Instability in turbulent stratified channel flow

Transcription

1 Under consideration for publication in J. Fluid Mech. 1 Instability in turbulent stratified channel flow L E N N O N Ó N Á R A I G H 1, O M A R K. M A T A R 1, P E T E R D. M. S P E L T 1 A N D T A M E R A. Z A K I 2 Departments of Chemical 1 and Mechanical 2 Engineering, Imperial College London, SW7 2AZ, United Kingdom Received 11 August 28) We consider the motion of a deformable interface that separates a fully-developed turbulent gas flow from a thin layer of laminar liquid. We outline a linear model to describe the interaction between the turbulent gas flow and the interfacial waves. This approach includes two steps. First, we derive a flat-interface base state. This comprises a velocity profile that takes account of the laminar sublayer present in the near-interfacial region of the gas, and a method for determining the wall and interfacial shear stresses as a function of the mean pressure gradient. The second step involves an Orr Sommerfeld analysis of the Reynolds-averaged Navier Stokes equations and necessitates the selection of a turbulent-stress closure scheme. This approach permits us to determine numerically the growth rate of the wave amplitude, as a function of the relevant dimensionless system parameters and turbulence closure relations. It also extends previous work by accounting for the effects of the thin liquid layer on the dynamics. The inclusion of this feature has a startling effect: by a judicious choice of viscosity or density contrast, or of surface-roughness parameter, we observe mode competition, in which the maximum growth rate of the instability derives its energy from internal effects in the liquid layer, rather than from interfacial effects. We have also classified the observed waves either as fast or slow, and compared our results with experimental data: excellent agreement is obtained in the slow-wave case, while only qualitative agreement is obtained for faster waves. This is consistent with our demonstration that the stationary turbulence model used is valid only for slow waves. 1. Introduction We investigate turbulent stratified flow in a horizontal channel, wherein liquid flows along the bottom wall, and the gas shears over the liquid; this situation is commonly encountered in oil-and-gas transport. A crucial modelling issue is the ability to predict whether the interface separating these fluids is stable, since an unstable interface can lead to the formation of waves, slugs and droplet entrainment, which can lead to a change in the flow regime. The framework we use has two facets: we derive a model velocity field describing the base turbulent flow in the system, and we then perform a stability study around this base state using an Orr Sommerfeld-type analysis. Before undertaking these steps, we place our work in context by outlining several studies that have been performed on similar gas-liquid systems in the past. The problem of computing the growth rate of an interfacial instability for a gas-liquid system was first considered by Phillips 1957), and by Miles 1957). Phillips studied a

2 2 L. Ó Náraigh, O. K. Matar, P. D. M. Spelt and T. A. Zaki resonant interaction between turbulent pressure fluctuations and capillary-gravity waves on an interface, while Miles focussed on a particular kind of shear inviscid flow that produces linear instability. In this paper, our focus will also be on linear stability theory. In the paper of Miles, in later works by the same author see Miles 1959, 1962), and in the work of Benjamin 1958), the liquid layer is neglected, and is replaced by a wavy interface, and the problem is reduced to computing the stress exerted on the interface by the turbulent gas flow. Energy transfer from the gas to the interface is governed by the second derivative of the mean flow. Indeed, the growth rate of the instability is determined by the sign of the second derivative of the mean flow at the critical height the height at which the mean flow and wave speed are equal. In these works, the turbulent nature of the gas flow is taken into account through the prescription of a logarithmic mean profile in the gas. However, the Reynolds stress terms that enter into the stability equations are ignored. This problem is rectified by Duin & Janssen 1992), and by Belcher and co-workers in a series of papers see Belcher & Hunt 1993; Belcher et al. 1994; Belcher & Hunt 1998). In these works, the authors describe the interfacial stability of a sheared air-water interface they specialize to an air-water system for oceanographical applications). With the exception of the paper of Belcher et al. 1994), the authors treat the interface in a manner similar to Miles. Furthermore, the papers by Belcher and co-workers take particular care in developing an understanding of the structure of the turbulent shear stresses inherent in the problem through the use of scaling arguments and a truncated mixing-length model. This is representative of an approximation of a Reynolds-averaged ensemble of realizations of the turbulent flow for a given phase of a small-amplitude interfacial wave. In this approach, the eddy viscosity is formulated in terms of the typical scale of a turbulent eddy, which depends on the distance between the eddy itself and the air-water interface. Far from the interface, the turbulent eddies are advected quickly over an interfacial undulation, and have insufficient time to equilibrate. In this case, rapid distortion theory as in Townsend 198) predicts that the Reynolds stresses in this region are negligible. Thus, the mixing length is truncated: it is a simple function of the vertical co-ordinate close to the interface, and is set to zero far from the interface. Using this approach, the growth rate of the interfacial disturbance is predicted. Furthermore, the papers of Belcher and co-workers identify a new mechanism of instability called non-separated sheltering, in which the turbulent stresses decelerate the air flow near the wave crest; this displaces the streamlines in an asymmetric way, which modifies the perturbed pressure, which in turn exerts a drag on the interface. This model is valid in the slow-wave limit, when the flow speed in the frame of the wavy interface) is small compared with the friction velocity U i. In a later paper, Cohen & Belcher 1998) refine this approach by introducing a damped mixing-length model, in which the simple mixing-length model transitions smoothly to zero far from the interface. They also extend the mixing-length model to a regime of fast waves. They compute the growth rate of the instability. For fast waves, the growth rate can become negative, indicating that the feedback of the turbulent air into the interface can actually damp the interfacial instability. This collection of papers gives a convincing explanation of the generation of waves by wind, and the importance of turbulent shear stress therein, although they consider only inviscid fluids. However, in the channel-flow situation we consider, and in particular, for thin liquid films, it is likely that the viscous nature of the fluids will become apparent. In this context, the wave propagation speed cannot be regarded simply as a parameter as in these studies involving capillary-gravity waves), rather it must be determined by the stability properties of the system. Furthermore, the critical-layer and sheltering

