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Experimental investigation on hydrodynamic
phenomena associated with a sudden gas expansion in a
narrow channel
Emanuele Semeraro
To cite this version:
Emanuele Semeraro. Experimental investigation on hydrodynamic phenomena associated with a sud-den gas expansion in a narrow channel. Analytical chemistry. Université Pierre et Marie Curie - Paris VI, 2014. English. �NNT : 2014PA066516�. �tel-01128095�
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Pierre et Marie Curie University
PhD Students School SMAER
Laboratory : CEA/DEN/DANS/DM2S/STMF/LATF
Experimental investigation on hydrodynamic phenomena
associated with a sudden gas expansion in a narrow channel
By Emanuele Semeraro
Supervised by Prof. Arnault Monavon
Présentée et soutenue publiquement le 08/12/2014
Presented in front of the jury composed of :
Arnault Monavon
Professor
Supervisor
Matteo Bucci
Research Scientist
Co-supervisor
Paolo Di Marco
Professor
Reader
Thomas McKrell
Research Scientist
Reader
Georges Berthoud
Research Scientist
Reader
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Acknowledgments
This work was supported by the CEA/RNR-Na grant. The grant officer, Mr. Martin, is sincerely acknowledged.
I would like to express my appreciation to my supervisor Prof. Monavon, my two tutors Dr. Bucci and Dr. Cariteau, and Mr. Magnaud, for encouraging my research and my growth as a scientist.
A special gratitude to Matteo Bucci for the opportunity of doing this three-year PhD, and for his friendship. A special thanks to the unconditional support of my family.
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Index
Abstract 7
1 Background and motivation of the present work 8
1.1 The DAC scenario 9
1.2 The two variants of the DAC scenario 10
1.3 Motivation of the present work 11
2 Stability of a surface 14
2.1 A review on hydrodynamic stability concepts 14
2.1.1 Basic concepts on hydrodynamic instability [7] 14
2.1.2 Surface instabilities [8] 16
2.1.3 Simplest explanation of Rayleigh-Taylor instability by Sudden [9] 18 2.2 Analytical models for the study of interface stability in DAC scenario 21
2.3 Liquid expulsion from a channel 21
2.3.1 Fluids system description 21
2.3.2 Initial state, t=0 21
2.3.3 Displacement statement, t>0 24
2.3.4 Definition of the reference flow 26
2.3.5 Dynamic of liquid expulsion 29
2.3.6 Gravity, acoustic and viscous effects 34
2.4 The linear Rayleigh-Taylor instabilities during gas expansion 35
2.4.1 Problem statement 36
2.4.2 Linear perturbation at the interface 39
2.4.3 Non-dimensional form 41
2.4.4 Initial state 43
2.4.5 Solution of equation system 46
3 A review of some previous experiments on RT-instabilities 51
4
3.2 Emmons et al. (1959)[19] 57
3.3 Ratafia (1973)[20] 59
3.4 Cole and Tankin (1973)[23] 61
3.5 Read (1984)[24] 63
3.6 Waddell et al. (2001)[26] 66
3.7 Concluding remarks 70
3.8 Rayleigh-Taylor instability in models with mass and heat transfer 72
3.8.1 Corradini (1978) [28] 72
3.8.2 Excobulle and Excobul program: Expansion and collapse of superheated two-phase
bubbles in cold liquid [29,30,31] 77
3.8.3 Berthoud et al. (1982)[32] 81
4 Experimental apparatus 84
4.1 Fluids system configuration 84
4.2 Analytical model 88
4.2.1 Initial instants 89
4.2.2 Displacement, t > 0 90
4.3 Dimensioning of the apparatus 93
4.3.1 Hydraulic and geometrical similarity with DAC scenario 93
4.3.2 Numerical values of the apparatus parameters 96
4.3.3 Last remarks on apparatus components 97
4.4 Operating procedure 98
4.5 Measurement system 100
5 Analysis of experimental results 103
5.1 Qualitative description of the interface evolution 103
5.2 Post-processing of images 109
5.2.1 Assembly of images of the top and bottom cameras 109 5.2.2 Processing of assembled images: interface length and gas/liquid volume 117
5
6 Phase of Rayleigh-Taylor instabilities 122
6.1 Linear phase 122
6.1.1 The mean interface dynamics 122
6.1.2 The reference disturbance: the mean amplitude h' of the interface profile 129 6.1.3 The amplitude analysis of the most advancing peak hp' 140
6.1.4 Spectral analysis of the interface profile 141
6.1.5 Interface length 150
6.1.6 Conclusive remarks: a global interpretation 153
6.2 Non-linear phase 158
6.2.1 Dynamics of the interface profile 160
6.2.2 Amplitudes analysis and estimate of the mixing zone 167
6.2.3 Length of the interface profile 176
6.2.4 Volumes and flow rate 180
6.2.5 Conclusive interpretation 189
7 Transition to disorderly: multi-structures flow 192
7.1 Transition phase 194
7.1.1 The main interface dynamics 196
7.1.2 Characterization of the dispersed phase 199
7.1.3 Interface length of the channel flow 204
7.1.4 Amplitude of the dispersed phase 207
7.1.5 Volume and Flow rate exiting from the channel 210
7.1.6 Conclusive remarks 213
7.2 Disorderly phase 214
7.2.1 The explosion of dispersed phase 215
7.2.2 Interface length of the dispersed phase 219
7.2.3 Amplitude of the dispersed phase in the channel 212
7.2.4 Analysis of the main structure dynamics 224
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7.2.6 Volume and flow rate exiting from the channel 231
7.3 Conclusive remarks 233
8 Effects of surface tension 235
8.1 Validity of hypothesis of the reference flow and instability theory 236
8.2 Rayleigh-Taylor instabilities 237
8.2.1 Linear phase 238
8.2.2 Non-linear phase 246
8.3 Multi-structures flow 251
8.3.1 Dispersed phase 251
8.3.2 Interface length increment 253
8.3.3 Evolution of volume fraction upon the channel 255
8.3.4 Final remarks 257
9 Conclusion 260
9.1 Characteristic explanation based on the interface length increment 260
9.2 Estimate of liquid film 264
9.3 Perspectives 265
Bibliography 267
List of Figures 270
List of Tables 277
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Abstract
This work aims at improving the understanding of hydrodynamic phenomena associated with the sudden vaporization of superheated liquid (postulated by the DAC scenario with vapor/liquid sodium). This phenomenon is suspected to be at the origin of the automatic shutdown for negative reactivity, occurred in the Phenix reactor at the end of the eighties.
An experimental apparatus has been designed and operated to reproduce the expansion of overpressurized air (6 liters), superposed to a water volume (1 m high) in a narrow vertical rectangular cross section channel (120 mm large, 2 mm deep, 1 m high). Air and water are used to simulate vapor and liquid sodium. The analysis is focused on hydrodynamic aspects. Thus heat and mass transfer phenomena have been omitted in the present investigation and air and water have been used to simulate sodium vapor and liquid.
When the gas expansion begins, the initial flat interface separating the two fluids becomes corrugated under the development of two-dimensional Rayleigh-Taylor instabilities (new analytical approach on RT-instabilities modeling). Since the channel is very narrow, RT-instabilities along the channel depth do not develop. Instead we observe the presence of a very thin liquid film pinned to the wall. During the gas expansion, the interface area increases significantly and may become even 50 times larger than the initial value (120 x 2 mm) at the end of the examined transient (60 cm of travelled distance by the mean interface). Moreover we observed the detachment of several secondary structures from the main interface. This contributes significantly to the increase of the interface area between the gas and liquid phase.
