Local similarity from Direct Numerical Simulations

In this work a steady, stably stratified, pressure driven smooth channel flow is studied by means of direct numerical simulation (DNS) of the Navier Stokes equations. The flow has a Reynolds number in the order ~7000. The same flow is also studied with a local-similarity model. Of course, the local similarity parameterization has its origin from atmospheric flow observations (with typical Reynold numbers O(10^8)). As such, the aim is to investigate if Reynolds-similarity holds. The simulations show a surprisingly close correspondence between the DNS and the local similarity model with respect to both the mean variable and the fluxes. This is a significant result because it shows that, even at those low Re-numbers, the DNS may be viewed as a useful surrogate of atmospheric turbulence under stably stratified conditions. With this result the DNS is used to compute the shape of the local similarity functions for the gradients of wind and temperature. This provides ‘benchmark MO-relationships' derived directly from fully controlled stationary homogeneous flow (see below). The dimensionless wind speed and temperature gradient both show a clear log-linear behaviour that is close to the ‘classical' 1+5 z/L (with L the local Obukhov length). As such our analysis is complementary to the earlier atmospheric findings by others that are often influenced by highly non-stationary and non-homogeneous effects. Finally, we show that analytical solutions of the particular flow under study are obtained that may facilitate future theoretical stability analysis. Fig. 1: Dimensionless gradients as a function of the stability parameter z/L , as obtained from the DNS. For comparison also the line 'PHI_m,h=1+5 z/L' is given.