3 Journal of Fluid Mechanics 3 mechanisms discussed so far are not the only means by which an interfacial wave is generated. It is therefore salutary to consider the paper of Boomkamp & Miesen 1996), in which the authors classify interfacial instabilities according to an energy-budget scheme. Here the contributions to the time-change in kinetic energy are derived from the linearized Navier Stokes equations, and several destabilizing factors are identified. Of interest in the present application are the Miles mechanism, the viscosity-contrast mechanism, and the internal mechanism in which the laminar flow in the bulk of the liquid film becomes unstable). In this approach of Boomkamp & Miesen 1996), and in similar work by Miesen & Boersma 1995); Özgen et al. 1998); Özgen 28), the viscous liquid layer and gas layers are modelled co-equally; these added complications necessitate a numerical solution of the equation. The interfacial wave speed is then determined, along with the growth rate, as the solution of an eigenvalue problem. Turbulence enters the problem only though the logarithmic mean-flow profile chosen, and the Reynolds stress terms are ignored. Although these papers are disparate in their approaches, their common thread is the use of a temporal stability analysis, in which the temporal growth rate is obtained for a given spatial wavenumber. This does not provide a complete picture of the stability properties of the system, since in general, the base flow will exhibit a spatio-temporal instability as in Criminale et al. 23). Indeed, in many experiments, the instability is observed to develop in space and time see for example Drazin & Reid 1981). However, Chomaz et al. 1991), Huerre & Monkewitz 199), and Chomaz 25) have shown that the maximum growth rate in the temporal analysis provides a global indicator of the instability in the system and the temporal analysis discussed here is therefore relevant. Furthermore, the Orr Sommerfeld framework developed here would form the basis of a spatial analysis. In justifying our approach, we must also take account of transient-growth effects. The normal-mode analysis described here involves singling out the eigenvalue-eigenvector pair corresponding to maximal growth for special study. Recent research Trefethen et al. 1993; South & Hooper 1999; Yecko & Zaleski 25) has also focussed on the phenomenon of transient growth. In general, the eigenfunctions of Orr Sommerfeld operator are nonorthogonal and the total energy associated with a disturbance flow can therefore be enhanced over that predicted by a single-mode analysis, thus giving rise to an enhanced transient) growth rate. Nevertheless, a normal-mode analysis, together with the associated energy-budget study is still an effective way of identifying the mechanisms at work in promoting instability. It also provides the theoretical framework for the transient-growth analysis. The framework proposed here is to unify the several approaches considered so far: using the Reynolds-averaged Navier Stokes RANS) equations, we take into account the Reynolds stresses in the problem in a manner similar to Belcher et al. and, continuing the approach of Miesen et al., we pay close attention to the dynamics of the liquid layer. Since the applications we have in mind involve transport in pipelines, we also focus in detail on the problem of modelling the mean flow in a channel. The turbulent two-phase flow profile for channels introduced by Biberg 27) will be helpful in this regard. The other papers mentioned here use a boundary-layer formalism that is inappropriate, at least for very long-wave disturbances, which we demonstrate in 7.2. Some progress has already been made in developing the linear stability formalism for a channel see Kuru et al. 1995), and here we develop a more detailed model of the base flow, and include the turbulent stress terms in the perturbation equations. By accessing parameter regimes far from the air-water paradigm, we shall demonstrate that the effects of these stresses can be substantial. This paper is organized as follows. In 2 we derive the base flow-profile and introduce

4 4 L. Ó Náraigh, O. K. Matar, P. D. M. Spelt and T. A. Zaki Figure 1. A schematic diagram of the base flow. The liquid layer is laminar while the gas layer exhibits fully-developed turbulence, described here by a Reynolds-averaged velocity profile. A pressure gradient in the x-direction drives the flow. the mixing-length model used in our stability calculations. We compare our model with profiles of direct numerical simulation found in the literature. We outline the turbulent Orr Sommerfeld equation necessary to describe the stability properties of the gas-liquid interface and introduce a numerical technique for the solution. In 3, we compare the numerical results with an asymptotic solution valid in a long-wave limit. In 4 we present the results of the stability analysis, and focus on obtaining dispersion curves and neutral curves as a function of the flow Reynolds number, the viscosity and density contrasts, and the channel aspect ratio. In 6 we describe the effect of finite surface roughness on our results, while in 7 we discuss some questions that arise from our treatment of the turbulent two-phase system, in particular, the classification of the waves as slow and the adequacy of our turbulent closure scheme. We also compare our predictions with experimental data. Finally in 8 we present our conclusions. 2. Theoretical formulation 2.1. The mean flow-profile We consider the dynamics of a two-layer system in a channel, described schematically in figure 1. The bottom layer is a thin, laminar, liquid layer, while the top layer is gaseous, turbulent and fully-developed. A pressure gradient is applied along the channel. The mean profile of the system is a uni-directional flow in the horizontal, x-direction. In the gas layer, near the gas-liquid interface and the gas-wall boundary, the flow is laminar, since here the viscous scale exceeds the characteristic length scale of the turbulence. In the bulk of the gas region, the flow possesses a logarithmic profile. We assume that the gas-liquid interface is smooth; the effect of finite surface roughness will be discussed in 6. The growth rate of the wave amplitude depends sensitively on the choice of mean flow. Therefore, it is necessary to derive a mean flow-profile that incorporates the characteristics of the flow observed in experiments. In this section, we generalize the model of Biberg 27) and accurately model the laminar sublayers found in two-phase turbulent flows. The functional form of the derived velocity profile enables us to express the wall and interfacial shear stresses as functions of the mean pressure gradient. This treatment also enables us to write down a turbulent closure scheme, wherein the eddy viscosity is constituted as a simple function of the vertical coordinate, z. Using the above-mentioned assumptions and approximations, we derive the mean flow in each layer.

