2023 | Book

### About this book

The properties and effects of flows are important in many areas of science and engineering - their prediction can be achieved through analytical, experimental and computational fluid mechanics. In this essential, Karim Ghaib introduces computational fluid dynamics. After an overview of mathematical principles, the author formulates the conservation equations of fluid mechanics and explains turbulence models. He describes the most important numerical methods and then gives types and evaluation criteria of computational meshes. This essential book is thus recommended to both the beginner and the user in the field of computational fluid dynamics.

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### Table of Contents

##### Frontmatter

##### 1. Mathematical Basics

Abstract

Knowledge of differential calculus, vector algebra and analysis is assumed in the field of computational fluid dynamics. In the first two sections of this chapter, the differential calculus for functions of both one variable and several variables, on which fluid mechanics is based, is discussed. The third section describes the power series expansion of a function used in the discretization of differential equations. Methods for solving systems of linear equations or discrete difference equations in fluid mechanics are explained in the fourth section. In the last section, topics of vector algebra and analysis such as differentiation of a vector with respect to a parameter, differentiation of fields and surface integrals are treated.

##### 2. Conservation Equations

Abstract

A flow problem is described mathematically by the conservation equations for mass, momentum and energy. In this chapter, the conservation equations are derived by deducing them from a general approach. The partial nonlinear differential equations are treated in a Cartesian reference frame, where an infinitesimally small volume element is considered as a balance space, whose edges are each parallel to the corresponding coordinate axes. The volume element is considered fixed in space. It is assumed that the fluid is homogeneous, while a distinction is made between compressible and incompressible flow.

##### 3. Turbulent Flows

Abstract

The conservation equations can be solved directly numerically. In the case of turbulent flows, however, a very high resolution of the flow is required in order to be able to capture even small eddies. This requires an extremely fine computational mesh, which leads to very high computational effort. If a coarse computational mesh is chosen, the numerical solution either does not converge at all or leads to a nonsensical result. In order to be able to calculate turbulent flows with less computational effort, models based on assumptions have been developed. The best known representatives of these methods are the RANS and LES methods. These are described in this chapter. Their advantages and disadvantages are mentioned.

##### 4. Discretization of the Conservation Equations

Abstract

Up to now, the conservation equations can only be solved analytically for special cases such as stationary one-dimensional flows. For more complex flows, they are solved approximately numerically. In the first two sections of this chapter, numerical solution methods for solving the conservation equations are presented. These convert the partial derivatives in the conservation equations into finite differences. Approximation errors of the methods are also discussed here. As a result of the numerical solution methods, a linear system of equations of difference equations is obtained whose matrices are sparse matrices. Solution methods for this are given in the third section. At the end of this chapter, the boundary conditions for flow problems are described.

##### 5. Computational Mesh

Abstract

A computational mesh is a set of surfaces in the computational domain that decompose it into subdomains for which the numerical solution is to be determined. The computational mesh influences the accuracy of the discretization procedure in space and time and the quality of the achievable results, because meshes with poor quality can falsify the results of a numerical simulation to the point of unusability. In this chapter, the computational mesh generation is discussed. The classes of meshes are mentioned, and their advantages and disadvantages are explained. A method for evaluating the mesh fineness is described. Criteria for the evaluation of the quality of a computational mesh are shown.