The gas expansion in a narrow channel can be divided into two main phases: Rayleigh-Taylor (linear and non-linear) and multi-structures (transition and disorderly) phases. The former is characterized by the dynamic of corrugated profile and the interface length results proportional to the amplitude of corrugation The latter is influenced by the behavior of the liquid structures dispersed in gas matrix and the interface length is mainly proportional to the number of liquid structures.
The distribution of the volume fraction suggests a model of channel flow consisting of three regions: the regular profile of peaks, the spike region and the structures tails.
We also investigated the sensibility to surface tension performing tests where water was replaced by a solution of water and ethanol (lower surface tension). The results confirm that with a lower surface tension, the fluids configuration is more unstable. The interface corrugations are more pronounced and more secondary structures are produced, leading to a higher increment of the interface area. Hence, when analyzing the DAC scenario (with vapor/liquid sodium), using the hydrodynamic correlations obtained for the adiabatic air/water case would maximize the re-condensation effects.
In conclusion this work could represent an experimental database to characterize the two-phase flow in sudden vapor expansion phenomena. On this purpose we provided suggestions and correlations to characterize the flow inside the channel. These observations and correlations could be fruitfully exploited to construct an interface area transport model to integrate in the previous computational models, in order to achieve a better description of the interface between vapor and liquid sodium, as expected in the DAC scenario.
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1
Background and motivation of the present work
In 1989 and 1990, four automatic shutdowns by negative reactivity insertion occurred at the Phénix reactor. They were called AURNs (Arrets d’Urgences par Réactivité Négative). These events occurred on August 6th, August 24th and September 14th 1989, and on September 9th 1990. The first three occurred at a core power of 580 MWth, the last one at 500 MWth. The insertion of negative reactivity was detected as a very fast and wide oscillation of the neutronic signal recorded by the neutron detectors, which were located beneath the reactor vessel.
Figure 1.1 shows the signals recorded during the first and the third AURN.
Figure 1.1: Neutronic signals recorded during the first and the third AURN.
Each signal is characterized by (see also Figure 1.1):
1. a sudden drop, down to a minimum reached after approximately 50 milliseconds, which triggered the automatic trip (control rod insertion);
2. a reactivity increase up to a maximum below the initial level; 3. a new drop, smaller in amplitude;
1 2
3
4
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4. a second positive peak, slightly exceeding the initial state, approximately 200 milliseconds after the beginning of the event;
5. a shutdown following the insertion of safety control rods.
1.1
The DAC scenario
A complete scenario consistent with observations has not been presently identified, however the insertion of negative reactivity seem to be engendered by a sort of sudden flowering of the core, consisting in a radial core deformation (radial expansion) which has been identified as the most likely mechanism for a fast decrease of the core reactivity.
Several scenarios have been postulated to cause the core flowering [1]. Recently, interest has been focused on the DAC scenario, originally proposed by Guidez et al. [2], and possible variants of it. This scenario is based on the coincidence of the four AURNs with the presence of experimental assemblies within the core, the DAC assemblies, containing moderator material (Figure 1.2 and Figure 1.3). Indeed the DAC assembly (Dispositif Assemblage Cobalt) consists of capsules containing cobalt pellets, which are surrounded by calcium hydride used usually as moderator for the neutron flux. At the same time, DAC assemblies were deployed near fertile assemblies. Thus these fertile assemblies near the DACs were partially exposed to a moderated neutron flux. As a consequence, fission reactions occurred inside the fertile assemblies providing an extra thermal power mostly inside those fertile assemblies having higher burn-up, richer in fissile materials. Despite boiling conditions were not attained in the fertile assemblies, a perturbed temperature field was possibly established in the vicinity of the DAC assemblies. Recently, it was suggested that the thermal-hydraulic design of the assembly was not completely accurate [3,4]. This was probably the cause of an unexpected high thermal power in the DAC assembly, involving a possible channel blockage. According to the DAC scenario, this channel blockage would be the reason for the formation of a large stagnating annular layer of superheated liquid sodium (see Figure 1.2 and
Figure 1.3). Hence the vaporization of this superheated liquid can start randomly, releasing a huge amount of energy that could cause the core flowering.
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DAC
Fertile
Void Cobalt CaH2 A AFigure 1.2: Axial cut view of the DAC assembly
Void
sodium
Cobalt
CaH2
Void
Figure 1.3: Cross cut view of the DAC assembly
1.2
The two variants of the DAC scenario
Depending on where the vaporization starts, two variants of the DAC scenario can be distinguished.
In the original one, the sudden vaporization occurs in the upper plenum (a two-phase bubble of mass at temperature 1250 and corresponding saturation pressure 4 ), where it causes the propagation of pressure and mass waves that interact with the assemblies and cause a modification in the core geometry, producing the expected insertion of negative reactivity. A further pressure load could be caused by the collapse of the vapor bubble. Nevertheless the effects of this first scenario are found not compatible with the core flowering postulated in the DAC scenario [5].
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In the second scenario, the vaporization occurs within the DAC itself (
Figure 1.4): a high pressure layer of sodium vapor (corresponding to 25 ) establishes and pressure waves propagate in the upper and the inter-assembly region, anticipating the expansion of the vapor bubble. These mass waves associated to the expansion of the vapor bubble could be the cause of pressure loads on the surrounding assemblies and therefore of the geometrical modifications of the core. It is in this second variant that we are interested in the present work.
Figure 1.4: Two-phase layer inside DAC assembly
This scenario involves the sudden vaporization of liquid sodium in a metastable state (superheated), confined in the bottom part of a vertical annular channel, and surmounted by a plug of cold liquid sodium. The return to equilibrium conditions is promoted by the vaporization of a part of superheated sodium, leading to the formation of a pressurized two phase mixture consisting of a large vapor volume and droplets of saturated liquid. Hence, the pressurized mixture pushes the cold liquid plug filling the upper part of the channel. As long as the cold liquid is expulsed, part of the vapor is condensed at the interface with the cold liquid. This process, as well as vapor expansion, tends to make the vapor pressure to decrease. However, as pressure decreases, liquid droplets dispersed in the vapor volume vaporize, limiting the pressure decrease. Therefore, to determine if the energy initially available as superheat is large enough to cause the expected core flowering it is necessary to characterize the dynamic of the vapor bubble.
1.3
Motivation of the present work
In the DAC scenario, a significant parameter influencing the vapor expansion phenomenon is condensation at the interface between the vapor and the liquid phase. An extensive analysis of vaporization and
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expansion phenomena within the DAC assembly has been carried out in a previous work [6]. It has been shown that the mixture pressure at the end of the expansion phase (when the cold liquid is completely expulsed from the DAC assembly) is close to the initial pressure (given by the saturation pressure at the initial temperature of the superheated sodium 1250 ). As a consequence, the cold liquid plug surmounting the two-phase mixture undergoes a quasi-constant upward acceleration, much larger than gravity. The modeling approach adopted in the aforesaid work [6] is anyway limited by a poor physical description of the interface between the vapor and the cold liquid: it was assumed that the interface is flat and the interface area coincides with the cross section area. This assumption entails an underestimation of condensation effect, because the interface area is minimized. In the actual expansion, the interface is expected to be rough, with the presence of Rayleigh-Taylor instabilities that evolve as far as the cold liquid is thrown out of the channel. To improve the modeling of the DAC scenario, the need is thus felt to achieve a better understanding and an appropriate quantification of these hydrodynamic instabilities, which determine the evolution of the interface available for the condensation of the vapor.