5 Journal of Fluid Mechanics 5 The liquid film: The Reynolds-averaged Navier Stokes RANS) equation appropriate for the liquid film is the following: 2 U L μ L z 2 p =, 2.1) x where U L and μ L denote the liquid mean flow velocity and viscosity, respectively; the pressure gradient, p/ x, contains partial derivatives because hydrostatic balance is imposed in the vertical direction, p j / z = ρ j g, in which j = L, G labels the phase. We integrate Eq. 2.1) and apply the following boundary conditions, which correspond, respectively, to continuity of tangential stress at the interface and no-slip at the channel bottom wall: U L μ L z = τ i, U L d L ) =, 2.2) z= where τ i is the interfacial stress and d L z is the domain of the liquid film. Later on we shall close the model by finding an expression for this stress in terms of the pressure gradient p/ x.) This yields the following relation for the mean flow velocity profile in the film, U L z) = 1 p z 2 d 2) + τ i z + d), d L z. 2.3) 2μ L x μ L Nondimensionalizing on the liquid scales P = μ 2 L /ρ Ld 2 L, U i = d L τ i /μ L gives the following dimensionless velocity profile where any decoration designating the use of dimensionless variables has been suppressed), U L z) = Az 2 + z + A + 1, A = d L p/ x 2τ i. 2.4) The gas laminar sublayer adjacent to the interface: The experiments of Liombas et al. 25) and the direct numerical simulations of Lombardi et al. 1995) and Solbakken & Andersson 24) suggest that it is appropriate to model the mean velocity field in the neighbourhood of a flat interface as a solid wall. Thus, the mean flow is laminar next to the interface so that U G z = τ i, 2.5) μ G where U G and μ G denote the mean flow gas velocity and viscosity, respectively. We can justify this by writing down the Reynolds-averaged Navier Stokes RANS) equation in this layer, which is given by μ G U G z + τ TSS = τ i + p z, 2.6) x where τ TSS = ρ G u w is the turbulent shear stress wherein ρ G denotes the gas density. Here, indicates the averaging, and u, w ) = u u, w w ) represent velocity fluctuations in the x and z directions, respectively. The scale on which this laminar sublayer exists is the viscous scale z i = tμ G ρ G τ i ) 1/2, where t is an order one constant in this work, we take t = 5). In the non-dimensional units discussed, the RANS equation is m U G z + τ TSS = 1 2Az, 2.7) τ i where m μ G /μ L and r ρ G /ρ L. Moreover, the scaled viscous sublayer height is given by the equation z i = mt rre L ) 1/2. 2.8)

6 6 L. Ó Náraigh, O. K. Matar, P. D. M. Spelt and T. A. Zaki This relation makes use of the liquid Reynolds number, defined as Re L = ρ LU i d L = τ i = 1 d L p μ L P 2 AP x = 1 P 2 A ; 2.9) note that we have also introduced the non-dimensional pressure gradient P = d L /P ) p/ x. For moderate pressure gradients, the term 2Az in Eq. 2.7) is negligible in the laminar sublayer. Furthermore, in this thin region, the turbulent velocity scale depends linearly on z, which is smaller than the length scale at which the viscous dissipation is important. This region z < z i is thus one in which τ TSS is unimportant and the RANS equation is then simply m U G / z) = 1. The boundary conditions are continuity of velocity and tangential stress at the interface: U L ) = U G ), U L μ L z ) = μ U G G ). 2.1) z The gas mean flow velocity in the laminar sublayer near the interface is thus U G = z m + A + 1, z z i. 2.11) The gas laminar layer near the wall: Near the wall z = h, the situation is analogous to that near the liquid interface. Thus, near the wall in non-dimensional form), the gas velocity is U G = 1 ) 1 Rm δ z, z w z 1 δ, 2.12) where δ is the channel aspect ratio d L /h, R = τ i τ w is the ratio of interfacial to wall shear stress, and where z w is defined as z w = δ 1 Rz i. 2.13) Note the non-trivial relation between the stresses τ w and τ i : they differ because of the applied pressure gradient, and two alternative integrations of the RANS equation show they are in the relation 1 2A τ i = τ w δ ) The turbulent core: In this region of the flow, the velocity profile must resemble a logarithmic function. To make progress, we introduce a new coordinate η = z z i z w z i, so that η = at the boundary between the core and the interfacial laminar viscous region, and η = 1 at the boundary between the core and the wall laminar viscous region. In this region, the eddy-viscosity model is appropriate. Thus, the TSS takes the form τ TSS = μ t U G z, 2.15) where μ t and U i, the eddy-viscosity and interfacial friction velocity, respectively, are expressed by μ t = κρ ) 1/2 GhU i τi G η), U i =, 2.16) R ρ G κ is the von Kármán constant, and where the function G η) is the product of a nondimensional mixing length and a non-dimensional turbulent velocity scale. Equation 2.16)

7 Journal of Fluid Mechanics 7 is the simplest possible turbulence closure scheme for a channel, and is a modified Prandtl zero-equation model. The dependence of the function G on the vertical co-ordinate ensures that the eddy viscosity reflects local conditions, which differ depending on whether one is close to the interface, or the wall. The paper of Biberg 27) provides the appropriate form for G η), η 1 η) η 3 + R 5/2 1 η) 3) G η) = R 2 1 η) 2 + R 1 η) η + η. 2.17) 2 This formula assumes that the interface possesses negligible surface roughness, and, with the exception of 6, it is this case that we consider. From the RANS equation μ G U G z + μ U G t z = τ i + p z, 2.18) x it follows that the velocity in the core region is ) z τ i + p x z U G z) = μ G + κρ R Ghu i + Const., 2.19) G η) z i where the constant of integration is chosen such that U G,laminar z i ) = U G,core z i ). So far we have been unable to obtain the parameter A hence the shear stress τ i ) as a function of the pressure gradient. This is now done by the matching condition U G,core z w ) = U G,laminar z w ). In nondimensional terms, this condition reads 1 z w z i ) [1 2Az i ) 2A z w z i ) η] dη m + κ δ r P η 1 η)η 3 +R 5/2 1 η) 3 ) 2 RA R 2 1 η) 2 +R1 η)η+η 2 ) +t 2A r P 1 1 R )+A+1 =. In this constraint equation for A, the parameters z i, z w, and R are themselves functions of A, and are given by 2.8), 2.13), and 2.14), respectively. The root of this equation fixes A as a function of the nondimensional parameters m, r, δ, P, κ =.41, and t = 5; this in turn fixes the interfacial stress and the liquid Reynolds number: τ i = U id L ρ L = Re L = 1 P P μ 2 L A. In summary, we have the following formula for the mean velocity profile. In the liquid, the velocity is given by U L = Az 2 + z + A + 1, 1 z, 2.2) and in the gas it is given by z m + A + 1, z z i, U G z) = z w z i ) η [1 2Az i ) 2Az w z i )η]dη z i m + A + 1, z i z z w, 1 Rm m+ κ δ r 2 P η 1 η)η 3 +R 5/2 1 η) 3 ) RA R 2 1 η) 2 +R1 η)η+η 2 ) + η= z z i zw z i 1 δ z), z w z 1 δ, 2.21) where A hence τ i /P ) is chosen so that the core velocity and the viscous sublayer velocity match at z = z w. In figure 2 we plot the relation between the pressure gradient and the Reynolds number for an air-water mixture in a channel with aspect ratio δ =.1. We then take a fixed pressure gradient P = 83 hence, Re L = 43.7), and plot the velocity field.

