Finite Reynolds number corrections of the 4/5 law for decaying turbulence

We examine finite Reynolds number contributions to the inertial range solution of the third order structure functions $D_{3,0}$ and $D_{1,2}$ stemming from the unsteady and viscous terms. Under the assumption that the second order correlations $f$ and $g$ are self-similar under a coordinate change, we are able to rewrite the exact second order equations as a function of a normalized scale $\tilde{r}$ only. We close the resulting system of equations using a power law and an eddy-viscosity ansatz. If we further assume K41 scaling, we find the same Reynolds number dependence as previously in the literature. We proceed to extrapolate towards higher Reynolds numbers to examine the unsteady and viscous terms in more detail. We find that the intersection between the two terms, where their contribution to the solution of the structure function equations is relatively small, scales with the Taylor scale $\lambda$.

Figure 1

Temporal evolution of kinetic energy $k$ and dissipation $\varepsilon$. Times used in the analysis below are indicated with the dotted vertical black lines. Dashed lines corresponds to (a) $k\sim t^{-n}$ and (b) $\varepsilon \sim t^{-n-1}$ with $n=1.45$.

Figure 2

Balance of (a) the longitudinal second order structure function, Eq. (24), and ${\text{Re}}_{\lambda }=121.39$ and (b) the transverse second order structure function equation, Eq. (25), for ${\text{Re}}_{\lambda }=121.39$. $◯$, unsteady term; $\square$, transport term; $\diamond$, dissipation; $▿$, viscous term.

Figure 3

Balance of (a) the longitudinal second order structure function equation, Eq. (24), and (b) the transverse second order structure function, Eq. (25), for ${\text{Re}}_{\lambda }=254.75$. $◯$, unsteady term; $\square$, transport term; $\diamond$, dissipation; $▿$, viscous term.

Figure 4

Third order structure functions (a) ${\tilde{D}}_{3,0}$ and (b) ${\tilde{D}}_{1,2}$ for ${\text{Re}}_{\lambda }=121.39$ as evaluated from DNS ($◯$) and compared to the power law closures, Eqs. (32) and (29) ($\square$), with parameters from Table 4.

Figure 5

Third order structure functions (a) ${\tilde{D}}_{3,0}$ and (b) ${\tilde{D}}_{1,2}$ for ${\text{Re}}_{\lambda }=254.75$ as evaluated from DNS ($◯$) and compared to the power law closures, Eqs. (32) and (29) ($\square$), with parameters from Table 4.

Figure 6

Third order structure functions (a) ${\tilde{D}}_{3,0}$ and (b) ${\tilde{D}}_{1,2}$ for ${\text{Re}}_{\lambda }=121.39$ as evaluated from DNS ($◯$) and compared to the eddy viscosity closure (solid line) with parameters from Table 3. Dashed lines correspond to model solutions without the unsteady terms.

Figure 7

Third order structure functions (a) ${\tilde{D}}_{3,0}$ and (b) ${\tilde{D}}_{1,2}$ for ${\text{Re}}_{\lambda }=254.75$ as evaluated from DNS ($◯$) and compared to the eddy viscosity closure (solid line) with parameters from Table 3. Dashed lines correspond to model solutions without the unsteady terms.

Figure 8

Normalized third order structure functions (a) ${\tilde{D}}_{3,0}$ and (b) ${\tilde{D}}_{1,2}$ extrapolated towards higher Reynolds numbers ${\text{Re}}_{\lambda }=625,1250,2500,5000,10000$ (lighter to darker color) using the eddy-viscosity closure.

Figure 9

Unsteady and viscous terms of (a) Eq. (24) and (b) Eq. (25) as evaluated for the extrapolated Reynolds numbers ${\text{Re}}_{\lambda }=625,1250,2500,5000,10000$ using the eddy-viscosity closure. Reynolds numbers are indicated by the same coloring as in Fig. 8.