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2009 | Buch

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Quasi-Gas Dynamic Equations

verfasst von: Tatiana G. Elizarova

Verlag: Springer Berlin Heidelberg

Buchreihe : Computational Fluid and Solid Mechanics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The monograph is devoted to modern mathematical models and numerical methods for solving gas- and ?uid-dynamic problems based on them. Two interconnected mathematical models generalizing the Navier–Stokes system are presented; they differ from the Navier–Stokes system by additional dissipative terms with a small parameter as a coef?cient. The new models are called the quasi-gas-dynamic and quasi-hydrodynamic equations. Based on these equations, effective ?nite-difference algorithms for calculating viscous nonstationary ?ows are constructed and examples of numerical computations are presented. The universality, the ef?ciency, and the exactness of the algorithms constructed are ensured by the ful?llment of integral conservation laws and the theorem on entropy balance for them. The book is a course of lectures and is intended for scientists and engineers who deal with constructing numerical algorithms and performing practical calculations of gas and ?uid ?ows and also for students and postgraduate students who specialize in numerical gas and ?uid dynamics.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Construction of Gas-Dynamic Equations by Using Conservation Laws

In this chapter, we recall physical principles serving as a base in deducing equations of the classical gas dynamics and the new quasi-gas-dynamic (QGD) and quasihydrodynamic (QHD) systems. As the Navier–Stokes equations, the QGD/QHD equations are a consequence of integral conservation laws, have a dissipative character, and can be obtained from the general system of conservation laws. A principal and substantial distinction of the QGD/QHD equations from the Navier–Stokes equations is the use of the time-spatial averaging procedure in order to find the main hydrodynamic quantities—the density, the velocity, and the temperature. The use of spatial averages leads to the Navier–Stokes system. For time-spatial averages, we propose two variants of closing the general system of equations, which lead to QGD/QHD systems. In this chapter, we present the expressions for the vectors of mass flux density, heat flux, and the tensor of viscous stresses for QGD/QHD systems without any deduction. Two methods for constructing these quantities for the QGD system are presented in Chap. 3. We discuss the physical meaning of the vector of mass flux. The presentation of this chapter is mainly based on [84, 181, 184, 190].

Tatjana G. Elizarova

Chapter 2. Elements of Kinetic Gas Theory

In this chapter, we present some aspects of kinetic theory. They will be used in deducing the quasi-gas-dynamic equations in Chap. 3, in constructing their generalizations (Chaps. 8 and 9), and in considering problems on the structure of a shock wave and the flow in microchannels (Appendices B and C). We present a schematic description of the kinetic DSMC algorithm,¹ which is widely used in numerical modelling of rarefied gas flows. Simulations within the framework of this approach were used for the verification of the QGD algorithm in modelling moderately rarefied flows. In the last section, we present a method for constructing kinetically consistent difference schemes whose differential analogs served as a basis for first variants of the QGD equations. The presentation in this chapter is based on [28, 51, 52, 54–56, 122, 127, 160, 184, 190].

Tatjana G. Elizarova

Chapter 3. Quasi-gas-dynamic Equations

In this chapter, we present two variants of constructing the quasi-gas-dynamic system, which allow us to obtain a concrete form of expressions for the vectors of mass flux density j _m, the viscous stress tensor Π, and the vector of heat flux q written early without deduction.

Tatjana G. Elizarova

Chapter 4. Quasi-gas-dynamic Equations and Coordinate Systems

In this chapter, the quasi-gas-dynamic equations are written in the tensor representations in arbitrary orthogonal coordinates, which allow one to use the QGD system in a coordinate system convenient for solving the problem considered. The written form of equations enlarges the class of problems solved to complicated computational domains, including three-dimensional domains, in which the coordinate grid can be given analytically or numerically. This allows one to construct homogeneous finite-difference schemes on quasi-orthogonal spatial grids in the transformed coordinate space (see [201]). In the last two sections, the QGD equations are written in Cartesian and cylindrical coordinate systems (see [80]).

Tatjana G. Elizarova

Chapter 5. Numerical Algorithms for Solving Gas-Dynamic Problems

In this chapter, we present numerical algorithms based on the quasi-gas-dynamic equations for solving gas-dynamic problems.

Tatjana G. Elizarova

Chapter 6. Algorithms for Solving the Quasi-gas-dynamic Equations on Nonstructured Grids

In this chapter, we generalize the proposed numerical algorithms to the case of nonstructured or irregular two-dimensional spatial grids. The use of irregular grids seems to be perspective for computing flows in domains with complicated boundaries. Moreover, the freedom of the calculator in choosing the location of nodes of the spatial grid allows him/her to approximate the flow zones with strong gradients in detail and diminish the dependence of the numerical solution from the direction of grid lines given a priori. This chapter is mainly based on the results of [71, 78].

Tatjana G. Elizarova

Chapter 7. Quasi-hydrodynamic Equations and Flows of Viscous Incompressible Fluids

This chapter is devoted to the study of the quasi-hydrodynamic system. This system was proposed and developed in the works of Yu. V. Sheretov (see, e.g., [180–182, 184]). In particular, Sheretov carried out a detailed study of this system, obtained it in the case of flows of viscous incompressible fluids in the Oberbeck–Boussinesq approximation, and constructed a number of exact solutions, which were compared with the corresponding solutions of the Navier–Stokes equations.

Tatjana G. Elizarova

Chapter 8. Quasi-gas-dynamic Equations for Nonequilibrium Gas Flows

In this chapter, we generalize the quasi-gas-dynamic equations to the case of gas flows with translational–rotational temperature nonequilibrium. Such a nonequilibrium is characteristic for moderately rarefied gases consisting of two-atom or poly-atom molecules [26–28, 122, 137, 168, 169, 212].

Tatjana G. Elizarova

Chapter 9. Quasi-gas-dynamic Equations for Binary Gas Mixtures

Numerical simulation of gas-mixture flows is of interest for many practical applications. In particular, effective modelling of non-reacting gas flows is a necessary stage preceding the construction of the gas model flows with chemical reactions (see [28, 99]).

Tatjana G. Elizarova

Backmatter

Titel: Quasi-Gas Dynamic Equations
verfasst von: Tatiana G. Elizarova
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-00292-2
Print ISBN: 978-3-642-00291-5
DOI: https://doi.org/10.1007/978-3-642-00292-2

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