Large-Eddy Simulation of Turbulent Flow and Plume Dispersion in a Spatially-Developing Turbulent Boundary Layer Flow

of the wind It shown that spanwise and vertical spreads of a 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. the model we compare our LES results of turbulent flow and plume dispersion in the near-wake region of a cubical building of concentration fluctuation of the LES model at the heights of z/H=0.1, 1.6 and 2.0 at the downstream position of x/H=3.5 with theoretical model. Concentration is normalized with the r.m.s. concentration. Concentration fluctuation intensity, C i , has values of 0.57, 1.3 and 2.2 at the heights of z/H=0.1, 1.6 and 2.0, respectively. For evaluating probability distributions of concentration fluctuation of each theoretical model, we use concentration statistics of n c , (cid:468) l , C i , (cid:461) 0 and (cid:468) 0 obtained by the LES model. c 99 , c 95 and c 90 are determined from 1-p(c)=0.99, 0.95 and 0.90, respectively. At z/H=0.1, the probability distribution of the LES model is almost the same as that of the log-normal type, while the model result of c 99 /c’ is much smaller than the exponential one. At z/H=1.6, the probability distribution of the LES model is similar to that of both the log-normal and exponential types. At z/H=2.0, the probability distribution of the LES model is consistent with that of the exponential type. Furthermore, the model results of c 99 /c’ are almost the same as those of the exponential one, while the log-normal probability distribution is different from that of the LES model. c 99 /c’ values obtained from the clipped-normal type are underestimated at each height. These facts The purpose of this book is to introduce researchers and graduate students to a broad range of applications of computational simulations, with a particular emphasis on those involving computational fluid dynamics (CFD) simulations. The book is divided into three parts: Part I covers some basic research topics and development in numerical algorithms for CFD simulations, including Reynolds stress transport modeling, central difference schemes for convection-diffusion equations, and flow simulations involving simple geometries such as a flat plate or a vertical channel. Part II covers a variety of important applications in which CFD simulations play a crucial role, including combustion process and automobile engine design, fluid heat exchange, airborne contaminant dispersion over buildings and atmospheric flow around a re-entry capsule, gas-solid two phase flow in long pipes, free surface flow around a ship hull, and hydrodynamic analysis of electrochemical cells. Part III covers applications of non-CFD based computational simulations, including atmospheric optical communications, climate system simulations, porous media flow, combustion, solidification, and sound field simulations for optimal acoustic effects.


Introduction
In one of local-scale dispersion problems, we have an important issue in the accurate prediction of airborne contaminant dispersion from industrial or nuclear facilities for safety and consequence assessments of nuclear facilities. For evaluating radiological consequences of radioactive materials, it is need to predict not only the material concentration in the air at the evaluation point for internal dose but also on the three-dimensional distribution of the plume and surface deposition for external dose. In a flat terrain, time-averaged concentration of a plume can be easily predicted by a conventional Gaussian plume model. However, in Japan, most nuclear facilities are located in complex coastal terrain. Therefore, it is important to predict the spatial distribution of concentrations considering effects from terrain and buildings. Another issue related to atmospheric dispersion in a local-scale is the potential problem that hazardous and flammable materials are accidentally or intentionally released into the atmosphere, either within or close to populated urban areas. For the assessment of human health hazards from such toxic substances, the existence of high concentration peaks in a plume should be considered because it is the instantaneous, not average, concentration that is fatal to humans. In such a situation, it is necessary to accurately predict the unsteady behavior of a plume, considering the effects of individual buildings. For the safety analysis of flammable gases, certain critical threshold levels should be evaluated. Therefore, in such a situation, not only the average levels but also instantaneous magnitudes of concentration should be accurately predicted. There are various methods for predicting plume dispersion in atmospheric boundary layers, e.g. wind tunnel experiments and Computational Fluid Dynamics (CFD). It is well known that wind tunnel experiments are a rational tool for predicting plume dispersion behavior under local topography and/or building conditions. For the case of accidental or intentional release of contaminated materials within urban areas, many studies using a wind tunnel have been made to investigate the spatial extent of contaminated areas and the characteristics of mean and fluctuating concentrations around an individual building. In the safety assessment for the construction of nuclear facilities, prediction of the spatial distribution of radionuclide concentrations over complex terrain containing buildings is required by using a wind tunnel (Meteorological Guide for Safety Analysis of Nuclear Power

