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Abstract
The neutral stability lines obtained from the viscous Kelvin-Helmholtz analysis and the inviscid analysis are quite different for the case of low liquid viscosities, whereas they are quite similar for high viscosity, contrary to what one would expect. This puzzling result is considered in this work. It is shown that the stability behavior regarding the amplification rate is actually almost the same for the two analyses for a wide range of liquid viscosities and for various pipe inclinations. The results obtained in the present work also support Barnea's interpretation of the viscous and inviscid analyses as a means for predicting various transitions from stratified flow.
... Using the geometric relations (A.6), the volumetric flow constraint (2), and assuming to be constant along , it can be shown that these terms can be written in conservative form [8]: ...
... This prevents the use of the basic model in its non-hyperbolic region. [2,3]. It maps the stability of steady states of the TFM, where friction is balanced by a constant driving pressure gradient (acting as a body force). ...
... Friction divides the well-posed region of Figure 3 into a region with damping and an unstable region (but with bounded growth rates). The stability boundary with friction is referred to as the viscous Kelvin-Helmholtz (VKH) boundary, while the ill-posedness boundary is referred to as the inviscid Kelvin-Helmholtz (IKH) boundary [2,3]. Table 1, plotted as a function of wavelength = 2 ∕ . ...
In this paper we present a complete framework for the energy-stable simulation of stratified incompress-ible flow in channels, using the one-dimensional two-fluid model. Building on earlier energy-conserving work on the basic two-fluid model, our new framework includes diffusion, friction, and surface tension. We show that surface tension can be added in an energy-conserving manner, and that diffusion and friction have a strictly dissipative effect on the energy. We then propose spatial discretizations for these terms such that a semi-discrete model is obtained that has the same conservation properties as the continuous model. Additionally, we propose a new energy-stable advective flux scheme that is energy-conserving in smooth regions of the flow and strictly dissipative where sharp gradients appear. This is obtained by combining, using flux limiters, a previously developed energy-conserving advective flux with a novel first-order upwind scheme that is shown to be strictly dissipative. The complete framework, with diffusion, surface tension, and a bounded energy, is linearly stable to short wavelength perturbations, and exhibits nonlinear damping near shocks. The model yields smoothly converging numerical solutions, even under conditions for which the basic two-fluid model is ill-posed. With our explicit expressions for the dissipation rates, we are able to attribute the nonlinear damping to the different dissipation mechanisms, and compare their effects.
... 4 We consider the incompressible form of the TFM, for (separated) stratified flow, assuming a hydrostatic balance between the fluids. 5 One reason this model is of interest, is its potential to dynamically simulate the transition from stratified flow to slug flow. 6,7 This stands in contrast to for example the drift-flux model, where the two velocities are not modelled separately: instead a closure relation is introduced for their difference. ...
... with ( ) ∶= 3 + 4 = + . (5) Since = ( ) and = ( , ), depends on through (but not on due to (4)). We can use these constraints to set up an equation for the pressure. ...
... and substituting (5) and using (4) to get ...
The pressure‐free two‐fluid model (PFTFM) is a recent reformulation of the one‐dimensional two‐fluid model (TFM) for stratified incompressible flow in ducts (including pipes and channels), in which the pressure is eliminated through intricate use of the volume constraint. The disadvantage of the PFTFM was that the volumetric flow rate had to be specified as an input, even though it is an unknown quantity in case of periodic boundary conditions. In this work, we derive an expression for the volumetric flow rate that is based on the demand for energy (and momentum) conservation. This leads to PFTFM solutions that match those of the TFM, justifying the validity and necessity of the derived choice of volumetric flow rate. Furthermore, we extend an energy‐conserving spatial discretization of the TFM, in the form of a finite volume scheme, to the PFTFM. We propose a discretization of the volumetric flow rate that yields discrete momentum and energy conservation. The discretization is extended with an energy‐conserving discretization of the source terms related to gravity acting in the streamwise direction. Our numerical experiments confirm that the discrete energy is conserved for different problem settings, including sloshing in an inclined closed tank, and a traveling wave in a periodic domain. The PFTFM solutions and the volumetric flow rates match the TFM solutions, with reduced computation time, and with exact momentum and energy conservation.
... where the parameter ψ(r αq ) is null for the Upwind scheme and ψ(r αq ) = 1 for the TVD van Leer scheme. Applying the same procedure for all equations, considering the staggered mesh, where φ e =φ P e iϕ / 2 , φ w = φ e e − iϕ and φ ww =φ w e − iϕ , the resulting matrix M for the discretized system of equations is Fig. 2. Comparison of the VKH differential and discrete growth rates with the results obtained by Barnea and Taitel (1993) for stratified air-water flow. Castello Branco et al. ...
... The methodology was verified by comparisons of the dispersion relations with results obtained by Barnea and Taitel (1993) with the viscous Kelvin-Helmholtz (VKH) approach, for horizontal stratified flows. The Two-Fluid Model was applied without the dynamic pressure term and with pressure jump (σ = 0.072), employing the standard momentum flux parameter for both phases equal to one (C L = C G = 1). ...
... For both cases, the TVD spatial scheme was used in the discrete analysis maintaining CFL G = 0.5, however, different mesh spacing were employed for each case. Fig. 2 and Fig. 3 show the comparison of the differential and discrete growth rates ( − ω i ) against the dimensionless wavelength (λ /D) with the results obtained by Barnea and Taitel (1993) and Montini (2011), respectively. In these figures, the reference data is represented by filled symbols, the discrete data by continuous lines and the present differential formulation by dashed lines. ...
For several industrial applications involving two-phase gas-liquid flows, the one-dimensional Two–Fluid Model equations are typically solved for flow simulation. To ensure reliable predictions, it is very important to assess the stability properties and grid dependency of one-dimensional formulations, which are known to be largely affected by the closure relations and numerical schemes employed. In this work, a stability analysis of the transient one-dimensional Two–Fluid Model was performed for vertical annular flows. A viscous approach of differential and discretized formulations was analysed. The influence of the momentum flux parameter, interfacial pressure jump due to surface tension and a dynamic pressure model were investigated. The analytical results were compared to those obtained by numerical solution of the model equations with the Finite Volume Method, for various experimental configurations taken from the literature. Results showed that closure models largely affect the wave frequencies and the growth rates captured by the model. The surface tension term introduced a cut-off frequency in the differential formulation rendering the model well-posed and effectively stabilizing short waves. The introduction of the dynamic pressure term considered here did not affect the cut-off values. However, the wave growth rates decreased compared to the case without this term. The liquid momentum flux parameter greater than 1 presented a stronger influence on the evolution of interfacial waves in a broadband of frequencies than the other closures and leads to more regular waves and a more uniform flow field but much higher values may excessively damp the solution, which blocks the natural formation of larger waves in vertical annular flows. The numerical solution of the model equations showed very good agreement with the discrete stability analysis performed and was able to capture the wave frequencies and associated (linear) growth rates in the wave formation region. Concerning the momentum flux parameter, the bandwidth of the most damped disturbances in the linear and nonlinear region was similar. This suggests that the frequency response in this case was not very different from linear to nonlinear wave regimes. The methodology proved to be an important tool for further development of closure models and numerical schemes applied to vertical annular flows.
... At high ReSL oscillations of the gas bubble produces energy due to which the plug length increases. The necessary condition for development of intermittent flow is that the liquid hold up must be above the value of 0.5 [8]. Barnea and Taitel [8] suggested the phenomena of development of plug/slug flow pattern in viscous and inviscid flow using linear stability analysis. ...
... This phenomenon is known as Kelvin-Helmholtz (K-H) stability. Barnea and Taitel [8] described that from the Bernoulli's equation, as the wave increases in amplitude the flow area of the gas reduces as in Fig.4. The reduction in gas flow path increases the velocity of gas but the pressure at upper surface of the wave decreases. ...
