Formation of power-law scalings of spectra and multiscale coherent structures in the near-field of grid-generated turbulence

We investigate the streamwise evolutions of energy and pressure spectra along the shear-layer region of very near-field grid-generated turbulence. The energy and pressure spectra evolve significantly in this near-field. The shear-layer and vortex-shedding frequencies appear immediately in different spectra but the shear-layer's spectral signature is very soon replaced by a broad power-law spectrum on both sides of that frequency. The spectra evolve further by filling the gap between the vortex shedding and the shear-layer frequencies eventually leading to near $-5/3$ power-law energy spectra at the point on the shear-layer region where the turbulence intensity reaches a maximum. The pressure spectrum reaches a $-7/3$ power-law shape significantly further downstream. These spectral scalings cover a range between the vortex-shedding frequency and frequencies larger than the shear-layer frequency. They are discussed in relation to turbulent coherent structures of various sizes, obtained by using Gaussian low-pass filtering of instantaneous turbulent flow fields. High enstrophy small-scale structures originate from the shear-layer instability whereas low enstrophy large-scale structures originate from the vortex shedding. The generation of near-field $-5/3$ energy and $-7/3$ pressure spectra involves cooperative interactions between these two different size structures.

Figure 1

Schematic of the grid used in this study. The length $L_0$ and the thickness $t_0$ are defined in the figure. This single square grid is placed on the inlet plane $x=0$ of the numerical domain of size $L_x\times L_y\times L_z=8L_0\times 2L_0\times 2L_0$. The $x$ axis is vertically upward with respect to the page. The red-double square denotes the starting point of the shear-layer line shown in Fig. 2.

Figure 2

The mean field of streamwise velocity obtained by subtracting the inflow velocity $U_{\infty }$ on the plane at $z=0$. The time average is taken over the period $6.94\le t/(L_0/U_{\infty })\le 55.56$. The black line denotes the locations which satisfy $\langle u_x\rangle -U_{\infty }=0$. The cross-section of the grid shown in Fig. 1 is illustrated by the black rectangle.

Figure 3

The streamwise evolution of root mean square velocity components and pressure along the shear-layer line as a function of the streamwise coordinate. The time average is taken for the period $6.94\le t/(L_0/U_{\infty })\le 55.56$.

Figure 4

Space-time diagrams of (a) $u_x^{\prime }$, (b) $u_y^{\prime }$, (c) $u_z^{\prime }$, and (d) $p^{\prime }$. The horizontal axis shows the streamwise coordinate ($0\le x/x_{*}\le 0.315$), and the vertical axis shows the time ($48.6\le t/(L_0/U_{\infty })\le 51.6$).

Figure 5

Logarithmic plots of the compensated power spectra (a) $E_x{\mathrm{St}}^{\frac{5}{3}}$, (b) $E_y{\mathrm{St}}^{\frac{5}{3}}$, (c) $E_z{\mathrm{St}}^{\frac{5}{3}}$, and (d) $E_p{\mathrm{St}}^{\frac{7}{3}}$, against the Strouhal number $\mathrm{St}=f{t}_0/U_{\infty }$, at the six different locations in the shear-layer line: $x/x_{*}=0$, $0.01$, $0.031$, $0.058$, and $0.10$.

Figure 6

Streamwise evolution of power spectra (a) $E_x$, (b) $E_y$, (c) $E_z$, and (d) $E_p$. The horizontal axis shows the streamwise coordinate ($0\le x/x_{*}\le 0.315$).

