**5.3 Dispersion field**

338 Computational Simulations and Applications

(a)

(b) Fig. 6. Streamwise variation of turbulence intensities. (a) Horizontal turbulence intensity. (b)

0

0 0.1 0.2 0.3

0 0.1 0.2 0.3


U/U∞

v'/U∞

0

0 0.1 0.2 0.3

0 0.1 0.2 0.3


U/U∞

v'/U∞

w'/U∞

1

2

3

z/H

4

5

0

1

2

3

z/H

4

5

0

1

2

3

z/H

4

5

w'/U∞

1

2

3

z/H

4

5

0

1

2

3

z/H

4

5

0

1

2

3

z/H

4

5

0 0.1 0.2 0.3

0 0.1 0.2 0.3

v'/U∞

w'/U∞

Figure 6 shows a comparison of our LES model results with the wind tunnel experimental data of the vertical profiles of (a) horizontal and (b) vertical turbulence intensities normalized by free-stream velocity obtained downstream at x/H=0.0, 1.5, 2.5 and 3.5. Just above the roof of the building at x/H=0.0, the LES model results of both horizontal and vertical turbulence intensities are underestimated. At the position located away from the

Vertical turbulence intensity.

0 0.1 0.2 0.3

0

1

2

3

z/H

4

5

0

1

2

3

z/H

4

5

0

1

2

3

z/H

4

5

WT (Sada, 2002) LES

0 0.1 0.2 0.3

WT (Sada, 2002) LES


WT (Sada, 2002) LES

U/U∞

v'/U∞

w'/U∞

0

1

2

3

z/H

4

5

0

1

2

3

z/H

4

5

0

1

2

3

z/H

4

5

Fig. 5. Streamwise variation of vertical profiles of mean wind velocity.


U/U∞

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.

Figure 8 shows a comparison of our LES model results and the wind tunnel experimental data (Sada & Sato., 2002) of the vertical profiles of mean (C) and r.m.s. (c') concentrations obtained downstream at x/H=1.5, 2.5, 3.5 and 5.0 in the near-wake region of the cubical building. The mean and r.m.s. concentrations are normalized by free-stream velocity, the building height and the source strength (Q). In both mean and r.m.s. concentration fields, the peak values of the LES model near the point source are about 50% smaller than the wind tunnel experimental results, while the model results are in good agreement with the experimental data, particularly at the position x/H=5.0, located away from the point source. 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

Large-Eddy Simulation of Turbulent Flow and

WT (Sada, 2002) LES

Plume Dispersion in a Spatially-DevelopingTurbulent Boundary Layer Flow 341

(a)

(b) Fig. 8. Streamwise variation of vertical profiles of mean concentrations in the near-wake of

0

0 2.5 5 7.5 10

0 2.5 5 7.5 10

CUH2/Q

c'UH2/Q

1

2

3

z/H

4

5

0

1

2

3

z/H

4

5

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, Li & Meroney (1983) 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 (c99, c95, c90) 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

> <sup>1</sup> ln <sup>1</sup> 2 2

*c n P c erf*

*c l*

, (15)

0

0 2.5 5 7.5 10

0 2.5 5 7.5 10

CUH2/Q

c'UH2/Q

1

2

3

z/H

4

5

0

1

2

3

z/H

4

5

the cubical building. (a) Mean concentration. (b) R.m.s. concentration.

0 2.5 5 7.5 10

0 2.5 5 7.5 10

CUH2/Q

c'UH2/Q

**5.4 Characteristics of the peak concentration** 

0

1

2

3

z/H

4

5

0

1

2

3

z/H

4

5

atmosphere as follows.

0

0 5 10 15 20

WT (Sada, 2002) LES

0 5 10 15 20

CUH2/Q

c'UH2/Q

1

2

3

z/H

4

5

0

1

2

3

z/H

4

5

Log-normal distribution function:

uniform profiles of mean and r.m.s. concentrations with downstream distance are similar to the experimental data.

(c) t\* =90

Fig. 7. Instantaneous plume dispersion field. The yellow areas on the isosurface indicate 0.01% of initial concentration.

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.

uniform profiles of mean and r.m.s. concentrations with downstream distance are similar to

(a) t\* =15

(b) t\* =45

(c) t\* =90 Fig. 7. Instantaneous plume dispersion field. The yellow areas on the isosurface indicate

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

the experimental data.

0.01% of initial concentration.

model gives satisfacotoly results.

Fig. 8. Streamwise variation of vertical profiles of mean concentrations in the near-wake of the cubical building. (a) Mean concentration. (b) R.m.s. concentration.

### **5.4 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, Li & Meroney (1983) 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 (c99, c95, c90) 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:

$$P(c) = \frac{1}{2} \left[ 1 + \text{erf} \left\{ \frac{\ln \left( c/n\_c \right)}{\sqrt{2} \sigma\_l} \right\} \right],\tag{15}$$

Large-Eddy Simulation of Turbulent Flow and

(b)z/H=0.16 and (c)z/H=2.0.

