Dynamic simulation of wavy-stratified two-phase flow with the one-dimensional two-fluid model

William D Fullmer, Purdue University

Abstract

The one-dimensional two-fluid model is the basis for the description of the transport of mass, momentum and energy in the thermal-hydraulic codes used for nuclear reactor safety analysis. Unlike other physical transport models, the one-dimensional two-fluid model suffers from the possibility of being ill-posed as an initial-boundary value problem depending on the flow conditions and the relevant physical closure laws. Typically, the ill-posedness is dealt with through either excessive numerical damping or the addition of unphysical closure laws designed for the sole purpose of hyperbolization. Unfortunately both methods eliminate the instability along with the problem of ill-posedness causing the model to undoubtedly lose some of its inherent dynamic capability. In this work, a one-dimensional two-fluid model for horizontal or slightly inclined stratified flow is developed. Higher order physical models that are often neglected, such as surface tension and axial viscous stress, are retained for their short-wavelength stability properties. Characteristic, dispersion and nonlinear analyses are performed to demonstrate that the resulting model is linearly well-posed and nonlinearly well-behaved. While it has been known that higher-order differential terms are able to regularize the short-wavelength problem of ill-posedness without removing the long-wavelength instability, the literature is relatively silent on the consequences of using a model under linearly unstable conditions. Using carefully selected conditions in an idealized infinite domain, it is demonstrated for the first time that the one-dimensional two-fluid model exhibits chaotic behavior in addition to limit cycles and asymptotic stability. The chaotic behavior is a consequence of the long-wavelength linear instability (energy source) the nonlinearity (energy transfer) and the short-wavelength dissipation (energy sink). Since the model is chaotic, solutions exhibit a sensitive dependence on initial conditions. This appears to result in non-convergence when particular solutions at a specific time are compared using different numerical discretizations. However, it is shown that the chaotic solutions exhibit an invariant spectrum in wavenumber space that can be used to assess the convergence of solutions. This concept is applied to a Kelvin-Helmholtz experiment of kerosene and liquid water in a tilted channel whereby many slightly different simulations are run and averaged to determine the mean behavior. Comparisons to experimental data are favorable; especially considering the limitations of applying a one-dimensional model to a dynamic simulation of wavy channel flow. When the analysis is extended to consider air-water flows, several additional challenges are encountered related to the long-wavelength inviscid Kelvin-Helmholtz instability, which is the instability inherent to the one-dimensional two-fluid model. The transition from stratified to wavy flow is significantly over-predicted, i.e., requires a larger velocity to become unstable than observed experimentally. The wave sheltering model of Brauner and Maron (1993) is included in the interfacial shear model and calibrated for flow in a rectangular channel. However, when the unstable flow regime is simulated a wavy flow pattern does not develop as in the liquid-liquid case. Due to the near absence of inertia in the lighter gas phase, viscosity and surface tension are unable to bound the growth of disturbances within the physical limitations of the channel geometry. Transitions to regions of single phase flow result, indicating a slug flow pattern where wavy flow should exist. A novel approach is taken where the instability mechanism, here the sheltering force, is adjusted based on local geometric conditions, namely the void fraction gradient. Comparison to data shows promising results, although a large degree of uncertainty in such an approach remains due to a lack of local experimental data.

Degree

Ph.D.

Advisors

Lopez de Bertodano, Purdue University.

Subject Area

Nuclear engineering

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