Multiscale analysis of the structure of homogeneous rotating turbulence

The structure of homogeneous rotating turbulence at moderate Reynolds number is investigated by analyzing the instantaneous statistics of the scale-dependent velocity gradient tensor perceived by a set of four fluid elements equally spaced. The relative orientations between dynamical vectors such as vorticity, rate-of-strain eigenframe, and vortex stretching vector, together with their orientations with the rotating frame, are measured by direct numerical simulation at different rotation rates. Measurements are performed in the entire inertial range of scales. The preferential orientation of turbulence with the rotating frame is found to be maximal at the scale of the horizontal large structures of the flow. The relative orientations between dynamical vectors exhibit a continuous and monotonic evolution with scale. Overall, the orientation properties reflect the Gaussianization and two-dimensionalization of turbulence under the effect of rotation. In particular, rotation suppresses some alignment properties valid in isotropic turbulence, which in turn induces a strong decrease of the enstrophy production and strain production rates. These results are found to be valid at all scales.

Figure 1

Instantaneous visualizations of the flow in horizontal planes for (a)–(d) $log\left(\right|\mathbf{\omega }|/{\langle {\mathbf{\omega }}^2\rangle }^{1/2})$ and (e) normalized vertical component of vorticity $\mathbf{\omega }·({\mathbf{\Omega }}_{\mathrm{rot}}/|{\mathbf{\Omega }}_{\mathrm{rot}}\left|\right)/{\langle {\mathbf{\omega }}^2\rangle }^{1/2}$: (a) isotropic turbulence, (b) ${\text{Ro}}^{\left(L\right)}=0.087$, (c) ${\text{Ro}}^{\left(L\right)}=0.051$, and (d) and (e) ${\text{Ro}}^{\left(L\right)}=0.027$. Snapshots (d) and (e) are plotted at the same time and in the same plane.

Figure 2

(a) PDF of the cosine of the angle between vortex stretching vector $\mathbf{W}\left(r_0\right)$ and rotation vector ${\mathbf{\Omega }}_{\mathrm{rot}}$ for $r_0\lesssim \eta$ and different values of the Rossby number. (b) Scale dependence of $\langle {cos}^2[\mathbf{W}\left(r_0\right),{\mathbf{\Omega }}_{\mathrm{rot}}]\rangle$ for different ${\text{Ro}}^{\left(L\right)}$. The horizontal line indicates the value $1/3$.

Figure 3

Scale dependence of $\langle {cos}^2[{\widehat{\mathbf{S}}}_i\left(r_0\right),{\mathbf{\Omega }}_{\mathrm{rot}}]\rangle$ for (a) $i=1$ (eigenvector associated with the largest rate of strain), (b) $i=2$ (intermediate eigenvector), and (c) $i=3$ (eigenvector associated with the smallest rate of strain). Blue squares denote ${\text{Ro}}^{\left(L\right)}=0.087$, red triangles ${\text{Ro}}^{\left(L\right)}=0.051$, and purple circles ${\text{Ro}}^{\left(L\right)}=0.027$. Horizontal lines indicate the value $1/3$.

Figure 4

(a) PDF of the normalized eigenvalues of strain $\widetilde{S_i}=S_i/{\left[\mathrm{Tr}\left({\mathbf{S}}^2\right)\right]}^{1/2}$ for different values of the Rossby number: isotropic (solid lines), ${\text{Ro}}^{\left(L\right)}=0.087$ (dashed lines), ${\text{Ro}}^{\left(L\right)}=0.051$ (dotted lines), and ${\text{Ro}}^{\left(L\right)}=0.027$ (dash-dotted lines), with $r_0\lesssim \eta$ in the main panel (in the inset $r_0\lesssim L)$. Also shown is the scale dependence of the mean normalized eigenvalues of strain with (b) $i=1$ (largest eigenvalue), (c) $i=2$ (intermediate eigenvalue), and (d) $i=3$ (smallest eigenvalue) for different values of the Rossby number: isotropic (open black circles), ${\text{Ro}}^{\left(L\right)}=0.087$ (blue squares), ${\text{Ro}}^{\left(L\right)}=0.051$ (red triangles), ${\text{Ro}}^{\left(L\right)}=0.027$ (closed purple circles). In the four panels the light gray lines indicate the values for a Gaussian ensemble of matrices $\mathbf{M}$.

