Three-Dimensional Shear-Flow Instability Saturation via Stable Modes

Abstract: Turbulence in three dimensions ($3$D) supports vortex stretching that has long been known to accomplish energy transfer to small scales. Moreover, net energy transfer from large-scale, forced, unstable flow-gradients to smaller scales is achieved by gradient-flattening instability. Despite such enforcement of energy transfer to small scales, it is shown here not only that the shear-flow-instability-supplied $3$D-fluctuation energy is largely inverse-transferred from the fluctuation to the mean-flow gradient, but that such inverse transfer is more efficient for turbulent fluctuations in $3$D than in two dimensions ($2$D). The transfer is due to linearly stable eigenmodes that are excited nonlinearly. The stable modes, thus, reduce both the nonlinear energy cascade to small scales and the viscous dissipation rate. The vortex-tube stretching is also suppressed. Up-gradient momentum transport by the stable modes counters the instability-driven down-gradient transport, which also is more effective in $3$D than in $2$D ($\mathrm{\approx} 70\% \mathrm{\,\, vs.\,\,}\mathrm{\approx} 50\%$). From unstable modes, these stable modes nonlinearly receive energy via zero-frequency fluctuations that vary only in the direction orthogonal to the plane of $2$D shear flow. The more widely occurring $3$D turbulence is thus inherently different from the commonly studied $2$D turbulence, despite both saturating via stable modes.