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3 Cartesian tensors Various levels of constitutive modeling are used in turbulent ﬂow. The simplest is to assume that the stress tensor is proportional to the rate-of-strain tensor. This is the linear eddy viscosity model; more correctly, it is a tensorally linear relation. Mathematically, if τij is the stress and Sij is the rate of strain, then the linear constitutive model is τij − 13 δij τkk = νT Sij , where νT is the eddy viscosity. Tensoral linearity means the free indices, i, j , on the right-hand side are subscripts of a single tensor, not of a matrix product.

Correlations can be functions of position and time, or of relative position in the case of two-point correlations. The type of models used in engineering computational ﬂuid dynamics are for single-point correlations. It will become apparent in Chapter 3 on the Reynolds averaged Navier–Stokes equation why prediction methods for engineering ﬂows are based solely on single-point correlations. For now, it can be rationalized by noting that, in a three-dimensional geometry, single-point correlations are functions of the three space dimensions, while two-point correlations are functions of all pairs of points, or three plus three dimensions – imagine having to construct a computational grid in six dimensions!

In constitutive modeling and in equilibrium analysis, the Reynolds stress is a tensor function of the rate-of-strain and rateof-rotation tensors. In other words, tensor-valued functions of tensor arguments arise: φij = Fij (akl , δkl ) and τij = Gij (Skl , kl ). Tensoral consistency demands that the free indices be i, j on both sides of these equations. The question that arises is this: What constraints can be placed on possible forms of the functions Fij and Gij ? The simple answer is that, if there are no hidden arguments, then they must be isotropic functions of their arguments.