Optimizing the Determination of Roughness Parameters for Model Urban Canopies

Abstract

We present an objective optimization procedure to determine the roughness parameters for very rough boundary-layer flow over model urban canopies. For neutral stratification the mean velocity profile above a model urban canopy is described by the logarithmic law together with the set of roughness parameters of displacement height d, roughness length $z_0$, and friction velocity $u_*$. Traditionally, values of these roughness parameters are obtained by fitting the logarithmic law through (all) the data points comprising the velocity profile. The new procedure generates unique velocity profiles from subsets or combinations of the data points of the original velocity profile, after which all possible profiles are examined. Each of the generated profiles is fitted to the logarithmic law for a sequence of values of d, with the representative value of d obtained from the minima of the summed least-squares errors for all the generated profiles. The representative values of $z_0$ and $u_*$ are identified by the peak in the bivariate histogram of $z_0$ and $u_*$. The methodology has been verified against laboratory datasets of flow above model urban canopies.

Appendix

The equations involved in the least-squares procedure for determining the friction velocity and roughness length are presented here. The friction velocity $u_*/U_H$ is obtained from the slope

$$\begin{aligned} \,\,m = \left[ \frac{\sum \left( \left( \ln \left( \frac{z-d}{H}\right) \right) \left( \frac{U(z/H)}{U_H}\right) \right) - \frac{\left( \sum \ln \left( \frac{z-d}{H}\right) \right) \left( \sum \frac{U(z/H)}{U_H}\right) }{k}}{{\sum \left( \frac{U(z/H)}{U_H}\right) ^2} - \frac{\left( \sum \frac{U(z/H)}{U_H}\right) ^2}{k}} \right] , \end{aligned}$$

(15)

and the non-dimensional friction velocity $u_*/U_H = \kappa /m$, thus

$$\begin{aligned} \frac{u_*}{U_H} = \kappa \left[ \frac{\sum \left( \left( \ln \left( \frac{z-d}{H}\right) \right) \left( \frac{U(z/H)}{U_H}\right) \right) - \frac{\left( \sum \ln \left( \frac{z-d}{H}\right) \right) \left( \sum \frac{U(z/H)}{U_H}\right) }{k}}{{\sum \left( \frac{U(z/H)}{U_H}\right) ^2} - \frac{\left( \sum \frac{U(z/H)}{U_H}\right) ^2}{k}} \right] ^{-1}, \end{aligned}$$

(16)

and the roughness length $z_0/H$ is obtained from the intercept

$$\begin{aligned} \,\,C=\left[ \frac{\sum \ln \left( \frac{z-d}{H}\right) }{k} - \frac{\kappa }{u_*/U_H} \frac{\sum \frac{U(z/H)}{U_H}}{k}\right] , \end{aligned}$$

(17)

and the non-dimensional roughness length $z_0/H=\exp (C)$, thus

$$\begin{aligned} \frac{z_0}{H}=\exp \left[ \frac{\sum \ln \left( \frac{z-d}{H}\right) }{k} - \frac{\kappa }{u_*/U_H} \frac{\sum \frac{U(z/H)}{U_H}}{k}\right] , \end{aligned}$$

(18)

where $\Sigma$ represents a summation, and k is the number of data points for a profile.