... In this context, several literature (Ghajar and Tang, 2007;Taitel and Dukler, 1976;Vaze and Banerjee, 2011;Sanchis et al., 2011; reported the onset of slug in horizontal gas-liquid two-phase flow. Some literature (Sanchis et al., 2011;Kordyban and Ranov, 1970;Wallis and Dodson, 1973;Lin and Hanratty, 1986;Barnea and Taitel, 1993;Hervieu and Seleghim, 1998;Woods and Hanratty, 1999;Vallée et al., 2008;Kadri et al., 2009;Dinaryanto et al., 2017;Saini et al., 2021;Saini and Banerjee, 2021b) reported the mechanism of slug formation from stratified/wavy flow. It is argued that beyond a critical wave amplitude, the waves become unstable due to the pressure component in the direction of wave profile which owing to the Bernoullis effect over the wave crest leads to slug formation. ...
... Some literature (Taitel and Dukler, 1976;Lin and Hanratty, 1986;Barnea and Taitel, 1993) reported models for development of slug flow from stratified-wavy flow using instability analysis based on inviscid and viscous theory. Since inviscid linear stability theory neglects the inertia effect on formation of slug flow, the results over predicted the transition line. ...
Slug is produced when the roll waves on the liquid surface undergo constructive interference and the resulting wave height increases beyond the critical height. Slug flow is associated with large aeration that develops a sudden surge in pressure and associated mechanical impact on pipe wall and bends. Such phenomena lead to flow accelerated corrosion (FAC) which results in degradation of pipe thickness with time resulting in cracks from where the gas and liquid leaks. Hence, to reduce the chances of failure in industrial pipes that are mostly opaque (flow pattern cannot be visualized) and associated hazard in nuclear power plant industries, it is important to develop mechanisms for detecting the occurrence of slug flow. The present research proposes a methodology for recognizing the onset of slug in opaque industrial pipes using the pressure signals measured from the pipeline. Recurrence analysis of pressure signal establishes different kinds of recurrence plots for stratified-wavy flow and slug flow regime. Also, the transition can be identified using the recurrence quantification parameters. The quantified parameters like recurrence rate, determinism, and entropy show sharp jump in value at the onset of slug from wavy regime.
... Unlike wind-waves observed in high channels, the transition in pipes from small-amplitude to large-amplitude waves, which are often termed roll waves, is usually associated with Kelvin-Helmholtz instability. [21][22][23][24] Roll waves that can be seen as shock wave-like disturbances are commonly observed not only in gas-liquid pipe flows, but also in downward inclined open-channel flows, such as aqueducts and spillways. A model for such waves was developed by Dressler 25 and later updated by Balmforth and Mandre 26 and by Richard and Gavrilyuk 27 that compared their results to Brock's [28][29][30] experiments. ...
... The stratified regime is bounded by intermittent flow at high liquid flow rates and by annular flow at high gas flow rates. The stratified-nonstratified transition boundaries in this map are determined by neutral stability curves resulting from the viscous Kelvin-Helmholz (VKH) and the inviscid Kelvin-Helmholz (IKH) instability analyses, Barnea and Taitel,23 and Barnea. 52 Each of the four sub-regions identified within stratified flowstratified smooth, inception of initial quasi-linear waves (ripples), roll waves, and pre-annular wavy flow-is denoted in Fig. 8 and in subsequent figures by its distinct symbol. ...
Different wavy regimes in stratified air–water pipe flow are determined for a wide range of gas and liquid flow rates in a 10 m long horizontal pipe with a diameter of 24 mm. Three sub-regions of wavy stratified flow are identified: ripples, roll waves, and pre-annular wavy flow. Statistical parameters, such as local mean film thickness and its higher moments (root-mean-square, skewness, excess kurtosis) as well as wave characteristics (mean heights and wave height distributions, lengths, propagation velocities, etc.), are measured and analyzed. It is demonstrated that ripples are essentially linear waves and their propagation velocities are described reasonably well by linear wave theory. High amplitude roll and pre-annular waves are substantially nonlinear, and their propagation velocities differ significantly from that of ripples. Transition to roll waves causes a sharp increase in higher statistical moments. Evolution of wave and statistical parameters characterizing each sub-region of stratified gas–liquid pipe flow is studied. Simplified models describing roll waves are presented; the model predictions are verified by experiments.
... Kelvin-Helmholtz for Viscous fluid (VKH) has been used by (Wallis, 1969, Lin and Hanratty, 1986, Wu et al., 1987, Andritsos et al., 1989, Barnea, 1991, Crowley et al., 1992 to determine the stability of the stratified two-phase flow considering the wall shear stress with the two fluids model. As reported by (Barnea and Taitel, 1993a), both IKH and VKH provided similar results for high viscous flow while a large discrepancy between IKH and VKH for the low viscous fluids was observed. Also, at the low amplification zone, (Barnea and Taitel, 1993a) confirmed that the slug flow or roll wave region was located between the neutral stability line of the VKH and IKH analysis. ...
... As reported by (Barnea and Taitel, 1993a), both IKH and VKH provided similar results for high viscous flow while a large discrepancy between IKH and VKH for the low viscous fluids was observed. Also, at the low amplification zone, (Barnea and Taitel, 1993a) confirmed that the slug flow or roll wave region was located between the neutral stability line of the VKH and IKH analysis. For the high amplification zone, either slug or annular flow regimes were situated above the IKH neutral stability line. ...
Precise prediction of the gas-liquid slug flow is crucial for proper design and operation at various industrial processes. This paper provides a comprehensive review of published experimental and numerical works in horizontal pipes, including a critical analysis of state-of-the-art parameters used to characterize the slug flow regime. The discussed slug characteristics are slug length, slug frequency, and slug velocity. Throughout the paper, an endeavor does devote to discerning the areas where modern development work is needed. Accordingly, some observed gaps in the literature, comprising hydrodynamic behavior at the transition zone and the interaction between slug flow and structural pipe. The lack of comprehensive laboratory experimental information for high liquid viscosity and the viscosity impact on the slug characteristic is also observed. On the contrary, robust and less empirically independent CFD models are required for accurately predicting the slug characteristics with a minimal computational cost.
... Vaze and Banerjee (2011) developed a flow pattern map based on the inlet flow condition ( gas (Re SG ) and liquid superficial Reynolds number (Re SL )) to account for geometrical properties of the pipe. In addition, Al-Sheikh et al. (1970), Taitel and Dukler (1976), Lin and Hanratty (1986) and Barnea and Taitel (1993) developed theoretical model for predicting flow pattern transition. ...
... Merging of the roll waves increases the liquid height as shown in Fig. 3b (iii). Due to roll wave coalescence (Sanchis et al. 2011) and K-H instability (Barnea and Taitel 1993), slug initiates in the pipe (shown in Fig. 3b (iii)). When wave amplitude reaches a critical height, it curls down in downstream direction leading to entrapment of gas bubble as shown in Fig. 3b (iv). ...
Aeration in slug and associated secondary flow leads to surging of pressure and erosion corrosion in pipe. Surging of pressure develops high mechanical impact on the pipe, and erosion corrosion reduces the thickness of internal wall, thereby resulting in pipe failure. To explore the phenomena of aeration in slug, flow visualization analysis is reported in this paper for intermittent flow sub-regimes and their transition. Analysis is reported for onset of slug, transition of plug to slug flow and development of aeration at the slug front. The visualized images and the motion pictures captured using high-speed photography in the present experiments are used to depict the process of air entrapment during the transition of wavy-stratified flow to slug flow as well as plug flow to slug flow. It is depicted for the first time through our visualization analysis that gas bubble entrapment in slug happens due to plunging kind of wave breaking mechanism. The captured images are also analyzed to describe the phenomena of augmentation of aeration in slug leading to the formation of highly aerated slug flow. Thorough understanding of aeration in slug will help in avoiding the chances of pipe failure.