Numerical model
The governing equations for LES of atmospheric flow are the filtered continuity equation, the Navier-Stokes equation, and where u i , t, p, ρ, , ιj and f i 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. δ ij , SGS , C s and f s 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, f i is incorporated into the Navier-Stokes equation to consider the building effects and can be assumed as the following expression; where and are negative constants. The stability limit is given by 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 C s 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. C s is set to 0.12 (Shirasawa et al., 2008).
where s j is the subgrid-scale scalar flux which is also parameterized by an eddy viscosity model. The model constant, Sc SGS , is the turbulent Schmidt number and it is set to a constant value of 0.5 . 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.

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,  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 .

Computational settings
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 certain method. 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 the recycle station as shown in Figure 1(b).

Recycle station Inlet
Tripping fence 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 1.0δ. The formulation of the method of Kataoka & Mizuno., (2002) is as follows.

Spatially-developing turbulent boundary layer
www.intechopen.com Here, inlt u and recy u are the instantaneous wind velocity at the inlet and the downstream position (the recycle station), respectively. inlt u is the specified mean wind velocity at the inlet.   i u is the averaged wind velocity in the horizontal plane.     is a damping function to control the transport of turbulent fluctuation into the free stream. a and b are constants. Calculations of both driver and main regions are done by the same model with different computational settings. As boundary conditions, the Sommerfeld radiation condition (Gresho., 1992) is imposed at the exit, a free-slip condition for streamwise and spanwise velocity components is imposed and the vertical velocity component is 0 at the top, a periodic condition is imposed at the side, and a non-slip condition for each velocity component is imposed at the ground surface. Here, in our LES model, we do not use wall functions as the boundary condition of the ground surface. Therefore, the resolution of a vertically stretched grid above the ground surface is set to 1.7 in order to resolve the viscous layer.
The size and the number of grid points for the driver region is 32.8H×10.0H×9.5H (H: height of the cubical building) and 410×120×70 in streamwise, spanwise and vertical directions, respectively. A tripping fence has a height of 0.45H. The size and number of grid points for the main region are 18.9H×10.0H×9.5H and 400×120×70 in streamwise, spanwise and vertical directions, respectively. The cubical building is resolved by 20×20×30 grids in the streamwise, spanwise and vertical directions, respectively. According to numerical experiments of Xie et al., (2006) and Santiago et al., (2008), a building should be resolved by at least 15-20 grid points in each dimension in order to capture complex turbulent behaviors. The mesh number of the building set up in our LES model is enough to accurately simulate turbulent flows around a building. At the inlet of the main region, the inflow turbulence data obtained near the exit of the driver region is imposed at each time interval. In a concentration field, zero gradient is imposed at all the boundaries. The release point of a tracer gas is located 1.5H upstream from the center of the building and elevated at height, H. According to the above-mentioned coordinates, the location of the release point of a plume corresponds to x/H=0.0 and z/H=1.0 as seen in Figure 2.  Figure 3 compares the LES model results with wind tunnel experimental data  of the vertical profiles of mean wind velocity (U), each component of turbulence www.intechopen.com intensities (u', v', w') and Reynolds stress in the driver region. The turbulence statistics is normalized by a free-stream velocity (U ∞ ). We see that the mean wind velocity profile obtained by our LES model is consistent with the experimental data. Strong turbulent fluctuations are produced from the ground surface to the upper part of the boundary layer and each component of the LES-generated turbulent intensity profiles is found to be in good agreement with the experimental data. The Reynolds stress profile obtained by our LES model is also in good agreement with the experimental data. Fig. 3. Turbulence characteristics of approach flow. Figure 4 shows mean velocity vectors by LES around a building. The reattachment lengths of recirculating flow behind the building normalized by the building height of the experiments and the LES model is L/H=1.2 and 1.35 (L: reattachment length), respectively; the latter is slightly larger. Figure 5 shows a comparison of our LES model results with the experimental data  of the vertical profiles of mean wind velocity obtained downstream at x/H=0.0, 1.5, 2.5 and 3.5. The LES model results of mean wind velocity are consistent with the wind tunnel experimental data at each downwind position.   building, x/H=3.5, the LES model results of both horizontal and vertical turbulence intensities around the building height are a little overestimated. Mochida et al., (1991) reported that the LES results of turbulence kinetic energy using the standard Smagorinsky model were overestimated around the upper edge of the recirculation zone of an obstacle in comparison with the wind tunnel experimental data. The overestimation of turbulence intensities of LES behind a building is due to the use of the standard Smagorinsky model. Although the standard Smagorinsky model has the above problems, the main characteristics are obtained in our simulation. They include a sharp peak behind the building due to the strong instability of separated shear layers and the formation of a uniform turbulent flow field with downstream distance due to the active turbulent motions almost the same as in the experimental data of . The slight overestimation of the reattachment length has also been reported in other LES calculations (Murakami et al., 1986). Therefore, our LES model with the conventional Smagorinsky model shows reasonable accuracy and satisfactorily results. Figure 7(a), (b) and (c) shows instantaneous plume dispersion fields around a building at times t* (=tU ∞ /H) = 15, 45 and 90 after the plume release. The yellow areas on iso-surface indicate 0.01% of initial concentration. It shows that the plume is passed above the building roof at first, and then the plume is entrained into the wake region of a building. After enough time passing, the plume is found to be widely dispersed behind a building due to the active turbulent motions. These large discrepancies near the point source are possibly due to a coarse grid resolution for the plume source. In our LES model, a plume source is provided in one grid-cell. Thus, the size of the point source is determined by the grid resolution. Michioka et al., (2003) examined the sensitivity of the grid resolution for the point source by LES of a plume dispersion released from the point sources corresponding to 1.0 and 10 times the real diameters of the point source. As a result, they found that the peak values of mean and r.m.s. concentrations near the point source in a coarse grid resolution were 80% smaller than the wind tunnel experimental data, while those in a fine grid resolution were consistent with the experimental data. The plume source diameter in our LES model is about 5.5 times that of the real one. Considering the discrepancy of the plume source diameter between the LES model and the wind tunnel experiments, our results have the same tendency to underestimate near the point source as the LES results by Michioka et al. Therefore, if the plume source size corresponding to the real one is properly set in our LES model, the model results near the point source should be improved. However, a fine grid resolution is not appropriate for our purpose and target scale considering its computational cost. Except for this discrepancy, the basic characteristics, such as a sharp peak just behind the cubical building and the formation of uniform profiles of mean and r.m.s. concentrations with downstream distance are similar to the experimental data.
In the present LES model, the point source diameter is larger than the real one because a numerical simulation with a fine grid resolution requires large computational time. However, as we explain above, our LES model presents almost the same patterns of concentration distributions as the wind tunnel experiments. This fact indicates that our LES model gives satisfacotoly results.

