Numerical simulation of air flow in an urban area with regularly aligned blocks

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Journal of Wind Engineering and Industrial Aerodynamics 67&68 (1997) 281-291

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Numerical simulation of air flow in an urban area with regularly aligned blocks P i n g H e a'*, T a d a h i s a K a t a y a m a b, T e t s u o H a y a s h i b, J u n - i c h i r o T s u t s u m i c, J u n T a n i m o t o b, I z u r u H o s o o k a b a Tsukuba Research Institute, Sanken Setsubi Kogyo Co., LM., Kinunodai 4-5-1, Yawara-mura, lbaraki 300-24, Japan UDepartment of Thermal Energy System, Kyushu University, Kasuga-shi, Fukuoka 816, Japan CDepartment of Civil Engineering and Architecture, University of the Ryukyus, Nishihara-cho, Okinawa 903-01, Japan

Abstract Numerical simulation of air flow distribution in a built-up area is an effective way to analyze and predict the urban thermal environment. A cyclic boundary conditions method for the numerical simulation of air flow around a block is used to model the unlimited spread of a built-up area. An equation for the calculation of the pressure difference between the windward and the leeward boundaries is proposed. Another simulation model which has 10 blocks aligned with the wind direction is used for comparison. The inflow boundary conditions are given by a wind tunnel test. The cyclic boundary conditions produced stable calculation results. The simulation results of the cyclic boundary conditions model are similar to those of the 10-block model in the cavity space. There is, however, a little difference between the results of these two models, and between them and the wind tunnel test in the higher area above the cavity and at the crossing point of the streets.

Keywords." Regularly aligned blocks; Air flow in an urban area; Cyclic boundary conditions

1. Introduction Air flow distribution in a built-up area has a strong influence on the urban climate [-1]. It is one of the essential factors to be investigated for a better understanding of the warming of u r b a n areas. The air flow in an urban area is directly influenced by the various factors of urban structure, e.g., scales of blocks and streets, heights of buildings and the whole size of the u r b a n area [2,3].

* Corresponding author. 0167-6105/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII $ 0 1 6 7 - 6 1 0 5 ( 9 7 ) 0 0 0 7 9 - 2

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Numerical simulation is one of the most effective methods to predict the air flow distribution. However, the specification of a simulation model is a serious problem, when numerical simulation is applied to the air flow in an urban area. If the simulation area is modeled after an actual urban area, it has to include a large area with an enormous number of calculation points, which makes the actual calculation very difficult. However, central districts in m a n y cities often have a similar style which consists of rectangular blocks separated by straight streets. If they are extremely simplified, they become districts with regularly aligned square blocks of constant size. This simplified urban model can easily be numerically simulated. Moreover, if the same blocks are aligned regularly, they are regarded as a repeat of a block. If this repeating is expressed by cyclic conditions, the central district of an urban area is represented by one or some block(s) with cyclic boundary conditions. When a numerical simulation model of one block with cyclic boundary conditions can express an unlimited spread of built-up area, it can reduce the space for simulation, computer memories and calculation time. The cyclic boundary conditions of a numerical simulation of air flow over an urban area as expressed by one block are proposed and tested in this paper. It is compared with another simulation model which consists of 10 blocks and a wind tunnel test.

