An analytical study on interfacial wave structure between the liquid film and gas core in a vertical tube

https://doi.org/10.1016/j.ijmultiphaseflow.2004.03.002Get rights and content

Abstract

A model has been derived for interfacial wave propagation for a liquid film on the wall of a vertical pipe and for a flowing gas in the central core. An analytical study is presented for the stability of a flat interface, and for traveling waves on the interface. Long wave theory is applied to both phases and the resulting conservation equations are of the same form as a two-fluid model. Two situations are examined: the interface between a Taylor bubble and the liquid film, where the gas velocity is small, and the interface for cocurrent annular flow where the gas velocity is relatively large. The interface between a Taylor bubble and a liquid film was found to be dominated by waves, which can be destabilized by the inertia of the liquid phase. For annular flow the interface is subject to a Kelvin–Helmholtz instability. When the gas flow rate is small, and surface tension is negligible, the traveling wave has a shape similar to that of a Taylor bubble except near the tip and trailing edge. When surface tension is dominant, the solution is a soliton. This region and the receding part of the soliton appears to be related to the ripple waves seen near the trailing edge of Taylor bubbles.

Introduction

It is important to understand the mechanisms of wave propagation on the interface between a liquid film on the wall of a conduit and the gas in a central core since many transient and steady-state two-phase flow phenomena are controlled by the dynamics of the interface. Examples include interface waves in annular dispersed flow, flow transition mechanisms between slug flow and annular flow, the interface between a Taylor bubble and the liquid film, the shape of Taylor bubbles, the entrainment of droplets in annular flow, and so on. Wave propagation solutions can also provide closure models for multi-dimensional, two-fluid, two-phase flow calculations, such as for interfacial area density and interfacial force density.

Fukano et al. (1985) analyzed the disturbance waves on a liquid film induced by gas flow in a horizontal rectangular duct. They showed that the region where disturbance waves are generated corresponds to the region where dynamic waves dominate kinematic waves, and the region where ripple waves are generated corresponds to the region where the kinematic waves dominate dynamic waves. However, they did not consider a wavy interface in a vertical channel.

Zabaras and Dukler (1986) measured the instantaneous local film thickness, wall shear stress, and pressure gradient for upward cocurrent gas-liquid annular flow, but they did not perform a thorough analysis of the dynamics of the wavy interface.

Dressler (1949) considered a roll wave in an inclined open channel. Discontinuous periodic solutions were constructed by joining together sections of a continuous solution through continuity shocks (i.e., “bores”). Neither gravity nor the effect of the gas phase was considered by Dressler. The gas phase does not affect the interface characteristics very much for Taylor bubbles, however, gravity may be important. Fukano et al. (1980) analyzed the shape of a Taylor bubble in a vertical channel using coordinates moving with the Taylor bubble. However, the effect of gas flow on the traveling wave was not included in this study.

Tilley et al. (1994) analyzed nonlinear long-waves in an inclined channel and derived a modified Kuramoto–Sivashinsky equation, which can exhibit chaotic phenomena. This theory is based on the work of Benney (1966), who applied long wave theory to a liquid film. However, the theory is based on the lubrication approximation, and is applicable only for very small liquid flow velocities in an inclined channel.

Void wave theory (Lahey, 1992) and flooding theory in annular two-phase flow (Fowler and Lisseter, 1992) can be very useful in the analysis of a wavy interface because the basic equations of the interface obtained using long wave theory are the same as the two-fluid model equations. Thus the method which can be used to analyze the wavy interface is similar to that used for void waves and flooding.

The wavy structure of an annular flow is important and many papers on this subject have been published recently. For example, Zapke and Kroger (2000), Vlachos et al. (2001), Vijayan et al., 2001, Vijayan et al., 2002, and Mouza et al. (2002) investigated flooding phenomena while Bugg et al. (1998), Polonsky et al. (1999), and Van Hout et al. (2002) considered the velocity field around a Taylor bubble. The interfacial shear stress and frictional pressure drop was investigated for annular flow by Fukano and Furukawa (1998), Fore et al. (2000), and Hajiloo et al. (2001). Waves on a falling liquid film were considered by Karimi and Kawaji (1999), Adomeit and Renz (2000), Takamasa and Kobayshi (2000), and Amvrosini et al. (2002), however, they did not consider the stability mechanisms of the wavy interface in detail.

