Three-dimensional instabilities and negative eddy viscosity in thin-layer flows

The stability of flows in layers of finite thickness $H$ is examined against small-scale three-dimensional (3D) perturbations and large-scale two-dimensional (2D) perturbations. The former provide an indication of a forward transfer of energy while the latter indicate an inverse transfer and the possibility of an inverse cascade. The analysis is performed using a Floquet-Bloch code that allows examination of the stability of modes with arbitrary large-scale separation. For thin layers, the 3D perturbations become unstable when the layer thickness $H$ becomes larger than $H>c_1{(\nu {\ell }_{{}_U}/U)}^{1/2}=c_1{\ell }_{{}_U}{\text{Re}}^{-1/2}$, where $U$ is the rms velocity of the flown, ${\ell }_{{}_U}$ is the correlation length scale of the flow, $\nu$ is the viscosity, and $\text{Re}={\ell }_{{}_U}U/\nu$ is the Reynolds number. At the same time, large-scale 2D perturbations also become unstable by an eddy viscosity mechanism when $\text{Re}>c_2$, where $c_1,c_2$ are order 1 nondimensional numbers. These relations define different regions in parameter space where 2D and 3D instabilities can (co)exist and this allows us to construct a stability diagram. Implications of these results for fully turbulent flows that display a change of direction of cascade as $H$ is varied are discussed.

Figure 1

The growth rate $\gamma$ as a function of the wave number $q$ for five different values of the the viscosity $\nu$. The dotted lines show fits to a quadratic power law $\gamma =a{q}^2$. Left panel: linear scale. Right panel: The absolute value of $\gamma$ in log-log scale.

Figure 2

The value of ${\nu }_{\mathrm{eddy}}+\nu$ as a function of $\nu$. When this quantity takes negative values, there is a large-scale instability. The dashed line indicates where ${\nu }_{\mathrm{eddy}}=0$. The doted line shows the fitting to the asymptotic result ${\nu }_{\mathrm{eddy}}=-0.5/\nu$. Large scales become unstable when $\nu <{\nu }^{*}\simeq 0.7$.

Figure 3

Energy spectrum of the unstable modes $e^{i\mathbf{q}·\mathbf{x}}\tilde{\mathbf{v}}(\mathbf{x},t)$ for different values of $q$ and $\nu =0.2$, and $k_{{}_U}=1$ is the wave number of the laminar flow. The value of $q$ is indicated by the furthest point to the left.

Figure 4

The ratio of the energy in the large-scale mode $E_0=\frac{1}{2}{\left|\langle \tilde{\mathbf{v}}\rangle \right|}^2$ to the total energy $E=\frac{1}{2}\langle |\tilde{\mathbf{v}}{|}^2\rangle$ as a function of $q$ for different values of $\nu$. Left, on a linear scale; right, in log-log scale.

Figure 5

Left panel: The energy growth rate as a function of $q_z$. Right panel: The critical wave number $q_z^{*}$ below which $q_z$ modes are unstable as a function of the viscosity.

Figure 6

Stability diagram for a 2D flow.

Figure 7

The growth rate $\gamma$ as a function of the wave number $q_x$ in linear scale (left panel) and logarithmic scale (right panel) for different values of $\nu$ and for $r=0.4$.

Figure 8

The ratio $E_0/E$ of the large energy $E_0=\frac{1}{2}{\left|\langle \tilde{\mathbf{v}}\rangle \right|}^2$ to the total energy of the perturbation $E=\frac{1}{2}\langle |\tilde{\mathbf{v}}{|}^2\rangle$ as a function of $q$, for different values of $\nu$ in linear scale (left panel) and logarithmic scale (right panel). The color shade of the lines is the same for both panels.

Figure 9

Left panel: Effective viscosity as a function of $\nu$ for the different values of $r$. Right panel: Small-scale instability growth rate $\gamma (q=0)$ as a function of $\nu$ for different values of $r$.

Figure 10

Stability diagram of the parameter space.