2. Simulation model A central part of a built-up area is often divided regularly by streets in m a n y cities. A cell surrounded by the streets which is called a block usually consists of several buildings and courtyards. However, it is sometimes regarded as one large building, if the buildings in a block are concentrated densely and their heights are all similar. If such a built-up area is very simply modeled, it may become a series of regularly aligned rectangular cubes. Fig. 1 shows a model of a built-up area that is a group of simple rectangular cubic blocks of the same shape placed on a two-dimensional regular grid. The gaps between the blocks are thought to be streets. The directions of the streets are supposed to be parallel or normal to the wind direction, and all the streets are assumed to be of the same breadth to simplify the model, which means that all intervals between adjoining blocks are of the same width. The plan of the block is a square of which a side is D that is used as the representative length, and the height is 3D/4. The interval between two blocks which means the width of the streets is 3D/5. X, Y and Z-axes are set to the wind direction, the horizontal direction normal to the wind and the vertical upward direction, respectively, as shown in Fig. 1. Such a model was actually used in the wind tunnel test. It is the aim of this paper to express such a group of the same blocks as a repeat of a block or some blocks by cyclic boundary conditions in numerical simulation. A line of some blocks in the X direction is supposed to be a simulation model. If the ordinary inflow and outflow boundary conditions are used on the boundaries normal to the X direction and the symmetrical boundary conditions are used on the boundaries normal to the Y direction, the simulation model expresses an urban area with a

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Fig. 1. Simplifiedmodel of a built-up area.

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length, and the height is 3D/4. The interval between two blocks which means the width of the streets is 3D/5. The upper boundary is set at a height of 3D. The boundaries normal to Y are set at the center of the streets going the X direction. There are an approaching area in the windward and a run-off area in the leeward of the blocks. The latter type is a one-block model shown in Fig. 3. This is a simulation model with one block and its surrounding area which is thought to be a part of the 10-block model. The broken line area in Fig. 1 indicates the one-block model. Cyclic boundary conditions are adopted on the boundaries not only normal to Y but also normal to X, which means that the simulation area is also unlimited in the X direction. The one-block model expresses a built-up area which spreads uniformly, two-dimensionally and unlimitedly. The vertical boundaries are placed at the center of the streets, which means the depth of the surrounding space is half of the street breadth. Calculation points of the 10-block model and the one-block model are set on structural grids which are shown in Figs. 2 and 3, respectively.

3. Numerical simulation

3.1. Calculation method The basic equations for the numerical simulation are shown in Table 1. The k-e two-equation model is used as a turbulence model. Forward differences are applied to the temporal differential terms, and central differences are applied to the spatial terms. However, the advective terms in the k and e transport equations are changed to up-wind differences. All physical quantities are normalized by the representative values. The representative length is the width of the block, D. The representative velocity is the mean wind speed at a height of D on the inflow boundary of the initial conditions.

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Table 1 Basic e q u a t i o n s a n d v a r i a b l e s for n u m e r i c a l s i m u l a t i o n

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The initial conditions of the mean wind speed and the turbulence intensity on the inflow boundary are given by the wind tunnel test which results are shown in Fig. 4. The mean wind speed can be fitted to the power law profile with the exponent ¼ which is used as the initial condition of the velocity component normal to the inflow boundary. The tangential velocity components are assumed to be zero on the inflow boundary. The values on the inflow boundary are fixed in the 10-block model, and the free-slip conditions are applied on the outflow boundary. The upper boundary is supposed to be the free-slip condition. The tangential velocity components on the ground and the surfaces of the blocks are supposed to be a power law profile with exponent of ¼. The boundary conditions of the other variables on the solid boundaries are free-slip. 3.2.

Cyclic boundary conditions

The lateral boundaries normal to Y of both the models are given by symmetrical boundary conditions. The scalar variables and the tangential velocity components at the symmetrical positions on both the boundaries are of the same values. Actually, the averages of these values on both sides are given as the boundary conditions. The normal velocity component is zero on these boundaries. The variables on the inflow and the outflow boundaries which are normal to X of the one-block model are given by cyclic boundary conditions. The cyclic boundary conditions on these boundaries mean that all the variables on the outflow boundary which are the results of the former calculation step are given to those on the inflow boundary in the next calculation step. The positions of the calculation points in the Y - Z plane are the same positions on these boundaries. However, if the pressure