The analysis given herein presents a theoretical foundation for the effects of interfacial stability, and considers the wave forms associated with traveling waves. In particular, the stability of a flat interface, as well as traveling waves, is considered by applying long wave theory to both phases. If the gas in the central core is assumed to have a relatively low velocity, the wavy interface between a Taylor bubble and the liquid film, and the Taylor bubble shape, can be predicted. For annular flow, the characteristics of the wavy interface and the application of long wave theory were considered when the gas velocity was relatively high. It should be noted that the long wave approximation which was used in this paper implies that the wave lengths are much larger than the liquid film thickness. In analyzing the behaviors of the resultant equations, a “short wave” length approximation was also made which implies that the wave length is on the order of the pipe diameter. These two approaches are fully consistent for the high void fractions annular flows that have been analyzed herein.

In Chapter 2, the basic equations, including interfacial and wall friction factors, are discussed. Linear stability of the interface is considered in Chapter 3, where the dispersion relation is investigated. In Chapter 4, a steady traveling wave is obtained for small curvature and surface tension dominant flows.

Section snippets

Application of shallow wave theory to both phases

As shown in Fig. 1, axisymmetric motion of the interface between two incompressible fluids in a vertical circular tube is considered. Phase-1 is a liquid film along the tube wall, and phase-2 is the gas flowing in the core. The radius of the tube is assumed to be R0. The conservation equations that govern this system are the continuity equation and the Navier–Stokes equation for each phase·uj=0,ρjujt+uj·uj=−pjjgj2ujwhere uj and pj are the velocity vector and pressure, and ρj and μj

Derivation of the linear equations and dispersion relation

This section summarizes the derivation of the linear equations which describe the liquid film. When the interface is disturbed from its steady-state, which is a liquid film with constant thickness, the perturbed variables are defined as follows:α=α0+ΔαandW1=W10+ΔW1W2=W20+ΔW2p1=p10+Δp1p2=p20+Δp2where, suffix 0 means the steady film flow at constant thickness.

Steady film flow at constant film thickness is described byp10z=−ξw0−ξc101Frc20ρFrwhereξw=1(1−α)λ4R0fw|W1|W1ξc1=1(1−α)(2R0

Steady traveling waves

The shape of steady traveling waves (i.e., solitons) is considered in this section. The velocity of a traveling wave is assumed to be c. A coordinate transformation is performed from stationary coordinates (t,z) to coordinates (τ,ζ), which move with traveling wave. The basic conservation equations, Eq. (9), are expressed in moving coordinates as follows:ddζ{(W1−c)(1−α)}=0,ddζ((W2−c)α)=0(W1−c)dW1dζ=−p1ζ+1Fr−ξw−ξc1ρ(W2−c)dW2dζ=−p1ζ+ρFrc21WeR0h138α−(5/2)αζ3+34α−(3/2)αζ2αζ21

Conclusions

A long wave model for the evolution of the interface between a liquid film on a vertical pipe and gas flow in the core was derived, and the results for stability of a flat interface and traveling waves were considered.

  • (1)

    When long wave theory was applied to both phases, the two-fluid model conservation equations are derived from this deterministic procedure without requiring averaging.

  • (2)

    For low gas flow rates, where Taylor bubbles could be generated, the inertia force on the gas phase and the

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      Waves appearing on the liquid film surface play important role in mass, momentum and energy heat transfer between the film and the gas core. Their nature has been investigated by many researchers both experimentally and numerically (Hall Taylor et al., 1963, 2014; Nedderman and Shearer, 1963; Hall Taylor and Nedderman, 1968; Azzopardi, 1986; Wolf et al., 1996; Inada et al., 2004; Sawant et al., 2008; Belt et al., 2010; Schubring et al., 2010; Zhao et al., 2013; Alekseenko et al., 2014, 2015; Pan et al., 2015; Yang et al., 2016). A wave that appears to roll over the surface of a liquid is called a “roll wave”.

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    This paper has been contributed to a special Festschrift edition of the International Journal of Multiphase flow (IJMF) in honor of professor George Yadigaroglu (ETH-Zurich). Dr. Yadigaroglu is a world-class multiphase researcher and was long-time editor of the IJMF. It is a pleasure to be able to honor him on the occasion of this retirement from ETH. We wish him many more happy, healthy and productive years.

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