A Hierarchy of Models for Two-Phase Flows

Summary. We derive a hierarchy of models for gas-liquid two-phase flows in the limit of infinite density ratio, when the liquid is assumed to be incompressible. The starting model is a system of nonconservative conservation laws with relaxation. At first order in the density ratio, we get a simplified system with viscosity, while at the limit we obtain a system of two conservation laws, the system of pressureless gases with constraint and undetermined pressure. Formal properties of this constraint model are provided, and sticky blocks solutions are introduced. We propose numerical methods for this last model, and the results are compared with the two previous models. Received April 20, 2000; accepted September 12, 2000 %%Online publication November 15, 2000 Communicated by Gérard Iooss     

Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model

A traffic flow model describing the formation and dynamics of traffic jams was introduced by Berthelin et al., which consists of a constrained pressureless gas dynamics system and can be derived from the Aw-Rascle model under the constraint condition ρ?ρ^? by letting the traffic pressure vanish. In this paper, we give up this constraint condition and consider the following form

     

Delta shock wave formation in the case of triangular hyperbolic system of conservation laws

We describe δ-shock wave generation from continuous initial data in the case of triangular conservation law system arising from “generalized pressureless gas dynamics model.” We use smooth approximations in the weak sense that are more general than small viscosity approximations.     

Propagation and conditional propagation of chaos for pressureless gas equations

We study the existence and uniqueness of a weak solution of a viscous d-dimensional system of pressureless gas equations. We construct a nonlinear diffusion by using the propagation and conditional propagation of chaos. The latter diffusion is associated with the above pressureless gas equations.Mathematics Subject Classification (2000):60H15, 35R60, 60H30     

Free/Congested Two-Phase Model from Weak Solutions to Multi-Dimensional Compressible Navier-Stokes Equations

We approximate a two–phase model by the compressible Navier-Stokes equations with a singular pressure term. Up to a subsequence, these solutions are shown to converge to a global weak solution of the compressible system with the congestion constraint studied for instance by Lions and Masmoudi. The paper is an extension of the previous result obtained in one-dimensional setting by Bresch et al. to the multi-dimensional case with heterogeneous barrier for the density.     

From gas dynamics to pressureless gas dynamics

This paper is devoted to the convergence of solutions of the compressible Euler equations towards solutions of the pressureless gas dynamics system, when the pressure tends to 0. The goal is to prove accurate uniform bounds for particular solutions of the Euler equations.

     

A Roe-type scheme with low Mach number preconditioning for a two-phase compressible flow model with pressure relaxation

We describe two-phase compressible flows by a hyperbolic six-equation single-velocity two-phase flow model with stiff mechanical relaxation. In particular, we are interested in the simulation of liquid-gas mixtures such as cavitating flows. The model equations are numerically approximated via a fractional step algorithm, which alternates between the solution of the homogeneous hyperbolic portion of the system through Godunov-type finite volume schemes, and the solution of a system of ordinary differential equations that takes into account the pressure relaxation terms. When used in this algorithm, classical schemes such as Roe’s or HLLC prove to be very efficient to simulate the dynamics of transonic and supersonic flows. Unfortunately, these methods suffer from the well known difficulties of loss of accuracy and efficiency for low Mach number regimes encountered by upwind finite volume discretizations. This issue is particularly critical for liquid-gasmixtures due to the large and rapid variation in the flow of the acoustic impedance. To cure the problem of loss of accuracy at low Mach number, in this work we apply to our original Roe-type scheme for the two-phase flow model the Turkel’s preconditioning technique studied by Guillard–Viozat [Computers & Fluids, 28, 1999] for the Roe’s scheme for the classical Euler equations.We present numerical results for a two-dimensional liquid-gas channel flow test that show the effectiveness of the resulting Roe-Turkel method for the two-phase system.     

