Equilibrium Atmospheric Boundary-Layer Flow: Computational Fluid Dynamics Simulation with Balanced Forces

Abstract

Forcing relationships in steady, neutrally stratified atmospheric boundary-layer (ABL) flow are thoroughly analyzed. The ABL flow can be viewed as balanced between a forcing and a drag term. The drag term results from turbulent stress divergence, and above the ABL, both the drag and the forcing terms vanish. In computational wind engineering applications, the ABL flow is simulated not by directly specifying a forcing term in the ABL but by specifying boundary conditions for the simulation domain. Usually, these include the inflow boundary and the top boundary conditions. This ‘boundary-driven’ ABL flow is dynamically different from its real counterpart, and this is the major reason that the simulated boundary-driven ABL flow does not maintain horizontal homogeneity. Here, first a dynamical approach is proposed to develop a neutrally stratified equilibrium ABL flow. Computational fluid dynamics (CFD) software (Fluent 6.3) with the standard $k$–$\varepsilon$ turbulence model is employed, and by applying a driving force profile, steady equilibrium ABL flows are simulated by the model. Profiles of wind speed and turbulent kinetic energy (TKE) derived using this approach are reasonable in comparison with the conventional logarithmic law and with observational data respectively. Secondly, the equilibrium ABL profiles apply as inflow conditions to simulate the boundary-driven ABL flow. Simulated properties between the inlet and the outlet sections across a fetch of 10 km are compared. Although profiles of wind speed, TKE, and its dissipation rate are consistently satisfactory under higher wind conditions, a deviation of TKE and its dissipation rate between the inlet and outlet are apparent (7–8 %) under lower wind-speed conditions (2 m s$^{-1}$ at 10 m). Furthermore, the simulated surface stress systematically decreases in the downwind direction. A redistribution of the pressure field is also found in the simulation domain, which provides a different driving pattern from the realistic case in the ABL.

Appendices

Appendix A: Variances in the Turbulent Velocity

According to Hanna (1982), the profiles of the standard deviations of turbulent velocities in the neutral atmospheric boundary layer are specified as,

$$\begin{aligned} \sigma _u&= 2u_*\exp (-3fz/u_*)=2u_*\exp (-3cz/h), \end{aligned}$$

(15)

$$\begin{aligned} \sigma _{v,w}&= 1.3u_*\exp (-2fz/u_*)=1.3u_*\exp (-2cz/h), \end{aligned}$$

(16)

where $h$ is the ABL height, taking the form of Eq. 13, and $c$ is a coefficient. As a result, the ratio of $k/u_*^2 =0.5(\sigma _u^2 +\sigma _v^2 +\sigma _w^2 )/u_*^2$ can be calculated in the ABL.

Appendix B: Standard $k$–$\varepsilon$ Model and Constants

The standard $k$–$\varepsilon$ turbulence model resolves two additional equations for turbulent kinetic energy $(k)$ and its dissipation rate $\varepsilon$, viz. (Launder and Spalding 1974),

$$\begin{aligned}&\frac{D(\rho k)}{Dt}=\tau _{ij} \frac{\partial \bar{{u}}_i }{\partial x_j }-\rho \varepsilon +\frac{\partial }{\partial x_j }\left( \left( \mu +\frac{\mu _T }{\sigma _k }\right) \frac{\partial k}{\partial x_j }\right) , \end{aligned}$$

(17)

$$\begin{aligned}&\frac{D(\rho \varepsilon )}{Dt}=C_{\varepsilon 1} \frac{\varepsilon }{k}\tau _{ij} \frac{\partial \bar{{u}}_i }{\partial x_j }-C_{\varepsilon 2} \rho \frac{\varepsilon ^{2}}{k}+\frac{\partial }{\partial x_j }\left( \left( \mu +\frac{\mu _T }{\sigma _\varepsilon }\right) \frac{\partial \varepsilon }{\partial x_j }\right) , \end{aligned}$$

(18)

where

$$\begin{aligned}&\tau _{ij} =-\rho \overline{{u}^{\prime }_i {u}^{\prime }_j } =\mu _T \left( \frac{\partial \bar{{u}}_i }{\partial x_j }+\frac{\partial \bar{{u}}_j }{\partial x_i }\right) -\frac{2}{3}\rho k\delta _{ij}, \end{aligned}$$

(19)

$$\begin{aligned}&\mu _T =C_\mu \rho \frac{k^{2}}{\varepsilon }. \end{aligned}$$

(20)

All symbols have their conventional meanings, and $\sigma _k,\sigma _\varepsilon ,C_{\varepsilon 1},C_{\varepsilon 2}$ and $C_\mu$ are model constants usually taking the standard values of 1.0, 1.3, 1.44, 1.92, and 0.09, respectively. In the surface layer, it is deduced that (RH1993; Apsley and Castro 1997; Detering and Etling 1985)

$$\begin{aligned}&k=\frac{u_*^2 }{\sqrt{C_\mu }}, \end{aligned}$$

(21)

$$\begin{aligned}&\kappa ^{2}=(C_{\varepsilon 2} -C_{\varepsilon 1} )\sigma _\varepsilon \sqrt{C_\mu }, \end{aligned}$$

(22)

where $\kappa$ is the von Karman constant. The relationship in Eq. 22 is usually applied to turbulence in the whole ABL. According to Detering and Etling (1985), the set of constants adjusted to fit observations of atmospheric TKE in our study can be taken as $(\sigma _k,\sigma _\varepsilon ,C_{\varepsilon 1},C_{\varepsilon 2},C_\mu ) = (1, 1, 1.13, 1.92, 0.0494)$. These constants satisfy Eq. 22, given the von Karman constant (0.4187) in Fluent.