8 8 L. Ó Náraigh, O. K. Matar, P. D. M. Spelt and T. A. Zaki [Shear Stress]/P τ i /P =Re L =1/2)S P τ w /P =1/2)S P /R P=d /P ) p/ x) L a) z/d L Gas Layer Liquid Layer u/u i b) τ TSS z c) Figure 2. Characteristics of the base profile for fixed parameter values m, r, δ) =.18,.12,.1). a) The relation between the pressure gradient and the interfacial and wall shear stresses τ i, τ w ); b) The mean velocity profile for turbulent gas flow in the liquid-gas system at a pressure gradient P = 83 Re L = 43.7); c) The Reynolds stress profile corresponding to the basic velocity in b). The plot extends to the gas-layer midpoint z = Model DNS, Andersson and Solbakken U y + Figure 3. Comparison with the results of Solbakken & Andersson 24) for a lubricated channel. The continuous curve gives the model profile across the channel, while the dashed curve describes the DNS results. The latter results only extend to the channel midpoint. Note that for fixed m, r, δ) as given in the caption, the relation between the pressure gradient and the Reynolds number is approximately quadratic, Re L ap 2 bp, where a and b are constants. We also plot the turbulent shear stress that corresponds to the displayed velocity field. The result is in agreement with the direct numerical simulations of Solbakken & Andersson 24); Fulgosi et al. 22) Validation and discussion We compare our model with the direct numerical simulations DNS) of Solbakken & Andersson 24) for lubricated channel flow. We take δ = 1/34, m = 1/2, r = 1, and P = m2 Re 2 τ ) , Re τ = H τ w /ρ = 18 μ G 2δ to agree with their results. Here τ w is the wall shear stress and H is the channel halfdepth. Furthermore, we take y + = z mre τ d L 1 + 1, U + = u [ 1 P U )]. 2δ i mare τ 2δ The results are shown in figure 3. There is excellent agreement, in particular near the interface. Thus, our model is an adequate base state for the stability analysis we now carry out.

9 Journal of Fluid Mechanics 9.1 z c ; Uz c )=2 1 z c ; Uz c )=1.8 z i.8 z i P a) P b) Figure 4. The location of the critical layer z c and the laminar sublayer z i for different wave speeds. The parameters r and m take the values.1 and.1 respectively, which is close to their air-water values; the aspect ratio δ is set to.1. Subfigure a) shows a slow wave, for which c/u i = 2 relative to the stationary bottom). The critical layer lies below the viscous sublayer height except at very high pressure gradients. On the other hand, subfigure b) is a similar plot for a faster wave, for which c/u i = 1. Here the critical layer moves outside the laminar sublayer at small pressure gradients. Our model also gives a way of predicting when the critical-layer instability is relevant. This is the height z c associated with an interfacial wave) at which the mean velocity U z) and the wave speed c are equal. In figure 4 we plot the root z c of the curve U z) c =, for two wave speeds thought to be typical of slow and fast waves at least for m, r) =.1,.1), which is close to the air-water mixture). For the slow wave, the critical layer lies inside the laminar sublayer at all but very large pressure gradients, while for the fast wave, the critical layer moves outside of the laminar zone for small pressure gradients. This suggests that the Miles mechanism of instability is unimportant for slow waves. In 4.1 we shall demonstrate that for channel flow over a thin liquid film, the interfacial waves are slow. It is important, however, to treat figure 4 only as an indication of the kind of results we might expect, since a central message of 4 is that the wave speed is not arbitrary, but rather is determined, along with the growth rate, by an Orr Sommerfeld type of analysis Linear stability analysis In this section we outline a linear stability analysis that determines the conditions under which the interface of the two-phase system becomes unstable. For viscous, parallel, incompressible flow, the stability analysis reduces to an eigenvalue problem in a single equation the Orr Sommerfeld equation). Following standard practice see for example Landahl 1975), we generalize this approach to turbulent stratified flow by performing a stability analysis on the flat-interface Reynolds-averaged Navier Stokes RANS) equations. The base state U j z) = U j z), ), j = L, G is a parallel flow given by the model equations in 2.1, while infinitesimal perturbations to this state are described by the linearized RANS equations, r j uj t + U j u j + u j U j ) = T j + T Rey,j r j Gẑ, 2.22a) u j =, 2.22b)