... 8 However, though it is a promising technique for the industry, annular flow can only be achieved under certain circumstances. 16 It is common to see transitions between annular flow and other flow patterns, such as stratified flow. Once the annular flow has transitioned to the stratified flow, the pressure drop increases sharply, which is a considerable challenge to the operation management of the system. ...
... Analyses of gas−liquid flows have been conducted using these two theories. 16 The theories have also been applied to oil−water flows. Trallero 17 used the VKH theory to predict the transition between stratified flow and unstable wavy stratified flow. ...
In the petroleum industry, oil–water flow appears in pipelines when connate water or injected water is present in the reservoir. As the characteristics of flow patterns differ, studying their transition behaviors helps to reduce the transportation risk caused by dramatic pressure drops. Additionally, pumping power can be significantly saved by transporting oil with water lubricating the pipe. In this paper, experiments were conducted in horizontal pipes to study viscous oil–water flow. Transparent acrylic pipes (7.5 m long and 26 mm internal diameter (ID)) were used, and water and viscous white oil (viscosity of 940 mPa·s) were adopted as the test fluids. The flow structures were recorded, and flow pattern maps were drawn. The characteristics of the transition flow were analyzed. The effects of oil viscosity, pipe ID, and interfacial tension on the transition behaviors were investigated by comparing the results obtained in this work with those from the literature. An empirical correlation for predicting the occurrence of annular flow was proposed based on the analyses of transition behaviors. The model results were found to be in good agreement with the experimental data.
... At high ReSL oscillations of the gas bubble produces energy due to which the plug length increases. The necessary condition for development of intermittent flow is that the liquid hold up must be above the value of 0.5 [8]. Barnea and Taitel [8] suggested the phenomena of development of plug/slug flow pattern in viscous and inviscid flow using linear stability analysis. ...
... This phenomenon is known as Kelvin-Helmholtz (K-H) stability. Barnea and Taitel [8] described that from the Bernoulli's equation, as the wave increases in amplitude the flow area of the gas reduces as in Fig.4. The reduction in gas flow path increases the velocity of gas but the pressure at upper surface of the wave decreases. ...
Gas-liquid two-phase flow is commonly observed in petroleum, chemical industries which are transported through pipelines. Due to intermittent characteristics of slug flow, it surges sudden pressure and mechanical impact on the piping system which enhances the fatigue stress and leads to pipeline failure at bends, T-and I-sections. Thus it is required to identify the formation of intermittent flow pattern which causes economic loss and hazards in industries. In this article transition of stratified to onset of intermittent flow pattern is visualized using high speed photography for different combination of gas (ReSG) and liquid (ReSL) superficial Reynolds number. Based on the observations of formation of intermittent flow pattern, a transition line is developed for different combination of superficial liquid (USL)and gas (USG) and compared with other experimental results.
... Экспериментальное исследование перехода расслоенного нефте-водяного потока в дисперсное течение проведено в работе [5]. В работах [6,7] исследовалась линейная устойчивость расслоенного режима течения на основании двухжидкостной модели, а также изучалась устойчивость стационарных решений. В работе [7] был получен критерий устойчивости расслоенного течения. ...
... В работах [6,7] исследовалась линейная устойчивость расслоенного режима течения на основании двухжидкостной модели, а также изучалась устойчивость стационарных решений. В работе [7] был получен критерий устойчивости расслоенного течения. Экспериментальное исследование дисперсного течения воды и нефти в наклонной трубе проведено в работе [8]. ...
The steady flow of a water-oil dispersion mixture in an oil producing well is simulated in the paper. On the basis of the equations of mechanics of multiphase systems, a model is proposed for predicting the phase costs in the ascending stream. For stabilized, almost vertical flows, a good agreement with the experimental data was obtained.
... A. Visualization The plug flow is observed at low gas superficial Reynolds number (ReSG) of 758 and liquid superficial Reynolds number (ReSL) above 1900. The plug flow is developed in the pipe due to K-H instability [7]. An increase in ReSL to 1850 at fixed ReSG of 758, the liquid height to pipe diameter ratio (hL/D) increases beyond 0.5. ...
... Further increase in ReSG to 2274, the bubble aeration increases which is classified as less aerated slug (LAS) flow pattern as in Fig 3(c). As shown in Fig 3( B. Pressure signals for intermittent flow regime As described in the aforementioned section III (A) the plug flow is developed due to the K-H instability [7]. The pressure in the plug flow slightly increases with increase in ReSL at fixed ReSG. ...
Gas-liquid two-phase flow in general exhibits separated, intermittent and dispersed flow patterns. Intermittent flow patterns (which include slug and plug flow) develops large pressure at the wall and bends and can lead to failure of pipe. Thus identification of such flows is very important. In this work, recurrence technique is proposed as a tool to identify intermittent flow patterns in industrial pipelines where visualization is not possible. For this, flow patterns are visualized and pressure signals are simultaneously measured by varying inlet gas and liquid superficial Reynolds numbers in a two-phase flow test rig [1]. Recurrence plot (RP) is developed from the pressure signals and recurrence quantification analysis (RQA) is carried out for the pressure signals associated with each flow pattern. It is established that the recurrence rate (RR) and entropy (ENT) associated with the flow patterns can be used for their identification in opaque pipes.
... The wall and interface stresses of the two-fluid model are typically modeled in the following manner (Taitel and Dukler, 1976): In this work we use the Taitel and Dukler friction model (Barnea and Taitel, 1993;Taitel and Dukler, 1976) ...
... Physics of Fluids ARTICLE pubs.aip.org/aip/pof This instability causes capillary waves at interface between two immiscible fluids due to shear stresses, and it is catastrophic when the interfacial tension and viscous effect are neglected (Barnea and Taitel, 1993). For this interfacial resonant sloshing case, multi-peaks behavior takes place at the free surface shown in Fig. 18(a) due to the fact that the dominant frequencies of the two liquids are different. ...
In the present study, hundreds of experiments have been conducted on the three-dimensional free-surface and interfacial sloshing in a vertical cylindrical tank containing two immiscible liquids. The bounds of different free-surface and interfacial wave regimes are determined by maintaining fixed excitation amplitude and slowly increasing excitation frequency until another type of wave regime began to appear. In general, three types of the free-surface wave regimes are observed when the excitation frequency is in the neighborhood of the lowest natural frequency of the free surface, i.e., planar gravity wave, chaotic gravity wave, and swirling gravity wave. Similarly, when the excitation frequency is near the lowest natural frequency of the internal interface, three types of interfacial wave regimes, i.e., planar gravity wave, chaotic gravity-capillary wave, and swirling gravity-capillary wave, are generated. Besides, it is worth pointing out that when the excitation frequency is near the lowest natural frequency of the internal interface as well as very close to a third of the lowest natural frequency of the free surface, large-amplitude rotating wave motion occurs at both the free surface and the internal interface. This is due to even though the excitation frequency is far away from the natural frequency of the free surface, the secondary resonance can still become dominant and lead to large-amplitude motion of the free-surface rotating wave and subsequently influences the internal interface. This paper reveals that the sloshing behaviors of two-layer liquid in the vertical cylindrical tank are much more complicated than those of single-layer liquid.
... The critical conditions corresponding to the onset of interfacial perturbations have been explored experimentally (e.g., [1][2][3][4][5]) and theoretically. To avoid complexity of rigorous stability analysis in circular pipe geometry, many theoretical studies were based on simplified mechanistic models, mainly in the framework of the Two-Fluid (TF) model (e.g., [6][7][8][9][10]). ...