To achieve this goal, this work presents an experimental and analytical analysis of the onset and the development of Rayleigh-Taylor instabilities, expected to occur in the DAC assembly. Attention has been focused on hydrodynamic phenomena, neglecting, for the time being, the effect of heat and mass transfer on the development of these instabilities. In this aim, an appropriate experimental apparatus has been designed and operated in similarity with the confined annular geometry of the DAC assembly. Pressurized air and water have been used to simulate vapor bubble and cold liquid sodium.
In chapter 2, the fundamentals of hydrodynamics stability of a surface are introduced and a simple physical description of Rayleigh-Taylor instabilities is reported. Then a simplified analytical analysis of DAC scenario is reported and the linear theory of hydrodynamic stability is analytically developed in the configuration of interest. Scaling parameters to design the experimental apparatus are identified and evaluated.
In chapter 3, a review of experiments performed in the last decades is carried out to identify the lack of understanding of the phenomenon and to guide the design of the experimental apparatus, which is described in chapter 4.
Chapter 5, 6, 7 and 8 are focused on the analysis of experimental results. Chapter 5 is intended to give an overall description of the phenomena associated with the expansion of the gas bubble and describe how the quantities of interest have been measured. Chapter 6 deals with the analysis of the linear and the non-linear phase of the Rayleigh-Taylor instability, whereas chapter 7 is focused on the disorderly phase originated by the break-up of the gas-liquid interface. Chapter 8 proposes a comparison between air-water and air-ethanol configuration, to investigate surface tension effects on the onset and the development of instabilities.
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A discussion about the applicability and the consequences of these results to the DAC scenario is finally proposed in chapter 9.
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2
Stability of a surface
This chapter is consecrated to the analysis of the hydrodynamics stability of a surface. It consists of two parts.
First, a review of hydrodynamic stability is reported in section 2.1. The basic concepts to be used in stability analysis of a hydrodynamic system are discussed in subsection 2.1.1. Then, an overview on some known surface instabilities is proposed, focusing the attention on Rayleigh-Taylor instabilities of interest for the present work (subsections 2.1.2 and 2.1.3).
The second part (sections 2.2, 2.3 and 2.4) focuses on the analytical description of the Rayleigh-Taylor instabilities. First of all, the analytical model of the liquid expulsion from an annular channel by gas expansion (supposed in DAC scenario) is reported in order to depict the reference flow in which RT-instabilities occur (section 2.3). Finally a new analytical approach on RT-instabilities model is reported in comparison to other approaches used in the past literature (section 2.4).
2.1
A review on hydrodynamic stability concepts
2.1.1
Basic concepts on hydrodynamic instability [7]
The equations of hydrodynamics, in spite of their complexity, allow some simple patterns of flow as stationary solutions. These can be realized for certain ranges of the parameters characterizing them. Outside these ranges, they cannot be realized because of their inherent instability. In others words they are unable to sustain themselves against small perturbations to which the examined physical system is subject. Suppose a hydrodynamic system to be in a stationary state defined by certain Xi parameters (
, . . . , with 1, . . . , ). They could be geometrical characteristics such as system dimensions; magnitudes of forces acting on the system such as pressure gradients or temperature gradients; parameters describing velocity fields; and others.
In considering the stability of a hydrodynamic system with a given set of parameters, we essentially seek to determine the reaction of the system to small disturbances. Specifically we can state that if the system is disturbed and if the disturbance gradually die down, the system is stable, while if the disturbance grows in amplitude in such a way that the system progressively departs from the initial state and never reverts to it, the system is unstable. Clearly the system must be considered unstable even if there is only one special mode of disturbance with respect to which it is unstable.
If all initial states are classified as stable or unstable, then a set of parameters exists, which separates the two classes of states and marks the passage from the one to the other. It defines the state of marginal stability of the system and, by this definition, a marginal state is a state of neutral stability. The
15
determination of this particular set of parameters Σ is one of the prime objects of an investigation on hydrodynamic stability. The mathematical treatment of a problem in stability generally proceeds along the following steps.
First we can distinguish perturbations of infinitesimal amplitude from the perturbations of finite amplitude. The first ones are object of study of the classical linear theory, while the second ones are analyzed by the non-linear theory.
In the linear theory, as it will be better explained in the next lines, we can assume a perturbation of infinitesimal amplitude and neglect all non-linear terms in equations. The linear theory can be then considered the starting point for the more complete non-linear theory.
We start from an initial flow which represents a stationary state of a system. After, supposing that the different physical variables describing the flow suffer small infinitesimal increments, we can obtain the equations governing these increments from the relevant motion equations. Here we can neglect all products and powers of the higher order to retain only liner terms.
Each reaction of the system to all possible disturbances must be examined for a complete investigation on stability, as above mentioned. For this reason, the disturbance must be expressed as an expansion of a suitable and complete set of normal modes. The stability of the system can be examined with respect to all these modes, since each mode is independent from the others.
Be the various modes indicated with the symbol " ". Several parameters may be needed to distinguish the different modes and the symbol is thus assumed to represent all the parameters that may be needed. Hence, if , denotes a typical amplitude describing the disturbance, we can write it as follows:
! " , # Eq. 2.1
The dependence on time can be eliminated by seeking a solution of the form
" , " $%&' Eq. 2.2
where ("is constant to be determined and the subscript emphasizes the connection of each ( value to a different normal mode .
The perturbation equations finally involve ("as a parameter. Solutions can be sought which satisfy certain necessary boundary conditions such as no slip at a rigid boundary or no tangential viscous stresses at a free boundary. Equations should allow no trivial solutions (vanishing everywhere) for any arbitrary ("value and lead directly to a characteristic value problem for (". The problem is thus reduced to determine (" for all possible modes.
The most general characteristic value (" is a complex number:
(" )$*("+ , -.*("+ Eq. 2.3
The condition for stability is that )$ (" is negative for all values. The states of neutral stability with respect to the disturbance mode is characterized by
16
)$*("+ 0 Eq. 2.4
This last relationship leads to a precise set of parameters Xi that satisfies the condition:
Σ" 0 Eq. 2.5
In most problems, the stability exchange principle1 is applied by imposing that even Im{nk} = 0. Otherwise, if
Im{nk}/0, periodic oscillations appear and the system is in the so-called overstability conditions.
2.1.2
Surface instabilities [8]
Interfacial instabilities that develop as a result of stratified flows of two liquids are important factors in natural phenomena and a wide variety of technical devices.
Apparently, the first historically consistent investigation of stability, in which the key role is played by surface phenomena, was the study of the jet disintegration that was launched by Rayleigh (see
Figure 2.1). The physical meaning of Rayleigh instability lies in the fact that, upon the appearance of a
random disturbance on the surface of a jet flowing from a channel, the disturbance tends to increase, as the most thermodynamically advantageous shape of liquid is a sphere (droplet) rather than a cylinder (jet). The disturbances rise until their amplitude achieves the value of a jet radius that, in a final analysis, leads to the disintegration of the jet into droplets. If ) is the cylindrical jet radius, the instable wavelength 0 of disturbance must be
0 1 4.5 )
Figure 2.1: Rayleigh instability: the disintegration of jets into droplets
Instabilities can also develop as a result of the flow of two liquids with a common interface. For example, the emergence of waves on the liquid surface under wind loading belongs to the classical type of such phenomena and was one of the favorite problems of 19th century.
