Towards an improved spatial representation of a boundary layer from the attached eddy model

This study compares the predicted synthetic flow fields generated based on the representative structures of the attached eddy model to experimental data captured using Particle Image Velocimetry of a turbulent boundary layer. To this end, wall-parallel and cross-stream planar fields are analyzed qualitatively and quantitatively by examining instantaneous flow features and by statistical two-point correlation functions, respectively. Our results reveal that although single-point flow statistics are in good agreement with the experimental data, a comparison of instantaneous flow fields and multipoint statistics between the attached eddy model and experiments shows differences in the spatial coherence. Based on these observations, a modification to the placement of the representative eddies in the attached eddy model is proposed that incorporates the meandering of these flow structures, which has been extensively reported in turbulent boundary layers. Our results reveal that this subtle modification provides a superior spatial representation of a turbulent boundary layer from the attached eddy model by reducing periodic effects and the overestimated spatial coherence. Similar improvements are also reported for the spatial representation of the spanwise velocity component.

Figure 1

Instantaneous streamwise velocity fluctuations $u^{+}$ on a wall-parallel plane ($xy$) at wall-normal height of $z/\delta \approx 0.06$ (within the logarithmic region). Left: Experimental data at ${\text{Re}}_{\tau }\approx 4200$. Right: Synthetic flow field generated based on the attached eddy model at ${\text{Re}}_{\tau }=3200$.

Figure 2

Schematic of a typical representative $\mathrm{\Lambda }$ eddy used to generate the synthetic flow fields from the attached eddy model, adapted from de Silva et al. [21]. The blue region corresponds to the low-momentum region that forms beneath the packet of $\mathrm{\Lambda }$ eddies and the red region corresponds to a higher-speed region.

Figure 3

Two-point correlation coefficient $R_{uu}$ computed on a wall-parallel plane at $z/\delta =0.06$. Left: Experimental database at ${\text{Re}}_{\tau }=4200$. Right: AEM synthetic flow fields at ${\text{Re}}_{\tau }=3200$. Solid lines represent positively correlated contour levels of $R_{uu}=0.1,0.3,0.5$ and the dotted lines show negative correlation contour at $R_{uu}=-0.05$.

Figure 4

Left: Schematic illustrating the meandering angle $\theta$ which is imposed on the representative AEM structures. Right: Probability distribution of $\theta$, where $\theta =0$ corresponds to the streamwise direction $x$. $\times$ symbols: Distribution based on results reported by Kevin et al. [24]. Solid line (-): Gaussian fit of the experimental data with similar standard deviation $\sigma$. Dashed line (- -): Gaussian fit of the experimental data with increased standard deviation $1.5\sigma$.

Figure 5

Probability distribution of the streamwise $u^{+}$ (left) and spanwise $v^{+}$ (right) fluctuations in a wall-parallel plane at $z/\delta =0.07$. Black line (-) corresponds to the original AEM. $\times$ symbols correspond to the modified AEM. $\circ$ symbols represent the experimental data at ${\text{Re}}_{\tau }=4200$.

Figure 6

Comparison of instantaneous streamwise velocity fluctuations fields $u^{+}$ based on the original AEM configuration (top), the modified AEM configuration (middle) and experiments (bottom). Left column: Wall distance $z/\delta =0.06$. Right column: Wall distance $z/\delta =0.23$.

Figure 7

Two-point correlation coefficient $R_{uu}$ for wall-parallel planes at $z/\delta =0.06$ (left column), $z/\delta =0.23$ (middle column), and $z/\delta =0.45$ (right column). Top row: original AEM configuration at ${\text{Re}}_{\tau }=3200$. Middle row: modified AEM configuration at ${\text{Re}}_{\tau }=3200$. Bottom row: Experiment at ${\text{Re}}_{\tau }=4200$. Solid lines represent contour levels of $R_{uu}=0.1,0.3,0.5$ and the dotted lines correspond to a contour level of $R_{uu}=-0.05$.

Figure 8

Comparison of the original AEM configuration ($\diamond$ symbols), the modified AEM configuration ($\times$ symbols), and the experimental data at ${\text{Re}}_{\tau }=4200$ ($\circ$). We note, $\times$ symbols correspond results computed based on the angle distribution $P\left(\theta \right)$ form Kevin et al. [24], while the $◃$ and $+$ symbols correspond to results computed from Gaussian distributions fitted to $P\left(\theta \right)$ with a standard deviation of $1\sigma$ and $1.5\sigma$, respectively. Left: Streamwise length L. Right: Spanwise width $W$. Length and width are calculated from the wall-parallel two-point correlations using a threshold of $R_{uu}=0.15$.

Figure 9

Instantaneous color contours of streamwise velocity $U/U_{\infty }$ on a cross-stream $yz$-plane. Left: Generated flow field based on the original AEM. Middle: Flow fields based on the modified AEM configuration. Right: Experimental data.

Figure 10

Two-point correlation coefficient $R_{uu}$ for a correlation point at $z/\delta =0.24$. Left: Original AEM configuration. Middle: Modified AEM configuration. Right: Experimental data. Solid lines represent positive correlated areas, dotted lines negative correlation.

Figure 11

A comparison of instantaneous spanwise velocity fluctuations $v^{+}$ based on the original AEM configuration (top), the modified AEM configuration (middle) and experiments (bottom). Left column: Wall distance $z/\delta =0.06$. Right column: Wall distance $z/\delta =0.23$.

Figure 12

Two-point correlation coefficient $R_{vv}$ for the spanwise velocity fluctuations computed on wall-parallel planes at $z/\delta =0.06$ (left column), $z/\delta =0.23$ (middle column) and $z/\delta =0.45$ (right column). Top row: Original AEM configuration at ${\text{Re}}_{\tau }=3200$. Middle row: Modified AEM configuration at ${\text{Re}}_{\tau }=3200$. Bottom row: Experiment at ${\text{Re}}_{\tau }=4200$. Solid lines represent contour levels of $R_{vv}=0.15,0.3,0.45$ and the dotted line corresponds to a contour level of $R_{vv}=-0.075$.

Figure 13

Conditioned two-point correlation coefficient $R_{vv}$ for a correlation point at $z/\delta =0.2$ (black) and $z/\delta =0.8$ (red). Top row: modified AEM configuration. Bottom row: Experimental data. Solid lines represent contour levels at $R_{vv}=0.3,0.4,0.5$. The left column represents all data, middle column conditioned in the correlation point on $v<-0.5{\sigma }_v$, right column conditioned on $v>0.5{\sigma }_v$.