A numerical framework for rigorous linear stability analysis of two-phase stratified flows of two immiscible fluids in horizontal circular pipes is presented. For the first time, three-dimensional disturbances, including those at the interface between two fluids, are considered. The proposed numerical framework is based on a finite-volume method and allows solving the problem numerically in bipolar cylindrical coordinates. In these coordinates, both the pipe wall and the unperturbed interface (of a constant curvature, e.g., plane interface, as considered in this work) coincide with the coordinate surfaces. Thereby, the no-slip as well as the interfacial boundary conditions can be imposed easily. It also enables investigation of the local behavior of the flow field and shear stresses in the vicinity of the triple points, where the interface contacts the pipe wall. The results obtained in the bipolar coordinates are verified by an independent numerical solution based on the problem formulation in Cartesian coordinates, where the pipe wall is treated by the immersed boundary method. Two representative examples of gas-liquid and liquid-liquid flows are included to demonstrate the applicability of the proposed numerical technique for analyzing the flow stability.
... The critical conditions corresponding to the onset of interfacial perturbations have been explored experimentally (e.g., Charles and Lilleleht, 1965;Yu and Sparrow, 1969;Kao and Park, 1972;Barnea et al., 1980;Kokal and Stanislav, 1989) and theoretically. To avoid complexity of rigorous stability analysis in circular pipe geometry, many theoretical studies were based on simplified mechanistic models, mainly in the framework of the Two-Fluid (TF) model (e.g., Andritsos et al., 1989;Brauner and Maron, 1991;Barnea and Taitel, 1993;Ullmann and Brauner, 2006;Kushnir et al., 2017). ...
A numerical framework for rigorous linear stability analysis of two-phase stratified flows of two immiscible fluids in horizontal circular pipes is presented. For the first time, three-dimensional disturbances, including those at the interface between two fluids, are considered. The proposed numerical framework is based on a finite-volume method and allows solving the problem numerically in bipolar cylindrical coordinates. In these coordinates, both the pipe wall and the unperturbed interface (of a constant curvature, e.g., plane interface, as considered in this work) coincide with the coordinate surfaces. Thereby, the no-slip as well as the interfacial boundary conditions can be imposed easily. It also enables investigation of the local behavior of the flow field and shear stresses in the vicinity of the triple points, where the interface contacts the pipe wall. The results obtained in the bipolar coordinates are verified by an independent numerical solution based on the problem formulation in Cartesian coordinates, where the pipe wall is treated by the immersed boundary method. Two representative examples of gas-liquid and liquid-liquid flows are included to demonstrate the applicability of the proposed numerical technique for analyzing the flow stability.
... The result is a dispersion relation that correlates the growth rates with a frequency or wavelength spectrum. In ill-posed systems, the growth rate will increase radically for critical frequencies/wavelengths. Barnea and Taitel (1993) devised inviscid and viscous formulations for the Kelvin-Helmholtz instability applied to stratified flow. They produced a stability map for varying liquid and gas superficial velocities, showing the neutral-stability regions for both formulations, and how they intersect with flow pattern transition. ...
... The result is a dispersion relation that correlates the growth rates with a frequency or wavelength spectrum. In ill-posed systems, the growth rate will increase radically for critical frequencies/wavelengths. Barnea and Taitel (1993) devised inviscid and viscous formulations for the Kelvin-Helmholtz instability applied to stratified flow. They produced a stability map for varying liquid and gas superficial velocities, showing the neutral-stability regions for both formulations, and how they intersect with flow pattern transition. ...
Multiphase flows are ubiquitous in nature and industry. In order to fully describe multiphase systems, simulation approaches must be able to handle the interfaces separating the different phases properly. The Volume of Fluid (VOF) approach is widely used by researchers and engineers due to its intrinsic ability to conserve volume and handle large interface topology changes. A common problem that occurs in VOF Methods relates to the calculation of interfacial tension forces in the momentum equations while resolving a sharp interface. Common methods based on the gradients of the volume fraction field may lack accuracy due to the curvature and normal vector estimates through an abrupt transition. To address these points, the present work introduces a new proposal, where the VOF method based on a cloud of points for interface curvatures computation (PC-VOF) is extended by the coupling with the sharp-interface advection algorithm isoAdvector, implemented in the open source suite OpenFOAM®. The coupled method performs better than the ones implemented in the original solvers in a number of benchmark cases from the literature. It is shown to significantly reduce the spurious currents and also presents more stable and accurate results, especially for irregular triangular meshes.
... The linear stability analysis of KHI can be used to determine whether a stratified flow is stable or not from the neutral stability lines. 5 For the KHI in a stable stratification, Thorpe 57 used a long rectangular test section that was rotated to induce the shear layer, and then rotated back to its original position. A non-dimensional number referred to as the Richardson number, Ri, which gives a measure of the ratio of potential to kinetic energy in the flow, was measured. ...
A numerical investigation of Rayleigh-Taylor instability (RTI) with different unstable thermal stratification and coupled Kelvin-Helmholtz (KH) and RTI (referred to as KHRTI) is performed by solving the compressible Navier-Stokes equation. Two air masses having temperature differences of $\Delta T^* = 21.75K$ and 46.5K (corresponding to Gay-Lussac numbers (Ga) of 0.073 and 0.156) are considered in an isolated box, initially separated by a non-conducting interface for studying RTI. For KHRTI, dimensionless tangential shear of $\Delta U = 0.92$ and 1.89, is additionally imposed on the two air masses with $\Delta T^* = 21.75K$. Onset, propagation and fully developed stages of the instabilities are explored via time-resolved and instantaneous temperature and vorticity. For RTI, lower $\Delta T^*$ case shows retarded growth of mixing layer and a set of interpenetrating bubbles. The higher $\Delta T^*$ case, shows an accelerated growth of mixing layer with alternating rows of spikes and bubbles. For KHRTI, flow is governed by KH dynamics at early times and RT dynamics at later times. To further understanding of interaction between RT and KH mechanisms, a compressible enstrophy transport equation in {\it A novel compressible enstrophy transport equation based analysis of instability of Magnus-Robins effects for very high rotation rates} - Suman, V. K. et al., Phys. Fluids, {\bf 34}, 044114 (2022), is used. Depending on Ga, either vortex stretching or compressibility contribution terms of the enstrophy transport are dominant for RTI. Depending on shear imposed, either baroclinic torque or viscous terms are dominant for KHRTI.
... A stability diagram can be made by computing the steady state of the two-fluid model for a range of liquid and gas velocities, and for each steady state computing whether the dispersion relation predicts complex angular frequencies. This is shown in Figure 4.4a and is similar to the map found in [137]. Alternatively, these steady states can be computed as a function of the given pressure gradient and hold-up -this is shown in Figure 4.4b. ...
Multiphase flows are described by the multiphase Navier-Stokes equations. Numerically solving these equations is computationally expensive, and performing many simulations for the purpose of design, optimization and uncertainty quantification is often prohibitively expensive. A cheaper, simplified model, the so-called two-fluid model, can be derived from a spatial averaging process. The averaging process introduces a closure problem, which is represented by unknown friction terms in the two-fluid model. Correctly modeling these friction terms is a long-standing problem in two-fluid model development.
In this work we take a new approach, and learn the closure terms in the two-fluid model from a set of unsteady high-fidelity simulations conducted with the open source code Gerris. These form the training data for a neural network (NN). The NN provides a functional relation between the two-fluid model's resolved quantities and the closure terms, which are added as source terms to the two-fluid model. With the addition of the locally defined interfacial slope as an input to the closure terms, the trained two-fluid model reproduces the dynamic behavior of high fidelity simulations better than the two-fluid model using a conventional set of closure terms.
... Assuming that there is no heat transfer between the gas and liquid phases when slug flow occurs, and there is no heat leakage from the pipe wall, the energy equation is not considered. It is assumed that the physical quantity on the cross-section of the tube is uniform, the mass conservation equations for the gas phase and liquid phase are listed respectively [38], ...