Some of the most significant types of instabilities that develop as a result of stratified flows are worth mentioning. Among these, the best-known is the Kelvin–Helmholtz instability. This situation is illustrated by the following example. A long pipe with a rectangular cross section that originates in a horizontal position is filled with water above a colored salt solution. Then, the pipe is suddenly tilted downward, thus
1
The stability exchange principle: in (control) linear systems, which is either dynamically stable or unstable depending
on the value of a parameter, the complex frequency varies with the parameter in such a way that its real and imaginary parts pass through zero simultaneously; the principle is often violated [7].
17
bringing both liquids into motion. The instability develops very rapidly at the interface and propagates into regular helical rolls (Figure 2.2.a).
Figure 2.2: Kelvin-Helmholtz instability
This type of instability plays is highly influential, particularly on atmospheric phenomena (Figure 2.2.b). The moving force of these effects lies in the differences between the densities and rates of the motion of layers. The condition of the onset of instability is expressed by the critical value of Richardson Number Ri. It is reverse to the Froude criterion, which is the measure of the ratio between potential and kinetic energies.
) 2 34
where 2 is the free-fall acceleration, 3 is the characteristic length, and 4 is the characteristic velocity. An instability of the considered type develops at ) < 0.25. It is significant that any additional stabilizing factor, particularly surface tension, prevents the development of this type of instability and increases the critical velocity at which it develops.
Another example of an analogous phenomenon occurring due to the same mechanism is the Richtmyer–
Meshkov instability, which consists of an impact on the side surface of the vessel that contains two layers
of liquid and, as a result of random non-uniformity in the acceleration distribution, leads to the development of instability while liquids penetrate one another. A similar effect appears upon the impact collision of the fluxes of two liquids with different densities (Figure 2.3).
18
A stratified flow of two viscoelastic liquids can cause an instability to appear due to differences in the elasticity and relevant differences in normal stresses between the layers of liquid. The jump of normal stresses at the interface always plays a destabilizing role. Wave motion initiated by surface forces also appears in a thin film flowing down the solid surface. In all real situations, different parameters, such as viscosity, elasticity, surface tension, and density difference, are imposed on one another, which substantially complicates the theoretical analysis and the elucidation of characteristic values of critical parameters.
A simplified description of Rayleigh-Taylor instability is instead presented in next section 2.1.3. The fluids density 5 and viscosity 6, the surface tension 7 and the system acceleration 8 are the parameter that influence the phenomenon.
NUMERICAL APPLICATION
Input data: 8 = 100 m/s2, 9 = 10 m/s2, 0 = 1 cm, 5= 103 kg/m3, 7 = 0.071 N/m, 6 = 1 mPa·s Results: Webber number: ;$ <=>?
@ 14
Bond number: AB <C?>
@ 140
Reynolds number: )$ <=?
D 100
The values of the three non-dimensional numbers, found by the given input data, define a inviscid flow where surface tension is unimportant.
2.1.3
Simplest explanation of Rayleigh-Taylor instability by Sudden [9]
The Rayleigh-Taylor instability is a fingering instability of an interface between two fluids of different densities. It can occur when a heavy fluid is superposed to a lighter one, or when a light fluid is accelerated against a heavy fluid. Below is a simple qualitative description of Rayleigh-Taylor instability, as reported by Sharp.
We can imagine the ceiling of a room plastered uniformly with water to a depth of 1 m (Figure 2.4.A). The layer of water will fall. However, it is not through the lack of support from air that water will fall. The pressure of atmosphere is equivalent to that of a column of water 10 m thick, quite sufficient to hold water against the ceiling. But in one respect, atmosphere fails as a supporting medium. It fails to constrain the air-water interface to flatness. No matter how carefully the air-water layer was prepared to begin with, it will deviate from planarity by some small amount (Figure 2.4.B). Those portions of the fluid, which lie higher than the average, experience more pressure than is needed for their support. They begin to rise, pushing aside water as they do so. A neighboring portion of the fluid, where the surface hangs a little lower than average, will require more than average pressure for its support. It begins to fall. The air cannot supply the variations in pressure from place to place necessary to prevent the interface irregularities from growing.
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The initial irregularities therefore increase in magnitude, exponentially with time at the beginning. The water which is moving downward concentrates in spikes. The air moves upward through the water in round topped columns (Figure 2.4 C). The water falls to the floor. The same layer of water lying on the floor would have been perfectly stable. Irregularities die out.
Figure 2.4: Evolution of a Rayleigh-Taylor instability (Figure from Sudden [9])
There is a complex phenomenology associated with the evolution of an unstable interface. This includes the formation of spikes, curtains and bubbles, the development of Helmholtz instabilities on the side of the spikes, competition among bubbles leading to their amalgamation, formation of droplets, entrainment and turbulent mixing, and a possible disorderly limit with a fractalized interface. It is helpful to organize a description of the growth of the instability into a number of stages as follows.
Stage 1. (Figure 2.4.B) If the initial perturbations in the interface or velocity field are extremely small, the early stages in the growth of the instability can be analyzed using the linearized form of the dynamic equations for the fluid. The result is that small amplitude perturbations of wavelength 0 grow exponentially with time. When the amplitude of the initial perturbation grows to a size of order 0, substantial deviations from the linear theory are observed.
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Stage 2. During the second stage, while the amplitude of the perturbation grows nonlinearly to a size of order0, the development is strongly influenced by three-dimensional effects and the value of the density ratio, or Atwood number, defined as
55EF 5G
E, 5G
where 5Eis the density of heavy fluid and 5G is the density of light fluid. If ~1, the light fluid moves into the heavy fluid in the form of round topped bubbles with circular cross sections. The heavy fluid will form spikes and walls or curtains between the bubbles, so that a horizontal section would show a honey-comb pattern. If ~0, one will instead find a pattern more like two sets of inter-penetrating bubbles (see section 3.3 and 3.6).
Stage 3. (Figure 2.4.C)The next stage is characterized by the development of structures on the spikes and interactions among the bubbles. These phenomena can originate from several sources. There is a nonlinear interaction among initial perturbations of different modified wavenumber. Also Helmholtz instability along the side of the spike can cause it to mushroom, increasing the effect of drag forces on the spike. This effect is more pronounced at low density ratios. There is some experimental evidence for bubble amalgamation, a process in which large bubbles absorb smaller ones, with the result that large bubbles grow larger and move faster. The presence of heterogeneities in various physical quantities can modify the shape and speed of bubbles and spikes to a degree which depends on the strength and length scale of the heterogeneity.
Stage 4. In the final stage, we encounter the breakup of the spike by various mechanisms, the penetration of a bubble through a slab of fluid of finite thickness and other complicated behavior that leads to a regime of turbulent or disorderly mixing of the two fluids.
Several factors influence the development of Taylor instability in a simple fluid. These include surface tension, viscosity, compressibility, effects of converging geometry, three-dimensional effects, time dependence of the driving acceleration, shocks, and a variety of forms of heterogeneity. In natural phenomena and technological applications, there are many other factors that can play an important role. For example, material properties and the equation of state of the fluids may be important. The fluids may conduct heat or diffuse mass. The material may change phase or consist of several components. Radiation often couples to hydro-dynamics.
There is a considerable body of literature which analyzes the initial stage in the growth of small amplitude Taylor instability, where the linearized form of the equations of fluid dynamics can be used. In the next
21
section, this technique is applied to evaluate the stability of the interface between a pressurized gas and liquid in a geometry representative of the DAC assembly.
