Linearly forced isotropic turbulence at low Reynolds numbers

We investigate the forcing strength needed to sustain a flow using linear forcing. A critical Reynolds number $R_c$ is determined, based on the longest wavelength allowed by the system, the forcing strength and the viscosity. A simple model is proposed for the dissipation rate, leading to a closed expression for the kinetic energy of the flow as a function of the Reynolds number. The dissipation model and the prediction for the kinetic energy are assessed using direct numerical simulations and two-point closure integrations. An analysis of the dissipation-rate equation and the triadic structure of the nonlinear transfer allows to refine the model in order to reproduce the low-Reynolds-number asymptotic behavior, where the kinetic energy is proportional to $R-R_c$.

Figure 1

(a) Time evolution of the box-averaged kinetic energy for several distinct Reynolds numbers (DNS results). (b) Steady-state values of the kinetic energy as a function of the Reynolds number $R$. All DNS results are measured as averages during the statistically steady state. Also shown are model prediction (9) and results from EDQNM closure integration. (c) Same plots as in (b) but in double-logarithmic representation, plotted as a function of $R-R_c$.

Figure 2

Steady-state kinetic energy spectra. (a) EDQNM results for Reynolds numbers $[1.01\le R\le 100]$. For the lowest Reynolds number the spectrum consists of a peak around $\kappa ={\kappa }_0=1$, while for the largest values of $R$ the beginning of an inertial range starts to emerge. (b) DNS results for Reynolds numbers $[2.5\le R\le 60]$.

Figure 3

Normalized dissipation rate as a function of the kinetic-energy-based Reynolds number $R_K$ (DNS results and EDQNM results). Also plotted is the approximation resulting from Model 1.

Figure 4

Comparison of the refined model, Model 2 [resulting from Eq. (45)] with the DNS and EDQNM results. Also shown are the results corresponding to Model 2a [Eq. (29), blue dashed line] and Model 2b [Eq. (43) black dash-dotted line].