Mechanism of mean flow generation in rotating turbulence through inhomogeneous helicity

Recent numerical simulations showed that mean flow is generated in the inhomogeneous turbulence of an incompressible fluid that is accompanied by helicity and system rotation. In order to investigate the mechanism of the phenomenon, we perform a numerical simulation of inhomogeneous turbulence in a rotating system. In the simulation, an external force is applied to inject inhomogeneous turbulent helicity and the rotation axis is perpendicular to the inhomogeneous direction. The mean velocity is set to zero in the initial condition of the simulation. The simulation results show that the mean flow directed to the rotation axis is generated and sustained only in the case with both the helical forcing and the system rotation. We investigate the physical origin of this flow-generation phenomenon by considering the budget of the Reynolds-stress transport equation. The results indicate that the pressure diffusion term significantly contributes to the Reynolds-stress equation and supports the generated mean flow. The results also reveal that a model expression for the pressure diffusion is expressed by the turbulent helicity gradient coupled with the angular velocity of the system rotation. This implies that inhomogeneous helicity plays a significant role in the generation of the large-scale velocity distribution in incompressible turbulent flows.

Figure 1

Computational domain and schematic profiles of turbulent energy and helicity. Here $K^{GS}$ $\left(=\langle {\overline{u}}_i^{\prime }{\overline{u}}_i^{\prime }\rangle /2\right)$ and $H^{GS}$ $\left(=\langle {\overline{u}}_i^{\prime }{\overline{\omega }}_i^{\prime }\rangle \right)$ denote the turbulent energy and helicity of the grid scale motions, respectively. An external forcing is applied only around the $y=0$ plane.

Figure 2

Time evolution of the axial mean velocity for run 6. The horizontal axis denotes the time, the vertical axis denotes the inhomogeneous direction $y$, and the color contour denotes the value of ${\langle {\overline{u}}_x\rangle }_S$.

Figure 3

Mean axial velocity of each run.

Figure 4

Reynolds stress $R_{xy}$ for run 6. The green line with squares denotes the directly evaluated value $R_{xy}=R_{xy}^{GS}-2\langle {\nu }_{sgs}{\overline{s}}_{xy}\rangle$ and the red line with crosses denotes the value estimated by the eddy-viscosity model that is given by Eq. (3.9) with $C_{\nu }=0.09$.

Figure 5

Budget of the transport equation for $R_{xy}^{GS}$ for run 6. The $y$ coordinate is limited to the region at $-2\le y\le 2$ where a high mean velocity exists.

Figure 6

Approximate evaluation of ${\mathrm{\Pi }}_{xy}^{GS}$ for run 6.