Generalised higher-order Kolmogorov scales

Abstract

Kolmogorov introduced dissipative scales based on the mean dissipation $\langle {\it\varepsilon}\rangle$ and the viscosity ${\it\nu}$, namely the Kolmogorov length ${\it\eta}=({\it\nu}^{3}/\langle {\it\varepsilon}\rangle )^{1/4}$ and the velocity $u_{{\it\eta}}=({\it\nu}\langle {\it\varepsilon}\rangle )^{1/4}$. However, the existence of smaller scales has been discussed in the literature based on phenomenological intermittency models. Here, we introduce exact dissipative scales for the even-order longitudinal structure functions. The derivation is based on exact relations between even-order moments of the longitudinal velocity gradient $(\partial u_{1}/\partial x_{1})^{2m}$ and the dissipation $\langle {\it\varepsilon}^{m}\rangle$. We then find a new length scale ${\it\eta}_{C,m}=({\it\nu}^{3}/\langle {\it\varepsilon}^{m/2}\rangle ^{2/m})^{1/4}$ and $u_{C,m}=({\it\nu}\langle {\it\varepsilon}^{m/2}\rangle ^{2/m})^{1/4}$, i.e. the dissipative scales depend rather on the moments of the dissipation $\langle {\it\varepsilon}^{m/2}\rangle$ and thus the full probability density function (p.d.f.) $P({\it\varepsilon})$ instead of powers of the mean $\langle {\it\varepsilon}\rangle ^{m/2}$. The results presented here are exact for longitudinal even-ordered structure functions under the assumptions of (local) isotropy, (local) homogeneity and incompressibility, and we find them to hold empirically also for the mixed and transverse as well as odd orders. We use direct numerical simulations (DNS) with Reynolds numbers from $Re_{{\it\lambda}}=88$ up to $Re_{{\it\lambda}}=754$ to compare the different scalings. We find that indeed $P({\it\varepsilon})$ or, more precisely, the scaling of $\langle {\it\varepsilon}^{m/2}\rangle /\langle {\it\varepsilon}\rangle ^{m/2}$ as a function of the Reynolds number is a key parameter, as it determines the ratio ${\it\eta}_{C,m}/{\it\eta}$ as well as the scaling of the moments of the velocity gradient p.d.f. As ${\it\eta}_{C,m}$ is smaller than ${\it\eta}$, this leads to a modification of the estimate of grid points required for DNS.