Roll Waves Research Articles

Coherent structures, self-similarity, and universal roll wave coarsening dynamics

Beyond their inception region from wide-banded inlet noise, roll waves on an inclined plane increase their amplitude and separation downstream in a scale-invariant linear manner by as much as one order of magnitude. Analysis of a set of dissipative hydraulic equations shows that this self-similar coarsening dynamics is driven by irreversible coalescence events between localized roll waves. By exploiting the localized structure of the roll waves and certain symmetries of the hydraulic equations to simple affine coordinate stretching by the substrate thickness, the time-averaged wave properties at every station and wave texture is shown to be self-similar. Moreover, due to weak variations in the substrate thickness and the sensitivity of the coarsening dynamics to it, the downstream evolution along this self-similar family can be captured by a generic cascade coalescence model for all wave separations. The linear coarsening rate of the average wave period is shown to be roughly (√−1) times the fraction of “excited” waves, which is only a weak function of a modified Froude number G based on the channel inclination angle.

Numerical analysis on heat transfer enhancement by waves on falling liquid film

Numerical simulations have been carried out for two dimensional wavy falling liquid films on a vertical wall. The algorithm of the simulation is based on MAC method and schemes for interfacial boundary conditions are modified. Small artificial perturbations given at the inflow boundary grow rapidly and then the amplitude of the waves approaches to developed waves. Effects of the disturbance frequency on the wave development behavior and heat transfer characteristics are especially investigated. For low frequency, a disturbance wave develops to a solitary wave consisted of a large amplitude roll wave and small amplitude capillary waves. Increasing the frequency, the wave amplitude decreases and the capillary wave disappears. For further high frequency, the disturbance amplitude reduces along down stream. The heat transfer coefficient is enhanced by the surface wave and has a maximum at a certain frequency. The streamlines and the temperature contours are shown for various frequency waves and the heat transfer enhancement mechanism is clarified.

Numerical analysis for a falling liquid film with interfacial waves on an inclined plate. Part 2: Effects of interfacial waves on flow dynamics and heat transfer

Flow and temperature fields in falling liquid films with interfacial waves have been obtained by means of a numerical simulation in which the algorithm is based on the HSMAC method and interfacial boundary conditions are treated with newly proposed methods. Small-amplitude disturbance waves at a low frequency develop into isolated solitary waves which are composed of a roll wave and capillary waves. Waves disturbed at a high frequency interfere with each other and develop into disordered waves. Circulation flow is observed in the roll wave, while there is no circulation flow in the disordered waves. Temperature fields in the wavy film are distorted by convection effects and differ greatly from those in the laminar film. The circulation in the roll wave has an especially strong effect on the temperature fields. The interfacial waves enhance the heat transfer by two kinds of effects: the variation of film thickness and convection in the film. The dominating effect depends on the Prandtl number. © 2000 Scripta Technica, Heat Trans Asian Res, 29(3): 233–248, 2000

Numerical analysis for a falling liquid film with interfacial waves on an inclined plate. Part 2: Effects of interfacial waves on flow dynamics and heat transfer

Flow and temperature fields in falling liquid films with interfacial waves have been obtained by means of a numerical simulation in which the algorithm is based on the HSMAC method and interfacial boundary conditions are treated with newly proposed methods. Small-amplitude disturbance waves at a low frequency develop into isolated solitary waves which are composed of a roll wave and capillary waves. Waves disturbed at a high frequency interfere with each other and develop into disordered waves. Circulation flow is observed in the roll wave, while there is no circulation flow in the disordered waves. Temperature fields in the wavy film are distorted by convection effects and differ greatly from those in the laminar film. The circulation in the roll wave has an especially strong effect on the temperature fields. The interfacial waves enhance the heat transfer by two kinds of effects: the variation of film thickness and convection in the film. The dominating effect depends on the Prandtl number. © 2000 Scripta Technica, Heat Trans Asian Res, 29(3): 233–248, 2000