Figure 7

Logarithmic plots of the power spectra (a) $E_x$, (b) $E_y$, (c) $E_z$, and (d) $E_p$, against the Strouhal number $\mathrm{St}=f{t}_0/U_{\infty }$. The spectra are computed at the six different streamwise locations in the shear-layer line: $x/x_{*}=0.058$, $0.1$, $0.13$, $0.16$, $0.2$, and $0.28$ from top to bottom. For clarity, the spectra shown are shifted downward along the vertical axis by multiplying by a factor of $5.0\times {10}^{-5}$, $5.0\times {10}^{-7}$, $5.0\times {10}^{-9}$, $5.0\times {10}^{-11}$, and $5.0\times {10}^{-13}$, with the exception of the ones on the top. The black-colored and blue-colored spectra belong to the production and decay regions, respectively. The red lines from top to bottom denote the slopes of power-law exponents (a), (b), (c) $-11/3,-7/3,-2,-5/3,-5/3,-5/3$ and (d) $-4,-3,-5/3,-5/3,-2,-7/3$.

Figure 8

Space-time evolution of coarse-grained enstrophy ${\tilde{Q}}_{\omega }$ along the shear-layer line. The cutoff wave numbers Eq. (11) used are (a) $k_c\left(2\right)$ and (b) $k_c\left(0.186\right)$. We observe the periodic behaviors with the period (a) about $0.1L_0/U_{\infty }$ in the production region at the scale $1/k_c\left(2\right)$ and (b) about $L_0/U_{\infty }$ in the decay region at the scale $1/k_c\left(0.186\right)$.

Figure 9

Visualizations of multiscale coherent vortices in the computational domain whose streamwise range and spanwise ranges are $0\le x/x_{*}\le 0.42$ and $-1\le y/L_0,z/L_0<1$. They are visualized by isosurfaces of (a) $Q$ and (b), (c) $\tilde{Q}$. The cutoff wave numbers used for low-pass filtering are (b) $k_c(\mathrm{St}=2.0)$ and (c) $k_c\left(0.186\right)$. The values of isosurfaces are (a) $12000{(U_{\infty }/L_0)}^2$, (b) $200{(U_{\infty }/L_0)}^2$, and (c) $1.5{(U_{\infty }/L_0)}^2$. (d) Superimposed visualization. The snapshots are taken at time $t=48.61L_0/U_{\infty }$. The grid width indicates $0.05x_{*}$.

Figure 10

Snapshot of multiscale coherent structures in the subdomain ($0\le x/x_{*}\le 0.315$, $-0.75\le y/L_0\le 0.25$). Red isosurfaces denote $Q=12000{(U_{\infty }/L_0)}^2$. Yellow isosurfaces denote $\tilde{Q}=120{(U_{\infty }/L_0)}^2$ with the cutoff wave number $k_c\left(2\right)$. Blue isosurfaces denote $\tilde{Q}=1.5{(U_{\infty }/L_0)}^2$ with the cutoff wave number $k_c\left(0.186\right)$. Transparent gray isosurface denote $u_x=U_{\infty }$. The snapshot is taken at time $t=48.89L_0/U_{\infty }$. The cuboid domains drawn with red and blue lines are used in Figs. 11 and 11, respectively. The grid width indicates $0.05x_{*}$.

Figure 11

Snapshot of coherent structures in the cropped domains shown in Fig. 10: (a) the cuboid domain with red lines ($0\le x\le 0.105$), and (b) the cuboid domain with blue lines ($0.157\le x\le 0.262$). $t=49.58L_0/U_{\infty }$. (a), (b) Red isosurfaces denote $Q=2000{(U_{\infty }/L_0)}^2$. Yellow isosurfaces denote $\tilde{Q}=120{(U_{\infty }/L_0)}^2$ with the cutoff wave number $k_c\left(2\right)$. (a) Transparent gray isosurface denote $u_x=U_{\infty }$. The grid width indicates $0.01x_{*}$.

Figure 12

Scale distribution of normalized spherically-averaged energy flux ${\mathrm{\Pi }}^{*}(\mathit{x},r_d)$ in physical space. $\mathit{x}$ is chosen at six different locations on the shear-layer line whose streamwise location is $x/x_{*}=0.031$, $0.058$, $0.10$, $0.16$, $0.20$, $0.28$.