2.2
Analytical models for the study of interface stability in DAC scenario
In the sodium vapor expansion of the DAC scenario, the interface is expected to be rough, with the presence of Rayleigh-Taylor instabilities that evolve as far as the cold liquid is thrown out of the channel. Before addressing the description of stability theory (section 2.4), we need to define the reference flow associated with the DAC scenario, that is an abrupt liquid expulsion under the force of a pressurized gas. Moreover, the analysis of the reference flow will shed light on hypotheses that can be used to find appropriate values of design parameters for the realization of the experimental apparatus, as illustrated in chapter 4. Indeed, the apparatus is aimed at duplicating the DAC scenario. Therefore, it aims at duplicating the expulsion of liquid from the channel due to the gas expansion, which, at the same time, lead interface to corrugate developing Rayleigh-Taylor instabilities. In any case, the two phenomena are required to be independent and not to interact with each other.
On this purpose the next section 2.3 presents the analytical model of a liquid volume expulsion from the DAC channel. Then the resulting reference flow will be introduced into the RT-instabilities model described in section 2.4.
2.3
Liquid expulsion from a channel
In this section we are going to characterize the reference flow of the liquid expulsion from a narrow channel, which will be destabilized by introducing a perturbation of small amplitudes in the section 2.4.
2.3.1
Fluids system description
Let us consider a fluids system as shown in Figure 2.5. The fluid domain is limited on the right channel by two rigid, cylindrical, annular and concentric walls of radius and , $ according its revolution axis and by two flat section distant IJ. The lower section is rigid. The system defines an annular space of large length.
22
b
e
g
z
p
ah
o interface gas liquid atmosphere free surfacer
H
oH
*z
p
Δp
p
a+ ρ
lg(H
o- h
o)
O
O
Figure 2.5: Initial configuration
The axis is oriented parallel to the gravity acceleration 2, uniform and constant in the whole physic space. The so defined space is described by cylindrical coordinates *K, , L, M+, centered on the revolution axis of the system. The origin K is place on the lower flat section and the axis KM is oriented upwards. By hypothesis, the following ratios are considered small parameters for the asymptotic methods:
NO $P 1 Eq. 2.6
NQ I
JP 1
Eq. 2.7 Physical properties of fluids, liquid and gas, are constant: both densities 5G and 5R, viscosities 6G and 6R, thermal conductivities G and R,the heat transfer coefficient at constant volume of liquid SG and the heat transfer coefficient at constant pressure of gas TR.
The great difference in density magnitude order suggest introducing the ratio U 55RV
G P 1
where 5RV is the gas density in the unperturbed state.
2.3.2
Initial state, t=0
This section is consecrated to the description of the initial state of the reference flow. The interface and the free surface are expressed by equations respectively
M 3V
23
obtaining
WV IVF 3V Eq. 2.8
At the initial state, two fluids are at rest: - Gas :
- Liquid :
The liquid domain is limited on the top (M IV at 0) by a free surface. Here it is in contact with an inviscid isobar atmosphere of pressure Z[. We are restricted to conditions (for asymptotic methods):
$ P 3V~IV P IJ
NE I$
VP 1
Eq. 2.9 Based on the relation IV P IJ, the domain is considered infinite in the upwards direction.
The velocity fields are null \R \G 0. Thus the hydrostatic pressure can be applied to the liquid volume. Ideally, for an incompressible fluid, speed of sound is infinite in the liquid and overpressure is soon established. It reads
ZGV M Z[, ]5G 2 ,∆ZW
V_ IVF M
Eq. 2.10 The overpressure ∆Z is an important input data since it represents the motor force of the flow. At M 3V
ZGV 3V ZV Z[, 5G 2 WV, ∆Z Eq. 2.11
Introducing firstly Eq. 2.8 and secondly Eq. 2.11 in Eq. 2.10, it gives ZGV M ZVF ]5G 2 ,∆ZW
V_ M F 3V
Eq. 2.12 In this reference case gas is supposed at uniform temperature V and the pressure at the interface level is
ZRV 3V ZGV 3V ZV Eq. 2.13
The static equation for a compressible fluid in the gravitational field is written #ZRV
#M F5RV 2 FZ) RV 2V
that, resolving with condition Eq. 2.13 and the following hypothesis, M ` 3V and 2 3) V
V P 1
gives that the volume forces produced by gas are unimportant
ZRV M c ZV Eq. 2.14
0 ` M ` 3V
24
2.3.3
Displacement statement, t>0
z
O
L
oH
H
oh
h
o gas liquid atmosphereδ
H
δ
h
Figure 2.6: Instant t>0The pressure field Eq. 2.12 is not fulfilled and liquid is submitted to the pressure gradient ∆Z/WV 1 0. It starts moving and gas can expand. Let us consider a generic displacement of e3 eI. Then we assume the two interfaces start curving and tensions begin rising on the strength of viscosity of two fluids. Flow is supposed laminar.
Variables
Variables of the problem become: - Pressure R ZV, ZR G ZVF 5G 2 , ∆Z/WV M F 3V , ZG - Temperature V, R V, G - Density 5RV, 5R 5G - Velocity \fR \fG
where the perturbation due to a generic displacement is thus represented by
- Gas :
- Liquid :
ZR, 5R, R, \fR
25
- Interface :
- Free surface : I F IV, ;gggf
Equations in the continuous domain
The system is composed of:
- a perfect, compressible, viscous, unsteady gas, where volume forces are unimportant (system Eq. 2.15);
- an incompressible, viscous and unsteady liquid (system Eq. 2.16).
The gas system, Eq. 2.15, is composed of state equations, as the gas perfect law and enthalpy , the mass conservation equation , the momentum equations # and the energy equation $.
. R h5RV, 5Ri ) V, R . 3R 3RV, TR R . j5j , k · l 5R RV, 5R \fRm 0 #. h5RV, 5Ri nj\fj , k\fR R· \fRo FkZR, 6RpΔ\fR,13 khk · \fRir $. h5RV, 5Ri nj3j , k3R R· \fRo jZj F kZR R· \fR, 26RstggfggfR F13 hk · \fRi u , R∆ R Eq. 2.15
The liquid system, Eq. 2.16, is composed of the intern energy equation , the mass conservation equation , the momentum equations and the energy equation #.
. 4G 4GV, SG G . k · \fG 0 . 5Gnj\fj , k\fG G· \fGo F∆ZW V$̂wF kZG, 6GΔ\fG #. 5G]j4j , k4G G· \fG_ 26GxtggfggfG: tggfggfGz , G∆ G Eq. 2.16
The gravity is eliminated by construction of equations and liquid is moved with the only acceleration of the initial state.
Boundary conditions at discontinuities
The mass, momentum and total energy jump relations are shown at the discontinuities of the interface and free surface.
- Interface
Neglecting the chemical reaction, the mass jump relation is written {.|} {*5 \f F ~ggf + • (€}
developing
26
.| 5Rh\fRF ~ggfi • (€ 5G \fGF ~ggf • (€ Eq. 2.17
where (€ is the normal at the interface from gas to liquid, ~ggf • (€ represents the normal displacement of interface. The flow rate is thus positive in condensation case.
