The Modeling And Numerical Simulation Of Gas Flow Networks

Traffic flow has been a continuous source of challenging mathematical problems. The following work is dedicated to recent questions in modeling, simulation and optimization of traffic flow networks. Mathematics can help to solve traffic problems in different ways. Modelling provides fundamental understanding of traffic dynamics and behaviour. Optimization yields solutions for complex situations and helps to organize traffic flow. During the last decade there has been intensive research in different fields of and related to traffic flow. One of the primary research activities focus on the development of new and more realistic models for traffic flow on a single road. Our work's primary focus is on models for networks. We provide new ideas on modelling flow in networks and solve different optimization problems analytically and numerically. The main result is the derivation of a hierarchy of models treating different situations with suitable traffic flow models. To each level of modeling we consider the optimal control problems and present techniques to address those problems. Furthermore, we derive an adjoint calculus for scalar hyperbolic equations with nonlinear boundary controls. The derived concepts fit for general network problems as well as they do for traffic flow issues. The principles of modeling and simplification can be applied to all kinds of network flows, like fluid flow in open channels or gas networks.

It is well established that non-equilibrium flows cannot properly be described by traditional hydrodynamics, namely, the Navier-Stokes-Fourier (NSF) equations. Such flows occur, for example, in micro-electro-mechanical systems (MEMS), and ultra vacuum systems, where the dimensions of the devices are comparable to the mean free path of a gas molecule. Therefore, the study of non-equilibrium effects in gas flows is extremely important. The general interest of the present study is to explore boundary value problems for moderately rarefied gas flows, with an emphasis on numerical solutions of the regularized 13--moment equations (R13). Boundary conditions for the moment equations are derived based on either phenomenological principles or on microscopic gas-surface scattering models, e.g., Maxwell's accommodation model and the isotropic scattering model.Using asymptotic analysis, several non-linear terms in the R13 equations are transformed into algebraic terms. The reduced equations allow us to obtain numerical solutions for multidimensional boundary value problems, with the same set of boundary conditions for the linearized and fully non-linear equations. Some basic flow configurations are employed to investigate steady and unsteady rarefaction effects in rarefied gas flows, namely, planar and cylindrical Couette flow, stationary heat transfer between two plates, unsteady and oscillatory Couette flow. A comparison with the corresponding results obtained previously by the DSMC method is performed. The influence of rarefaction effects in the lid driven cavity problem is investigated. Solutions obtained from several macroscopic models, in particular the classical NSF equations with jump and slip boundary conditions, and the R13--moment equations are compared. The R13 results compare well with those obtained from more costly solvers for rarefied gas dynamics, such as the Direct Simulation Monte Carlo (DSMC) method. Flow and heat transfer in a bottom heated square cavity in a moderately rarefied gas are investigated using the R13 and NSF equations. The results obtained are compared with those from the DSMC method with emphasis on understanding thermal flow characteristics from the slip flow to the early transition regime. The R13 theory gives satisfying results including flow patterns in fair agreement with DSMC in the transition regime, which the conventional Navier-Stokes-Fourier equations are not able to capture.

Includes following subjects: Solution of equations in Rn, Finite difference methods, Finite element methods, Techniques of scientific computing, Optimization theory and systems science, Numerical methods for fluids, Numerical methods for solids, Specific applications

The purpose of this book is to introduce researchers and graduate students to a broad range of applications of computational simulations, with a particular emphasis on those involving computational fluid dynamics (CFD) simulations. The book is divided into three parts: Part I covers some basic research topics and development in numerical algorithms for CFD simulations, including Reynolds stress transport modeling, central difference schemes for convection-diffusion equations, and flow simulations involving simple geometries such as a flat plate or a vertical channel. Part II covers a variety of important applications in which CFD simulations play a crucial role, including combustion process and automobile engine design, fluid heat exchange, airborne contaminant dispersion over buildings and atmospheric flow around a re-entry capsule, gas-solid two phase flow in long pipes, free surface flow around a ship hull, and hydrodynamic analysis of electrochemical cells. Part III covers applications of non-CFD based computational simulations, including atmospheric optical communications, climate system simulations, porous media flow, combustion, solidification, and sound field simulations for optimal acoustic effects.