Scalar mixing in an urban canyon

Abstract

The scalar dynamics within a unit-aspect-ratio street canyon are studied using large-eddy simulation. The key processes of ventilation and mixing are analysed with the canyon-averaged concentration, mean tracer age and variance. The results are sensitive to the source location and can be classified according to the streamline geometry. The canyon-averaged concentrations for the corner vortices, vortex sea and central vortex do not converge to the same value at large times, though the mean decay rates do. The variance measured with respect to the canyon average shows two distinct decay regimes: the early regime reflects large-scale straining and enhanced diffusion across streamlines, while the late regime is associated with escape from the canyon, i.e., ventilation. Analytical predictions for the variance-decay or mixing time scales are verified for the early regime. It is argued that the presence of an open boundary at the roof level suppresses rapid mixing of the scalar field and is responsible for differences with respect to scalar dynamics within closed domains.

Appendices

Appendix 1: Lyapunov exponents

The stretching of fluid trajectories or the divergence of particles trajectories is governed by the velocity-gradient tensor. It has elements $J_{ij}= \partial u_i/\partial x_j$. After discretising in time, (vector) perturbations evolve according to the map

$$\begin{aligned} dX_{n+1} = (J(\mathbf {U})\varDelta t)_n dX_n. \end{aligned}$$

(25)

For simplicity, $dX_0={\varvec{I}}$. The Lyapunov exponents are calculated following the algorithm of von Bremen et al. [6]. Applying QR-factorization to the matrix product $J_nJ_{n-1}\cdots J_1$ yields $R=\{R_0,\ R_1,\ \ldots \ ,\ R_{n-1},\ R_n\}$ where $n\in [0,n]$ and $R_i$ denotes the upper triangular matrix from the $i^\mathrm{{th}}$ QR-factorization. The Lyapunov exponents follow from

$$\begin{aligned} \lambda = \frac{1}{n\varDelta t}\sum \limits _{i=0}^n \log {(\varLambda R_i)}, \end{aligned}$$

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where $\varLambda$ denotes the diagonal elements of $R_i$ and $\lambda$ is a row vector, i.e. $\lambda = \lambda (\lambda _0,\lambda _1,\lambda _2)$ with $\lambda _0$ representing the maximum Lyapunov exponent. The associated time scale is just the reciprocal of $\lambda _0$. For initial conditions within $\mathcal {R}_i$ (cf. Fig. 5),

$$\begin{aligned} \tau _L=1/\langle \lambda _0 \rangle _{\mathcal {R}_i}, \end{aligned}$$

(27)

where $\langle \cdot \rangle _{\mathcal {R}_i}$ denotes the spatial average over $\mathcal {R}_i$.

Appendix 2: Variance-decay and Lyapunov time scales

Variance-decay time scales for the early and late regimes are listed in Tables 3 and 4, respectively. The corresponding Lyapunov time scales are shown in Table 5.

Table 3 Variance-decay time scales (s) for the late regime, $t\in [1000\,\mathrm{s}, 5000\,\mathrm{s}]$

Table 4 Variance-decay time scales (s) for the early regime (Fig. 12)

Table 5 Lypaunov time scales (s)

Appendix 3: Tangential velocity of the central vortex

The tangential velocity, $V(x,z)=\sqrt{u(x,z)^2+w(x,z)^2}$, is calculated by assuming that the vortex is circular and centred at the centre of the canyon O(0, H / 2) (see Fig. 5 of Duan and Ngan [16]). The coordinates of a particle are defined by

$$\begin{aligned} x = R\cos \theta , \quad z = R\sin \theta , \end{aligned}$$

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where R is the radius of the nominal vortex. Averaging in space and time

$$\begin{aligned} \langle \overline{V} \rangle = \frac{1}{A_{\ell }} \int \limits _{-L_y/2}^{L_y/2}dy \int \limits _{-\pi }^{\pi }Rd\theta \ \overline{{V}}(x,z), \end{aligned}$$

(29)

where $L_y$ is the canyon length in the spanwise direction and $A_{\ell }=2.0\pi RL_y$ is the surface area of the cylindrical vortex. For brevity, $\langle \overline{\cdot }\rangle$, is omitted from the text.