As a flexible-scale energy storage technology, underwater compressed gas energy storage is an emerging enabler of the transition to renewable and sustainable energy structure. Liquid accumulation often occurs in the process of underwater pipeline gas transmission, posing a challenge that must be overcome. In this study, an experimental investigation is presented to investigate the flow characteristics of liquid in a hilly-terrain tube with zero net liquid flow. By adjusting the gas velocity, pipe inclination, liquid accumulation volume, and measuring the expansion length of the liquid film in the horizontal pipe, the liquid accumulation movement is divided into three development stages, which are related to the emergence of the liquid slug. The liquid slug in the pipe is moved by absorption, consolidation, and outflow. The movement of the liquid slug is captured by a high-speed camera, and the formation mechanism of the liquid slug is divided into three categories. When the gas flow rate is low, a liquid slug is formed due to wave growth and convective extrusion. The liquid slug produced by wave merging occurs when the gas velocity is high. A theoretical model of the slugging velocity is established. The relationship between maximum growth factor, liquid film thickness, and gas velocity is analyzed. Furthermore, the critical slugging velocity is determined. The model predictions agreed with the experimental measurement results, with a maximum error of less than 10%. This work can help in furthering underwater compressed air energy storage technology.
... Most recently, based on cap-66 tured images, Saini and Banerjee, (2021 (a)) [35] reported the mechanism of onset of slug flow 67 through wave (plunging) breaking. 68 Some literature ([20]; [27]; [28]) reported models for development of slug flow from stratified-69 wavy flow using instability analysis based on inviscid and viscous theory. Since inviscid linear 70 stability theory neglects the inertia effect on formation of slug flow, the results over predicted 71 the transition line. ...
Slug is produced when the roll waves on the liquid surface undergo constructive interference and the resulting wave height increases beyond the critical height. Slug flow is associated with large aeration that develops a sudden surge in pressure and associated mechanical impact on pipe wall and bends. Such phenomena lead to flow accelerated corrosion (FAC) which results in degradation of pipe thickness with time resulting in cracks from where the gas and liquid leaks. Hence, to reduce the chances of failure in industrial pipes that are mostly opaque (flow pattern cannot be visualized) and associated hazard in nuclear power plant industries, it is important to develop mechanisms for detecting the occurrence of slug flow. The present research proposes a methodology for recognizing the onset of slug in opaque industrial pipes using the pressure signals measured from the pipeline. Recurrence analysis of pressure signal establishes different kinds of recurrence plots for stratified-wavy flow and slug flow regime. Also, the transition can be identified using the recurrence quantification parameters. The quantified parameters like recurrence rate, determinism, and entropy show sharp jump in value at the onset of slug from wavy regime.
... The instability that leads to viscous solitons fall in the second category, to which we restrict our attention in the following. A central issue in this problem is the relevance of the Kelvin-Helmholtz instability mechanism: While it is well established that this mechanism does not describe the wave generation in the low-viscosity case, including in the air-water configuration, it is is usually considered as the relevant mechanism in the large-viscosity case [15,16,[19][20][21][22]. The reason for this somewhat paradoxical result is that although the liquid viscosity affects the growth rate of the instability, it has no effect on the critical wind velocity nor on the most unstable wavelength, which remain governed by the inviscid Kelvin-Helmholtz predictions. ...
When wind blows at the surface of a liquid of sufficiently high viscosity, a wave packet of small amplitude is first generated, which sporadically forms large-amplitude fluid bumps that rapidly propagate downstream. These nonlinear structures, first observed by Francis [Philos. Mag. 42, 695 (1954)], have an almost vertical rear facing the wind and a weak slope at the front. We call them viscous solitons. We investigate their dynamics in a wind-tunnel experiment using silicon oil of kinematic viscosity 1000 mm 2 s −1 by means of laser sheet profilometry and particle image velocimetry. We give evidence of their subcritical nature: they are emitted in a region of large shear stress but, once formed, they are sustained by the wind and propagate in a region of lower stress. Their propagation velocity is given by the balance between aerodynamic drag in the air and viscous drag in the liquid. The stable soliton branch of the subcritical bifurcation diagram is reconstructed from the measured soliton amplitude at various wind velocities and distances along the channel. At large wind velocity, the emission frequency of solitons increases, resulting in a long-range screening of downstream mature solitons by newly formed upstream solitons, which limits their course.
... The instability that leads to viscous solitons fall in the second category, to which we restrict our attention in the following. A central issue in this problem is the relevance of the Kelvin-Helmholtz instability mechanism: While it is well established that this mechanism does not describe the wave generation in the low-viscosity case, including in the air-water configuration, it is is usually considered as the relevant mechanism in the large-viscosity case [17,18,[21][22][23][24]. The reason for this somewhat paradoxical result is that although the liquid viscosity affects the growth rate of the instability, it has no effect on the critical wind velocity or on the most unstable wavelength, which remain governed by the inviscid Kelvin-Helmholtz predictions. ...
When wind blows at the surface of a liquid of sufficiently high viscosity, a wave packet of small amplitude is first generated, which sporadically forms large-amplitude fluid bumps that rapidly propagate downstream. These nonlinear structures, first observed by Francis [Philos. Mag. 42, 695 (1954)], have an almost vertical rear facing the wind and a weak slope at the front. We call them viscous solitons. We investigate their dynamics in a wind-tunnel experiment using silicon oil of kinematic viscosity 1000 mm^2/s by means of laser sheet profilometry and particle image velocimetry. We give evidence of their subcritical nature: they are emitted in a region of large shear stress but, once formed, they are sustained by the wind and propagate in a region of lower stress. Their propagation velocity is given by the balance between aerodynamic drag in the air and viscous drag in the liquid. The stable soliton branch of the subcritical bifurcation diagram is reconstructed from the measured soliton amplitude at various wind velocities and distances along the channel. At large wind velocity, the emission frequency of solitons increases, resulting in a long-range screening of downstream mature solitons by newly formed upstream solitons, which limits their course.
... 1). В предположении изотермичности процесса и отсутствия массообмена, его динамика определяется законами сохранения массы и количества движения, записанными отдельно для газа и жидкости [6,7,8]: ...
The paper is devoted to numerical modeling of flow of gas-liquid flux in horizontal and inclined tubes by one-dimensional model. By the help of developed numerical algorithm the parametric study of transition from stratified flow of water-air flux to the slug flow is carried out. In the case of slug regime the influence of tube inclination on the length and frequency of liquid slugs is studied.
A numerical framework for rigorous linear stability analysis of two-phase stratified flows of two immiscible fluids in horizontal circular pipes is presented. For the first time, three-dimensional disturbances, including those at the interface between two fluids, are considered. The proposed numerical framework is based on a finite volume method and allows solving the problem numerically in bipolar cylindrical coordinates. In these coordinates, both the pipe wall and the unperturbed interface (of a constant curvature, e.g., plane interface, as considered in this work) coincide with the coordinate surfaces. Thereby, the no-slip as well as the interfacial boundary conditions can be imposed easily. It also enables investigation of the local behavior of the flow field and shear stresses in the vicinity of the triple points, where the interface contacts the pipe wall. The results obtained in the bipolar coordinates are verified by an independent numerical solution based on the problem formulation in Cartesian coordinates, where the pipe wall is treated by the immersed boundary method. Two representative examples of gas–liquid and liquid–liquid flows are included to demonstrate the applicability of the proposed numerical technique for analyzing the flow stability.
