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RELAXATION TWO-PHASE FLOW MODELS AND THE SUBCHARACTERISTIC CONDITION

The subcharacteristic condition for hyperbolic relaxation systems states that wave velocities of an equilibrium system cannot exceed the corresponding wave velocities of its relaxation system. This condition is central to the stability of hyperbolic relaxation systems, and is expected to hold for most such models describing natural phenomena. In this paper, we study a hierarchy of two-phase flow models. We consider relaxation with respect to volume transfer, heat transfer and mass transfer. We formally verify that our relaxation processes are consistent with the first and second laws of thermodynamics, and present analytical expressions for the wave velocities for each model in the hierarchy. Through an appropriate choice of variables, we prove directly by sums-of-squares that for all relaxation processes considered, the subcharacteristic condition holds for any thermodynamically stable equation of state.

On the Wave Structure of Two‐Phase Flow Models

We explore the relationship between two common two‐phase flow models, usually denoted as the two‐fluid and drift‐flux models. They differ in their mathematical description of momentum transfer between the phases. In this paper we provide a framework in which these two model formulations are unified. The drift‐flux model employs a mixture momentum equation and treats interphasic momentum exchange indirectly through the slip relation, which gives the relative velocity between the phases as a function of the flow parameters. This closure law is in general highly complex, which makes it difficult to analyze the model algebraically. To facilitate the analysis, we express the quasi‐linear formulation of the drift‐flux model as a function of three parameters: the derivatives of the slip with respect to the vector of unknown variables. The wave structure of this model is investigated using a perturbation technique. Then we rewrite the drift‐flux model with a general slip relation such that it is expressed in term...

Weakly Implicit Numerical Schemes for a Two-Fluid Model

The aim of this paper is to construct semi-implicit numerical schemes for a two-phase (two-fluid) flow model, allowing for violation of the CFL criterion for sonic waves while maintaining a high level of accuracy and stability on volume fraction waves. By using an appropriate hybridization of a robust implicit flux and an upwind explicit flux, we obtain a class of first-order schemes, which we refer to as weakly implicit mixture flux (WIMF) methods. In particular, by using an advection upstream splitting method (AUSMD) type of upwind flux [S. Evje and T. Flatten, J. Comput. Phys., 192 (2003), pp. 175--210], we obtain a scheme denoted as WIMF-AUSMD. We present several numerical simulations, all of them indicating that the CFL-stability of the WIMF-AUSMD scheme is governed by the velocity of the volume fraction waves and not the rapid sonic waves. Comparisons with an explicit Roe scheme indicate that the scheme presented in this paper is highly efficient, robust, and accurate on slow transients. By exploiting the possibility to take much larger time steps, it outperforms the Roe scheme in the resolution of the volume fraction wave for the classical water faucet problem. On the other hand, it is more diffusive on pressure waves. Although conservation of positivity for the masses is not proved, we demonstrate that a fix may be applied, making the scheme able to handle the transition to one-phase flow while maintaining a high level of accuracy on volume fraction fronts.

Wave Propagation in Multicomponent Flow Models

We consider systems of hyperbolic balance laws governing flows of an arbitrary number of components equipped with general equations of state. The components are assumed to be immiscible. We compare two such models: one in which thermal equilibrium is attained through a relaxation procedure, and a fully relaxed model in which equal temperatures are instantaneously imposed. We describe how the relaxation procedure may be made consistent with the second law of thermodynamics. Exact wave velocities for both models are obtained and compared. In particular, our formulation directly proves a general subcharacteristic condition: For an arbitrary number of components and thermodynamically stable equations of state, the mixture sonic velocity of the relaxed system can never exceed the sonic velocity of the relaxation system.

On Solutions to Equilibrium Problems for Systems of Stiffened Gases

We consider an isolated system of N immiscible fluids, each following a stiffened-gas equation of state. We consider the problem of calculating equilibrium states from the conserved fluid-mechanical properties, i.e., the partial densities and internal energies. We consider two cases; in each case mechanical equilibrium is assumed, but the fluids may or may not be in thermal equilibrium. For both cases, we address the issues of existence, uniqueness, and physical validity of equilibrium solutions. We derive necessary and sufficient conditions for physically valid solutions to exist, and prove that such solutions are unique. We show that for both cases, physically valid solutions can be expressed as the root of a monotonic function in one variable. We then formulate efficient algorithms which unconditionally guarantee global and quadratic convergence toward the physically valid solution.