Characteristics of the peak concentration
In case of accidental or intentional release of toxic or flammable gases into the atmosphere, it is important to estimate not only the mean but also the instantaneous and local peak concentrations near the surface of the ground. For example,  conducted wind tunnel experiments of gas dispersion of a plume released from the center roof vent of a cubical building and investigated the streamwise variation of vertical profiles of mean and r.m.s. concentrations, and various peak concentrations (c 99 , c 95 , c 90 ) defined as the values that are not exceeded for 99, 95, 90% of the cumulative probability density function in the nearwake region. In the theoretical studies of the probability distributions of concentration fluctuation have been studied by many researchers, Csanady (1973), Hanna (1984), and Lewellen & Sykes (1986) proposed theoretical models of the log-normal, exponential and clipped normal distributions for predicting concentration fluctuations of a plume in the atmosphere as follows. Log-normal distribution function:  1e x p c Clipped normal distribution function: Here, erf, n c , l , 0 and 0 are error function, the median concentration, the logarithmic standard deviation, the specified mean and the specified variance. According to Hanna (1984), I can be expressed as follows using C i which is the concentration fluctuation intensity defined as the ratio of r.m.s. concentration to mean concentration.
These theoretical models cannot predict the spatial distribution of concentration but can estimate peak concentrations at a stationary point. Sato and Sato., (2002) compared the lognormal, exponential and clipped normal probability distributions of concentration fluctuation in the near-wake region of a cubical building with those for wind tunnel experiments using concentration statistics of n c , l , C i , 0 and 0 obtained in the experiments. They showed that a peak concentration of c 99 could be predicted using the log-normal type for 0.3＜C i ＜1.0, the log-normal or the exponential types for 1.0＜C i ＜1.5, and the exponential type for C i ＞1.5, while peak concentration of the clipped-normal type was entirely underestimated.
Here, we first compare the probability distributions of concentration fluctuation of the LES model with those of the theoretical models and assess the prediction accuracy of the occurrences of instantaneous high concentrations in our LES model. Then, we examine the characteristics of not only peak concentration ratios of c 99 but also c 95 and c 90 in the nearwake region of the cubical building. .90, respectively. At z/H=0.1, the probability distribution of the LES model is almost the same as that of the log-normal type, while the model result of c 99 /c' is much smaller than the exponential one. At z/H=1.6, the probability distribution of the LES model is similar to that of both the log-normal and exponential types. At z/H=2.0, the probability distribution of the LES model is consistent with that of the exponential type. Furthermore, the model results of c 99 /c' are almost the same as those of the exponential one, while the lognormal probability distribution is different from that of the LES model. c 99 /c' values obtained from the clipped-normal type are underestimated at each height. These facts indicate that the characteristics of probability distributions of the LES model depending on the values of C i are consistent with those reported by . Therefore, the occurrences of instantaneous high concentrations are captured by LES and we conclude that the basic performance of the LES model for plume dispersion around the cubical building is nearly comparable to that for the wind tunnel experiments. Next, we examine various peak concentrations of c 99 , c 95 , and c 90 obtained by the LES model. Figure 10 shows vertical profiles of various peak concentration ratios at x/H=2.5 and 3.5. c 90 /c' values of the LES model have uniform distributions with a constant value of about 3.0 within the building height and gradually decrease above the building height. These tendencies are similar to the experimental data . c 95 /c' of the LES model also shows a constant value of about 4.0 within the building height and slightly decreases above the building height. c 99 /c' of the LES model provides uniform distributions with a constant value of about 5.0 at any position. This tendency is the same as the experimental data of Sato and Sato., 2002.