Graphical abstract
The present work reports an experimental characterization of linear and weakly nonlinear interfacial waves in stratified air-water horizontal pipe flow. An oscillating paddle was employed to generate controlled waves at the liquid interface. The driving signal of the oscillating paddle was controlled and synchronized with image acquisitions, enabling phase-locked measurements and the application of ensemble averaging techniques. Velocity field measurements in the liquid and gas phases were performed simultaneously using an off-axis Particle Image Velocimetry (PIV) set-up and Shadowgraphy. The combined techniques allowed us to extract the coherent part of flow fluctuations related to the excited waves. This was done for a range of flow rates and wave frequencies. The selected conditions are close to the transition from stratified to slug/plug flow regimes. In the presence of linear waves, the coherent disturbances in both phases were weakly dependent of near-wall disturbances. Flow changes in the presence of weakly nonlinear waves were also investigated. In these cases, noticeable modifications in the mean flow and in turbulence distribution were observed near the interface whereas close to the wall the flow was weakly affected. This investigation follows the work of Farias et al. (2023), where the threshold for linear and weakly nonlinear waves was studied. Here, a clear comparison between wave-induced disturbances in linear and weakly nonlinear regimes is reported in the literature for the first time for stratified turbulent gas-liquid pipe flows. The methodology proposed is relatively simple and can contribute to describe wave-related phenomena in stratified pipe flows.
The two-fluid model is widely used to describe two-phase flows in complex systems such as nuclear reactors. Although the two-phase flow was successfully simulated, the standard two-fluid model suffers from an ill-posed nature. There are several remedies for the ill-posedness of the one-dimensional (1D) two-fluid model; among those, artificial viscosity is the focus of this study. Some previous works added artificial diffusion terms to both mass and momentum equations to render the two-fluid model well-posed and demonstrated that this method provided a numerically converging model. However, they did not consider mass conservation, which is crucial for analyzing a closed reactor system. In fact, the total mass is not conserved in the previous models. This study improves the artificial viscosity model such that the 1D incompressible two-fluid model is well-posed, and the total mass is conserved. The water faucet and Kelvin-Helmholtz instability flows were simulated to test the effect of the proposed artificial viscosity model. The results indicate that the proposed artificial viscosity model effectively remedies the ill-posedness of the two-fluid model while maintaining a negligible total mass error.
This study investigates the slugging characteristics of the gas-liquid slug flow interface in horizontal pipes. Using air and water as the experimental media, an experimental system was established using double-parallel conductance probes in a pipe with an inner diameter of 5 cm. By capturing the transient development process of the gas-liquid interface, the slugging characteristics of the gas-liquid two-phase flow interface in different flow regions were revealed. The results show that the value of gas-phase superficial velocity has an important influence on the shape and development of the interface wave during the slugging process. When the gravity wave generated during the slugging process can propagate upstream, the slugging phenomenon is periodic, and when the gravity wave cannot propagate upstream, the slugging phenomenon is random. The experiment verified the correctness of the interface instability theory and the liquid slug stability theory, and clarified the definitions of h o and h s. In addition, the paper analyzed the influence of gas-liquid velocity on slugging distance, h o and h s, and liquid slug frequency.
We report hydrodynamic interaction between droplets and the interface of a pair of co-flowing immiscible streams in a microchannel revealing droplet migration and interfacial deformation. We find small droplets of confinement ratio, i.e., ratio of drop size to the suspending stream width, β<1, exhibit lateral migration—while smaller droplets of β<0.5 migrate towards the co-flow interface, unexpectedly larger droplets of 0.5<β<1 drift away from the interface. The size-based contrasting migration behavior is attributed to the interplay between the hitherto unexplored wall-directed negative lift force and the well-established center-directed noninertial lift force. We also find large droplets of β>1 cause deformation waves in an initially stable and flat interface that propagate downstream akin to traveling peristaltic waves. Numerical simulations reveal that interfacial deformation is a consequence of the distinctive pressure jumps across the co-flow interface at the upstream and downstream of a droplet. Our study reveals the amplitude of the deformation wave associated with a droplet grows spatially downstream but remains the same for a train of droplets crossing a fixed location, indicating convective instability. The effects of co-flow interfacial tension, viscosity contrast, confinement ratio, and droplet spacing on the deformation wave amplitude and its variation along the flow direction are studied. The experimental results are verified using a simple theoretical model. Our study presents an unexplored droplet-driven interfacial deformation wave that may find relevance in droplet microfluidics.
We investigate analytically the effect of thermal conduction on the Kelvin-Helmholtz instability (KHI) in a pipeline with different cross-sections. The results show that the relative tangential velocity of the interface between the upper and lower fluid in the pipe increases first and then decreases with the increase of the wave number. Furthermore, the smaller coefficient of interfacial heat conduction causes the larger decrease in the relative tangential velocity with the increase of the wave number, different from the straight pipeline with the same cross-section. In addition, the heat conduction increases the growth rate of KHI, in accordance with the straight pipeline with the same cross-section.
The description of geophysical granular flows, like avalanches and debris flows, is a challenging open problem due to the high complexity of the granular dynamics, which is characterized by various momentum exchange mechanisms and is strongly coupled with the solid volume fraction field. In order to capture the rich variability of the granular dynamics along the avalanche depth, we present a well-posed multilayer model, where various layers, made of the same granular material, are advected in a dynamically coupled way. The stress and shear-rate tensors are related to each other by the μ(I) rheology. A variable volume fraction field is introduced through a relaxation argument and is governed by a dilatancy law depending on the inertial number, I. To avoid short-wave instabilities, which are a well-known issue of the conditionally hyperbolic multilayer models and also of three-dimensional models implementing the μ(I) rheology, a physically based viscous regularization using a sensible approximation of the in-plane stress gradients is proposed. Linear stability analyses in the short-wave limit show the suitability of the proposed regularization in ensuring the model well-posedness and also in providing a finite cutoff frequency for the short-wave instabilities, which is beneficial for the practical convergence of numerical simulations. The model is numerically integrated by a time-splitting finite volume scheme with a high-resolution lateralized Harten–Lax–van Leer (LHLL) solver. Numerical tests illustrate the main features and the robust numerical stability of the model.
We report the effects of head loss, surface tension, viscosity and density ratio on the Kelvin-Helmholtz instability (KHI) in two typical pipelines, i.e., straight pipeline with different cross-sections and bend pipeline. The dynamic governing equations for upper and lower fluids in the two pipes are solved analytically. We find in the straight pipeline with different cross-sections that the relative tangential velocity of fluid decreases with the increase of the head loss, viscosity and density ratio of upper and lower fluids, but it increases with the surface tension; the amplification factor decreases with the increase of the head loss and surface tension but increases with the density ratio of upper and lower fluids; the higher the height of fluid interface is, the more both the relative tangential velocity of fluid and the amplification factor are depressed. In the bend pipeline, the critical tangential velocity of fluid is found to decrease with the increase of the head loss, viscosity and density ratio of upper and lower fluids, but it increases with the surface tension; the amplification factor increases with the head loss and density ratio of upper and lower fluids, but it decreases with the increase of the surface tension; when the elbow angle is close to 80∘, the head loss reaches its maximum. The results provide guidance for pipeline design and theoretical prediction for flooding velocity in different types of tubes.
Gas–liquid two-phase flow is commonly observed in petroleum and chemical industries. Stratified flow pattern is simpler form of gas–liquid two-phase flow in which higher density fluid flows under the lower density fluid with non-disturbed interface. Small alteration of flow rate, chemical and physical properties of phases and pipe orientation or geometry lead to transition from stratified to wavy and then intermittent flow. Intermittent flow is associated with sudden pressure surge, erosion-corrosion and fatigue stress in the pipeline. This causes pipe failure at bend, T- and I-sections. Such failures lead to hazards and economic losses for the industries. It is required to develop a realistic approach for predicting the transition of such patterns in order to avoid intermittent flow inside pipe. In current study, transition from stratified to wavy and then intermittent flow patterns has been identified experimentally using recurrence network analysis of recorded pressure fluctuations for different flow conditions. The recorded time series of instantaneous pressure fluctuations have been analyzed using traditional recurrence quantification and recurrence network analysis. The two coefficients, recurrence rate and entropy, have been used for differentiating the dynamics between these two-phase flow patterns.