012345678

LES Exponential type Log-normal type Clipped-normal type

c/c'

012345678

c90/c'

0.01

0.1

1-p(c)

1

(c)c99/c'.

0.0

0.5

1.0

z/H

1.5

2.0

data of Sato and Sato., 2002.

nearly comparable to that for the wind tunnel experiments.

Plume Dispersion in a Spatially-DevelopingTurbulent Boundary Layer Flow 343

indicate that the characteristics of probability distributions of the LES model depending on the values of *Ci* are consistent with those reported by Sato & Sada., (2002). 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

Fig. 9. Probability distribution functions of concentration fluctuation. (a)z/H=0.1,

012345678

LES Exponential type Log-normal type Clipped-normal type

c/c'

0.01

0.1

1-p(c)

1

0.01

0.0

0.5

1.0

z/H

1.5

2.0

012345678

LES Exponential type Log-normal type Clipped-normal type

c/c'

012345678

c99/c'

0.1

1-p(c)

1

Fig. 10. Vertical profiles of various peak concentration ratios of (a)c90/c', (b)c95/c' and

0.0

0.5

1.0

z/H

1.5

2.0

Next, we examine various peak concentrations of c99, c95, and c90 obtained by the LES model. Figure 10 shows vertical profiles of various peak concentration ratios at x/H=2.5 and 3.5. c90/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 (Sada & Sato., 2002). c95/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. c99/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

012345678

c95/c'

Exponential distribution function:

$$P(c) = 1 - I \exp\left(-I \frac{c}{C}\right) \tag{16}$$

Clipped normal distribution function:

$$P(c) = \frac{1}{2} \left| 1 + \operatorname{erf} \left( \frac{c - \mu\_0}{\sqrt{2} \sigma\_0} \right) \right| \tag{17}$$

Here, *erf, nc, σl, μ0* and *σ0* 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 *Ci* which is the concentration fluctuation intensity defined as the ratio of r.m.s. concentration to mean concentration.

$$I = \frac{2}{\overline{C}\_i^2 + 1} \,' \,. \tag{18}$$

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 *nc, σl, Ci, μ0* and *σ0* obtained in the experiments. They showed that a peak concentration of c99 could be predicted using the log-normal type for 0.3<*Ci*<1.0, the log-normal or the exponential types for 1.0<*Ci*<1.5, and the exponential type for *Ci*>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 c99 but also c95 and c90 in the nearwake region of the cubical building.

Figure 9 shows a comparison of probability distribution functions (1-p(c)) 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, *Ci*, 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 *nc, σl, Ci, μ0* and *σ0* obtained by the LES model. c99, c95 and c90 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 c99/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 c99/c' are almost the same as those of the exponential one, while the lognormal probability distribution is different from that of the LES model. c99/c' values obtained from the clipped-normal type are underestimated at each height. These facts

1 exp *<sup>c</sup> Pc I I*

<sup>0</sup>

2 2

Here, *erf, nc, σl, μ0* and *σ0* 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 *Ci* which is the concentration fluctuation intensity

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 *nc, σl, Ci, μ0* and *σ0* obtained in the experiments. They showed that a peak concentration of c99 could be predicted using the log-normal type for 0.3<*Ci*<1.0, the log-normal or the exponential types for 1.0<*Ci*<1.5, and the exponential type for *Ci*>1.5, while peak concentration of the clipped-normal type was

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 c99 but also c95 and c90 in the near-

Figure 9 shows a comparison of probability distribution functions (1-p(c)) 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, *Ci*, 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 *nc, σl, Ci, μ0* and *σ0* obtained by the LES model. c99, c95 and c90 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 c99/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 c99/c' are almost the same as those of the exponential one, while the lognormal probability distribution is different from that of the LES model. c99/c' values obtained from the clipped-normal type are underestimated at each height. These facts

*I C* 

 

<sup>1</sup> <sup>1</sup>

defined as the ratio of r.m.s. concentration to mean concentration.

*<sup>c</sup> P c erf*

*C*

0

, (16)

, (18)

, (17)

Exponential distribution function:

Clipped normal distribution function:

entirely underestimated.

wake region of the cubical building.

indicate that the characteristics of probability distributions of the LES model depending on the values of *Ci* are consistent with those reported by Sato & Sada., (2002). 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.

Fig. 9. Probability distribution functions of concentration fluctuation. (a)z/H=0.1, (b)z/H=0.16 and (c)z/H=2.0.

Fig. 10. Vertical profiles of various peak concentration ratios of (a)c90/c', (b)c95/c' and (c)c99/c'.

Next, we examine various peak concentrations of c99, c95, and c90 obtained by the LES model. Figure 10 shows vertical profiles of various peak concentration ratios at x/H=2.5 and 3.5. c90/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 (Sada & Sato., 2002). c95/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. c99/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.

Large-Eddy Simulation of Turbulent Flow and

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