Flow pattern transition,pressure gradient,hold-up predictions in gas/non-Newtonian power-law fluid stratified flow

This work focuses on gas/non-Newtonian power-law fluid stratified pipe flow. Two different theoretical approaches to obtain pressure gradient and hold-up predictions are presented: the steady fully developed two-fluid model and the pre-integrated model. The theoretical predictions are compared with experimental data available for horizontal and for slightly downward inclined air/shear thinning fluid stratified flow taken from literature. The predictions of the pre-integrated model are validated showing a good agreement when compared with experimental data. The criteria for the transition from the stratified flow pattern are applied to gas/non-Newtonian stratified flow. The neutral stability analysis (smooth/wavy stratified flow) and the well-posedness (existence region of stratified flow) of governing equations are carry out. The predicted transition boundaries are obtained using the steady fully developed two-fluid model and the pre-integrated model, where the shape factors and their derivatives are accounted for. A comparison between the predicted boundaries and experimental flow pattern maps is presented and shows a good agreement. A comment on the shear stress modeling by the pre-integrated model is provided.     

Two-phase flow modelling: the closure issue for a two-layer flow

The fully-developed, laminar flow of two fluid layers in a horizontal channel is studied by means of the height-averaged balance equations. The closure issue is addressed and the closure relations for the wall and interfacial shear stresses are given for some particular cases.     

On the Squire's transformation for stratified two-phase flows in inclined channels

The applicability of the Squire's transformation for stability analysis of stratified two-phase flow in horizontal and inclined channels is examined. It is shown that for the considered flow such a transformation requires some additional constraints on the change of the inclination angle and flow rates of each of the phases. While the Squire's theorem (on the two-dimensionality of the critical disturbances) rigorously holds for the horizontal two-phase flow, for the inclined flow an exact mathematical theorem cannot be formulated. Nevertheless, it has been proven that 2D perturbations are the critical ones also for the case of inclined channel, since the transformation of a 3D stability problem to its 2D analog is associated with a stabilizing effect of reducing the system inclination, in addition to the reduction of the phases flow rates as in the case of horizontal flows.     

Non-newtonian liquid-air stratified flow through horizontal tubes—ii

Non-Newtonian liquid gas stratified flow data were obtained using 0.052 and 0.025 m dia horizontal circular ducts. Unless the liquid velocity was very low, the flow pattern generally observed was non-uniform stratified flow having an interfacial level gradient between the two phases. The Heywood-Charles model is valid for predicting the pressure drop and liquid holdup in pseudoplastic (shear thinning) non-Newtonian liquid-gas uniform stratified flow. Two-phase drag reduction, which is predicted by the Heywood-Charles model did not occur because there was a transition to semi-slug flow before the model criteria were reached. Interfacial liquid and gas shear stresses were compared.     

Exact solutions of core-annular laminar inclined flows

An exact solution for laminar two-phase eccentric core-annular flows (CAF) in inclined pipes is derived. This solution complements the exact solutions that were obtained for inclined stratified flows with curved interfaces as to provide a set of solutions for two-phase laminar separated flows. A unified set of three dimensionless parameters for separated flows is defined and used to explore the effects of the system parameters and separated flow configurations on the velocity profiles and the resulting holdup, pressure gradient and pumping power requirement in horizontal and inclined concurrent and countercurrent flows. It is shown that similarly to stratified flows, also in CAF multiple solutions for the holdup and the associated flow characteristics can be obtained in inclined flows. The boundaries of the multiple solution regions are mapped and the effect of the core eccentricity and system parameters boundaries are demonstrated and discussed.The benefits of adding a lubricating phase for transportation of a viscous fluid in inclined CAFs is investigated. An adverse effect of the upward pipe inclination on the power savings in all of the separate flow configurations is demonstrated. Independently of the density of the lubricant, namely, whether it is lighter or heavier than the viscous fluid, the effect of hydrostatic pressure gradient always hinders the possibility of reducing the pumping requirement for transporting the viscous phase. However, surprisingly, a heavier lubricant is preferable form the view point of power saving. The implications of turbulent flow of the lubricating phase and the susceptibility to Ledinegg instability on the potential power savings are also considered and discussed. The application of the model for the analysis of experimental data of the holdup and pressure drop obtained in horizontal and inclined CAF is also demonstrated.     