Mechanism of slug formation in downwardly inclined pipes

This paper examines the effect of small downward inclinations on the formation of slugs. Experiments were conducted with air and water at atmospheric pressure, in a pipe with a diameter of 0.0763 m, a length of 23 m and inclinations of −0.2, −0.5 and −0.8°. Measurements of the variation of the interfacial displacement were made simultaneously at a number of locations. For low gas velocities in a horizontal configuration waves with lengths of 16–20 cm, grow until they reach of the top of the pipe. These waves evolve from smaller wavelength waves (8–10 cm) through a non-linear growth mechanism. At high gas velocities, the liquid height is not large enough for this mechanism to be operable. In these cases slugs evolve from the coalescence of roll waves. Surprisingly, the large amplitude small wavelength waves observed in horizontal flows, at the transition to slug flow, are damped in pipelines that are inclined slightly downward. The transition is associated with the initiation of long wavelength, small amplitude waves, whose appearance is predicted by a viscous long wavelength linear stability analysis. A local Kelvin–Helmholtz instability at the crest of a growing long wavelength wave is observed when a slug forms. The frequency of slugging is equal to frequency of these long wavelength waves.

Results from a two-dimensional turbulent diffusion-model for dispersion and deposition of droplets in horizontal annular dispersed gas/liquid flow

A previously used one-dimensional Turbulent Diffusion-model is extended to two dimensions. A quasi-stationary annular dispersed gas/liquid flow is simulated by adding the contributions for different particle sizes of a large number of annular line sources from roll wave tops in the upstream direction. One or more local maxima in the droplet concentration are predicted, if droplet sources are present all along the tube wall. A local maximum in the concentration has been found in experiments as well, but it was always solely attributed to the presence of a secondary gas flow. The droplet deposition flux as calculated in the two-dimensional model shows good agreement with a previously derived analytical expression for the droplet deposition flux. Finally we derive how the two-dimensional deposition flux depends on the Stokes- and Froude-numbers. For an arbitrary Froude-number it is predicted droplets up to which Stokes-number are able to deposit at the top of the tube.

Hyperbolic Systems with Supercharacteristic Relaxations and Roll Waves

We study a distinguished limit for general 2 × 2 hyperbolic systems with relaxation, which is valid in both the subcharacteristic and supercharacteristic cases. This is a weakly nonlinear limit, which leads the underlying relaxation systems into a Burgers equation with a source term; the sign of the source term depends on the characteristic interleaving condition. In the supercharacteristic case, the problem admits a periodic solution known as the roll wave, generated by a small perturbation around equilibrium constants. Such a limit is justified in the presence of artificial viscosity, using the energy method.

The structure of roll waves in two-layer flows

The structure of non-linear waves in a two-layer flow of an incompressible fluid in extended channels is investigated. Periodic discontinuous solutions, describing roll waves of finite amplitude, are constructred for the equations of two-layer shallow water. “Anomalous” waves of limited amplitude are found which correspond to the transition from stratified to slug flow conditions.

2436 R-113 垂直管外等温流下液膜 : 最小濡れ膜流量と液膜上波の特性について

The minimum wetting rate (MWR) of the isothermal R-113 falling film was examined. The amplitude, frequency, velocity and wave length on the film were also investigated. The MWR was not obtained even at extremely small flow rate of 0.38cc/min (Re_f=0.70). That flow rate was much smaller than the value expected for the smooth surface film although the surface observed was flat. In Re_f<&acd;300,waves were symmetrical between the front and the back side and behaved like theoretical small-perturbation waves which was postulated on the smooth-laminar film. When Re_f>&acd;300,the waves grew as proceeding to downstream, began to behave like roll waves and the characteristics deviated from those of the small perturbation waves.

Long-wavelength Cloud Rolls Over the Arctic Ice

Cloud bands observed over the Arctic sea ice are interpreted in terms of long-wavelength gravity-wave modes trapped between the ground and a vertically-thin over-reflecting layer in the mid-troposphere. The over-reflecting layer is neutrally-stratified and lies between the counter-streaming upper troposphere and lower troposphere in a strong high-pressure system. The Arctic cloud bands appear to share many features with the long-wavelength roll waves observed in more temperate latitudes.

Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state

Unsteady flow in collapsible tubes has been widely studied for a number of different physiological applications; the principal motivation for the work of this paper is the study of blood flow in the jugular vein of an upright, long-necked subject (a giraffe). The one-dimensional equations governing gravity- or pressure-driven flow in collapsible tubes have been solved in the past using finite-difference (MacCormack) methods. Such schemes, however, produce numerical artifacts near discontinuities such as elastic jumps. This paper describes a numerical scheme developed to solve the one-dimensional equations using a more accurate upwind finite volume (Godunov) scheme that has been used successfully in gas dynamics and shallow water wave problems. The adapatation of the Godunov method to the present application is non-trivial due to the highly nonlinear nature of the pressure–area relation for collapsible tubes.The code is tested by comparing both unsteady and converged solutions with analytical solutions where available. Further tests include comparison with solutions obtained from MacCormack methods which illustrate the accuracy of the present method.Finally the possibility of roll waves occurring in collapsible tubes is also considered, both as a test case for the scheme and as an interesting phenomenon in its own right, arising out of the similarity of the collapsible tube equations to those governing shallow water flow.

Numerical analysis on flow dynamics and heat transfer of falling liquid films with interfacial waves

Flow dynamics and heat transfer of falling liquid films with interfacial waves flowing on a vertical plate have been studied with originally proposed numerical simulation method. To discretize basic equations a staggered grid fixed on a physical space is employed. A small amplitude disturbance generated at inflow boundary develops to a solitary wave which consists of a large amplitude roll wave and small amplitude capillary waves. Instantaneous streamwise velocity profiles at the wave crest and trough are very different from a laminar flow. A circulation flow occurs in the roll wave and it affects temperature distributions, especially the strong effect is observed for high Prandtl number liquids. The interfacial wave enhances the heat transfer by two kinds of effects which are a film thinning effect and a convection effect. The dominating effect depends on the Prandtl number.

Roll Waves in High Gradient Channels

Vedernikov's number quantifies the allowable capacity that can be sustained as a stable uniform flow in a high gradient channel, but it does not provide any guidance to the design of freeboard when roll waves exist. Without considering roll waves, high gradient channels are undersized using the uniform flow approach and fail to maintain design integrity. This paper presents revised design curves to determine the stability of flow in high gradient channels. When the design condition exceeds the limiting condition, roll waves should be considered in design. This study applies the model of moving hydraulic jump to simulate roll waves. It provides estimations for wave heights when Froude number is greater than 1.50.

Two-Phase Zero-Net Liquid Flow in Upward Inclined Pipes: Experiment and Modeling

Summary The effect of pressure on two-phase zero-net liquid flow (ZNLF) in upward inclined pipes has been studied experimentally and theoretically. Experimental data were acquired for four upward inclination angles, namely, 1,2, 5, and 9°, and for three system pressures of 14.7, 44.7, and 64.7 psia. A sequence of flow configuration was observed with increasing gas flow rate as follows: slug flow for low superficial gas velocities; protoslug for medium velocities; and pulsating, small-amplitude waves or roll waves at high velocities. At a particular velocity of gas, termed the blowout velocity, all the liquid was blown out of the pipe. An analysis of equilibrium stratified flow, at the limit of very low and zero liquid flow rates, has been carried out. It was found that, under these conditions, the flow is always unstable, resulting in intermittent or pulsating stratified flow. A model has been developed for the prediction of the blowout velocity, as well as average liquid holdup and pressure gradient under ZNLF conditions. A very good agreement was observed between the predictions of the proposed model and the experimental data.

A numerical study of interfacial transport to a gas-sheared wavy liquid

Interfacial transport of a passive scalar from a gas to a thin liquid film is considered. The base flow in the liquid is laminar, with linear velocity profile produced by gas shear. The velocity disturbances imposed by waves are simulated by an inviscid, constant-vorticity model. Numerically computed roll waves are shown to enhance interfacial transport by modifying the convection pattern below the crest and by inducing a mixing effect on the substrate. It is argued that the entire wave spectrum is active in this enhancement of interfacial transport. † Tel.: 30 421 86226. Fax: 30 421 69787.

