Estimation of the Kolmogorov constant by large-eddy simulation in the stable PBL

Abstract

In this paper is evaluated the inertial subrange Kolmogorov constant $C_0$ in a stable boundary-layer. The importance of the constant $C_0$ is well known as predictions of turbulent dispersion by means of Lagrangian stochastic models depend upon its value. Different values of the $C_0$ constant has been proposed along the years, most of them determined at low Reynolds number and/or under different techniques. Here we estimate the constant $C_0$ by tracking an ensemble of Lagrangian particles injected in a stable planetary boundary-layer simulated with a large-eddy simulation model and analyzing the ensemble-averaged Lagrangian velocity structure function in the inertial subrange. Our estimative of $C_{0,w}$ is 3.7, which is consistent with values found in literature. The evaluation of $C_{0,u}$ and $C_{0,v}$ cannot be easily accomplished since it is difficult to identify an inertial subrange for the wind field horizontal components.

Introduction

Turbulent dispersion of scalars is most efficiently described in the Lagrangian framework as first suggested by the Taylor’s statistical theory of dispersion in homogeneous turbulence [1], [2]. Therefore, Lagrangian stochastic models constitute an important and effective tool to simulate the atmospheric dispersion of airborne pollutants. These models are based on the Langevin equation, which is derived from the hypothesis that the turbulent velocity is given by the combination between a deterministic and a stochastic term.

The second-order Lagrangian velocity structure function for the inertial subrange Kolmogorov scaling of turbulence assumes the form:

$$D_2^L\left(\tau \right)=〈{[u(t+\tau )-u\left(t\right)]}^2〉\equiv C_0\bar{\epsilon }\tau$$

for ${\tau }_{\eta }\le \tau \le {\tau }_L$ where $u\left(t\right)$ is the Lagrangian velocity, $〈〉$ denotes the ensemble average, $\tau$ is a time lag, $\bar{\epsilon }$ is the average dissipation rate of kinetic energy, ${\tau }_{\eta }$ is the Kolmogorov time scale (at which dissipation starts to be effective), ${\tau }_L$ is the Lagrangian integral time scale and finally $C_0$ is the Lagrangian Kolmogorov constant supposed to be universal at high Reynolds number for an isotropic turbulence. Comparing the Lagrangian velocity structure functions obtained from Langevin equation with Eq. (1), it is possible to determine the stochastic term of the Langevin equation as $b={\left(C_0\overline{\epsilon }\right)}^{1/2}$.

In this context it is evident that it is important to establish the numerical value of the constant $C_0$ as the predictions of turbulent dispersion by means of the Lagrangian stochastic models depend upon its value [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. In these models it is assumed that the evolution of a tracer particle’s state (velocity-position) is a Markovian process, so the particle’s trajectory can be statistically calculated. However, as $C_0$ is a Lagrangian quantity there are serious technical difficulties associated with its measurements. The numerical value of $C_0$ has been determined following a large number of techniques, like for example Lagrangian velocity measurements [13], [14], numerical experiments [15], observed dispersion of tracer particles in a flow [2] and Eulerian measurements [12]. This large variety of methodologies can in part explain the wide range of values found for the $C_0$ constant and the fact that its value is rather uncertain [12], [16], [17]. Typical values of $C_0$ obtained from both physical and numerical experiments vary within the range 2.0 to 7.0.

The objective of this work is to determine the ${C{}_0}_{}$value in a stable boundary-layer (SBL) using large-eddy simulation (LES) and the peak/plateau method [14], [15]. This method consists in tracking an ensemble of particle trajectories, obtained by considering the resolved velocity field from LES. In this context the Lagrangian structure function, $D_2^L\left(\tau \right)$, may be easily evaluated and the level of the peak/plateau of the $D_2^L\left(\tau \right)/\bar{\epsilon }\tau$ versus $\tau$ curve allows the determination of the constant $C_0$. The key point for this methodology is the presence of the inertial-subrange for the resolved velocity field generated by LES.