Role of the forcing dimensionality in thin-layer turbulent energy cascades

We investigate the transition from forward to inverse energy cascade in turbulent flows in thin layers, varying the functional form of the forcing and the thickness of the layer. We show that, as the forcing function becomes more three dimensional, the inverse cascade is suppressed and the critical height $h_c$, where the transition occurs, is decreased. We study the dependence of this critical height on a parameter $r$ which measures the dimensionality of the forcing and thus construct a phase-space diagram in the parameter space $r-h$. We discuss the effect of Reynolds number and domain size.

Figure 1

Value of $r_{\epsilon }$ (the fraction of energy injected in the 3D flow) as a function of the parameter $r$ for different values of $h$, for the set of simulations L1Re2.

Figure 2

Flow visualization in terms of vorticity for three different values of $r$. The color scale is the same for the three subplots with red for positive vorticity and blue for negative vorticity. The simulations were done with $h=1/8$, for the set of simulations L1Re2.

Figure 3

(a)–(c) Instantaneous energy spectra for different values of $r$, and for $h=1/8$, after quasiequilibrium has been reached. Thin straight lines show scalings of $k^{-5/3}$ or $k^2$ as indicated in the annotations. The set of simulations is L2Re2 for the left-hand and center panels, and L1Re3 for the right-hand panels. The dotted vertical line indicates $k=1/H$. (d)–(f) Spectral energy fluxes for the same simulations, averaged over the quasiequilibrium state. Fluxes are normalized by the energy injection rate ${\epsilon }_{\mathrm{inj}}$.

Figure 4

Strength of the inverse cascade for different values of $r$ and $h$, shown as a function of $h$ [panel (a)] and $r$ [panel (b)]. Also shown in panel (b) are the same values as a function of $r_{\epsilon }$ (dashed lines). All these values correspond to the set of simulations L2Re1. Thin dashed lines in panel (a) show how the extrapolation is made for an estimate of the critical value.

Figure 5

Schematic of the regions in $r-h$ space where an inverse cascade occurs or not. The dots show the critical values estimated from this study, and the line is a hypothesis for the shape of the diagram limit.

Figure 6

Strength of the inverse cascade for different values of $h$, shown as a function of $h$. Different line styles correspond to different sets of simulations, as indicated in the legends. All the simulations were done with the parameter $r=0.5$. (a) Behavior in the large-box limit. (b) Behavior in the low-viscosity limit.