**2. Numerical model**

332 Computational Simulations and Applications

*Plant Reactor*, Nuclear Safety Commission of Japan, 1982). Although reliable data can be obtained on air flow and plume dispersion, wind tunnel experiments are time consuming, costly, and have limited availability for these applications. For example, in the safety assessment, experimental results are only used to derive the effective stack height, which is applied for long term assessment using a Gaussian plume model, and the effective stack height is usually determined lower height than the actual height considering terrain and building effects in a way that provides a conservative evaluation. On the other hand, recently the CFD technique has been proposed for use as an alternative to wind tunnel experiments (Sada *et al*., 2009) developed a numerical model for atmospheric diffusion analysis and evaluation of effective dose for safety analysis and showed its effectiveness in

The CFD technique has been recognized as a helpful tool with the rapid development of computational technology. The CFD technique uses computers to numerically predict fluid flow, heat transfer and mass transfer by solving the governing equations. In particular, there are two different approaches, the Reynolds-Averaged Navier-Stokes (RANS) and Large-Eddy Simulation (LES) models, which are both effective for predicting turbulent flows. In RANS, a mean wind flow is computed, delivering an ensemble- or time-averaged solution, and all turbulent motions are modeled with a turbulence model. The main advantage of the RANS model is its efficiency in computing a mean flow field with relatively low computational cost. Sada *et al*., (2009) designed a practical numerical model based on the

Recently, LES has come to be regarded as an effective prediction method for environmental flows. LES resolves the large-scale turbulent motions and models only the smallest scale motions, which are usually more universal. Although the LES model requires larger computational costs than RANS model, it is no less useful the latter, considering the cost and limited availability of wind tunnels and the experimental time needed. Furthermore, LES can provide accurate predictions and detailed information about turbulence structures, and mean and fluctuating concentrations of a plume as well as wind tunnel experiments can provide them. Therefore, we have developed an LES dispersion model applicable to actual problems of atmospheric dispersion on a local scale. As a first step, we previously performed LES for turbulent flows and plume dispersion over a flat terrain (Nakayama & Nagai, 2009). When compared to experimental results of Fackrell & Robins., (1982), it was shown that turbulence structures, the characteristic mean and r.m.s. (root mean square) concentrations, turbulent concentration flux and peak concentration over a flat terrain are successfully simulated. These findings implied that our LES model could replace wind tunnel experiments for safety assessments of nuclear facilities and also provide detailed information for the consequence assessment of accidental and intentional releases of

For the second step, we apply our LES dispersion model to the complex behaviors of separated shear layers and large eddies in the near-wake of a building. First, we propose a scheme to generate a spatially-developing turbulent boundary layer flow with strong velocity fluctuations, which is applicable to various types of wind tunnel flows and perform an LES of plume dispersion around an isolated cubical building. Then, we examine basic performance of the model and scheme by comparing LES data of the turbulent structure and characteristics of mean and r.m.s. concentrations including peak concentration with

comparison with wind tunnel experiments.

radioactive materials into the atmosphere.

experimental data.

RANS model.

The governing equations for LES of atmospheric flow are the filtered continuity equation, the Navier-Stokes equation,

$$\frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_i} = \mathbf{0} \quad , \tag{1}$$

$$\frac{\partial \overline{u}\_i}{\partial t} + \overline{u}\_j \frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_j} = -\frac{1}{\rho} \frac{\partial \overline{p}}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_j} \nu \left( \frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_j} + \frac{\partial \overline{u}\_j}{\partial \mathbf{x}\_i} \right) - \frac{\partial}{\partial \mathbf{x}\_j} \mathbf{r}\_{ij} + f\_i \tag{2}$$

$$
\pi\_{ij} = \overline{u\_i \overline{u\_j}} - \overline{u}\_i \overline{u}\_j \tag{3}
$$

