Narrow Results

The main purpose of this work is to study the long-time behavior of a compressible gas–liquid model based on the drift-flux formulation. The model is composed of two mass conservation equations and one mixture momentum equation. The flow domain is closed at one end and involves a free gas–liquid interface at the other where both phases vanish. The model includes a slip between the gas and liquid phases, i.e. they move with different velocities. This is a main reason why the model is useful for many industrial applications. We introduce a reformulation of the model based on the gas velocity. This gives rise to new nonlinear terms in the mixture momentum equation which account for the difference in the gas and liquid velocity. New challenges in the mathematical analysis will then appear. In particular, under appropriate smallness conditions on initial data (initial energy) various time-independent estimates of gas and liquid masses, as well as fluid velocities, are obtained. Novel upper and lower bounds on masses are provided that contain precise information about the time-dependent decay rate. Hence, the long-time behavior can be directly extracted from these estimates.

In this paper, we consider the Dirichlet problem of a one-velocity viscous drift-flux model. One of the phases is compressible, the other one is weakly compressible. Under weak assumptions on the initial data, which can be discontinuous and large as well as involve transition to pure single-phase points or regions, we show existence of global bounded weak solutions. One main ingredient is that we employ a decomposition of the pressure term appearing in the mixture momentum equation into two components, one for each of the two phases. This paves the way for deriving a basic energy equality. In particular, upper bounds on the masses are extracted from the estimates provided by the energy equality. By relying on weak compactness tools we obtain an existence result within the class of weak solutions. An essential novel aspect of this analysis, compared to previous works on the same model, is that the solution space is large enough to allow for transition to single-phase flow without any constraints. In particular, one of the phases can vanish in a point while the other phase can persist. The key to achieve this result, which represents a major step forward compared to previous results for this model, is that we do not rely on any higher-order (i.e. derivatives in space) estimates on the masses or pressure, only low-order estimates provided by the energy equality and the uniform upper bounds on the liquid and gas mass.

In this paper, the large time behavior of the solution to the initial-boundary problems for the one-dimensional compressible gas–liquid drift-flux model with slip is studied. Under some suitable smallness conditions upon the initial data, the optimal pointwise upper and lower decay estimates on masses as well as the sharpest decay rates for the norms in terms of the velocity function are obtained. This result generalizes the one in [On the large time behavior of the compressible gas–liquid drift-flux model with slip, Math. Models Methods Appl. Sci.25 (2015) 2175–2215] by Evje and Wen. The key of the proof is to derive some new global-in-time weighted estimates. Our method can also be easily adopted to the study on the large time behavior of the solution to the one-dimensional compressible Naiver–Stokes equations.

This work presents computational simulations and analytical techniques for solving the drift-flux two-phase flow model. The model equations are formulated to describe the exact solution of the Riemann problem. The solution is constructed by solving the conservation of mass for each phase and the mixture conservation momentum equation of the two phases under isothermal conditions. Particular attention is given to address the expressions for jump relationships and the Riemann invariants. The performance of the developed Riemann solver is assessed with respect to different test cases selected from the literature. Comparisons with Godunov methods of centred-type are provided to demonstrate the use of the proposed exact and computational framework. Excellent agreement is observed between analytical results and numerical predictions.