The classical literature on tensor analysis, and hence a preponderance of the turbulence literature, study the properties of tensors via their components. In fact, in the standard turbulence literature dating back to the 1930's, the use of Cartesian coordinates is the rule. Nevertheless, certain problems in turbulence may warrant the use of non-Cartesian coordinate systems. For example, a problem with spherical symmetry may be more easily analyzed via the use of a spherical coordinate system.

As a tensor equation is valid in any coordinate system, formulating the invariance conditions as tensor equations is a potent analytical concept. In this seminar, the derivation of the invariance equation for the two-point, second-order correlation tensor is given along with a discussion of the resulting conditions for the tensor's components. The question of when the tensor components have the same invariance properties as the tensor itself will be addressed. It will be shown that this condition can only occur when the basis for the coordinate system have these very same invariance properties. For example, the adage, "For homogeneous turbulence, the components of the two-point correlation tensor are only a function of the displacement vector between the points," is true only for a translationally invariant coordinate system, e.g., a Cartesian one.

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