Handout (1.5 MB)
Turbulent flows over a rough surface exhibit a constant Reynoldsstress region in the proximity of the bottom solid boundary. This region is known as the inertial sublayer (ISL) that is commonly described by the logarithmic law of the wall (loglaw)
(1) 
where u_{*} is the friction velocity, k (= 0.41) the von Kármán constant, z the wallnormal distance, d the displacement height, and z_{0} the roughness length scale. The loglaw is well received by the community to parameterize the flows in the lower atmospheric boundary layer (ABL). While the size of roughness elements is not negligible compared with the boundary layer thickness, there exists another layer of enhanced mixing inbetween the ISL and the bottom surface that is characterized by the influence of individual (inhomogeneous) roughness elements. This layer is called the roughness sublayer (RSL) in which the velocity is more uniform, deviating from the conventional loglaw. Under this circumstance, parameterizing the flows in the lower ABL by Equation (1) must be applied with caution.
In view of the limitation of applying loglaw to the RSL, this study is conceived to examine the RSL flows and turbulence in response to various surfaceroughness configurations of different aerodynamic resistance. Idealized urbanroughness elements, in the form of identical ribs placed in cross flows, are used to model simplified urban morphology in our wind tunnel in isothermal conditions. The aerodynamic resistance of the rough surfaces is adjusted by the ribseparationtoheight ratio (pitch, k/h). Hotwire anemometry (HWA) is used in data sampling. A new analytical solution to the mean velocity profile over rough surfaces, including both RSL and ISL influence, is arrived as follows
(2) 
where z_{*} is the RSL thickness, γ (= 0.5772156649) the Euler constant, and μ an empirical constants based on the surface roughness. Figure 1 shows that the new analytical solution (rootmeansquare (RMS) error = 0.0756 to 0.226) agrees well with windtunnelmeasured wind speed than does loglaw (RMS error = 0.102 to 0.959). Equation (2) is then used together with the classic Ktheory, antigradient model to describe the vertical momentum flux, depicting the different mixing length scales and transport processes in the RSL and ISL. More detailed results will be reported in the symposium.
Figure 1. Velocity profiles of wind tunnel measurement (symbols) over rough surfaces of different friction factor f by adjusting the pitch (k/h). Also shown are the conventional loglaw (dashed lines), the analytical solution proposed in this paper (solid lines; Equation (2)), and the RSL/ISL.