The momentum jump equation is
•.| \f F Σff • (€‚ Fk · p7 x-ffF (€(€zr F7 ƒ (€
where 7 is the surface tension of the gas/liquid interface and ƒ is the mean curvature. It becomes p.| \fR, ] R,23 6Rk · \fR_ -ffF 26RtggfggfRr • (€ F x.| \fG, G-ffF 26GtggfggfGz • (€ F7 ƒ (€ Eq. 2.18
The energy jump relation is
•.| $ F x\f • Σff F „fz • (€‚ 0
where $ 4 , 1/2\f is the specific total energy. It is written
.| $RF p\fR• pF ] R,3 62 Rk · \fR_ -ff, 26RtggfggfRr , Rk Rr • (€ .| $GF p\fG• xF G-ff, 26GtggfggfGz , Gk Gr • (€
- Free Surface
The atmosphere is assimilated to a incompressible perfect fluid with velocity \f[ and pressure Z[. The mass transfer is null and chemical reactions are absent. The mass jump relation is
h\fGF ;gggfi • …† h\f[F ;gggfi • …† 0 Eq. 2.19
where …† is the normal at free surface towards the atmosphere and ;gggf • …† represents the normal displacement of the free surface.
The free surface is submitted to the atmosphere pressure and its surface tension 7[ and curvature ƒ[. The momentum jump relation is
GF Z[ …† F 26GtggfggfG• (€ F7[ ƒ[ …† Eq. 2.20
The energy jump relation is
p\fG• xF G-ff, 26GtggfggfGz , Gk Gr • …† FZ[\f[• …†
For walls and lower fixed section, we need to write adherence conditions.
2.3.4
Definition of the reference flow
The channel flow is strictly one-dimensional and composed of two perfect fluids (inviscid and uncompressible). The interfaces remain flat and no mass transfer occurs. The gas transformation is isentropic. The problem is reduced to an annular piston dynamic.
The velocity fields are the following \fR \R M, $̂w
27 Gas
Knowing the gas thermodynamic transformation, the total energy equation in system Eq. 2.15 becomes useless. Equations (Eq. 2.15.a,c,d) of the gas domain are rewritten
. ZRV, ZR h5RV, 5Ri‡ ZRV 5RV‡ . j5j ,R jM l 5j RV, 5R \Rm 0 . h5RV, 5Ri nj\j ,R j\jM \R Ro FjZjMR Eq. 2.21 where ZRV Z[, 5G 2 WV, ∆Z 5RV ZRV/ ) V Liquid
In absence of heat conduction, liquid will remain at its initial temperature V and the energy equation of system Eq. 2.16 is useless. Thus for the liquid we rewrite the Eq. 2.16 as follows
. \G \G . 5G#\# G ∆Z W VF jZG jM Eq. 2.22 Interface
By hypothesis the interface remains flat and its equation simply reads
M F 3 0 Eq. 2.23
Then from the mass and momentum relations (Eq. 2.17 and Eq. 2.18), we obtain respectively . \R \G ~ #3#
. ZR ZG F ]5G 2 ,∆ZW
V_ M F 3V
Eq. 2.24
Free surface
By hypothesis the free surface remains flat and its equation simply reads
M F I 0 Eq. 2.25
In absence of mass transfer and surface tension, the mass and momentum jump relation (Eq. 2.19 and Eq. 2.20) become
. \G[ \[ ; #I#
. ZG[ ]5G 2 ,∆ZW
V_ I F IV
28 Preliminary results: equations rearranging
From Eq. 2.24.a and Eq. 2.26.b, we obtain #
# I F 3 0 ˆ I F 3 IVF 3V WV Eq. 2.27
In absence of mass transfer, the liquid column height remains constant. It results also #\G # # 3# # I# Integrating Eq. 2.22.b ZG M, ]∆Z W VF 5G #\G # _ M , ƒ
Where the integration constant ƒ is determined by means of condition expressed by Eq. 2.26.b, obtaining
ZG M, 5G I F M #\# ,G W∆Z
V M F IV , 5G2 I F IV
and at the interface M 3 with Eq. 2.27, the liquid pressure is ZG 5G WV#\# FG ∆Z W
V‰ WVF I F IV Š , 5G2 I F IV
The expression of ZG can be obtained directly by applying the momentum equation to a water column
F5G WV#\# F 5G G2WV, G F G[ 0 Eq. 2.28
From Eq. 2.24.b, we have ZR 5G WV#\# F ∆ZG Hypothetical asymptotical state
In spite of the absence of dissipating phenomena, let us assume that system reaches an equilibrium state (index ∞) through an isentropic transformation at M I∞, when gas overpressure force is by now depleted
by the expansion. The hydrostatic pressure in liquid and gas pressure (as done for Eq. 2.10 and Eq. 2.11) are
G∞ M Z[, 5G2 I∞F M
R∞ 3∞ G∞ 3∞ Z[, 5G2WV
and the isentropic gas transformation and mass conservation produce
R∞
5R∞‡ RV
5RV‡
5R∞3∞ 5RV3V
Considering that the liquid volume in the channel remains constant,
I∞F 3∞ IVF 3V WV ˆ I∞F IV 3∞F 3V Eq. 2.29
29 I∞F WV IVF WV 3∞ 3V 5RV 5R∞ n RV R∞o /‡ ]1 ,Z ∆Z [, 5G2 WV_ /‡ Eq. 2.30 NUMERICAL APPLICATION.
Input data: Δ = 3 bar, Z[= 1 bar, WV= 1 m, IV= 1.5 m, = 2 cm, 5G= 103 kg/m3, 2= 10 m/s2. Results: 3∞= 1.28 m and I∞= 2.28 m
2.3.5
Dynamic of liquid expulsion
Quantities scales
Since the number of data is still significant, here we are going to simplify the problem by adopting some complementary hypothesis derived from the experience. Thus we assume that gas expansion starts from ambient conditions (Z[, V), using ordinary fluids (air and water), with a moderate overpressure (some bar). Hence we can write (for WV~1 m)
∆Z Œ Z[• 5G2WV Eq. 2.31
The heights of two fluids volume are of the same order
IV~3V~WV Eq. 2.32
The relation Eq. 2.30 becomes I∞F WV 3∞ 3V]1 ,
∆Z Z[, 5G2 WV_
/‡
and with Eq. 2.31 and Eq. 2.32, it provide the scales relation
I∞~IV~3∞~3V~WV Eq. 2.33
We could indifferently choose one of the five lengths as characteristic length scale. Nevertheless we distinguish them for the analysis of the particular case of next section 2.4.
