Water-channel estimation of Eulerian and Lagrangian time scales of the turbulence in idealized two-dimensional urban canopies

Lagrangian and Eulerian statistics are obtained from a water-channel experiment of an idealized two-dimensional urban canopy flow in neutral conditions. The objective is to quantify the Eulerian (TE) and Lagrangian (TL) time scales of the turbulence above the canopy layer as well as to investigate their dependence on the aspect ratio of the canopy, AR, as the latter is the ratio of the width (W) to the height (H) of the canyon. Experiments are also conducted for the case of flat terrain, which can be thought of as equivalent to a classical one-directional shear flow. The values found for the Eulerian time scales on flat terrain are in agreement with previous numerical results found in the literature. It is found that both the streamwise and vertical components of the Lagrangian time scale, TuL and TwL, follow Raupach’s linear law within the constant-flux layer. The same holds true for TwL in both the canopies analyzed (AR= 1 and AR= 2) and also for TuL when AR= 1. In contrast, for AR= 2 , TuL follows Raupach’s law only above z= 2 H. Below that level, TuL is nearly constant with height, showing at z= H a value approximately one order of magnitude greater than that found for AR= 1. It is shown that the assumption usually adopted for flat terrain, that β= TL/ TE is proportional to the inverse of the turbulence intensity, also holds true even for the canopy flow in the constant-flux layer. In particular, γ/ iu fits well βu=TuL/TuE in both the configurations by choosing γ to be 0.35 (here, iu= σu/ u¯ , where u¯ and σu are the mean and the root-mean-square of the streamwise velocity component, respectively). On the other hand, βw=TwL/TwE follows approximately γ/ iw= 0.65 / (σw/ u¯) for z> 2 H, irrespective of the AR value. The second main objective is to estimate other parameters of interest in dispersion studies, such as the eddy diffusivity of momentum (KT) and the Kolmogorov constant (C0). It is found that C0 depends appreciably on the velocity component both for the flat terrain and canopy flow, even though for the latter case it is insensitive to AR values. In all the three experimental configurations analyzed here, KT shows an overall linear growth with height in agreement with the linear trend predicted by Prandtl’s theory.