Internal intermittency

Internal intermittency is the fluctuating concentration of gradients, transfer and viscous dissipation inside turbulent fluid. It is distinct from an interface between turbulent and non-turbulent regions intermittently passing an observer. Its scale dependence is reflected in non-Gaussian increments and coarse-grained viscous dissipation moments.

Refined similarity writes $\Delta v=(r{ϵ}_r{)}^{1/3}V$ with universal normalized conditional statistics at high local Reynolds number. Then $S_p=C_p{r}^{p/3}⟨{ϵ}_r^{p/3}⟩$. The distribution of coarse-grained energy dissipation need not be universal. Refined similarity itself does not prescribe that distribution.

If $\log ({ϵ}_r/ϵ)$ is Gaussian with variance ${\sigma }_r^2$, normalization gives $⟨{ϵ}_r^q⟩/{ϵ}^q=\exp [q(q-1){\sigma }_r^2/2]$. Taking $\exp ({\sigma }_r^2)=B(\ell /r{)}^{\mu }$ predicts ${\zeta }_p=p/3-\mu p(p-3)/18$. Its eventual negative high-order exponents show that this extrapolation cannot hold for arbitrarily large orders with finite uniform velocity moments.

Coarse-grained energy dissipation is a spatial average of the local rate $2\nu S_{ij}{S}_{ij}$ over a region of size $r$. Its random value ${ϵ}_r$ has mean $ϵ$ in homogeneous turbulence. Refined similarity uses this local averaged viscous dissipation to scale velocity increments.

Integral-scale intermittency is modulation of the energy-containing motions and the energy supply on large lengths or long turnover times. Averaging velocity statistics over regions or realizations with differing viscous dissipation rates can invalidate universal coefficients based only on the global mean viscous dissipation.

Mixing turbulent populations with differing viscous dissipation gives $S_p={\beta }_p{r}^{p/3}⟨{ϵ}_a^{p/3}⟩$. Relative to the same power of the mean viscous dissipation, the coefficient changes with the intensity distribution except at $p=3$. Thus universality of all unconditional coefficients is incompatible with arbitrary integral-scale intermittency.