Classification of instabilities in parallel two-phase flow

There is extensive literature on the stability of parallel two-phase flow, both in the context of liquid-liquid as well as gas-liquid flow. Aimed at making this literature more transparent, this paper presents a classification,scheme for the various instabilities arising in parallel two-phase flow. To achieve such a classification, the equation governing the rate of change of the linetic energy of the disturbances is evaluated for relevant values of the physical parameters. This shows the existence of five different ways of energy transfer from the primary to the disturbed flow, which have their origin in density stratification, velocity profile curvature, viscosity stratification or shear effects. Each class is discussed on the basis of references covering the developments over the last 35 years

Forced and Unforced Flexural-gravity Solitary Waves

Flexural-gravity waves beneath an ice sheet are investigated. Forced waves generated by a moving load as well as freely propagating solitary waves are considered for the nonlinear problem as proposed by Plotnikov and Toland [2011]. In the unforced case, a Hamiltonian reformulation of the governing equations is presented in three dimensions. A weakly nonlinear analysis is performed to derive a cubic nonlinear Schrödinger equation near the minimum phase velocity in two dimensions. Both steady and time-dependent fully nonlinear computations are presented in the two-dimensional case, and the influence of finite depth is also discussed

Finite depth effects on solitary waves in a floating ice sheet

A theoretical and numerical study of two-dimensional nonlinear flexural-gravity waves propagating at the surface of an ideal fluid of finite depth, covered by a thin ice sheet, is presented. The ice-sheet model is based on the special Cosserat theory of hyperelastic shells satisfying Kirchhoff׳s hypothesis, which yields a conservative and nonlinear expression for the bending force. From a Hamiltonian reformulation of the governing equations, two weakly nonlinear wave models are derived: a 5th-order Korteweg–de Vries equation in the long-wave regime and a cubic nonlinear Schrödinger equation in the modulational regime. Solitary wave solutions of these models and their stability are analysed. In particular, there is a critical depth below which the nonlinear Schrödinger equation is of focusing type and thus admits stable soliton solutions. These weakly nonlinear results are validated by comparison with direct numerical simulations of the full governing equations. It is observed numerically that small- to large-amplitude solitary waves of depression are stable. Overturning waves of depression are also found for low wave speeds and sufficiently large depth. However, solitary waves of elevation seem to be unstable in all cases

Fully nonlinear interfacial waves in a bounded two-fluid system

We study the nonlinear flow which results when two immiscible inviscid incompressible fluids of different densities and separated by an interface which is free to move and which supports surface tension, are caused to flow in a straight infinite channel. Gravity is taken into consideration and the velocities of each phase can be different, thus giving rise to the Kelvin-Helmholtz instability. Our objective is to study the competing effects of the Kelvin-Helmholtz instability coupled with a stably or unstably stratified fluid system (Rayleigh-Taylor instability) when surface tension is present to regularize the dynamics. Our approach involves the derivation of two-and three-dimensional model evolution equations using long-wave asymptotics and the ensuing analysis and computation of these models. In addition, we derive the appropriate Birkhoff-Rott integro-differential equation for two-phase inviscid flows in channels of arbitrary aspect ratios. A long wave asymptotic analysis is undertaken to develop a theory for fully nonlinear interfacial waves allowing amplitudes as large as the channel thickness. The result is a set of evolution equations for the interfacial shape and the velocity jump across the interface. Linear stability analysis reveals that capillary forces stabilize short-wave disturbances in a dispersive manner and we study their effect on the fully nonlinear dynamics described by our models. In the case of two-dimensional interfacial deflections, traveling waves of permanent form are constructed and it is shown that solitary waves are possible for a range of physical parameters. All solitary waves are expressed implicitly in terms of incomplete elliptic integrals of the third kind. When the upper layer has zero density, two explicit solitary-wave solutions have been found whose amplitudes are equal to h/4 or h/9 where 2h is the channel thickness. In the absence of gravity, solitary waves are not possible but periodic ones are. Numerically constructed traveling and solitary waves are given for representative physical parameters. The initial value problem for the partial differential equations is also addressed numerically in periodic domains, and the regularizing effect of surface tension is investigated. In particular, when surface tension is absent it is shown that the system of governing evolution equations terminates in a singularity after a finite time. This is achieved by studying a 2 x 2 system of nonlinear conservation laws in the complex plane and by numerical solution of the evolution equations. The analysis shows that a sinusoidal perturbation of the flat interface and a cosine perturbation to the unit velocity jump across the interface, develop a singularity at time tc = ln 1/ε+0 (ln(ln 1/ε)) where ε is the initial amplitude of the disturbances. This result is asymptotic for small ε and is derived by studying the asymptotic form of the flow characteristics in the complex plane. We also derive the analogous three-dimensional evolution equations by assuming that the wavelengths in the principal horizontal directions are large compared to the channel thickness. Surface tension is again incorporated to regularize short-wave Kelvin-Helmholtz instabilities and the equations are solved numerically subject to periodic boundary conditions. Evidence of singularity formation is found. In particular, we observe that singularities occur at isolated points starting from general initial conditions. This finding is consistent with numerical studies of unbounded three-dimensional vortex sheets (see Introduction for a discussion and references). In the final part of this work we consider the vortex-sheet formulation of the exact nonlinear two-dimensional flow of a vortex sheet which is bounded in a channel. We derive a Birkhoff-Rott type integro-differential evolution equation for the velocity of the interface in terms of the vorticity as well as the evolution equation for the unnormalized vortex sheet strength. For the case of a spatially periodic vortex sheet, this Birkhoff-Rott type equation is written in terms of Jacobi\u27s functions. The equation is shown to recover the limits of unbounded and non-periodic flows which are known in the literature

