathematical model to analyze single-phase highly compressible supercritical fluid is initially presented. The salient feature of the model lies in its ability to calculate density directly from the mass conservation (continuity equation) without the need of any pressure-velocity coupling algorithm. This is achieved by directly incorporating the dependence of pressure on density and temperature in the momentum equation. The development of such a model is motivated by the eventual aim of modelling the continuous transition from supercritical to subcritical state and vice-versa. For the analysis of two phase flow in sub-critical state, the phase-field modelling approach is introduced at the end of the chapter, discussing the theoretical background and contemporary work in this field. One of the primary advantage of using phase field modelling approach is that it evades any geometrical reconstruction of the interfaces. Subsequently, a compressible phase-field model, which is in conjunction with prior developed compressible model, is described for isothermal systems.
Thus, the above set of equations (2.1.10), (2.1.14), (2.1.15) and (2.1.16) describe a model including all the important physics essential to investigate any thermo-fluidic system. Some of the prominent features of the model can be summarized as,
• The momentum equation (2.1.14) is completely autonomous as it does not contain any unknown pressure. This circumvents the need of any pressure-velocity coupling algorithm as required in usual solution methodologies.
• The density is calculated directly from the continuity equation which ensures mass conservation resolvable to the machine precision.
• The dependence of pressure on density and temperature is explicitly incorporated in the momentum equation using continuity and energy equation as described in (2.1.7) and (2.1.8).
The thermodynamic properties in the current work are evaluated using relations obtained from Renormalization Group Theory as described in [10]. However, in general these may also be obtained from property data base such as NIST [28].

Table of contents :

1.1 Fluids near the critical point
1.2 Behavior of the properties and critical exponents
1.3 Piston Effect: Heat transfer mechanism in near-critical fluids
1.4 Applications of near-critical fluids
1.5 Motivation and Scope of the current work
2.1 Single-phase compressible model
2.2 Phase field model
2.2.1 Basics of Phase-field approach
2.2.2 Contemporary work in phase-field modelling for fluid systems
2.2.3 Relevance to the current work
2.2.4 Mathematical description
2.2.5 Coupling to Navier-Stokes equation
2.2.6 Choice of phase-field parameter
2.2.7 Phase-field model with mass-fraction as order parameter
2.2.8 Final set of governing equations for isothermal sub-critical flow
2.3 Chapter summary
3.1 Basics of Finite Volume Method
3.2 Numerical code Thetis
3.3 Numerical model
3.3.1 Single-phase flow
3.3.2 Phase field Model
3.4 Validation of the Numerical Model
3.4.1 Single-phase model
3.4.2 Phase-field (Cahn-Hilliard) equation
3.5 Chapter Summary
4.1 Literature review
4.2 Problem Description
4.3 Comparison with experimental observations
4.4 Rayleigh-Vibrational Instabilities
4.4.1 Mechanism of Rayleigh vibrational instabilities
4.4.2 Critical Amplitude for Rayleigh vibrational instabilities
4.4.3 Analysis of wavelengths as a function of various parameters
4.4.4 Rayleigh vibrational number as a distance from critical point
4.5 Parametric instabilities
4.5.1 Mechanism of Parametric instabilities
4.5.2 Critical Amplitude for parametric instabilities
4.5.3 Effect of cell size on critical amplitude
4.5.4 Wavelength in parametric instabilities
4.6 Stability Analysis
4.7 Chapter Summary
5.1 Sink Zones: Regions with temperature below the boundary value
5.1.1 Contemporary prior work
5.1.2 Preliminary observations
5.1.3 Features of sink-zones
5.1.4 One-dimensional analysis with both walls quenched
5.1.5 One-dimensional analysis with one side adiabatic
5.1.6 Analysis of sink-zones in two-dimensional case
5.1.7 Parameters affecting onset time of sink-zones
5.2 See-saw motion of the thermal boundary layer
5.2.1 Forced Piston Effect
5.2.2 Relative thickness of the TBL
5.2.3 Mechanism of see-saw motion of the thermal boundary layer
5.2.4 Factors affecting FPE in vibration and see-saw motion
5.3 Chapter summary
6.1 Spurious currents in a stagnant bubble
6.2 Transformation of elliptical bubble to circular shape
6.3 Coalescence of two liquid drops
6.4 Separation of liquid and vapor phases
6.5 Chapter Summary
7.1 Conclusions
7.2 Perspectives
A. Propagation of a pressure wave in a 1D flow
B. Dispersion relation example
C. Error propagation equation
D. Numerical characteristics for second order central difference and implicit first order forward Euler in time numerical scheme
E. Free energy of a non-homogenous system
F. Coupling between phase-field (Cahn-Hilliard) and Navier Stokes


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