10 1 L. Ó Náraigh, O. K. Matar, P. D. M. Spelt and T. A. Zaki where G = gd L /Ui 2 is the nondimensional acceleration due to gravity, and T j is the stress tensor T zz,j = p j + 2m j Re L w j z, T xx,j = p j + 2m j Re L u j x, T xz,j = T zx,j = m j Re L uj z + w ) j. x We must also constitute the perturbation Reynolds stress. In general, this is given by the tensor T Rey,j = r j u j u j T Rey,j, where u is the pre-averaged, fully turbulent velocity field, and TRey,j is the Reynolds stress in the flat-interface state, modelled in 2.1. In the gas laminar and liquid layers, the Reynolds stresses are zero, hence T Rey =. In the turbulent gas layer, we take T Rey,xx,G = T Rey,zz,G =, T Rey,xz,G = T Rey,zx,G = rκu i δ R 1/2 U i G z) u G z, 2.23) where, m G = m = μ G /μ L, r G = r = ρ G /ρ L, m L = 1, r G = 1, and where Re L is the liquid Reynolds number, Re L = ρ LU i d L μ L = ρ Lτ i d 2 L μ 2 L = τ i P. Equation 2.23) is a simple closure for the Reynolds stress terms in which the normal stresses have been neglected the effects of the normal stresses are smaller than the dominant shear stresses included here; see Cohen & Belcher 1998; Townsend 1972). The tangential stress is determined by the formula T Rey,G,xz = μ t u G / z), with the eddy viscosity μ t given by the product of a turbulent velocity scale U i and a mixing length G z). For further discussion of such an approach, see Komen et al ) We re-express these equations in terms of the streamfunction u j, w j ) = Φ j / z, Φ j / x), and the Fourier decomposition Φ j = φ j z) e iαx ct). By rewriting the wave speed in terms of its real and imaginary parts, c = c r + ic i, the exponential component of the solution can be re-expressed as e αcit e iαx crt). Thus, positivity of c i indicates a growing instability; the constant λ = αc i is the growth rate, and this depends on the problem parameters, that is, λ = λ α, r, m, δ, Re L, S cap, F r). Here S cap and F r are the inverse Weber and Froude numbers respectively, defined below in 2.28) and 2.29). Using the streamfunction representation of the velocity field, the RANS equations 2.22) reduce to a pair of coupled Orr Sommerfeld-type equations, { iαre L [U L z) c] D 2 α 2) d 2 } U L φ L φ L 2 = D 2 α 2) 2 φl, 1 z <, 2.24a) { iα rre L /m) [U G z) c] D 2 α 2) d 2 } U G φ G φ G 2 = rre L /m) D 2 + α 2) τ + D 2 α 2) 2 φg, z δ 1, 2.24b) where we have taken τ to mean T Rey,xz,G and D to mean d/.

11 Journal of Fluid Mechanics 11 Since the base state has four distinct domains liquid, interfacial gas laminar, gas turbulent, and wall gas laminar), it is appropriate to solve the Orr Sommerfeld equation in each domain, and to apply matching conditions at the domain boundaries. These boundaries are the liquid-gas interface z = +[perturbations], and the virtual interfaces at z = z i + [perturbations], and at z = z w + [perturbations]. At these interfaces, the perturbation velocities and shear stresses must be continuous, and we apply the kinematic and normal stress conditions. Furthermore, we linearize these conditions on to the flat interfaces z =, z = z i, and z = z w. Thus, at z =, the streamfunction satisfies φ L = φ G,1, φ L Dφ L = Dφ G,1 + c U L dug,1 D 2 + α 2) φ L = m D 2 + α 2) φ G,1 + φ L c U L D 3 φ L 3α 2 Dφ L + iαre L [ c U L ) Dφ L + du L φ L 2.25a) du ) L, 2.25b) d 2 ) U G,1 m d2 U L 2, 2.25c) ] 2 iαre L c U L F r + α 2 S cap ) φl = m D 3 φ G,1 3α 2 Dφ G,1 ) + iαrrel [ c U L ) Dφ G,1 + du G,1 φ G,1 while at the virtual interfaces z = z i, z = z w, we have the conditions ], 2.25d) φ G,i = φ G,j, Dφ G,i = Dφ G,j + φ G,i c U G,i [ dug,j 2.26a) du ] G,i, 2.26b) D 2 + α 2) φ G,i = D 2 + α 2) φ G,j [ d + φ G,i 2 U G,j c U G,i 2 D 3 φ G,i 3α 2 Dφ G,i ) = D 3 φ G,j 3α 2 Dφ G,j ) Re L m dτg,i ) d2 U G,i dt Rey,j 2 dt )] Rey,i 2.26c) dτ ) [ G,j + 2α2 φ G,i dug,i du ] G,j, 2.26d) c U G,i where i, j) = 1, 2) for z = z i, and i, j) = 2, 3) for z = z w. These interfacial conditions are non-standard and are derived in Appendix A, while the concept of virtual interfaces is discussed in more depth by Boomkamp & Miesen 1996). Finally, to close the system of equations and to form an eigenvalue problem, we impose the no-slip conditions at the boundaries, φ L 1) = φ L 1) =, 2.27a) φ G,3 δ 1 ) = φ G,3 δ 1 ) =. 2.27b) Note that gravity and surface tension have been introduced to the problem through the interfacial condition 2.25d): here F r is the inverse Froude number F r = g ρ L ρ G ) d L ρ L U 2 i gd 3 = L ρ 2 ) L 1 r, 2.28) μ 2 L Re 2 L

