Structure interactions in a reduced-order model for wall-bounded turbulence

New reduced -order models (ROMs) are derived for sinusoidal shear flow (also known as Waleffe flow) and plane Couette flow in small periodic domains. A first derivation for Waleffe flow exploits Fourier modes that form a natural orthonormal basis for the problem. A ROM for such basis is obtained by a Galerkin projection of the Navier-Stokes equation. A large basis was reduced to 12 modes that contribute significantly in maintaining chaotic, turbulent dynamics. A key difference from earlier ROMs is the inclusion of two roll-streak structures, with spanwise wavelengths equal to $L_z$ and $L_z/2$, where $L_z$ is the spanwise length of the computational box. The resulting system was adapted to Couette flow by rewriting the Galerkin system for the same 12 modes, modified so as to satisfy no-slip conditions on the walls. The resulting dynamical systems lead to turbulence with finite lifetimes, in agreement with earlier ROMs and simulations in small domains. However, the present models display lifetimes that are much longer than in earlier ROMs, with differences of more than an order of magnitude. The Couette-flow model is compared to results of direct numerical simulation (DNS), with statistics displaying fair agreement. The inclusion of the $L_z$ and $L_z/2$ length scales is seen to be a key feature for longer turbulence lifetimes: Neglecting any of the roll modes, or their nonlinear interaction, leads to drastic reductions of turbulence lifetimes. The present ROMs thus highlight some of the dominant nonlinear interactions that are relevant in maintaining turbulence for long lifetimes.

Figure 1

Sketch of geometry, coordinate system (red) and laminar solutions (blue lines and arrows) for Waleffe and Couette flow.

Figure 2

Sample model results for $\mathrm{Re}=200$. The temporal coefficients of selected modes, $a_1$ (mean flow), $a_4$ (streaks), and $a_{11}$ ($L_z/2$ streaks) are shown.

Figure 3

Waleffe-flow model lifetime statistics, taken from 1000 simulations with random initial conditions with norm equal to 0.3.

Figure 4

Turbulence lifetime of the Waleffe-flow model following an initial disturbance given by $(a_1-1)=a_2=\cdots =a_{12}=A$. Simulations carried out until $t=5000$.

Figure 5

Couette-flow model lifetime statistics, taken from 1000 simulations with random initial conditions with norm equal to 0.3.

Figure 6

Turbulence lifetime of the Couette-flow model following an initial disturbance given by $a_1=a_2=\cdots =a_{12}=A$. Simulations carried out until $t=5000$.

Figure 7

Comparison between statistics of model (full lines) and DNS (symbols).

Figure 8

Sample cross section of the flow as predicted by the model (top row) and extracted from filtered (middle row) and full DNS (bottom row). Colors show the instantaneous streamwise velocity $u$. Left and right columns refer to two different sample time steps. See the Supplemental Material for an animated version of this figure [51].

Figure 9

Energy budgets for $\text{Re}=500$. “Visc.” stands for losses related to the viscous term, whereas the other bars show energy contributions from each of the other linear and nonlinear terms in the equations, here represented by the mode coefficients in each term.

Figure 10

Median lifetimes of the model for Couette flow, neglecting the interaction between wall-normal vortices $a_6$ and $a_7$. Results from the full model, from Fig. 5, are repeated here for comparison.

Figure 11

Effect of computational domain on turbulence lifetimes. Results from Fig. 5 are compared to median lifetimes with other domain sizes, calculated from 1000 simulations starting from random initial condition with norm equal to 0.3.