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I am trying to calculate the second and third invariants for a turbulent flow. I have the second order statistics (both transient and averaged). i.e $uu$, $vv$, $ww$, $uv$, $vw$ and $uw$. These are supposed to be expanded as

$$II = b_{ij} / 2 \, ,$$

and

$$III = b_{ij} b_{jk} b_{ki}/3 \, ,$$

but I am stuck at this point on how to expand them further to get the second and third invariants.

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    $\begingroup$ Should the second invariant be also a scalar? $\endgroup$
    – nicoguaro ♦
    Commented Aug 12, 2014 at 12:55

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Besides fixing the second invariant, you need to know that repeated indicies imply summation. So $A_{ii}=\sum_{k=1}^d [A]_{kk}$. This is known as the Einstein Summation Convention. Each of the second-order statistics is a rank 3 tensor that you can sum up at each point of your domain using this convention in order to compute the invariants at those locations.

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