The turbulence dissipation constant is proportional to the number of large-scale stagnation points and is therefore not universal

Bos et al (PoF 19, 045101, 2007) showed how the turbulence dissipation constant $C_{\epsilon}$ can differ between stationary and decaying homogeneous isotropic turbulence (HIT) and Mazellier \& Vassilicos (PoF 20, 015101, 2008) showed how this constant is in fact proportional to the third power of the number of large-scale zero-crossings of a 1D velocity component signal sampled from a 3D HIT. Their result implies and quantifies the non-universality of $C_{\epsilon}$ and was obtained by application of the Rice theorem and the 2/3 scaling-range scaling of the number density of zero crossings (Davila \& Vassilicos PRL 91, 144501, 2003). We generalise the Rice theorem to stagnation points and use it in conjunction with the exponent 2 scaling-range scaling of the number density of stagnation points (Davila \& Vassilicos 2003) to show that $C_{\epsilon}$ is proportional to the number of large-scale stagnation points in HIT. We run DNS of HIT with different low wavenumber energy spectra and show that different values of $C_{\epsilon}$ result from these different simulations, and that these different values are well accounted for by the differences in stagnation point structure of the different HIT flows. Our formula linking $C_{\epsilon}$ to this stagnation point structure allows to collapse all data into a single $Re_{\lambda}$-dependence curve and explains, quantitatively, the non-universality of $C_{\epsilon}$.