Figure 5

(a) PDF of the cosine between vorticity $\mathbf{\Omega }\left(r_0\right)$ and the eigenvectors of the rate of strain ${\widehat{\mathbf{S}}}_i\left(r_0\right)$ for $r_0\lesssim \eta$ and different values of the Rossby number: isotropic (solid lines), ${\text{Ro}}^{\left(L\right)}=0.087$ (dashed lines), ${\text{Ro}}^{\left(L\right)}=0.051$ (dotted lines), and ${\text{Ro}}^{\left(L\right)}=0.027$ (dash-dotted lines). (b) Scale dependence of $\langle {cos}^2(\mathbf{\Omega },{\widehat{\mathbf{S}}}_i)\rangle$ for different Rossby numbers: isotropic (open circles), ${\text{Ro}}^{\left(L\right)}=0.087$ (squares), ${\text{Ro}}^{\left(L\right)}=0.051$ (triangles), and ${\text{Ro}}^{\left(L\right)}=0.027$ (closed circles). The horizontal line indicates the value $1/3$.

Figure 6

Scale dependence of (a) $\langle {cos}^2[\mathbf{\Omega }\left(r_0\right),{\widehat{\mathbf{S}}}_2\left(r_0\right)]\rangle$ and (b) $\langle {cos}^2[\mathbf{\Omega }\left(r_0\right),{\widehat{\mathbf{S}}}_3\left(r_0\right)]\rangle$ conditioned on the magnitude of ${cos}^2[\mathbf{\Omega }\left(r_0\right),{\mathbf{\Omega }}_{\mathrm{rot}}]$ and of ${cos}^2[{\widehat{\mathbf{S}}}_i\left(r_0\right),{\mathbf{\Omega }}_{\mathrm{rot}}]$, for ${\text{Ro}}^{\left(L\right)}=0.027$. Horizontal lines indicate the value $1/3$.

Figure 7

PDF of the cosine between vorticity $\mathbf{\Omega }\left(r_0\right)$ and vortex stretching vector $\mathbf{W}\left(r_0\right)$ for (a) $r_0\lesssim \eta$ and different values of the Rossby number and (b) different values of $r_0$ (increasing values from red to blue) and ${\text{Ro}}^{\left(L\right)}=0.027$ in the main panel (or without rotation in the inset). The light gray lines are the distributions expected for a Gaussian ensemble of matrices $\mathbf{M}$. (c) Scale dependence of $\langle cos[\mathbf{\Omega }\left(r_0\right),\mathbf{W}\left(r_0\right)]\rangle$ for different values of ${\text{Ro}}^{\left(L\right)}$.

Figure 8

Scale dependence of the low-order moments of the perceived velocity gradient tensor $\mathbf{M}$ for different values of ${\text{Ro}}^{\left(L\right)}$: (a) second-order moments $\langle {\mathbf{\Omega }}^2\rangle /2$ (lines) and $\langle \text{Tr}\left({\mathbf{S}}^2\right)\rangle$ (symbols), (b) third-order moments $\langle \mathbf{\Omega }\mathbf{S}\mathbf{\Omega }\rangle$ (lines) and $-(4/3)\langle \text{Tr}\left({\mathbf{S}}^3\right)\rangle$ (symbols), (c) third-order moments $\langle \mathbf{\Omega }\mathbf{S}\mathbf{\Omega }\rangle$ (lines) and $2\langle {\mathbf{\Omega }}_{\mathrm{rot}}\mathbf{S}\mathbf{\Omega }\rangle$ (symbols), and (d) normalized enstrophy production $\langle \mathbf{\Omega }\mathbf{S}\mathbf{\Omega }\rangle /\left[\langle {\mathbf{\Omega }}^2\rangle {\langle \text{Tr}\left({\mathbf{S}}^2\right)\rangle }^{1/2}\right]$. In (a) and (b) the straight lines indicate the slopes $-4/3$ and $-2$, respectively. In all panels are the following values of the Rossby number: isotropic (open black circles), ${\text{Ro}}^{\left(L\right)}=0.087$ (blue squares), ${\text{Ro}}^{\left(L\right)}=0.051$ (red triangles), and ${\text{Ro}}^{\left(L\right)}=0.027$ (closed purple circles).