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on the outflow boundary is given to the pressure on the inflow boundary, the pressure difference between the inflow and the outflow boundaries becomes zero, which means that the source power for the wind is lost and the numerical simulation cannot be continued. There are only few studies [4] on the pressure difference of the cyclic boundary conditions, in which an empirical constant value is given as the pressure difference between the inflow and the outflow boundaries. Therefore, an equation for the calculation of the pressure difference is proposed in this paper. This is made from the momentum equation of the X direction, since the inflow and the outflow boundaries are normal to the X-axis. The pressure gradient on the inflow and the outflow boundaries is calculated as follows: AH .4

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P. He et al./J. Wind Eng. Ind. Aerodyn. 67&68 (1997) 281-291

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changes of the pressure gradient on the boundaries more mild than it is calculated at each point. These equations give the average pressure gradient between the outside and the inside cells adjoining the inflow or the outflow boundary. The pressure of the outside cell is calculated from this pressure gradient and the pressure of the inside cell. The initial values on the inflow boundary of both the models are given by the wind tunnel test. As for the one-block model, preparatory calculations of 200 steps are carried out without using the cyclic boundary conditions at first, and then the cyclic boundary conditions are applied to the inflow and the outflow boundaries.

4. Wind tunnel test

The outline of the experimental model for the wind tunnel test is shown in Fig. 1, which has the same geometry as the numerical simulation models. The actual size of the representative length, D, is 75 mm. The model blocks are aligned in 12 lines parallel to the wind direction, and there are 10 blocks in each line. The intervals between two lines and two blocks in a line are constantly 45 mm that is 3D/5. Each line of the model blocks, especially the line near the center of these lines is equivalent to the 10-block model. The wind tunnel used here is a closed circuit type. The test section is 8 m in length, 1.5 m in width and 1 m in height. The model blocks are directly fixed on the floor of the test section. The experimental wind speed is 6 m/s at a height of 600 mm from the floor. The approach mean wind speed and turbulence intensity profiles are shown in Fig. 4. A power law with the exponent of ¼ fits the mean wind speed profile up to a height of 300 mm. These profiles are used for the inflow boundary conditions in the numerical simulation of the 10-block model and they are also used for the initial conditions for the one-block model.

5. Results and discussion 5.1. Pressure difference on the cyclic boundary

The fluctuation of the pressure difference of the one-block model after the preparatory calculation is shown in Fig. 5. The pressure difference means the difference between the average pressure on the inflow boundary and that on the outflow boundary. The pressure difference changes violently at the beginning of the calculation. Then, it is stable at the simulation time t = 20, and the value then decreases gradually to 0.01 at t = 80. The calculation method proposed here creates a stable pressure difference between the inflow and the outflow boundary. 5.2. Mean wind speed profile

The change of mean wind speed profile with simulation time for the one-block model is shown in Fig. 6. These profiles are calculated from the simulation results at

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Fig. 6. Change of mean wind speed profile in the one-block model with simulation time.

the central points of the intersection of the streets a n d the cavity area between the blocks. The center of the intersection is the corner of the calculational area of the one-block model. The cavity area m e a n s the street n o r m a l to the wind direction. At the center of the intersection, the m e a n wind speed decreases below the height of the block a n d increases over the height of the block as the s i m u l a t i o n time progresses. The same tendency is observed in the change of the profiles at the center of the cavity. The wind speed below the height of the block at the center of the cavity space is

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Fig. 7. C o m p a r i s o n of mean wind speed and turbulence energy profiles.

