Transition from non-swirling to swirling axisymmetric turbulence

Strictly axisymmetric turbulence, i.e., turbulence governed by the Navier-Stokes equations modified such that the flow is invariant in the azimuthal direction, is a system intermediate between two- and three-dimensional turbulence. We investigate statistically steady states of this system by direct numerical simulation using a forcing protocol, which allows the injected energy in the toroidal and in the poloidal directions to be tuned independently. A sharp transition between a two-dimensional two-component (2D2C, non-swirling) flow and a two-dimensional three-component (2D3C, swirling) flow is observed. We derive a statistical model which reproduces this transition.

Figure 1

Time evolution of the poloidal and toroidal energy components for: (a) $c_{\mathrm{P}}=3$, $c_{\mathrm{T}}=1$ ($c_{\mathrm{T}}/c_{\mathrm{P}}=1/3$) and (b) $c_{\mathrm{P}}=1$, $c_{\mathrm{T}}=3$ ($c_{\mathrm{T}}/c_{\mathrm{P}}=3$). Time is normalized by $\tau =R/{(2E/3)}^{1/2}$, where $E$ is the total kinetic energy of the flow, time averaged in the statistically steady state.

Figure 2

(a) Steady-state values of the poloidal and toroidal energies as a function of $c_{\mathrm{T}}/c_{\mathrm{P}}$, for the complete set of simulations. (b) Ratio between the energy components time averaged during the steady state, $\gamma =E_{\mathrm{T}}/E_{\mathrm{P}}$, plotted as a function of the forcing parameter ratio.

Figure 3

Contour plots of the time-averaged poloidal stream function $\psi$ defined as $u_r=-{\partial }_z\psi /r$, $u_z={\partial }_r\psi /r$. The average is performed during the statistically steady state. The time interval over which $\psi$ is averaged is (a) $4<t/\tau <16$ and (b) $9<t/\tau <22$. Forcing parameters are (a) $c_{\mathrm{P}}=3$, $c_{\mathrm{T}}=1$ ($c_{\mathrm{T}}/c_{\mathrm{P}}=1/3$) and (b) $c_{\mathrm{P}}=1.2$, $c_{\mathrm{T}}=3.6$ ($c_{\mathrm{T}}/c_{\mathrm{P}}=3$).

Figure 4

Time-averaged (a, b, c) energy spectra of the total, poloidal and toroidal energy, and (d, e, f) energy flux. Values of the forcing parameters: (a, d) $c_{\mathrm{P}}=3$, $c_{\mathrm{T}}=1$ ($c_{\mathrm{T}}/c_{\mathrm{P}}=1/3$); (b, e) $c_{\mathrm{P}}=2.77$, $c_{\mathrm{T}}=3.13$ ($c_{\mathrm{T}}/c_{\mathrm{P}}=1.1$); (c, f) $c_{\mathrm{P}}=1.2$, $c_{\mathrm{T}}=3.6$ ($c_{\mathrm{T}}/c_{\mathrm{P}}=3$). The vertical dashed lines indicate the range of wave numbers in which forcing is applied. The solid black line in panel (c) indicates a scaling proportional to $k$.

Figure 5

(a) Steady-state values of the poloidal and toroidal dissipation rates as a function of the forcing ratio $c_{\mathrm{T}}/c_{\mathrm{P}}$, for the complete set of simulations. (b) Value of the energy injection rates $F_{\mathrm{P}},F_{\mathrm{T}}$, plotted as a function of the energies $E_{\mathrm{P}},E_{\mathrm{T}}$.

Figure 6

Transfer from the toroidal to the poloidal energy components, $\mathcal{T}=\overline{〈u_{\theta }^2{u}_r/r〉}$, plotted as a function of (a) $c_{\mathrm{T}}/c_{\mathrm{P}}$ and (b) $\gamma$.

Figure 7

(a) Comparison of the model solution (22) to the data, for an optimal fit of the constants. Also shown is the asymptotic value obtained in the model for large values of $\zeta$ (dashed line). The dashed box is zoomed upon in panel (b). Here we see that the interval in which hysteresis is predicted by the model is very narrow (vertical dashed lines). This hysteresis is not observed in the simulations. The dashed black line represents the unstable branch of the solution of the model.