Studies on two-phase co-current air/non-Newtonian shear-thinning fluid flows in inclined smooth pipes

In this work, co-current flow characteristics of air/non-Newtonian liquid systems in inclined smooth pipes are studied experimentally and theoretically using transparent tubes of 20, 40 and 60 mm in diameter. Each tube includes two 10 m long pipe branches connected by a U-bend that is capable of being inclined to any angle, from a completely horizontal to a fully vertical position. The flow rate of each phase is varied over a wide range. The studied flow phenomena are bubbly flow, stratified flow, plug flow, slug flow, churn flow and annular flow. These are observed and recorded by a high-speed camera over a wide range of operating conditions. The effects of the liquid phase properties, the inclination angle and the pipe diameter on two-phase flow characteristics are systematically studied. The Heywood–Charles model for horizontal flow was modified to accommodate stratified flow in inclined pipes, taking into account the average void fraction and pressure drop of the mixture flow of a gas/non-Newtonian liquid. The pressure drop gradient model of Taitel and Barnea for a gas/Newtonian liquid slug flow was extended to include liquids possessing shear-thinning flow behaviour in inclined pipes. The comparison of the predicted values with the experimental data shows that the models presented here provide a reasonable estimate of the average void fraction and the corresponding pressure drop for the mixture flow of a gas/non-Newtonian liquid.     

One-dimensional two-fluid equations for horizontal stratified two-phase flow

Advanced computer codes for water reactor loss-of-coolant analysis are based on the use of the two-fluid model of two-phase flow, in which conservation equations are solved for the gas and liquid phases separately. The standard two-fluid equations, however, sometimes predict the growth of instabilities in the flow, and occasionally become improperly posed. These difficulties have in the past led to the proposal of several different forms for the conservations equations.To help resolve these uncertainties a widely accepted form of the one-dimensional two-fluid equations is used to calculate wave propagation speeds, and stability limits, for the illustrative case of a frictionless horizontal stratified gas-liquid flow. Calculated propagation velocities are shown to agree with the appropriate limit of an exact solution, and the predicted stability limits are found consistent with available observations on the stability of the stratified flow regime.These comparisons help improve confidence in the ability of the two-fluid equations to analyse more complex problems in transient two-phase flow.     

Effects of bottom shear stresses on the wave-induced dynamic response in a porous seabed：PORO-WSSI （shear） model

When ocean waves propagate over the sea floor,dynamic wave pressures and bottom shear stresses exert on the surface of seabed.The bottom shear stresses provide a horizontal loading in the wave-seabed interaction system,while dynamic wave pressures provide a vertical loading in the system.However,the bottom shear stresses have been ignored in most previous studies in the past.In this study,the effects of the bottom shear stresses on the dynamic response in a seabed of finite thickness under wave loading will be examined,based on Biot’s dynamic poro-elastic theory.In the model,an "u-p" approximation will be adopted instead of quasi-static model that have been used in most previous studies.Numerical results indicate that the bottom shear stresses has certain influences on the wave-induced seabed dynamic response.Furthermore,wave and soil characteristics have considerable influences on the relative difference of seabed response between the previous model(without shear stresses) and the present model(with shear stresses).As shown in the parametric study,the relative differences between two models could up to 10% of p0,depending on the amplitude of bottom shear stresses.     

Numerical simulation of the onset of slug initiation in laminar horizontal channel flow

Results are presented for the initiation of slug-type structures from stratified 2D, two-layer pressure-driven channel flow. Good agreement is obtained with an Orr–Sommerfeld-type stability analysis for the growth rate and wave speed of very small disturbances. The numerical results elucidate the non-linear evolution of the interface shape once small disturbances have grown substantially. It is shown that relatively short waves (which are the most unstable according to linear theory) saturate when the length of the periodic domain is equally short. In longer domains, coalescence of short waves of small-amplitude is shown to lead to large-amplitude long waves, which subsequently exhibit a tendency towards slug formation. The non-uniform distribution of the interfacial shear stress is shown to be a significant mechanism for wave growth in the non-linear regime.     

