Velocity and energy profiles in two- versus three-dimensional channels: Effects of an inverse- versus a direct-energy cascade

In light of some recent experiments on quasi two-dimensional (2D) turbulent channel flow we provide here a model of the ideal case, for the sake of comparison. The ideal 2D channel flow differs from its three-dimensional (3D) counterpart by having a second quadratic conserved variable in addition to the energy and the latter has an inverse rather than a direct cascade. The resulting qualitative differences in profiles of velocity $V$ and energy $K$ as a function of the distance from the wall are highlighted and explained. The most glaring difference is that the 2D channel is much more energetic, with $K$ in wall units increasing logarithmically with the Reynolds number ${\text{Re}}_{\tau }$ instead of being ${\text{Re}}_{\tau }$ independent in 3D channels.

Figure 1

(Color online) Mean velocity profiles as a function of distance from the wall, in wall units, in three- and two-dimensional channels, for four values of the Reynolds number ${\text{Re}}_{\tau }$, shifted up by five units for clarity. The (gray) symbols in the left panel represent numerical simulations [16]. Note that the log-law (dashed lines) in 3D with an invariant von-Kármán constant is increasing its range of validity whereas in 2D there is only an apparent log-law with a variable “constant” $\kappa$ (see insets).

Figure 2

(Color online) Left and middle panels: Profiles of the turbulent kinetic energy $K^{+}=K/\left(p^{\prime }L\right)$ in 3D and 2D channels, respectively. Symbols in the left panel are from numerical simulations [16]. Right panel: Reynolds shear stress $W^{+}=W/\left(p^{\prime }L\right)$ as a function of $y/L$ in three (solid lines) and two dimensions (dashed lines), respectively, for four different values of ${\text{Re}}_{\tau }$. The symbols are numerical simulations [16]. The inset in the left panel compares the profiles of $K$ in three and two dimensions at ${\text{Re}}_{\tau }=2003$ normalized to their maximal values.

Figure 3

(Color online) Wall units. Left panel: the energy balance between production, diffusion, and (minus) dissipation for 3D (solid lines) and 2D (broken lines) channels at ${\text{Re}}_{\tau }=2003$. Middle panel: the ${\text{Re}}_{\tau }$ dependence of typical values of the kinetic energy in 2D channels. Note the agreement with the estimate in Eq. (12). Right panel: averaged turbulent kinetic energy over the channel half-width $L$: $K_{\text{average}}=L^{-1}{\int }_0^{L}K\left(y\right)dy$. Inset: the quick saturation of $K_{\text{average}}^{+}$ in 3D channels.