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

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


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

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

  • 1
    $\begingroup$ Should the second invariant be also a scalar? $\endgroup$
    – nicoguaro
    Commented Aug 12, 2014 at 12:55

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.