Chapter 2 Fluid Dynamics

“Stands on shifting sands, the scales held in her hands. The wind it just whips her and wails and fills up her brigantine sails.”

The Stone Roses, Waterfall (1989).

This chapter introduces fluid dynamics for CFD. It describes: governing equations, i.e. conservation of mass, momentum and energy; and, associated physical models, e.g. for viscosity, heat conduction and thermodynamics.

The equations describe fluid motion, forces and heat in time and three-dimensional (3D) space. Vector notation provides a mathematical framework to present the equations in a compact form. It enables the equations to be presented independently of any co-ordinate system, e.g. Cartesian (eqn/eqn/eqn) or spherical (eqn/eqn/eqn). It includes a standard set of algebraic operations, e.g. the inner (dot) and outer products.

The notation helps to ensure that the terms in equations are unchanged, or invariant, under a co-ordinate system transformation. Without invariance, a flow solution, e.g. along a pipe, would be dependent on the orientation of the pipe with respect to the co-ordinate system. Logically this dependence cannot exist; the laws of motion are the same in all “inertial frames”.1

PICT\relax \special {t4ht=

The derivation of the governing equations uses a control volume eqn bounded by a surface eqn, presented using the two-dimensional (2D) illustration above. We use eqn and eqn to describe an infinitesimally small volume and surface, respectively, and eqn is the unit normal vector for each increment of surface eqn, discussed in Sec. 2.1 . It is important to note in any derivation whether the volume is defined as fixed in space or moving with the fluid.

Each derivation generally begins with a definite integral of some quantity, e.g. eqn, over the volume eqn denoted by

Z dV: V \relax \special {t4ht=
If this notation is unfamiliar, understand it to mean a summation for all increments of volume eqn that make up the total volume eqn. The summed values are eqn, where eqn is the value in the respective eqn.

The derivations also use integrals over the surface eqn, e.g.

Z Z n dS or (dS ); S S \relax \special {t4ht=
where eqn. Volume and surface integrals are connected through Gauss’s Theorem, introduced in Sec. 2.4 .
1Galileo Galilei, Dialogo sopra i due massimi sistemi del mondo, 1632.

Notes on CFD: General Principles - Chapter 2 Fluid Dynamics