Spectral proper orthogonal decomposition analysis of the turbulent wake of a disk at Re = 50 000

The coherent structures in the turbulent wake of a disk at a moderately high Reynolds number ($\text{Re}$) of $50000$ are examined using spectral proper orthogonal decomposition (SPOD) which considers all three velocity components in a numerical database. The SPOD eigenvalues at a given streamwise ($x$) location are functions of azimuthal wave number ($m$), frequency ($\text{St}$), and SPOD index ($n$). By $x/D=10$, two specific modes dominate the fluctuation energy: (i) the vortex shedding (VS) mode with $m=1,\text{St}=0.135,n=1$, and (ii) the double helix (DH) mode with $m=2,\text{St}\to 0,n=1$. The VS mode is more energetic than the DH mode in the near wake but, in the far wake, it is the DH mode which is dominant. The DH mode, when scaled with local turbulent velocity and length scales, shows self-similarity in eigenvalues and eigenmodes while the VS mode, which is a global mode, does not exhibit strict self-similarity. Modes $m=0$, 3, and 4, although subdominant, also make a significant net contribution to the fluctuation energy, and their eigenspectra are evaluated. The reconstruction of turbulent kinetic energy (TKE) and Reynolds shear stress, $\langle u_x^{\prime }{u}_r^{\prime }\rangle$, is evaluated by varying $(m,\text{St},n)$ combinations. Higher SPOD modes contribute significantly to the TKE, especially near the centerline. In contrast, reconstruction of $\langle u_x^{\prime }{u}_r^{\prime }\rangle$ requires far fewer modes: $\left|m\right|\le 4, \left|\text{St}\right|\le 1$, and $n\le 3$. Among azimuthal modes, $m=1$ and 2 are the leading contributors to both TKE and $\langle u_x^{\prime }{u}_r^{\prime }\rangle$. While $m=1$ captures the slope of the shear-stress profile near the centerline, $m=2$ is important to capture $\langle u_x^{\prime }{u}_r^{\prime }\rangle$ at and near its peak. SPOD is also performed in the vicinity of the disk to describe the modal transition to the principal contributors in the wake. The leading SPOD modes show a high-frequency shear-layer peak close to the disk and the vortex shedding mode commences its initial dominance of the wake at the end of the recirculation region.

Figure 1

Isosurfaces of $Q$ criterion. $Q=0.001$ of the filtered velocity for (a) $0<x/D<30$ and for (b) $30<x/D<100$ at a given time instant; (c) $Q=0.05$ of the residual field for $30<x/D<100$.

Figure 2

Two-dimensional contours of $Q$ criterion of the filtered velocity fields at the same time instant as in Fig. 1 at various streamwise locations: (a) $x/D=0.05$, (b) $x/D=0.5$, and (c) $x/D=1.05$.

Figure 3

Ratio of centerline rms velocity fluctuations and $|\langle u_x^{\prime }{u}_r^{\prime }\rangle {|}_{\max }^{1/2}$ to $U_d$ as a function of $x/D$.

Figure 4

TKE profiles scaled with $K_o$ at different streamwise locations ($20<x/D<120$) with radial direction scaled by (a) $D$ and (b) $L_k$.

Figure 5

Normal turbulent stresses [(a), (b), (c)] and $\langle u_x^{\prime }{u}_r^{\prime }\rangle$ (d) profiles at different streamwise locations ($20<x/D<120$) scaled by their maximum values at the respective locations. The radial direction is scaled by TKE-based wake width $L_k$.

Figure 6

SPOD contour maps showing energy contained in leading SPOD mode, ${\lambda }^{\left(1\right)}$, as a function of azimuthal wave number $m$ and frequency $\text{St}$ at different locations: (a) $x/D=20$, (b) $x/D=40$, (c) $x/D=80$, and (d) $x/D=100$. The color-bar limits are set according to the maximum values of ${\lambda }^{\left(1\right)}$ over all ($m, \text{St}$) pairs at the respective $x/D$ locations.

Figure 7

Frequency-integrated eigenspectrum as a function of azimuthal mode number $m$ at different locations: (a) $x/D=20$, (b) $x/D=40$, (c) $x/D=80$, and (d) $x/D=100$. Three leading SPOD modes (${\lambda }^{\left(1\right)}, {\lambda }^{\left(2\right)}$, and ${\lambda }^{\left(3\right)}$) at each $m$ are shown in terms of their percentage contributions to the area-integrated TKE.

Figure 8

SPOD eigenspectra of 25 modes (dark to light shade corresponds to high to low energy eigenvalues): (a) $m=1, x/D=20$; (b) $m=1, x/D=80$; (c) $m=2, x/D=20$; and (d) $m=2, x/D=80$.

Figure 9

Evolution of leading SPOD mode of $m=1, \text{St}=0.135$ and $m=2, \text{St}\to 0$ as $x/D$.

Figure 10

${\lambda }^{\left(1\right)}$ of (a) $m=1$ and (b) $m=2$, scaled by ${\left(K_o^{1/2}{L}_k\right)}^2$ for $20<x/D<100$. Here $K_o$ is the centerline value of TKE and $L_k$ is the TKE-based wake width.

Figure 11

Instantaneous snapshots of the $u_x^{\prime }$ field showing the imprint of the $m=1$ (bottom row) and $m=2$ (top row) modes: [(a), (d)] at $x/D=10$, [(b), (e)] at $x/D=40$, and [(c), (f)] at $x/D=80$.