smaller than that at the center of the intersection, and the change of the wind speed at the center of the cavity space with the simulation time is also smaller than that at the center of the intersection. When the simulation time is 20, the wind speed profiles at these points over the height of the block are similar to the initial conditions on the inflow boundary that is given by the wind tunnel test. Therefore, the wind speed profile at t = 20 is similar to the approach wind speed profile of the wind tunnel test. This may be one reason why there is a short flat step in the fluctuation of the pressure difference shown in Fig. 5. The simulation results after t = 20 show a different tendency from those before t = 20. The air flows after t = 20 perhaps indicate the air flow over a built-up area spread unlimitedly. The difference between the wind speed profile at t = 20 and that after t = 20 is thought to be the effect of the cyclic boundary conditions. The result at t = 20 of the one-block model is thought to be suitable for comparison with the results of the 10-block model and the wind tunnel test. The wind speed profiles at the two points mentioned above by the one-block model, the 10-block model and the wind tunnel test are compared in Fig. 7a. The wind tunnel profile at the center of the cavity space is omitted. The wind direction in the cavity space is very complicated and it is very difficult to measure the correct speed. The results of the 10-block model and the wind tunnel test indicated in Fig. 7a are the data between the seventh and the eighth blocks, since the wind speed profile is stable downwind of this point. The wind speed profiles in the one-block model, the 10-block model and the wind tunnel test are a little different from each other at the center of the intersection. The distribution patterns of these profiles resemble each other. At the center of the cavity,

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P. He et al./J. Wind Eng. Ind. Aerodyn. 67&68 (1997) 281-291

Vertical section at the center of that block

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Horizontal section at the center of the block height (a) One-block model

(b) 10-block model

Fig. 8. Distribution of wind velocity vectors in two numerical simulation models.

the wind speed profiles in the one-block model and the 10-block model show a similar curve below the height of the block. However, these profiles over the height of the block are a little different from each other, and each profile is similar to that at the center of the intersection of each model.

5.3. Turbulence energy profile The profiles of turbulence energy for the one-block model and the 10-block model are shown in Fig. 7b. The profiles at the center of the intersection are quite different from each other except in the lower layer. The values of the turbulence energy for the 10-block model are 5 times larger than those for the one-block model. However, these profiles are similar below the height of the block at the center of the cavity space, and each profile is close to that at the center of the intersection of each model over the height of the block.

5.4. Distributions of mean velocity vector The distributions of the mean velocity vectors for the numerical simulation results of the one-block model and the 10-block model are shown in Fig. 8. In the results of

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both simulation models, circulating secondary flow is observed in the vertical section at the center of the cavity space. In the horizontal section at a half height of the block, the distributions of the mean velocity vectors for both simulation models indicate symmetrical secondary circulations. The distribution patterns of the wind velocity vectors for the one-block model and the 10-block model are similar.

6. Conclusions Two types of numerical simulation and a wind tunnel test of air flow distributions in a built-up area with regularly aligned blocks were carried out, and the possibility of practical use of the cyclic boundary conditions proposed here was examined. The proposed equation for the pressure difference on the inflow and the outflow boundaries produces a stable result. Mean wind speed profiles in the one-block model at an appropriate simulation time are fairly similar to those in the 10-block model and in the experiment model of the wind tunnel test, which means that the one-block model with the cyclic boundary conditions is useful for a numerical simulation of air flow in an unlimited built-up area. The mean wind speed profile in the one-block model changes with the simulation time, and the profile at the later simulation time is a little different from that in the 10-block model. The reason is that the length in the X direction of the one-block model is unlimited while that of the 10-block model is limited. The turbulence energy in the one-block model decreases with the simulation time, which is the problem in the cyclic boundary conditions to be solved.

References [1] T. Katayama, A. Ishii, M. Nishida, J. Tsutsumi et al., Cooling effects of a river and sea breeze on the thermal environment in a built-up area, Energy and Buildings 15-16 (1991) 973. [2] J. Tsutsumi, T. Katayama, M. Nishida, Wind tunnel tests of wind pressure on regularly aligned buildings, J. Wind Eng. Ind. Aerodyn. 41~14 (1992) 1799. [3] B.E. Lee, M. Hussain, B. Soliman, Prediction natural ventilation forces upon low-rise buildings, ASHRAE J. (1980) 35. [4] S. Murakami, K. Hibi, A. Mochida, Three dimensional analysis of turbulent flow field around street blocks by means of large eddy simulation (Part 1), J. Archt. Plann. Environ. Eng. AIJ 412 (1990) 1 (in Japanese).