On the interfacial roughness scale in turbulent stratified two-phase flow: 3D lattice Boltzmann numerical simulations with forced turbulence and surfactant

Numerical 3D simulations of turbulent, stratified two-phase shear flow with a surfactant laden interface were used to test and develop a phenomenological interfacial roughness scale model where the energy required to deform the interface (buoyancy, interfacial tension, and viscous work) is proportional to the turbulent kinetic energy adjacent to the interface.The turbulence was forced in the upper and lower liquids in the simulations, to emulate the interfacial dynamics without requiring (prohibitively) large simulation domains and Reynolds numbers. The addition of surfactant lead to an increased roughness scale (for the same turbulent kinetic energy) due to the introduction of interfacial dilatational elasticity that suppressed horizontal motion parallel to the interface, and enhanced the vertical motion.The phenomenological roughness scale model was not fully developed for dilatational elasticity in this work, but we proposed a source term that represents surfactant induced pressure fluctuations near the interface. This source term should be developed further to account for the relation between surfactant density fluctuations and turbulence adjacent to the interface. We foresee that the roughness scale model can be used as a basis for more general interfacial closure relations in Reynolds averaged turbulence models, where also mobile surfactant is accounted for.     

An accurate model for the thin film flow

This paper deals with the linear stability of a liquid film flowing down an inclined plane. The Navier-Stokes equations were reduced into four evolution equations that describe the development of the film depth, the flow rate, the free surface velocity, and the wall shear stress, using the Karman-Polhausen boundary layer integral method. Thus, we were able to determine the stability threshold and approach well the critical wave number for long waves. The obtained results were found to be in good agreement with the experiments of Liu et al.     

Nonlinear convection induced by inclined thermal and solutal gradients with mass flow

The energy method is developed for the convection problem induced by inclined thermal and solutal gradients, with horizontal mass flow, in a horizontal layer of a saturated porous medium. A non-linear stability analysis is performed and compound matrix method is employed for numerical calculations. For representative parameter values the critical vertical thermal Rayleigh number and wave number are calculated. It is noted that the effect of horizontal thermal or solutal gradient is to switch from stabilizing to destabilizing as their magnitude increases, for zero or small values of mass flow rate. For higher values of mass flow rate the effect is always destabilizing. It is also noted that the horizontal concentration gradient has a stronger destabilizing effect as compared to the horizontal temperature gradient. Received March 18, 2000     

Generation of Periodic Motions by a Disk Performing Torsional Oscillations in a Viscous, Continuously Stratified Fluid

The solution of the problem of nonlinear generation of periodic internal waves by a boundary flow on a vertical cylinder or a horizontal disk performing torsional oscillations in an exponentially stratified fluid is constructed. The calculations are in satisfactory agreement with the results of experiments in which both horizontal and inclined disks of various diameters and a model propeller performing periodic torsional oscillations, including oscillations against a background of uniform rotation, are used as perturbation sources. The experiments were carried out over a wide range of parameters including the laminar, transition, and turbulent flow regimes. The limits of applicability of the proposed analytic theory of wave radiation are determined.     

Numerical modelling of cavities on axisymmetric bodies at zero and non-zero angle of attack

The flow about submerged, fully cavitating axisymmetric bodies at both zero and non-zero angle of attack is considered in this paper. A cavity closure model that relates the point of detachment, the angle that the separating streamline makes with the body and the cavity length is described. The direct boundary element method is used to solve the potential flow problem and to determine the cavity shape. A momentum integral boundary layer solver is included in the formulation so that shear stresses can be incorporated into the drag calculations. The numerical predictions based on the proposed closure model are compared with water tunnel measurements and photographs.     

Calculation and measurement of conical beams of three-dimensional periodic internal waves excited by a vertically oscillating piston

The structure of the wave field given by an exact solution of the linearized problem of radiation of three-dimensional periodic internal waves in a continuously stratified viscous fluid is analyzed numerically. The waves are generated by a piston, i.e., a disk lying on a fixed horizontal plane and oscillating in the vertical direction. The flow fields and the wave displacements are compared with the data of shadow visualization and measurements of the wave amplitudes made using a contact sensor. The calculated and observed wave patterns are in satisfactory agreement and the displacement distributions coincide correct to a fitting coefficient 0.7 < K < 1.1 characterizing the role of the nonlinear effects and other factors neglected in this model.     