Duality solutions for pressureless gases,monotone scalar conservation laws,and uniqueness

We introduce for the system of pressureless gases a new notion of solution, which consist in interpreting the system as two nonlinearly coupled linear equations. We prove In this setting existence of solutions for the Cauchy Problem, as well as uniqueness under optimal conditions on initlaffata. The proofs rely on the detailed study of the relations between pressureless gases, tie dynamics of sticky particles and nonlinear scalar conservation laws with monotone initial data. We prove for the latter problem that monotonicit implies uniqueness. and a generalization of Oleinik's entropy condition     

A Navier–Stokes–Fourier system for incompressible fluids with temperature dependent material coefficients

We consider a complete thermodynamic model for unsteady flows of incompressible homogeneous Newtonian fluids in a fixed bounded three-dimensional domain. The model comprises evolutionary equations for the velocity, pressure and temperature fields that satisfy the balance of linear momentum and the balance of energy on any (measurable) subset of the domain, and is completed by the incompressibility constraint. Finding a solution in such a framework is tantamount to looking for a weak solution to the relevant equations of continuum physics. If in addition the entropy inequality is required to hold on any subset of the domain, the solution that fulfills all these requirements is called the suitable weak solution. In our setting, both the viscosity and the coefficient of the thermal conductivity are functions of the temperature. We deal with Navier’s slip boundary conditions for the velocity that yield a globally integrable pressure, and we consider zero heat flux across the boundary. For such a problem, we establish the large-data and long-time existence of weak as well as suitable weak solutions, extending thus Leray [J. Leray, Sur le mouvement d’un liquide visquex emplissant l’espace, Acta Math. 63 (1934) 193–248] and Caffarelli, Kohn and Nirenberg [L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35 (6) (1982) 771–831] results, that deal with the problem in a purely mechanical context, to the problem formulated in a fully thermodynamic setting.     

Robust assortment optimization using worst-case CVaR under the multinomial logit model

We consider robust assortment optimization problems with partial distributional information of parameters in the multinomial logit choice model. The objective is to find an assortment that maximizes a revenue target using a distributionally robust chance constraint, which can be approximated by the worst-case Conditional Value-at-Risk. We show that our problems are equivalent to robust assortment optimization problems over special uncertainty sets of parameters, implying the optimality of revenue-ordered assortments under certain conditions.     

Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process

Using the weak asymptotic method, we approximate a triangular system of conservation laws arising from the so‐called generalized pressureless gas dynamics by a diagonal linear system. Then, we apply the usual method of characteristics to find approximate solution to the original system. As a consequence, we shall see how the delta shock wave naturally arises along the characteristics. Also, we propose a procedure that could be applied to more general systems of conservation laws. Copyright © 2009 John Wiley & Sons, Ltd.     

Microfoundations for diffusion price processes

We study microeconomic foundations of diffusion processes as models of stock price dynamics. To this end, we develop a microscopic model of a stock market with finitely many heterogeneous economic agents, who trade in continuous time, giving rise to an endogeneous pure-jump process describing the evolution of stock prices over time. When the number of agents in the market is large, we show that the price process can be approximated by a diffusion, with price-dependent drift and volatility coefficients that are determined by small excess demands and trading volume in the microscopic model. We extend the microscopic model further by allowing for non-market interactions between agents, to model herd behavior in the market. In this case, price dynamics can be approximated by a process with stochastic volatility. Finally, we demonstrate how heavy-tailed stock returns emerge when agents have a strong tendency towards herd behavior.     

Error estimates of a linear decoupled Euler–FEM scheme for a mass diffusion model

We present error estimates of a linear fully discrete scheme for a three-dimensional mass diffusion model for incompressible fluids (also called Kazhikhov–Smagulov model). All unknowns of the model (velocity, pressure and density) are approximated in space by C ⁰-finite elements and in time an Euler type scheme is used decoupling the density from the velocity–pressure pair. If we assume that the velocity and pressure finite-element spaces satisfy the inf–sup condition and the density finite-element space contains the products of any two discrete velocities, we first obtain point-wise stability estimates for the density, under the constraint lim_(h,k)→0 h/k = 0 (h and k being the space and time discrete parameters, respectively), and error estimates for the velocity and density in energy type norms, at the same time. Afterwards, error estimates for the density in stronger norms are deduced. All these error estimates will be optimal (of order O(h+k){\mathcal{O}(h+k)}) for regular enough solutions without imposing nonlocal compatibility conditions at the initial time. Finally, we also study two convergent iterative methods for the two problems to solve at each time step, which hold constant matrices (independent of iterations).     

