Spectral-scaling-based extension to the attached eddy model of wall turbulence

Two-dimensional (2D) spectra of the streamwise velocity component, measured at friction Reynolds numbers ranging from 2400 to $26000$, are used to refine a model for the logarithmic region of turbulent boundary layers. Here we focus on the attached eddy model (AEM). The conventional AEM assumes the boundary layer to be populated with hierarchies of self-similar wall-attached (Type A) eddies alone. While Type A eddies represent the dominant energetic large-scale motions at high Reynolds numbers, the scales that are not represented by such eddies are observed to carry a significant proportion of the total kinetic energy. Therefore, in the present study, we propose an extended AEM that incorporates two additional representative eddies. These eddies, named Type ${\mathrm{C}}_{\mathrm{A}}$ and Type SS, represent the self-similar but wall-incoherent low-Reynolds-number features and the non-self-similar wall-coherent superstructures, respectively. The extended AEM is shown to better predict a greater range of energetic length scales and capture the low- and high-Reynolds-number scaling trends in the 2D spectra of all three velocity components. A discussion on spectral self-similarity and the associated $k^{-1}$ scaling law is also presented.

Figure 1

(a) Two-dimensional spectrum of the streamwise velocity ($u$) at $z^{+}=2.6{{\mathrm{Re}}_{\tau }}^{1/2}$ for ${\mathrm{Re}}_{\tau }\approx 26000$; the blue contour lines represent $k_x{k}_y{\varphi }_{uu}/U_{\tau }^2=0.25,0.35$, and 0.45. (b) 2D spectrum obtained from attached eddy model with Type A eddies at same $z^{+}$ and ${\mathrm{Re}}_{\tau }$. The red solid and dashed lines denote the ${\lambda }_y/z\sim {\lambda }_x/z$ scaling and ${\lambda }_y/z\sim {({\lambda }_x/z)}^{1/2}$ relationship, respectively.

Figure 2

[(a) and (c)] Inner-flow scaling and outer-flow scaling, respectively, of the constant energy contour $k_x{k}_y{\varphi }_{uu}/U_{\tau }^2=0.15$, and [(b) and (d)] inner-flow scaling and outer-flow scaling, respectively, of the energetic ridges. The shaded regions in (a) and (b) represents the wall-coherent scales as per Baars et al. [30]. The boundary layer thicknesses $\delta$ at ${\mathrm{Re}}_{\tau }=2400$ and $26000$, calculated by fitting the velocity profile to the composite profile of Chauhan et al. [41], are 0.069 m and 0.337 m, respectively, and the friction velocities $U_{\tau }$ at ${\mathrm{Re}}_{\tau }=2400$ and $26000$ are 0.545 m/s and 1.231 m/s, respectively.

Figure 3

(a) Geometry of Type A, Type ${\mathrm{C}}_{\mathrm{A}}$, and Type SS representative packet eddies considered in the extended model with their respective contribution to the streamwise velocity. The packet eddies are constructed by aligning geometrically scaled $\mathrm{\Lambda }$-shaped hairpin vortices in the streamwise direction at a growth angle of ${10}^{\circ }$. $\mathcal{L},\mathcal{W}$, and $\mathcal{H}$ are the characteristic length, width, and height of the eddies, respectively. [(b), (c), and (d)] Schematics showing the organization of Type A, Type ${\mathrm{C}}_{\mathrm{A}}$, and Type SS eddies, respectively. Type A (wall attached) and Type ${\mathrm{C}}_{\mathrm{A}}$ (wall detached) follow a self-similar hierarchical organization, where the size and the probability density of eddies are directly and inversely proportional to $z$, respectively. The largest Type ${\mathrm{C}}_{\mathrm{A}}$ eddy is detached from the wall by $h_o$. Type SS is organized as a single hierarchy with $\mathcal{H}={\delta }_E$.

Figure 4

[(a), (e), and (i)] Two-dimensional spectra of $u$, [(b), (f), and (j)] inner-flow scaling, [(c), (g), and (k)] outer-flow scaling, and [(d), (h), and (l)] profile of turbulence intensity of Type A, Type SS, and Type ${\mathrm{C}}_{\mathrm{A}}$ representative eddies, respectively. Line contours represent a constant energy of $\max (k_x{k}_y{\varphi }_{uu}/U_{\tau }^2)/3$. Dark shade to light shade is $z/\delta =2.6{\mathrm{Re}}_{\tau }^{-1/2}$ to 0.15. The blue dashed and solid line contours in (i) are from the $h_o/\mathcal{H}=0$ (attached) case and the $h_o/\mathcal{H}=0.15$ case, respectively. The black solid and dashed lines in (a), (e), and (i) denote the ${\lambda }_y/z\sim {\lambda }_x/z$ scaling and ${\lambda }_y/z\sim {({\lambda }_x/z)}^{1/2}$ relationship, respectively.

Figure 5

Comparison of 2D spectra of $u$ from the extended AEM with experiments at $z^{+}=2.6{\mathrm{Re}}_{\tau }^{1/2}$ for [(a) and (b)] ${\mathrm{Re}}_{\tau }\approx 26000$ and [(c) and (d)] ${\mathrm{Re}}_{\tau }\approx 2400$. The line contour represents $\max \left(k_x{k}_y{\varphi }_{uu}^{+}\right)/4$. In (b) and (d) black, red, green, and blue contours represent composite, Type A, Type SS, and Type ${\mathrm{C}}_{\mathrm{A}}$ spectra, respectively, and the gray solid and dashed lines are the references for ${\lambda }_y/z\sim {\lambda }_x/z$ scaling and ${\lambda }_y/z\sim {({\lambda }_x/z)}^{1/2}$ relationship, respectively.

