Turbulent Flows

From vector calculus we are familiar with scalars and vectors. A scalar has a single value, which is the same in any coordinate system. A vector has a magnitude and a direction, and (in any given coordinate system) it has three components. With Cartesian tensors, we can represent not only scalar and vectors, but also quantities with more directions associated with them. Specifically, an Nth-order tensor (N ≥ 0) has N directions associated with it, and (in a given Cartesian coordinate system) it has 3N components. A zeroth-order tensor is a scalar, and a first-order tensor is a vector. Before defining higher-order tensors, we briefly review the representation of vectors in Cartesian coordinates.

Cartesian coordinates and vectors

Fluid flows (and other phenomena in classical mechanics) take place in the three-dimensional, Euclidean, physical space. As sketched in Fig. A.1, let E denote a Cartesian coordinate system in physical space. This is defined by the position of the origin O, and by the directions of the three mutually perpendicular axes. The unit vectors in the three coordinate directions are denoted by e1, e2, and e3. We write ei to refer to any one of these, with the understanding that the suffix i (or any other suffix) takes the value 1, 2, or 3.

The basic properties of the unit vectors ei are succinctly expressed in terms of the Kronecker delta δij.