The critical transition conditions from the stratified smooth flow to the 2-D wave and K-H wave flow in a horizontal pipe are analyzed on the basis of the viscous linear stability analysis in consideration of the interfacial shear stress. By interpreting the available experimental data in water-air and water-carbon dioxide flow, it is found that the wavelength at the transition conditions is between the one specified at the neutral stability and the one at the maximum amplification factor. Based on this fact, the wavelength is then correlated with four dimensionless parameters. To further evaluate the feasibility of the correlation, the experiments on the liquid nitrogen-vaporous nitrogen horizontal flow are conducted. The calculated results agree reasonably well with the experimental data in this work and the water-air data available in the literature. Furthermore, the effects of the liquid viscosity and the surface tension on the transition condition are also briefly discussed.
The evolution of interfacial waves is investigated for a stratified laminar-laminar flow in a plane channel using numerical simulations based on the Volume of Fluid (VOF) method. Different nonlinear instability mechanisms that can promote saturation or rapid amplification of interfacial waves are investigated. Controlled disturbances are introduced at the interface between the two fluids, to assess five different scenarios of nonlinear interactions including harmonic excitation, subharmonic resonance, interaction between a short and a long wave, and two kinds of modulated wavepackets. Present simulations suggest that the same non-resonant type of wave interaction dominates the nonlinear stages of evolution in all the scenarios, for the fluid properties and parameters covered in this work. This mechanism involves the amplification of harmonics by quadratic nonlinearities and leads to the finite-amplitude saturation of interfacial waves, which results in a wavy flow. Reduced order models based on interaction of few wavelengths and described by the Stuart–Landau equation are proposed for the prediction of the nonlinear flow dynamics of laminar liquid-liquid two-phase flows
The linear stability of stratified two-phase flows in rectangular ducts is studied numerically. The linear stability analysis takes into account all possible infinitesimal three-dimensional disturbances and is carried out by solution of the associated eigenproblem. The neutral stability boundary and the corresponding critical wave number are obtained for liquid – liquid and air – water systems. Depending on the problem parameters, the instability sets in owing to short, intermediate, or long wave most unstable perturbations. Patterns of the most unstable disturbances are reported and discussed. It is shown that the instability arises due to shear, or interfacial mechanisms. Effects of the surface tension and of width/height aspect ratio are also studied. The results support the premise that the stability analysis of stratified two-phase flow in the simpler geometry of two-infinite plates can provide a reasonable estimation of the conditions for which this flow pattern can be considered to be linearly stable.
The linear stability of stratified two-phase flows in rectangular ducts is studied numerically. The linear stability analysis takes into account all possible infinitesimal three-dimensional disturbances and is carried out by solution of the associated eigenproblem. The neutral stability boundary and the corresponding critical wave number are obtained for liquid - liquid and air - water systems. Depending on the problem parameters, the instability sets in owing to short, intermediate, of long wave most unstable perturbations. Patterns of the most unstable disturbances are reported and discussed. It is shown that the instability arises due to shear, or interfacial mechanisms. Effects of the surface tension and of width/height aspect ratio are also studied. The results support the premise that the stability analysis of stratified two-phase flow in the simpler geometry of two-infinite plates can provide a reasonable estimation of the conditions for which this flow pattern can be considered to be linearly stable.
The flow regime for two-phase flow plays an important role in nuclear safety analysis, because different physical models for two-phase flow are used depending on the flow regime. This paper deals with the flow regime criteria for horizontal stratification, which frequently occurs in the hot or cold leg during a loss-of-coolant accident (LOCA). Although there are several horizontal stratification criteria based on the Kelvin-Helmholtz theory, the condition for the onset of waves leading to a wavy-stratified flow or a non-stratified flow is still unclear. This study approached the horizontal stratification criteria from the perspective of viscous/inviscid flow instability theories. The criterion between smoothly stratified flow and transition region was adopted with viscous Kelvin-Helmholtz (VKH) analysis. The onset of ill-posedness for the governing equation set, which is related to the inviscid Kelvin-Helmholtz (IKH) analysis, was used to determine the transition criterion between the transition region and non-stratified flow. These criteria were incorporated into the SPACE thermal-hydraulic system code. Simulation results show good agreements with experimental data, and the predictability of horizontal stratification is thereby further improved.
When wind blows at the surface of a liquid of sufficiently high viscosity, a wave packet of small amplitude is first generated, which sporadically forms large-amplitude fluid bumps that rapidly propagate downstream. These nonlinear structures, first observed by Francis [J. R. D. Francis, Philos. Mag. 42, 695 (1954)], have an almost vertical rear facing the wind and a weak slope at the front. We call them viscous solitons. We investigate their dynamics in a wind-tunnel experiment using silicon oil of kinematic viscosity 1000mm2s−1 by means of laser sheet profilometry and particle image velocimetry. We give evidence of their subcritical nature: They are emitted in a region of large shear stress but, once formed, they are sustained by the wind and propagate in a region of lower stress. Their propagation velocity is given by the balance between aerodynamic drag in the air and viscous drag in the liquid. The stable soliton branch of the subcritical bifurcation diagram is reconstructed from the measured soliton amplitude at various wind velocities and distances along the channel. At large wind velocity, the emission frequency of solitons increases, resulting in a long-range sheltering of downstream mature solitons by newly formed upstream solitons, which limits their course.
Oil and gas operators rely on accurate flow rate measurements to optimize production and generate more from their reservoirs, particularly in wet gas fields. A cost-effective solution for these flow measurements is the use of single-phase measurement technologies with an over-reading correction that corrects the gas flow rate for the presence of the liquid phase. Traditional flow measurement technologies in wet gas fields are Venturi meters and orifice plate meters that involve differential pressure measurements. Over the years, a higher installed base of ultrasonic flow meters is observed in wet gas fields. Ultrasonic flow meters have advantages over conventional wet gas technologies; however, an over-reading correction method for this measurement technology has not yet been derived. The current work is a first attempt to devise a correction method based on a large data set of ultrasonic measurements in horizontal configuration at conditions comparable to field applications. The correction method is a physical model for the gas void fraction and is based on the dominant dimensionless numbers in wet gas flows that originate from the fundamental equations of multiphase flow dynamics. This approach leads to the definition of the over-reading correction in different flow regimes in terms of these dimensionless numbers and is supported by an extensive set of measurement data and evidence from visual observation of the flow patterns. The correction method is capable of correcting the ultrasonic over-reading with a resulting uncertainty of about 4% for a 95% confidence interval for a range of conditions relevant to the oil and gas industry.
Graphical abstract
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For gas condensate flow at high pressure in a large diameter pipe theoretical and experimental studies were carried out to determine the transition from a stratified to a non stratified flow regime. The theoretical study reveals that for the high pressure situation the transition according to a one dimensional linear wave stability analysis differs markedly from that of the widely used Taitel-Dukler flow regime map. The one dimensional theory satisfactorily describes the transition to non stratified flow measured in a horizontal 8 inch pipe with natural gas and live condensate at a pressure of 75 bar. Experimentally, at the theoretical transition to unstable waves, stratified flow shift to a flow regime featuring pipe wall wetting and liquid entrainment, while intermittent flow occurs at higher liquid loadings. (A)
In a closed channel, where the presence of the top wall enhances the pressure variation over the waves, the Kelvin-Helmholtz instability becomes important, perhaps not so much for the initial formation of waves, but rather as the cause for the instability of high waves as they approach the top of the channel. This instability is considered responsible for the formation of slugs. In order to understand properly the onset of slugging, it is necessary to study the behavior of waves as they approach the Kelvin-Helmholtz instability. This paper then represents a study of various characteristics of high waves in a closed channel with particular consideration of their proximity to the instability.