Figure 6

Inner-flow scaling of 2D spectra of $u$: (a) Experiments at ${\mathrm{Re}}_{\tau }\approx 26000$. (b) Composite spectra from the extended AEM at ${\mathrm{Re}}_{\tau }\approx 26000$. [(c), (d), and (e)] Highlighting Type ${\mathrm{C}}_{\mathrm{A}}$ (blue), Type A (red) and Type SS (green) contributions to the composite 2D spectra and [(f), (g), and (h)] highlighting Type ${\mathrm{C}}_{\mathrm{A}}$, Type A, and Type SS contributions to the composite 1D streamwise spectra. The line contours represent a constant energy of $\max \left(k_x{k}_y{\varphi }_{uu}^{+}{|}_{z^{+}=125}\right)/4$. The blue solid and dashed lines in (b) are the references for ${\lambda }_y/z\sim {\lambda }_x/z$ scaling and ${\lambda }_y/z\sim {({\lambda }_x/z)}^{1/2}$ relationship, respectively.

Figure 7

Outer-flow scaling of 2D spectra of $u$ when the wavelengths are normalized by the boundary layer thickness $\delta$ (in experiments) or the height of the largest eddy ${\delta }_E$ (in AEM). Details of the plots are the same as in Fig. 6.

Figure 8

[(a)–(d)] Comparison of the extended AEM with DNS of Lee and Moser [57] at ${\mathrm{Re}}_{\tau }=5200$ and $z^{+}=2.6{\mathrm{Re}}_{\tau }^{1/2}$; [(a) and (b)] spectra of spanwise velocity ($v$); and [(c) and (d)] spectra of wall-normal velocity ($w$). [(e) and (f)] Spectra of $v$ and $w$, respectively, as predicted by the extended AEM for ${\mathrm{Re}}_{\tau }=26000$. The line contours in all panels represent a constant energy of $\max \left(k_x{k}_y{\varphi }^{+}\right)/6$. In (b), (d), (e), and (f) black, red, green, and blue contours represent composite, Type A, Type SS, and Type ${\mathrm{C}}_{\mathrm{A}}$ spectra, respectively. The gray solid and dashed lines denote ${\lambda }_y/z={\lambda }_x/z$ and ${\lambda }_y/z\sim {({\lambda }_x/z)}^{1/2}$, respectively.

Figure 9

Inner-flow scaling of 2D spectra of $v$ from (a) the DNS of Lee and Moser [57] and [(b), (c), and (d)] extended AEM highlighting the contributions from Type ${\mathrm{C}}_{\mathrm{A}}$ (blue), Type A (red), and Type SS (green) to the $v$ spectra, respectively, at $z^{+}=150,2.6{\mathrm{Re}}_{\tau }^{1/2}$, and $3.9{\mathrm{Re}}_{\tau }^{1/2}$ (dark to light shade, respectively). [(e)–(h)] Similar plots in outer-flow scaling. The line contours represent a constant energy of $\max \left(k_x{k}_y{\varphi }_{vv}^{+}{|}_{z^{+}=150}\right)/6$. The gray solid and dashed lines denote ${\lambda }_y={\lambda }_x$ and ${\lambda }_y\sim {\left({\lambda }_x\right)}^{1/2}$, respectively.

Figure 10

Inner-flow scaling of 2D spectra of $w$ from (a) the DNS of Lee and Moser [57] and [(b), (c), (d)] extended AEM highlighting the contributions from Type ${\mathrm{C}}_{\mathrm{A}}$ (blue), Type A (red), and Type SS (green) to the $w$ spectra, respectively, at $z^{+}=150,2.6{\mathrm{Re}}_{\tau }^{1/2}$, and $3.9{\mathrm{Re}}_{\tau }^{1/2}$ (dark to light shade respectively). [(e)–(h)] Similar plots in outer-flow scaling. The line contours represent a constant energy of $\max \left(k_x{k}_y{\varphi }_{ww}^{+}{|}_{z^{+}=150}\right)/6$. The gray solid line denotes ${\lambda }_y={\lambda }_x$.

Figure 11

(a) Variation of $m$ versus ${\mathrm{Re}}_{\tau }$ at $z^{+}\approx 150$ from experiments and the extended AEM. The solid black curve is the empirical fit of the form $m=1-C_1\exp (-{\mathrm{Re}}_{\tau }/C_2)$ from Chandran et al. [31], where $C_1=0.5$ and the value of $C_2$ is simply fitted to the data. [(b) and (c)] Two-dimensional spectrum and the associated 1D spectra at ${\mathrm{Re}}_{\tau }=2400$ and ${\mathrm{Re}}_{\tau }=60000$, respectively, from the extended AEM. The energy contribution of Type ${\mathrm{C}}_{\mathrm{A}}$ (blue), Type A (red), and Type SS (green) motions are plotted in (b) and (c).

Figure 12

Spectra of $u$ at $z/{\delta }_E\sim {10}^{-4}$ for an asymptotic high Reynolds number such that $z\gg \nu /U_{\tau }$. A decade of $k^{-1}$ plateau in both streamwise and spanwise spectra is highlighted. The small-scale and large-scale bounds of the $k^{-1}$ region is indicated in the plots. The energy contribution of Type ${\mathrm{C}}_{\mathrm{A}}$ (blue), Type A (red), and Type SS (green) motions are highlighted.