12 12 L. Ó Náraigh, O. K. Matar, P. D. M. Spelt and T. A. Zaki and S cap is the inverse Weber number, ) σ σρl d L 1 S cap = ρ L Ui 2d = L μ 2 L Re 2, 2.29) L where we have made use of the gravitational and surface tension constants g and σ. In experiments these constants remain fixed while the liquid and gas flow rates, and therefore the Reynolds number are made to vary. Thus, in the parametric study that follows, we allow the inverse Froude and Weber numbers to vary as a function of the Reynolds number, keeping the prefactors in 2.28) and 2.29) fixed. Having now outlined the model problem, we discuss a numerical method for its solution Numerical method and validation In each domain liquid, interfacial gas laminar, turbulent gas, wall gas laminar), we propose an approximate solution to the Orr Sommerfeld OS) equations 2.24) in terms of Chebyshev polynomials, such that N L φ L z) a n T n η L ), n= φ G,j z) N G,j n= b n,j T n η G,j ), 2.3) where we have mapped the domains of the various layers into the interval η L, η G,j [ 1, 1]. To reduce the problem 2.24) to a matrix form, we substitute the approximate solution 2.3) into the OS equations 2.24) and evaluate the resulting identity at the collocation points ) qπ η L = cos, q = 1,..., N L 3, N L 2 ) qπ η G,j = cos, q = 1,..., N G,j 3. N G,j 2 In the four-layer application that we have in mind, this gives N L 3) + N G,1 3) + N G,2 3) + N G,3 3) = N L + N G,1 + N G,2 + N G,3 12 equations in N L +N G,1 +N G,2 +N G,3 +4 unknowns. We must therefore find an additional 16 equations to close the system. These are provided by the four boundary conditions and the 12 interfacial conditions four interfacial conditions at the real interface, and four conditions at each of the two virtual interfaces). Thus, we are reduced to solving for the wave speed c, given N L + N G,1 + N G,2 + N G,3 + 4 linear equations in as many unknowns. In symbols, we have a generalized eigenvalue problem Lx = cmx. 2.31) The solution of 2.31) is best found using a linear algebra package see for example Trefethen 2). Then, a built-in eigenvalue solver automatically balances the matrices L and M, an important issue here since these matrices can be very badly conditioned, owing to the high-order derivatives of the Chebyshev polynomials that appear in the expansion of the streamfunction as in the work by Boomkamp et al. 1997)). Furthermore, a specialized eigenvalue solver takes into account the appearance of zero rows in the matrix M hence infinite eigenvalues), and thus gives an accurate answer. Following standard practice, we modify the number of collocation points until convergence is achieved. Before carrying out a parametric study on 2.31)r, we validate our numerical code by comparing a dispersion relation obtained from our solver with that obtained by Miesen & Boersma 1995). We solve the eigenvalue problem for the case of a turbulent air flow in

13 Journal of Fluid Mechanics λ=imαc) α Figure 5. A comparison with the paper of Miesen & Boersma 1995). The stability problem is solved for a boundary-layer profile in which the velocity field is given by piecewise function. The mixture is air and water. The growth rate agrees exactly with that obtained by Miesen and Boersma Figure 7, p. 192), validating the implementation of our numerical scheme. an unbounded domain, shearing over a film of water. The flow profile is piecewise linear. We plot the growth rate in figure 5, and note that it is in perfect agreement with the result of Miesen and Boersma. Finally, one additional way to check the validity of the numerical code is by carrying out a long-wave analysis of the Orr Sommerfeld equation and by comparing the numerical solution with the semi-analytical, long-wave solution. It is to this calculation that we now turn. 3. Long-wave analysis We carry out a long-wave analysis of the Orr Sommerfeld equations 2.24) in the framework of a piecewise linear-constant approximation to the velocity profile. Although this represents a radical simplification to the true profile, we use it for the following reasons. First, its simple form enables the computation of explicit solutions in the longwave limit; second, the previous works of Miesen & Boersma 1995) and Boomkamp et al. 1997) involving a boundary-layer setup have shown that a piecewise linear profile may yield qualitative answers about instability. To write down the approximate velocity profile for a fixed pressure gradient P, we match the parameters of the piecewise profile with the lin-log profile, taking the interfacial and wall stresses from the latter profile as given. We also make use of the mean velocity δ 1 U G,m = δ U G z), from the full model. Then, the approximate profile is the following: z m A, z b i, U G z) = U G,m, b i z b w, 1 δ z), b w z 1 δ, where 1 mr b i = m U G,m 1 A), b w = 1 δ mru G,m.

14 14 L. Ó Náraigh, O. K. Matar, P. D. M. Spelt and T. A. Zaki a) b) Figure 6. The imaginary and real inset) parts of the wave speed for small values of α, and for m, r) =.1,.1). The solid lines show the wave speed obtained from the perturbation analysis, while the datapoints show numerical values. These analytical values coincide with the the numerical curves as α ; b) The real part of the wave speed for small values of α and for variable m, r). Here the wave speed increases with decreasing viscosity ratio m. In both figures, we have taken δ =.1 and P = 1. Following Yih 1967), we expand the streamfunctions in powers of α, φ L = φ ) L + αφ1) L +..., φ G,j = φ ) G,j + αφ1) G,j +..., c = c ) + αc 1) +... At lowest order, we have the following solutions to the Orr Sommerfeld equation, φ ) L = A) L + B) L z + C) L z2 + D ) L z3, φ ) G1 = A) G1 + B) G1 z + C) G1 z2 + D ) G1 z3, η φ ) G2 = A) G2 + B) G2 η + d η η φ ) G1 = A) G3 + B) G3 z + C) G3 z2 + D ) G3 z3, d η C ) G2 + D) G2 η 1 + τ Re G G η ), T where x = A ) L,..., D) G3) is a vector of constants. These are are determined by the interfacial and boundary conditions through a generalized eigenvalue problem A x = c ) B x. The eigenvalues c ) depend on the parameters Re L, r, m, δ), and on the shape functions Σ 1 = 1 ds 1 s1 ds τ Re G G s 2 ), Σ 2 = 1 ds 1 s1 s 2 ds τ Re G G s 2 ), 1 ds 1 Σ 3 = 1 + τ Re G G s), Σ sds 4 = 1 + τ Re G G s), τ = rκu i. δ R 1/2 U i This dependence is encoded in a long formula that we do not report here. We find that c ) is real, and thus, at lowest order in this expansion, the interface is stable.