$$
\sigma\_{\vec{\eta}} - \frac{1}{3} \delta\_{\vec{\eta}} \tau\_{kk} = -\nu\_{\text{SGS}} \overline{S\_{\vec{\eta}}} \quad \nu\_{\text{SGS}} = \left( \mathbb{C}\_{s} f\_{s} \overline{\Delta} \right)^{2} \left( 2 \overline{S\_{\vec{\eta}}} \overline{S\_{\vec{\eta}}} \right)^{\frac{1}{2}} \tag{4}
$$

$$\overline{S\_{ij}} = \left(\widehat{\alpha}\overline{u\_i} \Big/ \widehat{\alpha}\mathbf{x}\_j + \widehat{\alpha}\overline{u\_j} \Big/ \widehat{\alpha}\mathbf{x}\_i\right) / \text{ }\mathbf{2}\text{ }\tag{5}$$

and

$$
\overline{\Delta} = \left( \overline{\Delta\_x} \overline{\Delta\_y} \overline{\Delta\_z} \right)^{\frac{1}{3}} \tag{6}
$$

where *ui, t, p, ρ, ν, τιj* and *fi* are the wind velocity, time, pressure, density, kinematic viscosity, subgrid-scale Reynolds stress and external force term, respectively. The subscript *i* stands for coordinates (1, streamwise; 2, spanwise; and 3, vertical direction). Over bars, (¯) denote application of the spatial filter. *δij, νSGS, Cs* and *fs* are the Kronecker delta, the eddy viscosity coefficient, the model constant of the flow field and Van Driest damping function (Van Driest., 1956), respectively. denotes the grid-filter width. In this LES model, the external force term proposed by Goldstein *et al*., (1993) is applied because of its computational stability for turbulent flow around a bluff body. The force term, *fi* is incorporated into the Navier-Stokes equation to consider the building effects and can be assumed as the following expression;

$$\int\_{\gamma} f\_i = \alpha \int\_0^t \mu\_i(t')dt' + \beta \mu\_i(t), \quad \alpha < 0, \ \beta < 0\tag{7}$$

where *α* and *β* are negative constants. The stability limit is given by <sup>2</sup> <sup>2</sup> *<sup>k</sup> t* 

$$\frac{-\beta - \sqrt{\left(\beta^2 - 2ak\right)}}{a}$$

where *k* is a constant of order 1. The most commonly used sub-grid scale models are the standard and dynamic type Smagorinsky models (Smagorinsky., 1963; Germano *et al*., 1991; Lilly *et al*., 1992; Meneveau *et al*., 1996). Although *Cs* should be estimated depending on the flow type, the standard Smagorinsky model (Smagorinsky., 1963) with the Van Driest damping function is used instead in our LES model because of its simplicity and low computational cost. *Cs* is set to 0.12 (Shirasawa *et al*., 2008).

Large-Eddy Simulation of Turbulent Flow and

the recycle station as shown in Figure 1(b).

**Spatially-developing turbulent boundary layer**

method of Kataoka & Mizuno., (2002) is as follows.

 

(a) (b)

**4. Computational settings** 

certain method.

**Recycle Inlet station**

building.

**Tripping fence**

Plume Dispersion in a Spatially-DevelopingTurbulent Boundary Layer Flow 335

In wind tunnel experiments, a neutral atmospheric turbulent boundary layer is simulated mainly using various types of obstacle, such as spires, tripping fences and roughness blocks. Therefore, various wind tunnel flows that have different turbulence characteristics can be obtained depending on the wind tunnel facility. In an LES study of turbulent flow in the atmosphere, an approach flow with turbulent fluctuations as the inlet boundary condition of the model domain should be generated depending on the target wind tunnel flow by a

In our LES model, the driver region for generating a spatially-developing turbulent boundary layer flow and the main region for simulating of plume dispersion around a cubical building immersed in a fully-developed turbulent boundary layer are set up. In this scheme, first a thick turbulent boundary layer flow is generated by incorporating the existing inflow turbulence generation method, that is, the method of Kataoka & Mizuno., (2002) into an upstream small fraction of the driver region as shown in Figure 1(a). Then, a strong wind velocity fluctuation is produced by a tripping fence placed downstream from