Assuming the absence of acoustic waves, only a time scale exists Ž •
Then the quantities are expressed by the following scales: - Gas M 3∞ M• 5R e5R 5•R ZR eZR Z•R \R 9 \•R - Liquid M 3V, I∞F 3V M̂ IVF WV, eI , WV M̂ ZG eZG Z•G \G 9 \•G - Interface
30
3 3V, 3∞F 3V 3‘ 3V, e3 3‘
- Free Surface
I IV, I∞F IV I’ IV, eI 3‘
Only one velocity scale (9) is taken into account since a priori the maximum gas velocity is that of the interface (equal to liquid velocity). The density variations are unknown (e5R) as well as the variations of pressure and temperature (eZG, eZR and e R). It is even imposed
eI e3 I’ 3‘ “V e33 V “” 3e3 ” Eq. 2.34 Equations
Introducing quantities expressed through their own scales, equations are rewritten (non-dimensional scales groups are gathered inside the two vertical brackets |·|)
- Gas (from Eq. 2.21) . 1 , –eZR RV– Z•R n1 , – e5R 5RV– 5•Ro ‡ . –3∞ WV WV 9Že55RVR– j5•R j • ,jM• sn1 , –j e55RVR– 5•Ro \•Ru 0 . n1 , –e55 R RV– 5•Ro n— 3∞ WV WV 9Ž—j\•j ,R j\•jM• \•R Ro F –5eZRV9 –R jZ•R jM• Eq. 2.35
- Liquid (from Eq. 2.22) p1 , —eIW V— I’ F ]1 , eI WV_ M̂r — WV 9Ž—#\•# •G —5eZG9 — Z•G G, — ∆Z 5G9 — p1 F ]1 , — eI WV—_ M̂r F — 2WV 9 eIWV— I’ Eq. 2.36
31 . M• —33V ∞— , — e3 3∞— 3‘ . M̂ —W eI V, eI— I’ . \•R \•G —e3W V WV 9Ž—#3‘j • —eIWV WV 9Ž—#I’j • #. –U5eZR RV9 – Z•R — eZG 5G9 — Z•G F — ∆Z 5G9 ]1 , 5G2WV ∆Z _e3WV— 3‘ Eq. 2.37
- Free Surface (from Eq. 2.25 and Eq. 2.26) . M• —IIV ∞— , — eI I∞— I’ . M̂ —WWV, eI V, eI— I’ . \•R \•G —e39Ž—#3‘j • —eI9Ž—#I’j • #. —U5eZG G9 — Z•G[ — ∆Z 5G9 ]1 , 5G2WV ∆Z _eIWV— I’ Eq. 2.38
- Lower Fixed Section . M• 0
. \•R 0
Eq. 2.39
Solution
At the initial instant, hydrostatic pressure is established instantaneously inside incompressible liquid and a homogeneous pressure is present in the gas by means of acoustic waves quickly travelling inside gas volume. The liquid column is then moved by gas expansion.
Let us pay attention on the expression of Z•G rewriting Eq. 2.36 on the interface F —˜™š™›9Ž—WV #\•# •G %œ•' [ žV•Ÿœ , —5∆Z G9 —]1 F — eI WV— I’_ F — 2eI 9 — I’ ˜™™™™™™™™™š™™™™™™™™™› SVG¡¢œ žV•Ÿœ , —5eZG G9 —Z•G ˜™™š™™› ¡•ž[Ÿœ žV•Ÿœ 0 Eq. 2.40
In Eq. 2.40 on one side the function I’ is null and grows afterwards, on the other side the perturbation Z•G is null at the beginning. The liquid column is moved by volume forces while inertia forces begin appearing. It follows that in the first instant of transient, the magnitude order of the inertia forces is given by
WV
9Ž 5∆ZG9 Eq. 2.41
and Eq. 2.40 is rewritten F#\•# • , 1 FG eIW
V ]1 ,
5G2WV
32
From Eq. 2.37.c and Eq. 2.29, we obtain
9Ž eI e3 Eq. 2.43
From Eq. 2.41 and Eq. 2.43, it gives the velocity and time scales 9 £eI ∆ZW
V 5G
Ž £ 5G eI W∆Z V
Eq. 2.44
Eq. 2.45
As expected velocity and time are respectively proportional and inversely proportional to the square root of overpressure.
NUMERICAL APPLICATION.
Input data: Δ = 3 bar, WV= 1 m, eI= 20 cm, 5G= 103 kg/m3, 2= 10 m/s2. Results: 9= 7.75 m/s and Ž= 26 ms
To be unsteady, the mass balance Eq. 2.35.b imposes e5R 5RV 9Ž 3∞ eI 3∞
and with Eq. 2.35.a, it becomes eZR RV e5R 5RV eI 3∞ Eq. 2.46 This two last relation determine the values of density and pressure scales e5R and eZR.
The momentum balance Eq. 2.35.c and the interface condition Eq. 2.37.d, with Eq. 2.29 and Eq. 2.41, become respectively n1 , –e55 R RV– 5•Ro n j\•R j , j\•R jM• \•Ro F – eZR 5RV9 – jZ•R jM• Eq. 2.47 –U5eZR RV9 – Z•R — eZG 5G9 — Z•G F ]1 , — 5G2WV ∆Z —_e3WV3‘ Eq. 2.48
The gas pressure scale in Eq. 2.48, after using Eq. 2.46 with the condition Eq. 2.14, then Eq. 2.41 and Eq. 2.43, can be estimated eZR 5RV9 1 U3W∞V]1 , Z[, 5G2WV ∆Z _ • 1 Eq. 2.49
This result shows that in momentum equation, pressure gradient is predominant and pressure is homogeneous. Nevertheless in order to define its scale value, Eq. 2.48 imposes
U5eZR RV9 WV 3∞]1 , Z[, 5G2WV ∆Z _ sup §5eZG9 , 1¨G Eq. 2.50
33
On one hand, the left term is equal to the unity and on the other hand this last expression is the only relation that joins gas and liquid motion. These remarks impose the simplest definition that allows estimating the eZG scale
eZG
5G9 1 © eZG 5G9
eI WV ∆Z
And through Eq. 2.49, the gas pressure Z•R from Eq. 2.47 is jZ•R
jM• 0 Eq. 2.51
Thus Eq. 2.42 is rewritten #\•G
# • 1 , —eI3∞ 1 , ª' — Z•R
Eq. 2.52 with
ª' ª[, ªR ∆Z ,Z[ 5G∆Z2WV Z[, 5∆ZG2WV
After this scaling analysis, the equations system is composed of the following simplified relations: - gas perfect law Eq. 2.35.a,
- the gas mass conservation Eq. 2.35.b, - the gas momentum equation Eq. 2.51,
- the liquid momentum equation Eq. 2.52, - the interface equation Eq. 2.37.c. They are respectively:
. 1 , “∞Z•R h1 , “∞5•Ri ‡ . j5•j • ,R jM• lh1 , “j ∞5•Ri\•Rm 0 . jZ•jM•R 0 #. #\•# •G 1 , 1 , ª' “∞Z•R $. \•R 3«, • #I’# • Eq. 2.53
with boundary conditions
I’ 0 0 #I’# • 0 0 5•R M•, 0 0 Z•R M•, 0 0
\•R M•, 0 0 \•R 0, • 0 \•G M•, 0 0
Through Eq. 2.53.c and Eq. 2.53.a, homogeneous pressure and density are established. They are respectively
Z•R Z•R •
34
Taking into account the condition of the lower fixed section (Eq. 2.39.b), Eq. 2.53.b can be integrated to obtain j\•R jM• F1 , “1∞5•R #5•R # • © \•R M•, • F1 , “1∞5•R #5•R # • M• Eq. 2.54
Being the gas density derivative negative, the velocity field varies linearly from the lower section. With Eq. 2.34 and Eq. 2.54, Eq. 2.53.e can be integrated, obtaining the relation between 3‘ and 5•R.
3∞ 3V #3‘ # • ,1 , “1∞5•R #5•R # • h1 , “V3‘i 0 © 1 , “V3‘ 1 , “1∞5•R
where initial conditions of subsection 2.3.1 and 2.3.2 have been used.
Finally Eq. 2.53.d provides the solution of the system in the form of differential equation in 3‘ # 3‘
# • 1 F 1 , ª' ¬h1 , “V3‘i
-‡
F 1® Eq. 2.55
Eq. 2.55 will be integrated in section 4 (see Eq. 4.9) and results shown in Figure 4.7 and Figure 4.8. The second member of Eq. 2.55 vanishes and changes sign for
3‘ 1 or 3 3”
Assuming the liquid column motion finishes with oscillations of small amplitude, the second member can be developed
# 3‘
# • c 1 F ¯ 1 , ª' “V3‘ Eq. 2.56
and frequency ° is estimated ° 2± ² 1 , ª¯ ' “V
NUMERICAL APPLICATION.