15 Journal of Fluid Mechanics 15 Symbol Meaning Numerical value m m = μ G/μ L.1 r r = ρ G /ρ L.1 P Nondimensional pressure gradient 1 Re L Liquid Reynolds number U Lμ L/ρ Ld L Re G Gas Reynolds number rre L/m κ Von Kármán constant.41 t Viscous layer thickness 5 S 1 Base velocity parameter d L p/ x /2τ i.13 δ Channel aspect ratio d L/h.1 F r Inverse Froude number S cap Inverse Weber tension r)/re 2 L 1 4 /Re 2 L Table 1. Summary of numerical values used in 4 At first order, the streamfunction has the form φ 1) L = A1) L + B1) L z + C1) L z2 + D 1) L z3 + φ PI,L z), φ 1) G1 = A1) G1 + B1) G1 z + C1) G1 z2 + D 1) G1 z3 + φ PI,G1 z), η φ 1) G2 = A1) G2 + B1) G2 η + where again x 1 = d η η d η C 1) G2 + D1) G2 η 1 + τ Re G G η ) + φ PI,G2 z), φ 1) G3 = A1) G3 + B1) G3 z + C1) G3 z2 + D 1) G3 z3 + φ PI,G3 z), A 1) 1 φ PI,j z) is the particular integral of the equation [ D 4 φ 1) j + δ G2,j τ Re G D 2 G z) D 2 φ 1) j,..., D1) G3) T is a vector of constants, and where the function ] [ = ire j U j z) c )) D 2 φ ) j U j z) φ ) j in the j th layer here δ G2,j is the Kronecker delta). Notice that the polynomial velocity profile gives rise to polynomial particular integrals; more complicated velocity profiles would not produce such a simple solution. The interfacial and boundary conditions provide sixteen equations in the seventeen unknowns x 1 and c 1). The system is closed by taking A 1) 1 = without loss of generality; this is accomplished by demanding that the constant solution to the Orr Sommerfeld equation in the liquid layer be equal to unity. This gives an inhomogeneous linear problem, A 1 c 1) x 1 ], ) = b ) The asymptotic growth rate c 1) is a function of the parameters Re L, r, m, δ, F r) and is obtained from the solution of the linear equation 3.1). The real wave speed c and the growth rate are shown in figure 6. This figure also demonstrates that the numerical and analytical results are in agreement. The involved nature of this calculation shows the necessity for the choice of the simplified velocity profile in this long-wave analysis. However, if only the real part of the velocity is desired, this calculation can be repeated with the full profile. We shall make use of this fact in 5. In any case, this long-wave analysis has confirmed the validity of the numerical method, and we can now proceed to extract more detailed information from the model.

16 16 L. Ó Náraigh, O. K. Matar, P. D. M. Spelt and T. A. Zaki λ=imαc) A B Full profile Piecewise profile α a) λ=imαc) C Full profile Piecewise profile α b) D Figure 7. Analysis of the OS equation: growth rates for a) the interfacial mode, and b) internal mode. The parameter values for these dispersion curves are given by table 1. The dotted line gives the growth rate for the piecewise linear-constant profile, and the solid line gives the growth rate for the finite-curvature profile introduced in 2. DISS G DISS L REY G REY L T URB NOR T AN Interfacial, full A) Interfacial, piecewise B) Internal, piecewise C) Internal, full D) Table 2. The energy budget corresponding to the points A to D in figure 7. The parameter values are given in table 1. The energy terms have been normalized so that T AN = Instability and mode competition In this section we present the results of the Orr Sommerfeld OS) analysis. The results comprise dispersion curves and neutral curves. To begin, we carry out a stability analysis based on the values in table 1. Note in particular that we have chosen the inverse Froude number F r = r) Re 2, where the prefactor is calculated based on a thin film of water of height O 1 4) m. We compare and contrast the piecewise and full profiles of 2.1. The growth rate of the most unstable mode is shown in figure 7 a). The piecewise and full profiles are substantially different, except at short wavelengths, which suggests the necessity of using the full profile for accurate growth-rate predictions. The maximum growth rate occurs for α = O 1), that is, for a wavelength comparable to the film thickness. In figure 7 b), the piecewise profile yields a positive growth rate for another mode, which we shall justify calling the internal mode. This mode is stabilized by switching to the full profile; however, by a judicious parameter choice m, r, δ, P ), this mode can be made unstable again; we focus on the problem of choosing such a parameter set in 4.2. Note that for the internal mode and the full profile, αc i is nonzero at α =. This is not unusual: Yiantsios & Higgins 1988) have observed similar modal behaviour in two-phase laminar Poiseuille flow in a channel. Other unstable modes are observed but we do not report them here since they correspond to the virtual interfaces and are therefore artifacts of the piecewise nature of the basic velocity field. To understand the instability mechanisms at work in figure 7, we perform an energy decomposition following the method described in Appendix B see also Boomkamp & Miesen 1996). The energy budgets associated with the points A, B, C, and D in figure 7 are shown in table 2. The modes A and B derive their instability from the

17 Journal of Fluid Mechanics T xz z,z=)/maxt xz x,z=)) ηx)/maxηx)) x [,2π/α] a) b) Figure 8. a) The phase shift between the viscous shear stress at the interface, T xz x, z = ) and the interface shape η x) for the point A discussed in the text, and in table 2. The shift is small and resides in the range [ π 2, π 2 ]. Thus the tangential term in the energy budget is positive destabilizing), as required by 4.1); b) The behaviour of the viscous shear stress across the interface. tangential contribution. Following the argument of Boomkamp & Miesen 1996), and using the details provided in Appendix B, we recast the T AN term as T AN = l dx [u L u G,1 ) T xz ] z=, l = 2π/α, since the disturbance shear stress T xz is continuous across z =. The kinematic condition gives for the jump u L u G,1, u L u G,1 = η ) U G,1 ) U L = ηu 1 L m 1, on z =. Thus, the tangential term is non-zero due to the viscosity ratio m 1, ) 1 l T AN = U L ) m 1 dx η x) T xz x, z = ), 4.1) and we conclude that the interfacial instability is due to the viscosity contrast. Note that for m < 1, positivity of this term relies on the phase between the interfacial shape η x) and the disturbance stress T xz x, z = ) lying in the range [ π 2, π 2 ] see figure 8). Points C and D in the energy budget contain two positive contributions to the instability: one from the liquid Reynolds term REY L, and one from the tangential term. This is the so-called internal mode that Boomkamp & Miesen 1996) associated with the thin liquid film becoming unstable as it is sheared by a turbulent gas layer. Superficially, the energy budgets C piecewise profile) and D full profile, with turbulent shear stress model) are similar, although, as indicated by figure 7, C is unstable while D is stable. Indeed, table 2 suggests the following interpretation of the role of the turbulent shear stress: since the terms DISS L and DISS G are a measure of the flow s gradient content, and since these become large once the turbulent shear stress is considered, this implies that the additional stress term amplifies gradients in the problem. The large gradients stabilize the change in total kinetic energy through dissipation, and destabilize it through the interfacial contribution, and these effects compete. For the interfacial mode, the destabilizing interfacial term wins, and the effect of the augmented T AN term is greater than that of the augmented dissipation. On the other hand, for the internal mode and the full profile, the stabilizing dissipation dominates over the in-