It is proposed in this work that the transition to slug flow occurs due to Kelvin-Helmholtz instability, which, in this case, is enhanced by the proximity of the upper wall and becomes wave-amplitude dependent. Since the surface waves possess a limiting amplitude, the transition can be predicted by examining whether the highest possible waves are unstable. The theoretical prediction is in good agreement with the authors’ experimental results. It also agrees reasonably well with Baker’s and Schicht’s flow pattern charts for strictly horizontal channels, but it exhibits large differences when the channels deviate somewhat from the horizontal.
When air flows concurrently with a thin liquid film in an enclosed channel, a number of different wave forms are generated depending, principally, on the velocity of the air and the flow rate of the liquid. This paper summarizes progress that has been made in explaining these transitions. The approach taken is to solve the linear momentum equations to determine whether small amplitude wavelike disturbances at the interface will grow or decay. In carrying out this analysis it is convenient to consider separately the gas and liquid flows.
The stability of steady-state cocurrent and countercurrent gas—liquid annular flow is considered. Transient formulation, based on the two-fluid model, results in a Kelvin—Helmholtz type of instability. Thus, this formulation is not suited for analyzing the steady-state stability of annular flow, since steady-state annular flow is inherently unstable with respect to the interfacial structure. Various ways of transient formulations which neglect the Kelvin—Helmholtz contribution to instability are proposed, resulting in a simple and useful criterion for the stability of steady-state annular flow.
Experimental measurements of flow patterns for gas-liquid flow in inclined pipes are reported. The results compare well with a recently published theory for the prediction of flow patterns in horizontal and inclined pipes (Taitel & Dukler 1976).
A linear stability analysis, which takes into account the shear stresses, is presented and applied to the stability of stratified flow at various angles of inclinations. A combined model which uses the viscous and the inviscid Kelvin—Helmholtz stability analyses is suggested for the determination of the transition to slug and annular flows. The current results are compared with the simpler Taitel and Dukler model.
A criterion for the onset of a slug flow in a horizontal duct is derived theoretically. A potential flow analysis is carried out by considering waves of finite amplitude. The stability criterion is obtained by introducing the wave deformation limit and the ″most dangerous wave″ concept in the stability analysis. The present theoretical criterion for slug formation shows very good agreement with a large number of experimental data and with some empirical correlations.
Air flowing over a liquid surface encounters an increased resistance if waves are present. The relation of this increased resistance to the properties of the waves has been studied. Air and a liquid flowed co-currently in an enclosed channel which is 12 in. wide and 1 in. high and which is long enough so that flow in the air and the liquid and the interfacial structure are fully developed. The drag on interfaces with three-dimensional wave structures was found to increase with the square of the gas velocity and to depend more on the height of the waves than on other parameters characterizing the interface. The ratio of the equivalent sand roughness to the root-mean-square of the fluctuations in the height of the liquid film is approximately equal to 3 [surd radical]2. The velocity profiles in the gas were found to be different from what has been reported for flows over sand roughened surfaces.
A semi‐mechanistic model for two phase gas‐liquid slug flow proposed recently by Dukler and Hubbard has been modified and extended to apply to the entire intermittent flow regime. Flow predictions of the model proposed in this paper are compared with detailed experimental data recently obtained for an air‐oil system.
The model requires the use of empirical correlations for the slug velocity and the in situ liquid volume fraction in the slug.
In addition, either the slug frequencies or length corresponding to the given design conditions must be known. However, calculated values of average pressure gradient and in situ liquid volume fraction are relatively insensitive to these latter parameters, and in fact, good results are obtained assuming a constant slug length.
The paper includes a discussion of the limitations of the proposed model and the expected direction of further study required to extend its mechanistic aspects.
Measurements of the effect of liquid viscosity on the initiation of roll waves in a horizontal gas-liquid flow are presented. These results are interpreted by an analysis based on the calculation of the growth of long wavelength disturbances.
Models are presented for determining flow regime transitions in two-phase gas-liquid flow. The mechanisms for transition are based on physical concepts and are fully predictive in that no flow regime transitions are used in their development. A generalized flow regime map based on this theory is presented.
Experiments were conducted with air and liquid flowing in horizontal pipelines, 2.52 and 9.53 cm dia, to determine the interfacial instabilities that exist in a stratified flow. The liquid viscosity was varied from 1 to 80 cP. Three types of instabilities are defined: regular 2-D waves are associated with pressure variations in phase with the wave slope, irregular large-amplitude waves and atomization of the liquid are associated with pressure variations in phase with the wave height. Linear stability theory is used to provide a physical interpretation and to predict conditions for the initiation of these instabilities.
A one-dimensional wave model for incompressible flows predicts the transition between the stratified and slug flow regimes in pipes. The one-dimensional wave theory contains less empiricism than the commonly used Taitel-Dukler model for this transition. The empiricism embodied in the Taitel-Dukler analysis leads to the underprediction of the transition velocity at high gas density in large pipes in particular. This paper presents a complete solution methodology for the one-dimensional wave approach for this transition and validates the method by comparison with a wide range of flow regime data at large pipe diameters, at high gas density and in horizontal or inclined pipes. The analysis is extended, by using the method of characteristics, to model wave growth, decay and interaction. Since all waves usually propagate downstream we are led to question the Taitel-Dukler model for slug frequency and to suggest that the inlet characteristics, including compliance, play a role.
The effect of liquid viscosity on the initiation of slug flow was studied in horizontal 2.52 and 9.53 cm pipelines. The results show the stabilizing effect of viscosity predicted by Lin & Hanratty, and are at variance with analyses which use a long-wavelength inviscid approximation. For very viscous liquids a stability analysis which recognizes that slugs originate from a train of small-wavelength sinusoidal waves seems consistent with the measurements.
New results are presented on interfacial patterns observed for air and water flowing in horizontal 2.54 and 9.53 cm pipelines close to atmospheric conditions. This work differs from previous studies in that measurements of pressure fluctuations at two locations separated in the streamwise direction are used to detect slugs. The liquid flow needed to initiate slugs at low gas velocities is strongly affected by pipe diameter and appears to depend on a linear instability. At high gas velocities the transition is approximately independent of pipe diameter and is explained by a nonlinear mechanism associated with the coalescence of roll waves. The initiation of slugs in the annular flow regime is determined to occur at much lower liquid flows than had been reported by previous investigators. The transition from stratified to annular flow is different in smaller-diameter pipes than in larger pipes because wave wetting plays a more important role.
This paper explores the application of linear stability theory to explain the onset of slugging. It is shown that the inviscid Kelvin-Helmholtz theory correctly predicts stability of a stratified flow only for very large liquid viscosities. In general, however, inviscid theory is in error because it ignores the destabilizing effect of liquid inertia. Good agreement is noted between the linear stability analysis and observations of the initiation of slugs in 2.54- and 9.53-cm horizontal pipes at superficial gas velocities less than 3.3 m/s.
Intermittent two phase flow in horizontal pipes: predictive models A model for prediction of flow regime transitions in horizontal and near horizontal gas-liquid flow
- M K Nicholson
- K Aziz
- G A Gregory
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- Y Taitel
- A E Dukler
NICHOLSON, M. K., AZIZ, K. & GREGORY, G. A. 1978 Intermittent two phase flow in horizontal pipes: predictive models. Can. J. Chem. Engng 56, 653~63. TAITEL, Y. & DUKLER, A. E. 1976 A model for prediction of flow regime transitions in horizontal and near horizontal gas-liquid flow.,,fiChE Jl 22, 47-55.
Effect of pipe diameter on the interfacial configurations for air-water flow in horizontal pipes
- Lin