Finite-Reynolds-number effects on the structure functions in turbulent flow with external intermittency

High-resolution direct numerical simulation data of a turbulent plane jet at Reynolds numbers

and

10 Superscript 5 10 5 $10^5$

, based on the nozzle width, are employed to investigate the behaviour of the structure functions in the flow region with external intermittency. In the intermittent turbulent region, the conditional second-order longitudinal and transverse structure functions exhibit self-similar behaviour at small and intermediate scales. Moreover, as the order increases from the second to the sixth, the conditional higher even-order structure function profiles progressively deviate from self-similarity and the predictions of Kolmogorov’s universal equilibrium theory. The conditional third-order structure function only displays self-similarity within the small-scale dissipation range, albeit the range of self-similarity extends to progressively larger values of

r divided by eta Superscript upper T r / η T $r/\eta ^T$

for a higher Reynolds number, where

eta Superscript upper T η T $\eta ^T$

denotes conditional Kolmogorov length scale and

r r $r$

is the separation distance. As the intermittency factor decreases, the unsteady term in the Kármán–Howarth equation becomes more significant, leading to a larger deviation from Kolmogorov’s

4 divided by 5 4 / 5 $4/5$

law. The extrapolation results based on the empirical formula for the structure function

left angle bracket delta u Superscript prime n Baseline right angle bracket left parenthesis r right parenthesis ⟨ δ u ′ n ⟩ ( r ) $\langle \delta u^{\prime n}\rangle (r)$

indicate that the finite-Reynolds-number effect on the structure function may differ between the intermittent and fully turbulent regions. The structure functions in the intermittent region may follow the predictions of K41 theory, i.e.

left angle bracket delta u Superscript prime n Baseline right angle bracket left parenthesis r right parenthesis tilde r Superscript n divided by 3 ⟨ δ u ′ n ⟩ ( r ) ∼ r n / 3 $\langle \delta u^{\prime n}\rangle (r) \sim r^{n/3}$

, which is consistent with the results observed in the fully turbulent region. However, the realisation of Kolmogorov’s predictions is more difficult in the intermittent region than in the fully turbulent region and requires a much higher local Reynolds number. It is further found that the conditional third-order structure function profiles recover self-similarity at intermediate scales when the local Reynolds number exceeds

10 Superscript 5 10 5 $10^5$

. These findings provide valuable insights into the understanding and modelling of mixing transition problems.