18 18 L. Ó Náraigh, O. K. Matar, P. D. M. Spelt and T. A. Zaki P=-1 P=-1 2 P=-1 3 P=-1 4 B C D λ.4.2 A α Figure 9. The growth rate as a function of the wavenumber and the nondimensional pressure gradient P. Solid line: P = 1 Re L = 5); dashed line: P = 1 Re L = 487); dotted line: P = 1 Re L = 4643); dashed line: P = 1 Re L = 4797). Increasing the pressure gradient has a destabilizing effect on the interface, both in the sense of increasing the maximal growth rate and the cutoff wavenumber. Here we have set m =.1, r =.1, and δ =.1. terfacial energy term and the internal mode becomes stable. However, this mode can be destabilized by a judicious choice of parameters, as discussed in the next section. In table 1 we have chosen a set of parameters that are comparable in magnitude to those describing an air-water system. However, we wish to quantify the stability properties of the system in full generality, and we therefore investigate the implications of varying the pressure gradient, the viscosity and density contrasts, the channel aspect ratio, and the eddy viscosity The interfacial mode In figure 9 we plot the growth rate for various nondimensional pressure gradients P or, equivalently, for various Reynolds numbers. We use a logarithmic scale because of the strong dependence of the cutoff wavenumber on the pressure gradient. Increasing the pressure gradient has a destabilizing effect: not only is the range of wavenumbers for which the system is unstable increased, but so is the maximum growth rate. The destabilizing effect of the large pressure gradient is felt through an increase in the liquid Reynolds number, which diminishes the stabilizing effects of dissipation, gravity, and surface tension, as indicated by 2.28) and 2.29). To understand the results in figure 9 in more depth, the energy budgets associated with the points A, B, C, and D are studied in table 3. We see again that the T AN term provides the only positive contribution to the kinetic energy growth; hence, the instability is of interfacial type. Furthermore, table 3 indicates that the critical layer lies inside the laminar sublayer for all but the highest pressure gradients, confirming the marginality of the Miles mechanism in this study. Indeed, even at point D in the table, when the critical layer lies outside the laminar sublayer, the interfacial instability is still due to the viscosity-contrast mechanism. Note also that the apparent shoulder in this dispersion curve with P = 1 is an artifact of the logarithmic scale, and that the instability is tangential for all points along the dispersion curve. We have verified this by obtaining the energy budget for numerous points along the curve; at each point, it is the tangential term that is dominant. To understand the effects of the parameters m, r, and δ on the interfacial stability, we plot growth rates for different values of these parameters. In figure 1 a) we investigate the effects of varying the viscosity contrast by holding the other parameters fixed: r =.1, δ =.1, and P = 1. The interface becomes more unstable as the viscosity

19 Journal of Fluid Mechanics 19 Point P DISS G DISS L REY G REY L T URB NOR T AN c r z c /z i A B C D Table 3. The energy budget for the most dangerous mode at various pressure gradients. The energy terms have been normalized so that all contributions sum to unity. The points A to D correspond to those points indicated in figure 9. In each case, it is the tangential contribution that destabilizes the interface. For the pressure gradients A to C, the critical layer lies inside the laminar sublayer. In case D, the critical layer lies outside the laminar sublayer, although the instability is still due to the tangential term. 6 4 m=.1 m=.1 m=.1 A 6 4 r=.1 r=.1 r= δ=.1 δ=.1 δ=1 λ 2 B 2 λ λ α a) α b) α c) Figure 1. a) Effect of viscosity variations on the growth rate of the interfacial mode. We fix r =.1, δ =.1, P = 1, and vary the viscosity ratio m. The growth rate is enhanced as the viscosity ratio m decreases; b) Effect of density variations on the interfacial growth rate. We fix m =.1, δ =.1, P = 1, and vary the density ratio r. The growth rate is enhanced as the density ratio r increases; c) Effect of the channel aspect ratio on the growth rate of the interfacial mode. We fix m =.1, r =.1, P = 1, and vary the channel aspect ratio δ = d L /h. The deeper channel configurations smaller δ-values) produce a more unstable interface. Point m DISS G DISS L REY G REY L T URB NOR T AN c r z c /z i A B Table 4. The energy budget for the most dangerous mode for two regimes of viscosity contrast. The energy terms have been normalized so that all contributions sum to unity. The points A and B correspond to those points indicated in figure 1 a). In each case, it is the tangential contribution that destabilizes the interface, although the m =.1 case also enjoys a contribution from the REY G term. In each case, P = 1, r =.1, and δ =.1. of the turbulent gas layer is decreased relative to the liquid layer. The instability is enhanced both through an increase in the cutoff wavenumber, and through an increase in the maximal growth rate. Because of the relevance of the m =.1 curve to the oil-gas transport problem, we single it out for special study and show the energy budget of the most dangerous mode in table 4, in comparison to the energy budget of the m =.1 case. The m =.1 case is straightforward, and the instability is due solely to interfacial effects the T AN term). On the other hand, the m =.1 energy budget enjoys destabilizing contributions from the gas Reynolds term and from the tangential term the latter is dominant). Note, however, that the critical layer is confined to the laminar sublayer for both values of m. To understand the origin of the positive gas Reynolds stress, we study