Large time step TVD schemes for hyperbolic conservation laws

Large time step explicit schemes in the form originally proposed by LeVeque have seen a significant revival in recent years. In this paper we consider a general framework of local 2k + 1 point schemes containing LeVeques scheme (denoted as LTS-Godunov) as a member. A modified equation analysis allows us to interpret each numerical cell interface coefficient of the framework as a partial numerical viscosity coefficient. We identify the least and most diffusive TVD schemes in this framework. The most diffusive scheme is the 2k + 1-point Lax-Friedrichs scheme (LTS-LxF). The least diffusive scheme is the Large Time Step scheme of LeVeque based on Roe upwinding (LTS-Roe). Herein, we prove a generalization of Hartens lemma: all partial numerical viscosity coefficients of any local unconditionally TVD scheme are bounded by the values of the corresponding coefficients of the LTS-Roe and LTS-LxF schemes. We discuss the nature of entropy violations associated with the LTS-Roe scheme, in particular we extend the notion of transonic rarefactions to the LTS framework. We provide explicit inequalities relating the numerical viscosities of LTS-Roe and LTS-Godunov across such generalized transonic rarefactions, and discuss numerical entropy fixes. Finally, we propose a one-parameter family of Large Time Step TVD schemes spanning the entire range of the admissible total numerical viscosity. Extensions to nonlinear systems are obtained through the Roe linearization. The 1D Burgers equation and the Euler system are used as numerical illustrations.

Review of two-phase flow models for control and estimation

Abstract Most model-based control and estimation techniques put limitations on the structure and complexity of the models to which they are applied. This has motivated the development of simplified models of gas-liquid two-phase flow for control and estimation applications. This paper reviews the literature for such models with a focus on applications from the field of drilling. The models are categorized in terms of complexity and the physical interpretation of the simplifications employed. A simulation study is used to evaluate their ability to qualitatively represent dynamics of 3 different gas-liquid scenarios encountered in drilling, based on which conclusions are drawn.

ON THE DISPERSIVE WAVE DYNAMICS OF 2 × 2 RELAXATION SYSTEMS

We consider hyperbolic conservation laws with relaxation terms. By studying the dispersion relation of the solution of general linearized 2 × 2 hyperbolic relaxation systems, we investigate in detail the transition between the wave dynamics of the homogeneous relaxation system and that of the local equilibrium approximation. We establish that the wave velocities of the Fourier components of the solution to the relaxation system will be monotonic functions of a stiffness parameter φ = eξ, where e is the relaxation time and ξ is the wave number. This allows us to extend in a natural way the classical concept of the sub-characteristic condition into a more general transitional sub-characteristic condition. We further identify two parameters β and γ that characterize the behavior of such general 2 × 2 linear relaxation systems. In particular, these parameters define a natural transition point, representing a value of φ where the dynamics will change abruptly from being equilibrium-like to behaving more like the homogeneous relaxation system. Herein, the parameter γ determines the location of the transition point, whereas β measures the degree of smoothness of this transition.

CFL-Violating Numerical Schemes for a Two-Fluid Model

In this paper we propose a class of linearly implicit numerical schemes for a two-phase flow model, allowing for violation of the CFL-criterion for all waves. Based on the Weakly Implixit Mixture Flux (WIMF) approach [SIAM J. Sci. Comput., 26 (2005), pp. 1449–1484], we here develop an extension denoted as Strongly Implicit Mixture Flux (SIMF). Whereas the WIMF schemes are restricted by a weak CFL condition which relates time steps to the fluid velocity, the SIMF schemes are able to break the CFL conditions corresponding to both the sonic and advective velocities. The schemes possess some desirable features compared to current industrial pressure-based codes. They allow for sequential updating of the momentum and mass variables on a nonstaggered grid by solving two sparse linear systems. The schemes are conservative in all convective fluxes and consistency between the mass variables and pressure is formally maintained. Numerical experiments are presented to shed light on the inherent differences between the WIMF and SIMF families of schemes. In particular, we demonstrate that the WIMF scheme is able to give an exact resolution of a moving contact discontinuity. The SIMF schemes do not possess the “exact resolution” property of WIMF, however, the ability to take larger time steps can be exploited so that more efficient calculations can be made when accurate resolution of sharp fronts is not essential, e.g. to calculate steady state solutions.

Wave dynamics of linear hyperbolic relaxation systems

We consider linear hyperbolic systems with a stable rank 1 relaxation term and establish that the characteristic polynomial for the individual Fourier components of the solution can be written as a convex combination of the characteristic polynomials for the formal stiff and non-stiff limits. This allows us to provide a direct and elementary proof of the equivalence between linear stability and the subcharacteristic condition. In a similar vein, a maximum principle follows: The velocity of each individual Fourier component is bounded by the minimum and maximum eigenvalues of the non-stiff limit system.