Input data: Δ = 3 bar, WV= 1 m, 3V= 0.3 m, eI= 20 cm, 5G= 103 kg/m3, 2= 10 m/s2, Z[= 1 bar, ¯=1.4. Results: °= 0.21 Hz
2.3.6
Gravity, acoustic and viscous effects
The solution Eq. 2.55 is limited by same hypothesis recalled below that have been used in the conception of the experimental apparatus (chapter 4). Let us use one length scale WV according to Eq. 2.33:
I∞~IV~3∞~3V~WV
Gravity
Gravity effects can be neglected during the entire interface motion pressure forces
gravity forces 5G RV2WV
35 RV 5G 2WV Z[, 5G2WV,ΔZ 5G 2WV ΔZ 5G 2WV • 1 Acoustic
The gas-liquid interface is followed by the emission of plane waves which propagate in the gas with velocity
[V ¯) V ¯ RV/5RV
Thus we have to compare the time that these waves take to travel the gas column with the characteristic time of gas expansion
Ž[
Ž h¯ RV/5WVRVi / £5∆ZG WV £¯U∆ZRV £U¯ P 1 Viscosity
Viscous boundary layers take a time Ž¼ $½
to diffuse over the whole annular width. Therefore viscous effects can be consider unimportant if the viscous time scale is much larger than the characteristic time of gas expansion
Ž ŽS
½
$ £5∆ZGWV £½ 5$¾∆Z P 1GWV
The non-dimensional number under the square root sign is a Reynolds number.
NUMERICAL APPLICATION.
Input data: Δ = 3 bar, WV= 1 m, 5G= 103 kg/m3, 5RV= 1 kg/m3, 2= 10 m/s2, e= 2 mm, ¯= 1.4, ½=1.1 mm2 /s Results: ¿T <À RÁÂ= 30 , ÃÄ Ã= 7·10 -4 and à ÃÅ= 2.7·10 -2
As shown by the results of the numerical application, gravity, acoustic and viscosity effects all have negligible influence on the development of the reference flow.
2.4
The linear Rayleigh-Taylor instabilities during gas expansion
This section is consecrated to the analytical analysis of stability of the interface M 3 seen in the reference flow of the last section 2.3. Most of the scientists in the literature used a system composed of mass and momentum equations to find a solution for the continuous fluid domain (far from the interface). Then the equations system was completed by boundary conditions (on the interface) as a kinematic condition,
36
pressure drop at the interface and the integration of the unsteady Bernoulli equation through the interface [10,11,12].
Here we propose a new approach made of only mass and momentum equations system applied to two continuous phases, gas and liquid, and at the interface.
2.4.1
Problem statement
General equations
The statement of the problem is defined by perturbing the reference flow of section 2.3. It is supposed a fluctuation depending on space and time develops in the neighbourhood of the interface. The interface thus ceases to be flat (see Figure 2.9)
3 , L, M, 3• ,3TÆ , L, with 3TÆ P 3•
where the index and Z refer to the reference and perturbed flow. The generic mass and balance equation, written below,
j5
j , k · 5\f 0 j 5\f
j , k · 5\f\f F kZ , 52f
are expressed in cylindrical coordinates, as follows (\É 4, \• \, \w ~) j5
j ,1 jjL 54 ,1 jj 5\ ,jM 5~j 0 5 nj4j ,4 j4jL F4 , \j4j , ~j4jMo F 1 jZjL
5 ]j\j ,4 j\jL ,4\, \j\j , ~j\jM_ F jZj 5 ]j~j ,4 j~jL , \j~j , ~j~jM _ F jZjM F 52
Approximation of tangent planes
Since the gap width e is much smaller than the radius b, the tangent plane approximation is used
• $ Eq. 2.57
Two tangent planes to the inner and the outer cylinder surfaces are drawn to describe the local annular region by means of Cartesian coordinates, as shown in Figure 2.7. Therefore, the idealized domain consists of a rectangular cross section channel having the same width $ than the annular cross section channel, and approximately the same cross section area and the same hydraulic diameter (see Figure 2.8). To obtain the differential equations for the idealized geometry, the following substitutions have been introduced in the governing equations:
37 , Ê L Ë M M # #Ê #L #Ë #M #M Eq. 2.58 with 0 ` Ë ` 2± 0 ` Ê ` $
b
b
θ
0
y
x
r
e
Figure 2.7: Sketch of the annular cross section channel
2πb 0
y
x e
Figure 2.8: Sketch of the idealized rectangular cross section channel
38 . j5j , , ÊjË 54 ,j , Ê1 jÊ ‰ , Ê 5\Š ,j jM 5~j 0 . 5 nj4j , , Ê 4j4jË , \j4jÊ F 4, Ê , ~j4jMo F , ÊjZjË . 5 ]j\j , , Ê 4j\jË , \j\jÊ , 4\, Ê , ~j\jM_ F jZjÊ #. 5 ]j~j , , Ê 4j~jË , \j~jÊ , ~j~jM _ F jZjM F 52 Eq. 2.59
At the interface, the mass and momentum jump equations are . {5 \f F \f · (€ } {.|} 0
. Ì5\f \fF\f · (€ , -ff· (€ Í F7 ƒÆ(€ Eq. 2.60
Reference solution
Since the entire system is moved by gas expansion, we preferred to consider the relative reference system displaced by velocity \fœ \œ $̂w. The only momentum balance equation on z-direction Eq. 2.59.d changes in
5 ]j~j , , Ê 4j~jË , \j~jÊ , ~j~jM _ F jZjM F 5 2 , \|œ
and equally the z-coordinate becomes Î M F 3
where Î 0 is on the unperturbed interface (see Figure 2.9).
v
eh’
(x,t)
λ
gas liquidZ
O
2
π
b
x
Figure 2.9: Perturbed interface
The characteristic of the reference flow resulted in the last section 2.3 is recalled below: - Liquid
39 G ZVF 5G 2 , ∆Z/WV M F 3V , ZG - Gas \R \R M, $̂w R ZV, ZR - Interface \R \G \
Where ZV Z[, 5G 2 WV, ∆Z. This solution, illustrated in section 2.3.5, is recalled here in dimensional form: # 3 # • 5∆ZG WVÏ1 , 1 , ª' s] 3V 3 _ ‡ F 1uÐ \R ]1 ,Î3_ #3# F \œ 5R 5RV 3V 3 n RVRo /‡ 5GWV\|œ RV]33 _V ‡ F Z[, 5G2WV G Z[, s RV]33 _V ‡ F Z[u ]1 FWÎ V_
2.4.2
Linear perturbation at the interface
In all analysis of linear stability, the perturbation form can be freely chosen on the purpose to determine if that perturbation form can exist as such. The wavelength 0 of perturbation is assumed larger than the gap width $ of the annular channel so that it can be considered independent from the y-coordinate
0 1 $ Eq. 2.61
Therefore we can write the perturbed quantities \Æ \ ÑÒ Ë, Î, $̂Ñ, \wÒ Ë, Î, $̂w 5Æ 5Æ Ë, Î, ZÆ ZÆ Ë, Î, ÎÆ 3Æ Ë, Eq. 2.62
The perturbed quantities 3Æ are assumed to be much smaller in comparison to the reference quantities 3 (as said at the beginning of section 2.4.1). This will allows to linearize the equations system.
Now we can simplify the system Eq. 2.59 by using the assumption Eq. 2.61 on the perturbation over the Ê-direction and the approximation of tangent planes Eq. 2.57. Thus the generic system becomes