> **Turbulent flow**

Fig. 1. Schematic of numerical model. (a) Driver region for generating turbulent boundary layer flow. (b) Main region for turbulent flow and plume dispersion around a cubical

In the method of Kataoka & Mizuno., (2002), the fluctuating part of the velocity at the recycle station is recycled and added to the specified mean wind velocity at each time interval by assuming that boundary layer thickness is constant within the driver section. This method requires a driver section with a length of about 1.0*δ*. The formulation of the

> *uinlt y*, , *zt u z u inlt*

> > *winlt y*, , *zt w*

2 1 2

*v yzt v yzt inlt* , , 

 <sup>1</sup> 1 tanh tanh

*a b*

*b b*

**Turbulent inflow**

> **Fully-developed turbulent boundary layer**

> > **Building**

*recy y*, , *zt u z* , (11)

*recy* , , , (12)

, (14)

*recy y*, , *zt w z* , (13)

*a*

**Plume dispersion Source**

The LES of plume dispersion is also computed by using the standard Smagorinsky model. The spatially filtered scalar conservation equations are presented by

$$\frac{\partial \overline{\overline{c}}}{\partial t} + \overline{\mu}\_{j} \frac{\partial \overline{\overline{c}}}{\partial \mathbf{x}\_{j}} = -\frac{\partial}{\partial \mathbf{x}\_{j}} \mathbf{s}\_{j} \tag{8}$$

$$s\_j = \overline{u\_j}\overline{c} - \overline{u}\_j\overline{c} \tag{9}$$

and

$$s\_j = -\frac{\nu\_{SGS}}{\text{Sc}\_{SGS}} \frac{\partial \overline{\mathcal{C}}}{\partial \mathbf{x}\_j} \tag{10}$$

where *sj* is the subgrid-scale scalar flux which is also parameterized by an eddy viscosity model. The model constant, *ScSGS*, is the turbulent Schmidt number and it is set to a constant value of 0.5 (Sada & Sato., 2002).

The coupling algorithm of the velocity and pressure fields is based on the Marker and Cell (Harlow & Welch, 1965) method with the Adams-Bashforth scheme for time integration. The Poisson equation is solved by the Successive Over-Relaxation method that is an iterative method for solving a Poisson equation for pressure. For the spatial discretization in the governing equation of flow and the tracer transport, a second-order accurate central difference is used. For only the advection term in the dispersion field, the Cubic Interpolated Pseudo-particle (CIP) method (Takewaki *et al*., 1985; Yabe & Takei., 1988) is imposed in order to prevent a numerical instability. The CIP is a very stable scheme that can solve generalized hyperbolic equations in space. The Reynolds number based on the cubical building height and wind speed at the building height is almost 5,000.

### **3. Wind tunnel experiments for evaluating the model performance**

Over the past few decades many wind tunnel experiments have been conducted on the dispersion characteristics of a plume in the near-wake of a cubical building. For example, Sada & Sato., (2002) conducted experiments under neutral atmospheric stratification in the wind tunnel of Central Research Institute of Electric Power Industry. The wind tunnel test section was 20m long, 3m wide and 1.5m high. An approaching flow with strong velocity fluctuations was generated using roughness elements with L-shaped cross sections placed on the floor at the entrance of the wind tunnel. It was shown that spanwise and vertical spreads of a plume corresponded to the Pasquill-Gifford stability class D. A plume was released from an elevated point source located upstream from the cubical building and concentration of the plume is measured by a fast-response flame ionization detector. The vertical profiles of mean wind velocity, turbulence intensity, mean and r.m.s. concentrations and peak concentration in the near-wake region of the cubical building were all obtained from the experiments. In this wind tunnel experiment, the building Reynolds numbers based on the cubical building height and wind speed at the building height is about 13,000. In the present paper, in order to evaluate the model performance, we compare our LES results of turbulent flow and plume dispersion in the near-wake region of a cubical building with the wind tunnel experimental data of Sada & Sato., (2002).