Figure 12

$r-t$ plot of the real part of the azimuthally decomposed velocity field: (a) $m=1$ at $x/D=10$ and (b) $m=2$ at $x/D=80$. Power spectra of time series: (c) $r/D=1, x/D=10$ for $m=1$, and (d) $r/D=2, x/D=80$ for $m=2$, with the $r/D$ locations shown by dashed black lines in panels (a) and (b), respectively.

Figure 13

SPOD eigenspectra of 25 modes (dark to light shade corresponds to high to low energy eigenvalues) for $m=0$ (left), $m=3$ (middle), and $m=4$ (right). Top row shows $x/D=20$ and bottom row shows $x/D=80$.

Figure 14

Leading SPOD eigenvalue (${\lambda }^{\left(1\right)}$), scaled by ${\left(K_o^{1/2}{L}_k\right)}^2$, is plotted over $20<x/D<100$ for the following modes: (a) $m=0$, (b) $m=3$, and (c) $m=4$, Here $K_o$ is the centerline value of TKE and $L_k$ is the TKE-based wake width.

Figure 15

Modulus of eigenmode shapes for all three velocity components corresponding to ${\lambda }^{\left(1\right)}$ of the DH and VS structures: Panels (a), (c), and (e) correspond to $u_r, u_{\theta }, u_x$ eigenmodes respectively of the DH structure and panels (b), (d), and (f) correspond to $u_r, u_{\theta }, u_x$ eigenmodes respectively of the VS structure. Inset figures show the two-dimensional structure of real part of corresponding eigenmodes at $x/D=50$.

Figure 16

Reconstruction of TKE from a low-order truncation that comprises the leading three SPOD modes of azimuthal modes $\left(\right|m|\le 4)$ with energy summed over $-1\le \text{St}\le 1$: (a) $x/D=20$, (b) $x/D=40$, (c) $x/D=80$, and (d) $x/D=100$. The radial direction is scaled with TKE-based wake width $L_k$ and TKE is scaled with centerline TKE $K_o$.

Figure 17

Reconstruction of TKE at $x/D=40$ using (a) $\left|m\right|\le 4, \left|\text{St}\right|\le 1, n\le 3$; (b) $\left|m\right|\le 10, \left|\text{St}\right|\le 1, n\le 3$; (c) $\left|m\right|\le 10, \left|\text{St}\right|\le 2, n\le 3$; and (d) $\left|m\right|\le 10, \left|\text{St}\right|\le 2, n\le 10$. Radial direction is scaled with TKE-based wake width $L_k$ and TKE is scaled with centerline TKE $K_o$. In each subplot from (b) to (d), one parameter is changed relative to the preceding subplot and that parameter is boldfaced in the legend.

Figure 18

Reconstruction of $\langle u_x^{\prime }{u}_r^{\prime }\rangle$ from the leading three SPOD modes of various azimuthal modes $\left(\right|m|\le 4)$ summed over $-1\le \text{St}\le 1$ at different streamwise locations: (a) $x/D=20$, (b) $x/D=40$, (c) $x/D=80$, and (d) $x/D=100$. Radial direction is scaled with TKE-based wake width $L_k$ and $\langle u_x^{\prime }{u}_r^{\prime }\rangle$ is scaled with ${\langle -u_x^{\prime }{u}_r^{\prime }\rangle }_{\max }$.

Figure 19

Reconstruction of $\langle u_x^{\prime }{u}_r^{\prime }\rangle$ at $x/D=40$ using (a) $\left|m\right|\le 4, \left|\text{St}\right|\le 1, n\le 3$; (b) $\left|m\right|\le 2, \left|\text{St}\right|\le 1, n\le 3$; (c) $\left|m\right|\le 2, \left|\text{St}\right|\le 0.5, n\le 3$; and (d) $\left|m\right|\le 2, \left|\text{St}\right|\le 0.5, n=1$. Radial direction is scaled with TKE-based wake width $L_k$ and $\langle u_x^{\prime }{u}_r^{\prime }\rangle$ is scaled with ${\langle -u_x^{\prime }{u}_r^{\prime }\rangle }_{\max }$. In each subplot from (b) to (d), one parameter is changed relative to the preceding plot and that parameter is boldfaced in the legend.

Figure 20

SPOD contour maps showing energy contained in leading SPOD mode, ${\lambda }^{\left(1\right)}$, as a function of azimuthal wave number $m$ and frequency $\text{St}$ at different locations near the disk: (a) $x/D=0.1$, (b) $x/D=1$, (c) $x/D=2$, and (d) $x/D=5$. The color-bar limits are set according to the maximum values of ${\lambda }^{\left(1\right)}$ over all ($m, \text{St}$) pairs at the respective $x/D$ locations.

Figure 21

Frequency-integrated eigenspectrum as a function of azimuthal mode number $m$ at different locations in the near wake: (a) $x/D=0.1$, (b) $x/D=1$, (c) $x/D=2$, and (d) $x/D=5$. Three leading SPOD modes (${\lambda }^{\left(1\right)}, {\lambda }^{\left(2\right)}$, and ${\lambda }^{\left(3\right)}$) at each $m$ are shown in terms of their percentage contributions to the area-integrated TKE.

Figure 22

SPOD eigenspectra of 25 modes (dark to light shade corresponds to high- to low-energy eigenvalues): (a) $m=0, x/D=0.1$; (b) $m=1, x/D=0.1$; (c) $m=2, x/D=0.1$; (d) $m=0, x/D=2$; (e) $m=1, x/D=2$; and (f) $m=2, x/D=2$.