Nonlocal modulation of the energy cascade in broadband-forced turbulence

Classically, large-scale forced turbulence is characterized by a transfer of energy from large to small scales via nonlinear interactions. We have investigated the changes in this energy transfer process in broadband forced turbulence where an additional perturbation of flow at smaller scales is introduced. The modulation of the energy dynamics via the introduction of forcing at smaller scales occurs not only in the forced region but also in a broad range of length scales outside the forced bands due to nonlocal triad interactions. Broadband forcing changes the energy distribution and energy transfer function in a characteristic manner leading to a significant modulation of the turbulence. We studied the changes in this transfer of energy when changing the strength and location of the small-scale forcing support. The energy content in the larger scales was observed to decrease, while the energy transport power for scales in between the large and small scale forcing regions was enhanced. This was investigated further in terms of the detailed transfer function between the triad contributions and observing the long-time statistics of the flow. The energy is transferred toward smaller scales not only by wave numbers of similar size as in the case of large-scale forced turbulence, but by a much wider extent of scales that can be externally controlled.

Figure 1

Broadband forcing in spectral space. Large-scale forcing $k\le k_0$ with an additional high-$k$ forced region $k_1<k\le k_2$.

Figure 2

Schematic triadic interaction that occurs between wave vector $\mathbf{k}$ in shell ${\mathbb{S}}_k$ and wave vectors $\mathbf{p}$, $\mathbf{q}$ taken from regions $\mathbb{P}$ and $\mathbb{Q}$ of spectral space that each consist of four shells with central wave numbers $k_p$ and $k_q$.

Figure 3

Time-averaged total kinetic energy $\widehat{E}$ (solid), total energy dissipation rate $\widehat{\epsilon }$ (dotted), and Taylor-Reynolds number $R_{\lambda }$ (dashed) for forced turbulence with different locations of the second band at $\mathrm{Re}=1061$.

Figure 4

Compensated shell- and time-averaged energy spectrum $E_c\left(k\right)$ for two-band forcing: ${\mathbb{S}}_1$ and ${\mathbb{K}}_{p,p+3}$. Large-scale forcing ${\mathbb{S}}_1$ (solid), additional forcing in the second band ${\mathbb{K}}_{5,8}$ (dashed), ${\mathbb{K}}_{9,12}$ (dash-dotted), ${\mathbb{K}}_{17,20}$ $(▷)$, ${\mathbb{K}}_{25,28}$ $(◇)$ at $\mathrm{Re}=1061$.

Figure 5

Time-averaged total energy $\widehat{E}$ (solid), total energy dissipation rate $\widehat{\epsilon }$ (dotted), and Taylor-Reynolds number $R_{\lambda }$ (dashed) for two-band forced turbulence with varying strength in the second band ${\epsilon }_{w,2}$ at $\mathrm{Re}=1061$.

Figure 6

Compensated shell- and time-averaged energy spectrum $E_c\left(k\right)$ for forced turbulence in the band ${\mathbb{K}}_{17,20}$ at different strengths of forcing ${\epsilon }_{w,2}$ and $\mathrm{Re}=1061$. Large-scale forcing only (solid), additional second band forcing with ${\epsilon }_{w,2}=0.07$, 0.15, 0.30, 0.45, 0.60, 0.75, 0.90 denoted as $◻$, dotted, dashed, dash-dotted, $▷$, $◇$, $엯$, respectively. In each case ${\epsilon }_{w,1}=0.15$.

Figure 7

Normalized accumulated energy $S_E\left(k\right)$ for two-band ${\mathbb{S}}_1$ and ${\mathbb{K}}_{17,24}$ forced turbulence at $\mathrm{Re}=1061$. Large-scale forcing at ${\epsilon }_{w,1}=0.15$ (solid) with additional second band forcing at ${\epsilon }_{w,2}=0.15$, 0.30, 0.45, 0.60, 0.75, 0.90 is denoted by the dotted, dashed, dash-dotted, $▷$, $◇$, and $엯$ curves, respectively.

Figure 8

Time-averaged energy transfer $T\left(k\right)$ for two-band ${\mathbb{S}}_1$ and ${\mathbb{K}}_{17,20}$ forced turbulence for different strengths of forcing in the second band ${\epsilon }_{w,2}$ at $\mathrm{Re}=1061$. Large-scale forcing (solid) with additional second band forcing at ${\epsilon }_{w,2}=0.07$, 0.15, 0.30, 0.45, 0.60, 0.75, 0.90 denoted by $◻$, dotted, dashed, dash-dotted, $▷$, $◇$, and $엯$ curves, respectively.

Figure 9

Time-averaged transport power spectra for broadband forced turbulence in the ${\mathbb{K}}_{17,24}$ band at $\mathrm{Re}=1061$. Large-scale forcing (solid) with additional second band forced at ${\epsilon }_{w,2}=0.15$, 0.30, 0.45, 0.60, 0.75, 0.90 denoted as dotted, dashed, dash-dotted, $▷$, $◇$, $엯$, curves, respectively. The supplementary forced region $k_1<k\le k_2$ is denoted with dotted lines.

Figure 10

Time-averaged transport power spectra for Reynolds number $\mathrm{Re}=4243$ and forcing in the ${\mathbb{K}}_{17,20}$ band. Large-scale forcing (solid), additional broadband forcing in the ${\mathbb{K}}_{17,20}$ band with ${\epsilon }_{w,2}=0.30$ (dashed).

Figure 11

(Color online) The normalized triad energy transfer function ${\overline{T}}_{pq}(k,k_p,k_q)$ for $k∕\left(2\pi \right)=14$, 42, 82 in (a)–(c), respectively. The dashed lines correspond to the lower $\left(k_1\right)$ and upper $\left(k_2\right)$ wave numbers used in the broadband forcing at $\mathrm{Re}=4243$. The contour levels are $\pm 1∕2^n$, $n=0,\dots ,18$ that are the same for all three pictures.

Figure 12

(Color online) The normalized triad energy transfer function ${\overline{T}}_{pq}(k,k_p,k_q)$ for $k∕\left(2\pi \right)=46$ at $\mathrm{Re}=4243$. The upper (bottom) rectangle presents broadband (large-scale) forced turbulence, respectively. The contour levels are $\pm 1∕2^n$, $n=0,\dots ,24$.

Figure 13

(Color online) The normalized triad energy transfer function ${\overline{T}}_p(k,k_p)$ at $\mathrm{Re}=4243$ (run 19). The contour levels are $\pm 1∕2^n$, $n=0,\dots ,18$. The supplementary forced region $k_1<k\le k_2$ denoted with dashed lines.

Figure 14

The triad energy transfer function $T_p(k,k_p)$ at $k_p∕\left(2\pi \right)=30,34,\dots ,94$ and $\mathrm{Re}=4243$ (run 19).

Figure 15

The triad energy transfer function $T_p(k,k_p)$ at $k_p∕\left(2\pi \right)=30$ and $\mathrm{Re}=1061$. Large-scale forcing (solid) with additional second band forced at ${\epsilon }_{w,2}=0.07$, 0.15, 0.30, 0.45, 0.60, 0.75, 0.90 denoted as $◻$, dotted, dashed, dash-dotted, $▷$, $◇$, $엯$, respectively.