Conclusion
In this study, we performed LES of turbulent flow and plume dispersion around a cubical building and investigated the basic performance of the LES model in comparison with experimental data of , Sato and Sato., (2002). The scheme to generate a spatially-developing turbulent boundary layer flow in the driver region was proposed by incorporating an existing inflow turbulence generation method into an upstream small fraction of the driver region with a tripping fence placed at the ground surface. With this scheme, we successfully simulated a turbulent boundary layer flow corresponding to that in wind tunnel experiments of  in the driver section. The findings implied that our LES model could simulate various types of wind tunnel flow by incorporating the existing inflow turbulence generation method and moderately setting up roughness obstacles in the driver region. Turbulence intensities behind a cubical building around the building height were slightly overestimated in comparison with experimental data of . However, the main characteristics of turbulent flow such as a sharp peak behind the building and the formation of a uniform turbulent flow field with downstream distance, corresponding to the wind tunnel experiment, are obtained. Also, dispersion characteristics such as a sharp peak close to the point source and the formation of concentration profiles with downstream distance were successfully simulated in comparison with the experimental data. These indicated that the difference in turbulence characteristics between the experiments and our LES model was not significant. The basic performance of our LES model for turbulent flow and dispersion fields could be recognized as comparable to the wind tunnel experiment. Dependence of the patterns of probability distributions of concentration fluctuation of the LES model on the values of concentration fluctuation intensity were consistent with those of the wind tunnel experimental results of . Focusing on various peak concentration ratios, we saw c 90 /c' and c 99 /c' values of the LES model were in good agreement with the experimental data. From these facts, we considered that the occurrences of high concentrations were captured by our LES model. From the above results, we considered that the basic performance of our LES model was almost comparable level to that obtained by the wind tunnel experimental techniques.
Here, we discuss the applicability of the present LES model for flow and dispersion around an isolated building to the problem of plume dispersion within a group of buildings. Within an urban canopy, three typical flow patterns, isolated flow, wake interference flow and skimming flow within building arrays are formed depending on the ratio of the buildings height to the street width and the flow filed is highly complex (Oke., 1998). Considering a surface geometry of urban canopy is composed of a group of isolated buildings and obstacles, numerical simulation with moderate grid points for each building and obstacle can capture such complex flow patterns with reasonable accuracy. Therefore, from the validation of our LES model for turbulent flow and plume dispersion around an isolated building, it is considered that the present model with moderate grid points for each building can apply to a simulation of plume dispersion within urban canopy.