Determination of yield stress fluid behaviour from inclined plane test

The aim of this paper is to determine precisely under which conditions an inclined plane can be used as a rheometer, which could represent a practical and rapid technique for various types of industrial or natural viscoplastic coarse suspensions. We first examine its efficiency and relevancy for determining fluid yield stress in a straight way by measuring the deepest fluid layer able to stay on the inclined plane. We have made experiments with different materials (clay-water suspensions) whose yield stress ranged from 35 to 90 Pa, using 1 m long open rectangular channels with a slope ranging from 10 to 30° and a width ranging from 5 to 25 cm. Our procedure involved measuring the final fluid depth far from edges a long time after the end of the slow gravity-induced emptying of a dam placed upstream. The fluid yield stress was also estimated independently by fitting a Herschel-Bulkley model to simple shear rheometry data obtained within a relatively wide shear rate range. A good agreement between inclined rectangular channel tests and independent usual rheometrical tests is obtained even for aspect ratios (flow depth to channel width ratio) as large as 1 when one assumes that, when the fluid has stopped, the side and bottom wall shear stresses are equal to the fluid yield stress. These results prove the efficiency of the inclined plane test for determining yield stress when appropriate experimental precautions are taken for both tests. In addition we examine the possibility of determining the simple shear flow curve of a mud suspension from fluid depth, velocity and discharge measurements of different steady flows in a wide open channel (8 m long; 60 cm wide) equipped with a recirculating system. The results obtained from inclined plane tests are in good agreement with independent rheometrical data (with torsional geometries). However it is technically difficult to cover a wide shear rate range from the inclined plane technique since this requires a rather wide channel flow rate range.     

Linear Dynamics of Perturbations in Flows with Constant Horizontal Shear

The dynamics of perturbations in shallow water and incompressible stratified fluid flows with constant horizontal shear are described using the nonmodal analysis. It is shown that the shear flow perturbations can be divided into two classes on the basis of the potential vorticity: rapidly oscillating wave perturbations with zero potential vorticity and slow vortex perturbations with nonzero potential vorticity. In the cases of weak and strong shear the main features of the dynamics of wave and vortex perturbations are studied analytically (using the WKBG method) and numerically. It is shown that for large times the wave perturbation energy increases linearly, i.e., the shear flow is algebraically unstable due to the growth of rapid wave perturbations. This instability can be of importance in processes of turbulence development and surface and internal wave generation.     

Long waves between a subsonic gas flow-film interface flowing down an inclined plate

The present work deals with temporal stability properties of a falling liquid film down an inclined plane in the presence of a parallel subsonic gas flow. The waves are described by evolution equation previously derived as a generalization of the model for the Newtonian liquid. We confirm linear stability results of the basic flow using the Orr–Sommerfeld analysis to that obtained by long wave approximation analysis. The non-linear stability criteria of the model are discussed analytically and stability branches are obtained. Finally, the solitary wave solutions at the liquid–gas interface are discussed, using specially envelope transform and direct ansatz approach to Ginzburg–Landau equation. The influence of different parameters governing the flow on the stability behavior of the system is discussed in detail.     

Direct Numerical Simulation of Turbulence in a Stably Stratified Fluid and Wave-Shear Interaction

Turbulence decay in a strongly stratified medium is simulated by a direct pseudo-spectral code solving the three-dimensional equations of motion under the Boussinesq approximation. The results are compared to non-stratified simulations results. We focus on the production of mean shear energy observed in the stratified case. We then simulate the decay of stratified turbulence when affected by an initial horizontal mean flow and show that this mean flow is the major component remaining at large t. Next, we give some analytical elements on wave-shear interaction by using a simple refraction calculation with WKB hypothesis. This calculation is illustrated by simulating the interaction between one monochromatic internal wave and a vertical shear profile. We conclude that the existence of singularities in the mean shear production term in the presence of internal gravity waves may be one of the possible mechanisms involved within stratified turbulent shear flows.     

Pore pressure generation in a saturated sand layer subjected to a cyclic horizontal acceleration at its base

A compaction theory for saturated sand, of differential type, is applied to shear wave propagation through a layer induced by cyclic horizontal acceleration of its base. The compaction relation incorporates dependence on the pore fluid pressure, and in turn the pore pressure is related to the compaction through the elastic compression relations. An assumption of undrained flow, which will yield the maximum pore pressures, allows straightforward numerical solution of the simplified dynamic equations. A variety of results are computed for comparison with alternative predictions.