Stability of stratified two-phase flows in horizontal channels

Abstract

Linear stability of stratified two-phase flows in horizontal channels to arbitrary wavenumber disturbances is studied. The problem is reduced to Orr-Sommerfeld equations for the stream function disturbances, defined in each sublayer and coupled via boundary conditions that account also for possible interface deformation and capillary forces. Applying the Chebyshev collocation method, the equations and interface boundary conditions are reduced to the generalized eigenvalue problems solved by standard means of numerical linear algebra for the entire spectrum of eigenvalues and the associated eigenvectors. Some additional conclusions concerning the instability nature are derived from the most unstable perturbation patterns. The results are summarized in the form of stability maps showing the operational conditions at which a stratified-smooth flow pattern is stable. It is found that for gas-liquid and liquid-liquid systems the stratified flow with a smooth interface is stable only in confined zone of relatively low flow rates, which is in agreement with experiments, but is not predicted by long-wave analysis. Depending on the flow conditions, the critical perturbations can originate mainly at the interface (so-called "interfacial modes of instability") or in the bulk of one of the phases (i.e., "shear modes"). The present analysis revealed that there is no definite correlation between the type of instability and the perturbation wavelength

Stability of stratified two-phase channel flows of Newtonian/non-Newtonian shear-thinning fluids

“…In fact, gravity effects are negligible in this region, since it corresponds to very high superficial velocity of the viscous liquid (and the air). Such a stable narrow region was predicted also for a zerogravity air-water system (see Barmak et al, 2016a). Note that the operational window of the experiments conducted by Picchi et al (2015) is beyond the obtained stability boundary, and indeed, stratified flow was not observed.…”

“…The temporal linear stability is studied by solving the system of differential equations (17), (18) and (19)-(24) assuming an arbitrary wavenumber for each given set of the other parameters. The time increment is defined as a complex eigenvalue The stability problem is solved by applying the Chebyshev collocation method (with 50 N  collocation points for each sublayer) for discretization of the Orr-Sommerfeld equations and the boundary conditions (see details in Barmak et al, 2016a) and by using the QR algorithm (Francis, 1962) for the computation of the eigenvalues and eigenvectors. The numerical solution was verified by a comparison with the solution for the flow of two Newtonian fluids (presented in Barmak et al, 2016a, b) and by assuring the numerical convergence (independency of the results on the truncation number, N ).…”

Non-modal stability analysis of stratified two-phase channel flows

“…Another interesting test case is horizontal air-water flow in a larger (20cm-height) channel. For this case, the modal stability analysis predicts that low air flow rates stabilize the corresponding single-phase water flow, whereby the critical water superficial velocity for destabilization of the flow is much higher than that predicted by the single-phase flow laminar limits (see Barmak et al, 2016a). The superficial phase velocity corresponding to the single-phase flow laminar limit (dashed lines for the gas and liquid phases in Fig.17) is determined by the critical Reynolds number, Re 5772 Cr  (Orszag, 1971).…”

“…Gravity is known to change dramatically the stability boundaries of stratified two-phase flow (see, e.g., Barmak et al, 2016a). However, its effect on non-modal stability has not been investigated in the literature.…”

Hydrodynamic instabilities in miscible fluids

“…The same kind of regularization via viscous dissipation can be found for the Rayleigh-Taylor instability [51,19]. Due to the general and ubiquitous nature of stratified flows, a lot of effort [52,53,54,55,50,56] has been devoted to the stability problem in more realistic situations, e.g. two fluids with different viscosities flowing between two rigid plane-parallel boundaries separated by a finite distance.…”