Dispersion of a Passive Scalar Fluctuating Plume in a Turbulent Boundary Layer. Part III: Stochastic Modelling

Abstract

We analyze the reliability of the Lagrangian stochastic micromixing method in predicting higher-order statistics of the passive scalar concentration induced by an elevated source (of varying diameter) placed in a turbulent boundary layer. To that purpose we analyze two different modelling approaches by testing their results against the wind-tunnel measurements discussed in Part I (Nironi et al., Boundary-Layer Meteorology, 2015, Vol. 156, 415–446). The first is a probability density function (PDF) micromixing model that simulates the effects of the molecular diffusivity on the concentration fluctuations by taking into account the background particles. The second is a new model, named VP$\varGamma$, conceived in order to minimize the computational costs. This is based on the volumetric particle approach providing estimates of the first two concentration moments with no need for the simulation of the background particles. In this second approach, higher-order moments are computed based on the estimates of these two moments and under the assumption that the concentration PDF is a Gamma distribution. The comparisons concern the spatial distribution of the first four moments of the concentration and the evolution of the PDF along the plume centreline. The novelty of this work is twofold: (i) we perform a systematic comparison of the results of micro-mixing Lagrangian models against experiments providing profiles of the first four moments of the concentration within an inhomogeneous and anisotropic turbulent flow, and (ii) we show the reliability of the VP$\varGamma$ model as an operational tool for the prediction of the PDF of the concentration.

Appendix

We report here the formulation of the micromixing time scale $\tau _m$ presented in Cassiani et al. (2005a). In isotropic turbulence $\tau _m$ is assumed as depending on the time scale $\tau _r$ of the relative dispersion, i.e. the spreading of the plume around its centre of mass,

$$\begin{aligned} \tau _m = \mu _t\tau _r = \mu _t\frac{\sigma _r}{\sigma _{ur}} \end{aligned}$$

(20)

where $\mu _t$ is an empirical constant to be set, $\sigma _r$ is the relative plume spread around the plume’s centroid, and $\sigma _{ur}= \sqrt{\overline{u_r^2}}$ is the r.m.s of the relative velocity fluctuations. The term $u_r$ represents the difference between a turbulent velocity component and the corresponding velocity component of the instantaneous centre-of-mass (meandering process). We model $\sigma _r$ as,

$$\begin{aligned} \sigma _{ur}^2 = \sigma _u^2\left( \frac{\sigma _r}{L} \right) ^{2/3} \end{aligned}$$

(21)

where $\sigma _u^2$ is the variance of the turbulent velocity, and L represents the Eulerian integral length scale parametrized assuming the stationarity of the energy cascade (Sawford and Stapountzis 1986),

$$\begin{aligned} L = \frac{\left( {3\sigma _u^2}/{2}\right) ^{3/2}}{\varepsilon }. \end{aligned}$$

(22)

When $\sigma _r = L$ the meandering process becomes negligible with respect to the relative dispersion and all the energy contributes to the expansion. For this reason, we imposed the constraint $\sigma _{ur} = \sigma _u$, if $\sigma _r>L$, and parametrized $\sigma _r$ as follows,

$$\begin{aligned}&\displaystyle \sigma _r^2 = \frac{d_r^2}{1+\left( d_r^2-\sigma _0^2\right) / \left( \sigma _0^2+2\sigma _u^2T_Lt\right) }, \end{aligned}$$

(23)

$$\begin{aligned}&\displaystyle d_r^2 = C_r\varepsilon (t_0+t)^3, \end{aligned}$$

(24)

where $t_0 = \left( {\sigma _0^2}\big /{C_r\varepsilon } \right) ^{1/3}$ is the inertial formulation for a dispersion from a finite source size (Franzese 2003), $\sigma _0$ is the source size, and $T_L={2\sigma _u^2}\big /{C_0\varepsilon }$ is the Lagrangian time scale. Following Cassiani et al. (2005a) the formulation of the micromixing time scale in non-homogeneous and non-isotropic turbulence requires defining the local variance $\sigma _u^2$ as the average of the variances of the three velocity components. Equation 24 is discretized in time as follows,

$$\begin{aligned}&\displaystyle d_r^2 (t+\Delta t) = d_r^2(t) + 3C_r\varepsilon (t_0+t)^2\Delta t \end{aligned}$$

(25)

$$\begin{aligned}&\displaystyle d_r^2(t=0) = \sigma _0^2 \end{aligned}$$

(26)

where it is worth noting that a Lagrangian stochastic model associated with these micromixing models (PMM and VPA) requires three parameters to be set: $\mu _t$, $C_r$, and $C_0$. The term $C_0$ influences the averaged dispersion and its value has to be fixed irrespectively of the used micromixing model (if the micromixing model respects the criterion of not altering the mean concentration field, e.g. Pope 1998; Sawford 2004). For this reason, we evaluate $C_0$ as the best-fit between the numerical and experimental values of ${\overline{c}}$ and we found $C_0 = 4.5$. This value is in the range generally accepted in the literature, $2\le C_0 \le 8$ (Du et al. 1995; Lien and D’Asaro 2002; Rizza et al. 2006).

The evaluation of $C_r$ is performed by comparing the numerical solutions of the concentration variance with the corresponding experimental values. As reported in Table 1, the best-fit is obtained with $C_r=0.3$. According to Franzese and Cassiani (2007), $C_r$ should be equal to $C_0/11$. Since $C_0=4.5$, the value $C_r = 0.3$ is therefore close to the former theoretical prediction. Finally, $\mu _t$ is an empirical constant.