On the Mean Flow Behaviour in the Presence of Regional-Scale Surface Roughness Heterogeneity

Abstract

A suite of large-eddy simulations of the neutral atmospheric boundary layer is conducted to study the mean flow response to the presence of surface roughness heterogeneity at regional scales (surface roughness heterogeneity on the scale of several boundary-layer heights). The roughness heterogeneity is imposed using alternating rough wall patches with numerically resolved rectangular roughness elements of different packing densities. The flow near the surface is found to adjust rapidly, reaching equilibrium conditions at distances on the order of a single inter-roughness element spacing. Despite the regional heterogeneity in surface roughness, it is often desirable to parametrize the entire rough wall using one single effective roughness height. To develop such a parametrization the model of Bou-Zeid et al. [Water Resources Research 40(2):1, 2004] is extended to incorporate the displacement height, d. Predictions from this parametrization are compared with the simulations, with reasonably good agreement.

Appendix: Fitting for \(z_o\), d

Both the effective roughness height $z_o$ and the zero-plane displacement d must be fitted for the cases XXXX. The friction velocity $u_\tau$ is directly measured from the LES model. With knowledge of $u_\tau$, the zero-plane displacement d is swept in $0<d<h$. For each d, a $z_o$ can be fitted. The zero-plane displacement is the d that yields the least-squares error between $U/u_\tau$ and ($1/\kappa ) \ln [(z-d)/z_o]$ in the fitted region. The corresponding $z_o$ is the effective roughness height. The fitting error is evaluated according to

$$\begin{aligned} err=\frac{1}{h}\int _{1.5h}^{2.5h}\left[ \frac{U(z)}{u_\tau }-\frac{1}{\kappa }\ln \left( \frac{z-d}{z_o}\right) \right] ^2dz, \end{aligned}$$

(10)

where $z=1.5h$ to $z=2.5h$ is the vertical range within which the logarithmic law is fitted.

We fit for $z_o$ and d for the case 1111 as an example. The drag D within $3\delta _0<x<9\delta _0$ is directly measured, the friction velocity is $u_\tau =\sqrt{D/\rho A}=0.079U_0$, where A is the planar area and $\rho$ is the fluid density, $U_0$ is the freestream velocity. d is swept from 0 to h and the square error in logarithmic fitting is computed from $z=1.5h$ to $z=2.5h$. The von Karman constant $\kappa$ is taken to be 0.4. Figure 13 shows the square error of the fitting. The minimum is located at $d=0.64$. Compared to Fig. 6, Fig. 13 is a more direct way to check the quality of the logarithmic law fitting. The corresponding $z_o=0.052h$.

Fig. 13

Square error (err) in logarithmic law fitting using d from 0 to h