The Theory of Turbulence by Edward A. Spiegel

Author:Edward A. Spiegel
Language: eng
Format: epub
Publisher: Springer Netherlands, Dordrecht

Or, since ξ j is not in general zero,

(10.28)

This gives the relation between Q 1 and Q 2 on whose account we may conclude that Q ij can be expressed in terms of one defining scalar.

10.3.2 Further Manipulations

If we take ∂ T ijk /∂ x k , we get an expression for a second order tensor, with symmetry in the indices. There result two equations, so that for a solenoidal isotropic tensor, only one independent defining scalar is required. We shall return to this matter presently. But first we elaborate on the nature of the second order tensor.

As we have seen, the isotropic tensors may be represented by a defining scalar, and so it is clear that the tensor equations governing isotropic turbulence may be transformed into scalar equations governing these defining scalars. The immediate problem is to pass to the scalar equations from the tensor equations. This requires a direct way of specifying the defining scalar. This may be done most directly for solenoidal tensors by introducing a ‘tensor potential’. That is, one can express a tensor solenoidal in the index j by the curl of a tensor with respect to that index. Clearly, this potential tensor must be skew-symmetric, as its curl must be an isotropic tensor. In the case of first order tensors, we have seen that there exist no non-vanishing isotropic vectors. We go then to second order tensors.

If q ij is a skew tensor, the isotropic, solenoidal tensor is

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