Characteristics of endwall and sidewall boundary layers in a rotating cylinder with a differentially rotating endwall

The flow in a rotating cylinder driven by the differential rotation of its top endwall is studied by numerically solving the time-dependent axisymmetric Navier–Stokes equations. When the differential rotation is small, the flow is well described in terms of similarity solutions of individual rotating disks of infinite radius. For larger differential rotations, whether the top is co-rotating or counter-rotating results in qualitatively distinct behaviour. For counter-rotation, the boundary layer on the top endwall separates, forming a free shear layer and this results in a global coupling between the boundary layer flows on the various walls and a global departure from the similarity flows. At large Reynolds numbers, this shear layer becomes unstable. For a co-rotating top, there is a qualitative change in the flow depending on whether the top rotates faster or slower than the rest of the cylinder. When the top rotates faster, so does the bulk of the interior fluid, and the sidewall boundary layer region where the fluid adjusts to the slower rotation rate of the cylinder is centrifugally unstable. The secondary induced meridional flow is also potentially unstable in this region. This is manifested by the inflectional radial profiles of the vertical velocity and azimuthal vorticity in this region. At large Reynolds numbers, the instability of the sidewall layer results in roll waves propagating downwards.

Potential Effects of Translatory Waves on Estimation of Peak Flows

During the afternoon of August 19, 1971, an intense thunderstorm a few miles southwest of Wikieup, Arizona, produced one of the largest known flood peaks for a 49.2-square-km drainage basin. Initial computations of the peak discharge assumed stable flow conditions and a four-section slope area measurement indicated that discharge was 2,082 m3/s. Recent findings based on free-surface instability characteristics at the site suggest that gravitational forces exceeded boundary retarding forces, and flow in the wide sand channel was unstable. Computations for roll or translatory waves indicate that waves crashed into the highway bridge at velocities of as much as 12.5 m/s. The close agreement of free surface instability results, translatory wave computations, estimates of the steady flow on which the translatory waves traveled, and an eyewitness account of the translatory waves suggest the total peak discharge could have been 2,742 m3/s or 32% greater than the published discharge. The occurrence of translatory...

Prediction in stratified gas-liquid co-current flow in horizontal pipelines

Abstract The predictive performance of existing models based on momentum balances has been shown to be generally unsatisfactory. An iterative procedure was developed for the prediction of pressure drop and holdup that incorporated new relationships for the interfacial and liquid friction factors in the solution of the phase momentum balance equations for two phase horizontal co-current flow. The method adequately predicted data for the film plus droplet, annular roll wave and stratified type regimes. Successful performance was achieved regardless of fluid properties or pipe diameter. For gas flow rates, where non-uniform stratified flow with an interfacial level gradient occurred, this method of prediction was inaccurate and open channel flow theory recommended. Thus the model did not apply to the intermittent or non-uniform flow regimes.

Numerical Analysis for a Falling Liquid Film with Interfacial Waves on an Inclined Plate. 2nd Report. Effects of Interfacial Waves on Flow Dynamics and Heat Transfer.

Flow and temperature fields in falling liquid films with interfacial waves have been obtained by means of numerical simulation in which the algorithm is based on the HSMAC method and interfacial boundary conditions are treated with newly proposed methods. Small amplitude disturbance waves at a low frequency develop into isolated solitary waves which are composed of a roll wave and capillary waves. Waves disturbed at a high frequency interfere with each other and develop into disordered waves. Circulation is observed in the roll wave, while there is no circulation in the disordered waves. Temperature fields in the wavy film are distored by the convection effects and differ greatly from those in the laminar film. The circulation in the roll wave has an especially strong effect on the temperature fields. The interfacial waves enhance the heat transfer by two kinds of effects : the variation of the film thickness and the convection in the film. The dominating effect depends on the Prandtl number.

Numerical Analysis for a Falling Liquid Film with Interfacial Waves on an Inclined Plate. 1st Report. Numerical Methods for Boundary Counditions at the Free Surface.

Numerical simulations have been carried out for a two-dimensional falling liquid film with interfacial waves. The staggered grid fixed on a physical space is employed and basic equations, continuity and Navier-Stokes equations, as well as equations for boundary conditions are discretized on the staggered grid without omitting any term in the equations. Neglecting the interfacial shear stress is only an assumption for the present analysis. New numerical methods are proposed for the interfacial boundary conditions and for the change of the film thickness, derived from the continuity equation. Simulation results agree well with experimental results in which waves are formed by artificial periodic disturbances. For both the simulation and the experiment, the developed waves disturbed at low frequency have a structure combined of a roll wave and capillary waves, while there is no capillary wave for high-frequency disturbance. The adyantages of the proposed methods are also presented.