A Chebyshev Collocation Method for Solving Two-Phase Flow Stability Problems

Boomkamp, P.A.M.; Boersma, Bendiks Jan; Miesen, R.H.M.; Beijnon, G.V.

doi:10.1006/jcph.1996.5571

Cited by 58 publications

(56 citation statements)

References 21 publications

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“…But the results obtained by both of these approaches are exactly the same. Boomkamp et al (1997) developed a collocation method based on Chebyshev polynomials to solve the eigenvalue problem governing the stability of parallel two-phase flow, which is similar to the methodology developed by Orszag (1971) to investigate the stability of Poiseuille's flow. In our approach, we begin by expanding the eigenfunction ϕ(z) as a truncated series of Chebyshev polynomials, T i (z) (defined on the interval [−1, 1]),…”

Section: Appendix a Numerical Methods For The Os Problemmentioning

confidence: 99%

Absolute and convective instabilities in counter-current gas–liquid film flows

2014

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Absolute and convective instabilities in counter-current gas-liquid film flows. Mechanics, 763, Additional Information: Journal of Fluid• This is the accepted manuscript of an article subsequently published in We consider a thin liquid film flowing down an inclined plate in the presence of a countercurrent turbulent gas. By making appropriate assumptions, Tseluiko & Kalliadasis (2011) developed low-dimensional non-local models for the liquid problem, namely a long-wave model and a weighted integral-boundary-layer model, that incorporate the effect of the turbulent gas. By utilising these models, along with the Orr-Sommerfeld problem formulated using the full governing equations for the liquid phase and associated boundary conditions, we explore the linear stability of the gas-liquid system. In addition, we devise a generalised methodology to investigate absolute and convective instabilities in the nonlocal equations describing the gas-liquid flow. We observe that at low gas flow rates, the system is convectively unstable with the localised disturbances being convected downwards. As the gas flow rate is increased, the instability becomes absolute and localised disturbances spread across the whole domain. As the gas flow rate is further increased, the system again becomes convectively unstable with the localised disturbances propagating upwards. We find that the upper limit of the absolute instability region is close to the 'flooding' point associated with the appearance of large-amplitude standing waves, as obtained in Tseluiko & Kalliadasis (2011), and our analysis can therefore be used to predict the onset of flooding. We also find that an increase in the angle of inclination of the channel requires an increased gas flow rate for the onset of absolute instability. We generally find good agreement between the results obtained using the full equations and the reduced models. Moreover, we find that the weighted integral-boundary-layer model generally provides better agreement with the results for the full equations than the longwave model. Such an analysis is important for understanding the ranges of validity of the reduced model equations. In addition, a comparison of our theoretical predictions with the experiments of Zapke & Kröger (2000a) shows a fairly good agreement. We supplement our stability analysis with time-dependent computations of the linearised weighted integral-boundary-layer model. To provide some insight into the mechanisms of instability, we perform an energy budget analysis.

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Section: Appendix a Numerical Methods For The Os Problemmentioning

confidence: 99%

Absolute and convective instabilities in counter-current gas–liquid film flows

2014

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“…(10) and (11), only half of the channel is considered, y ∈ [1/2, 1]. This domain is decomposed into three regions, 1/2 ≤ y ≤ 1/2 + h, 1/2 + h ≤ y ≤ 1/2 + h + q and 1/2 + h + q ≤ y ≤ 1, and the eigenfunctions in each region are then expanded using Chebyshev polynomials through a spectral method [53,59,60]. The decomposition of the domain endows the edges of the mixed layer with more points than its interior, thereby enhancing the resolution of the numerical solutions where the base state concentration and its derivatives must be continuous [53].…”

Section: Linear Stability Analysismentioning

confidence: 99%

Linear stability analysis and numerical simulation of miscible two-layer channel flow

Sahu¹,

Ding²,

Valluri³

et al. 2009

Physics of Fluids

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The stability of miscible two-fluid flow in a horizontal channel is examined. The flow dynamics are governed by the continuity and Navier-Stokes equations coupled to a convective-diffusion equation for the concentration of the more viscous fluid through a concentration-dependent viscosity.Our analysis of the flow in the linear regime delineates the presence of convective and absolute instabilities and identifies the vertical gradients of viscosity perturbations as the main destabilizing influence in agreement with previous work. Our transient numerical simulations demonstrate the development of complex dynamics in the nonlinear regime, characterized by roll-up phenomena and intense convective mixing; these become pronounced with increasing flow rate and viscosity ratio, as well as weak diffusion. *

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“…Modern computational facilities allow us to solve this eigenvalue problem entirely numerically for almost any prescribed primary profile u/O') and a large range of the dimensionless parameters ~, R, m, r, n, S, F and ft. Following the method described in the papers by Miesen and Boersma (1995) and Boomkamp et al (1997) we solve the problem by means of a spectral technique, based on an expansion of the functions Oj(y) in Chebyshev polynomials and on point collocation. Subsequent solution of the resulting generalized eigenvalue problem with the QZ-algorithm (Molar and Stewart 1973;NAG 1988) then provides the dispersion relation c = c(~, R, m, r, n, S, F,//), [10] and the corresponding eigenfunctions ~j(),).…”

Section: Formulation Of the Stabilio Problemmentioning

confidence: 99%