Detection of small-scale/large-scale interactions in turbulent wall-bounded flows

The structure of turbulent wall-bounded flows is comprised by a complete spectrum of scales. Studies showed that the large-scale structures of the outer layer influence near-wall small-scale structures by several mechanisms, namely, superposition, amplitude and frequency modulation, and distortions by sweeps and ejections. Most analyses focus on one specific aspect of these phenomena, which means that the full interaction scenario is not comprehensively analyzed. In this paper, a method is presented which targets the combined effect of superposition, amplitude modulation, and distortions on the near-wall dynamics. It is validated against synthetic signals to provide a reliable tool for further analysis including the scale separation technique and the detection and elimination of large-scale influences. For the sake of clarity, the proposed method is applied to one-dimensional signals. Besides synthetic data, one-dimensional time-dependent streamwise velocity fluctuations of a near-wall and an outer-layer location extracted from a three-dimensional direct numerical simulation of a turbulent channel flow at ${\text{Re}}_{\tau }\approx 1000$ are analyzed. The paper shows that scale separation by the empirical mode decomposition approach and by conventional spectral filtering yields similar conclusions about the interaction mechanisms. However, the empirical mode decomposition is advantageous since it does not require any a priori knowledge about the analyzed flow and, thus, preprocessing and error sources are reduced. Furthermore, an approach for the investigation of the amplitude modulation is presented. Compared to existing techniques, it is superior in revealing this phenomenon due to a preceding removal of the influences of superposition and distortions and a different approach to determine the near-wall modulation. The near-wall and the outer-layer large-scale amplitudes are determined from the analytical large-scale signals, which are obtained by the Hilbert transform. Unlike conventional methods, this approach reveals the existence of a significant amplitude modulation across the channel half-height.

Figure 1

Synthetic near-wall signal $u_{\text{NW}}^{\prime }$.

Figure 2

The sifting process for an arbitrary 1D signal. (a) First iteration. (b) Second iteration.

Figure 3

Flowchart of the analysis of the interaction phenomena.

Figure 4

Scale separation of the synthetic near-wall signal using EMD. (a) IMF 1. (b) IMF 2. (c) Residual.

Figure 5

Interaction by superposition and distortions by sweeps and ejections (EMD). (a) Near-wall $u_{\text{NW,LS}}^{\prime }$ and imposed large-scale signals $u_{\text{OL,LS}}^{\prime }$, $u_{\text{OL,D}}^{\prime }$. (b) Near-wall signal after having removed the influences of superposition and distortions $u_{\text{NW,woSD}}^{\prime }$.

Figure 6

Interaction by AM (EMD); note the different scaling of the figures. (a) Near-wall large-scale amplitude ${\widehat{u}}_{\text{NW,SS-LS}}^{\prime }$ and imposed large-scale signals $u_{\text{OL,LS}}^{\prime },u_{\text{OL,D}}^{\prime }$ (AM1). (b) Near-wall ${\widehat{u}}_{\text{NW,LS2}}^{\prime }$ and imposed large-scale ${\widehat{u}}_{\text{OL,LS}}^{\prime }$ amplitudes (AM2).

Figure 7

Near-wall signal after having removed the influence of AM using $\text{AM2}{u}_{\text{NW,woSDAM2}}^{\prime }$ (EMD).

Figure 8

Near-wall $u_{\text{NW}}^{\prime }\left(y_{\text{NW}}^{+}\right)$ and outer-layer $u_{\text{OL}}^{\prime }\left(y_{\text{OL,E}}^{+}\right)$ DNS signals.

Figure 9

(a) Near-wall $u_{\text{NW,LS}}^{\prime }$ and outer-layer large-scale signals $u_{\text{OL,LS}}^{\prime }$ and (b) their correlation $R_{\text{SD}}$.

Figure 10

Near-wall signal after removing the influences of superposition and distortions $u_{\text{NW,woSD}}^{\prime }$.

Figure 11

Interaction by AM: (a), (b) near-wall large-scale amplitudes ${\widehat{u}}_{\text{NW,SS-LS}}^{\prime }$ and outer-layer large-scale signals $u_{\text{OL,LS}}^{\prime },u_{\text{OL,D}}^{\prime }$ (AM1) and (c), (d) near-wall ${\widehat{u}}_{\text{NW,LS2}}^{\prime }$ and outer-layer large-scale amplitudes ${\widehat{u}}_{\text{OL,LS}}^{\prime }$ (AM2). (a) EMD (AM1). (b) Spectral filtering (AM1). (c) EMD (AM2). (d) Spectral filtering (AM2).

Figure 12

Correlation of (a) near-wall large-scale amplitudes and outer-layer large scales $R_{\text{AM1}}$ (AM1) and (b) near-wall and outer-layer large-scale amplitudes $R_{\text{AM2}}$ (AM2).

Figure 13

Near-wall signal after having removed the influence of AM $u_{\text{NW,woSDAM}}^{\prime }$; note the different scaling. (a) AM1. (b) AM2.

Figure 14

Wall-normal evolution of correlation of near-wall and outer-layer large-scale amplitudes ${\overline{R}}_{\text{AM2,max}}$ using AM2.

Figure 15

Evolution of the synthetic near-wall signal with imposed modulations. (a) Universal small scales $u_{\text{NW,SS}}^{\prime }$. (b) Universal small scales with imposed intrawave FM $u_{\text{NW,SS,FM}}^{\prime }$. (c) Universal small scales with imposed intrawave FM and AM $u_{\text{NW,SS,FM,AM}}^{\prime }$. (d) Universal small scales with imposed intrawave FM, AM, and superposition $u_{\text{NW,SS,FM,AM,S}}^{\prime }$. (e) Universal small scales with imposed intrawave FM, AM, superposition, and distortions $u_{\text{NW}}^{\prime }$.

Figure 16

Scale separation of the synthetic near-wall signal using spectral filtering. (a) Scale separation of the synthetic near-wall signal. (b) Near-wall $u_{\text{NW,LS}}^{\prime }$ and imposed large-scale signals $u_{\text{OL,LS}}^{\prime },u_{\text{OL,D}}^{\prime }$.

Figure 17

Interaction by AM (spectral filtering); note the different scaling of the figures. (a) Near-wall large-scale amplitude ${\widehat{u}}_{\text{NW,SS-LS}}^{\prime }$ and imposed large-scale signals $u_{\text{OL,LS}}^{\prime },u_{\text{OL,D}}^{\prime }$ (AM1). (b) Near-wall ${\widehat{u}}_{\text{NW,LS2}}^{\prime }$ and imposed ${\widehat{u}}_{\text{OL,LS}}^{\prime }$ large-scale amplitudes (AM2).

Figure 18

(a) Near-wall and imposed large-scale signals and (b) amplitudes (AM2) with inappropriate cutoff frequencies.

Figure 19

Correlation of (a) near-wall and outer-layer large scales $R_{\text{SD}}$ and (b) near-wall and outer-layer large-scale amplitudes $R_{\text{AM2}}$.