Numerical investigation of pressure loss reduction in a power plant stack

The strengthened environmental laws require the power plants to reduce the emissions. Flue gas desulphurization and deNO_x involve adding chemicals to the flow stream, thereby resulting in increased mass flow. This problem could be overcome by reducing the pressure drop in the duct work and stack combination, so that a higher flow at reduced pressure drop can be handled by the existing fans. In this study, a power plant stack model of 1:40 was investigated numerically. The pressure reduction was achieved by introduction of baffles with various orientations and turning vanes at the inlet of the stack. The flows were modeled and analyzed using commercial computational fluid dynamics (CFD) software Fluent 6.2. The numerical results were validated with the experimental data. The 30° baffle without turning vanes was found to be the optimum baffle angle in terms of the pressure loss reduction. Variation of axial velocity, swirling component and turbulence kinetic energy along the axis of the stack was analyzed to understand the mechanism of the pressure loss reduction in a power plant stack. Guidelines for further pressure loss reduction were provided based on the insight gained from the simulation results.     

基于正弦型光滑打磨函数对0-1规划问题的连续化求解方法

传统的求解0-1规划问题方法大多属于直接离散的解法.现提出一个包含严格转换和近似逼近三个步骤的连续化解法:(1)借助阶跃函数把0-1离散变量转化为[0,1]区间上的连续变量;(2)对目标函数采用逼近折中阶跃函数近光滑打磨函数,约束条件采用线性打磨函数逼近折中阶跃函数,把0-1规划问题由离散问题转化为连续优化模型;(3)利用高阶光滑的解法求解优化模型.该方法打破了特定求解方法仅适用于特定类型0-1规划问题惯例,使求解0-1规划问题的方法更加一般化.在具体求解时,采用正弦型光滑打磨函数来逼近折中阶跃函数,计算效果很好.     

Option pricing under a normal mixture distribution derived from the Markov tree model

We examine a Markov tree (MT) model for option pricing in which the dynamics of the underlying asset are modeled by a non-IID process. We show that the discrete probability mass function of log returns generated by the tree is closely approximated by a continuous mixture of two normal distributions. Using this normal mixture distribution and risk-neutral pricing, we derive a closed-form expression for European call option prices. We also suggest a regression tree-based method for estimating three volatility parameters σ, σ⁺, and σ⁻ required to apply the MT model. We apply the MT model to price call options on 89 non-dividend paying stocks from the S&P 500 index. For each stock symbol on a given day, we use the same parameters to price options across all strikes and expires. Comparing against the Black–Scholes model, we find that the MT model’s prices are closer to market prices.     

A scaling and energy equality preserving approximation for the 3D Navier-Stokes equations in the finite energy case

We discuss a new model (inspired by the work of Vishik and Fursikov) approximating the 3D Navier-Stokes equations, which preserves the scaling as in the Navier-Stokes equations and thus allows the study of self-similar solutions. Using some energy estimates and Leray’s limiting process, we show the existence of a solution of this model in the finite energy case, and the energy equality and inequality fulfilled by it. This approximation can be shown to converge to the Navier-Stokes equations using a mild approach based on the approximated pressure, and the solution satisfies Scheffer’s local energy inequality, an essential tool for proving Caffarelli, Kohn and Nirenberg’s regularity criterion. We also give a partial result of self-similarity satisfied by the approximated solution in the infinite energy case.     

Two compressible immiscible fluids in porous media

We consider a model of flow of two compressible and immiscible phases in a three-dimensional porous media. The equations are obtained by the conservation of the mass of each phase. This model is treated in its general form with the whole nonlinear terms. The only assumption concerns the dependence of densities on a global pressure. We obtain the existence of weak solutions under different kinds of degeneracies of the capillary terms.     

The Cahn–Hilliard equation with time-dependent constraint

We propose an abstract variational inequality formulation of the Cahn–Hilliard equation with a time-dependent constraint. We introduce notions of strong and weak solutions, and prove that a strong solution, if it exists, is a weak solution, and that the existence of a unique weak solution holds under an appropriate time-dependence condition on the constraint. We also show that the weak solution is a strong solution under appropriate assumptions on the data. Our abstract results can be applied to various concrete problems.