Some aspects of dispersive horizons: lessons from surface waves

Hydrodynamic surface waves propagating on a moving background flow experience an effective curved space-time. We discuss experiments with gravity waves and capillary-gravity waves in which we study hydrodynamic black/white-hole horizons and the possibility of penetrating across them. Such possibility of penetration is due to the interaction with an additional "blue" horizon, which results from the inclusion of surface tension in the low-frequency gravity-wave theory. This interaction leads to a dispersive cusp beyond which both horizons completely disappear. We speculate the appearance of high-frequency "superluminal" corrections to be a universal characteristic of analogue gravity systems, and discuss their relevance for the trans-Planckian problem. We also discuss the role of Airy interference in hybridising the incoming waves with the flowing background (the effective spacetime) and blurring the position of the black/white-hole horizon.Comment: 29 pages. Lecture Notes for the IX SIGRAV School on "Analogue Gravity", Como (Italy), May 201

Investigation of long-wave model equations for a liquid CO2-seawater interface in deep water

Masteroppgave i anvendt og beregningsorientert matematikkMAB39

One-dimensional modelling of mixing, dispersion and segregation of multiphase fluids flowing in pipelines

The flow of immiscible liquids in pipelines has been studied in this work in order to formulate a one-dimensional model for the computer analysis of two-phase liquid-liquid flow in horizontal pipes. The model simplifies the number of flow patterns commonly encountered in liquid-liquid flow to stratified flow, fully dispersed flow and partial dispersion with the formation of one or two different emulsions. The model is based on the solution of continuity equations for dispersed and continuous phase; correlations available in the literature are used for the calculation of the maximum and mean dispersed phase drop diameter, the emulsion viscosity, the phase inversion point, the liquid-wall friction factors, liquid-liquid friction factors at interface and the slip velocity between the phases. In absence of validated models for entrainment and deposition in liquid-liquid flow, two entrainment rate correlations and two deposition models originally developed for gas-liquid flow have been adapted to liquid-liquid flow. The model was applied to the flow of oil and water; the predicted flow regimes have been presented as a function of the input water fraction and mixture velocity and compared with experimental results, showing an overall good agreement between calculation and experiments. Calculated values of oil-in-water and water-in-oil dispersed fractions were compared against experimental data for different oil and water superficial velocities, input water fractions and mixture velocities. Pressure losses calculated in the full developed flow region of the pipe, a crucial quantity in industrial applications, are reasonably close to measured values. Discrepancies and possible improvements of the model are also discussed. The model for two-phase flow was extended to three-phase liquid-liquid-gas flow within the framework of the two-fluid model. The two liquid phases were treated as a unique liquid phase with properly averaged properties. The model for three-phase flow thus developed was implemented in an existing research code for the simulation of three-phase slug flow with the formation of emulsions in the liquid phase and phase inversion phenomena. Comparisons with experimental data are presented