Energy transport due to pressure diffusion enhanced by helicity and system rotation in inhomogeneous turbulence

It is known that turbulent energy is rapidly transferred in the direction of the rotation axis in a rotating system, in comparison with the nonrotating case. In this study, this phenomenon is investigated as a problem of energy diffusion expressed by the Reynolds averaged Navier-Stokes (RANS) model. The conventional gradient-diffusion approximation for the turbulent energy flux cannot account for the enhanced energy transport observed in rotating inhomogeneous turbulence. In experiments, inhomogeneity of turbulence is modeled with an oscillating grid, leading to the turbulent energy falling off away from the grid. To adequately describe the phenomenon, we propose a new model for the energy flux due to the pressure associated with the rotational motion of a fluid. The model of the energy flux is expressed to be proportional to the turbulent helicity, which is the statistically averaged value of the inner product of the velocity and vorticity fluctuations. This property is closely related to the group velocity of inertial waves in a rapidly rotating fluid. The validity of the model is assessed using a direct numerical simulation of inhomogeneous turbulence under rotation where the flow configuration is similar to the oscillating-gird turbulence. It is shown that most of the turbulent energy transport enhanced by the system rotation is attributed to the pressure diffusion term. The spatial distribution of the energy flux due to the pressure related to the system rotation is similar to that of the turbulent helicity with negative coefficient. Hence, the new model which is proportional to the turbulent helicity is able to qualitatively account for the enhanced energy flux due to the system rotation. Finally, the helical Rossby number is proposed to estimate the relative importance of the energy flux enhanced by the turbulent helicity and the rotation, in comparison to the conventional gradient-diffusion approximation.

Figure 1

Schematic diagram for oscillating-grid turbulence. ${\mathrm{\Omega }}^{\mathrm{F}}$ denotes the angular velocity of the system rotation.

Figure 2

Spatial distribution of turbulent energy at each time for (a) run 0 and (b) run 1.

Figure 3

Spatial distribution of the turbulent energy at $2{\mathrm{\Omega }}^{\mathrm{F}}t=2$: (a) comparison between four runs of the rotating cases with the linear inviscid solution (linear inv.) and (b) comparison between the numerical solution of run 1, the linear inviscid solution, and the solution of run 1 obtained from the Navier-Stokes equation without the nonlinear term (linear vis.).

Figure 4

The growth of the the width of the turbulence region against time. In (b), time is normalized by the angular velocity of the system rotation. The result of the linear inviscid case is also plotted.

Figure 5

The budget of the turbulent energy transport equation for (a) run 0 at $t=1$, (b) run 1 at $t=1(2{\mathrm{\Omega }}^{\mathrm{F}}t=2)$, and (c) run 5 at $t=0.2(2{\mathrm{\Omega }}^{\mathrm{F}}t=2)$.

Figure 6

Comparison between the total energy flux due to turbulence, $\langle u_z^{\prime }{p}^{\prime }\rangle +\langle u_z^{\prime }{u}_i^{\prime }{u}_i^{\prime }/2\rangle$, and the gradient-diffusion approximation given by Eq. (11) for (a) run 0 and (b) run 1 at $t=1$ and $2(2{\mathrm{\Omega }}^{\mathrm{F}}t=2$ and 4 for run 1).

Figure 7

Comparison between the energy flux due to the nonlinearity, $\langle u_z^{\prime }{p}^{\mathrm{N}}{}^{\prime }\rangle +\langle u_z^{\prime }{u}_i^{\prime }{u}_i^{\prime }/2\rangle$, and the gradient-diffusion approximation given by the right-hand side of Eq. (11) for (a) run 08 and (b) run 1 at $2{\mathrm{\Omega }}^{\mathrm{F}}t=2$ and 4.

Figure 8

Spatial distribution of the turbulent helicity for (a) run 0 and (b) run 1, and (c) the rotational pressure flux for run 1 at each time.

Figure 9

Comparison between the rotational energy flux and its model given by Eq. (15) for (a) run 08 and (b) run 1 at $2{\mathrm{\Omega }}^{\mathrm{F}}t=2$ and 4.

Figure 10

Comparison between the rotational pressure flux and Eq. (28) for (a) run 08, (b) run 1, and (c) run 2 at $2{\mathrm{\Omega }}^{\mathrm{F}}t=2$ and 4.

Figure 11

Spatial distribution of the helical Rossby number given by Eq. (29) for each run at $2{\mathrm{\Omega }}^{\mathrm{F}}t=2$.

Figure 12

Spatial distribution of the simplified helical Rossby number given by Eq. (30) for each run at $2{\mathrm{\Omega }}^{\mathrm{F}}t=2$.