4th International Conference On Building Energy, Environment

Modelling Urban Airflows by a New Parallel High-Order Semi-Lagrangian 3D Fluid Flow Solver

M. Mortezazadeh, L. Wang

Centre for Zero Energy Building Studies, Department of Building, Civil and Environmental Engineering, Concordia University,

1455 de Maisonneuve Blvd. West, Montreal, Quebec, H3G 1M8, Canada

SUMMARY A fast and high-order 3D CFD solver based on semi-Lagrangian method is developed for solving urban airflows. Recently, semi-Lagrangian methods have been widely applied to modelling airflows in built environment. Limited studies have been conducted for large scale urban airflows, for which semi-Lagrangian methods are expected to perform well owing to their unconditional stability and simplicity, especially with large time steps. In this work, the airflow around the downtown Montreal is simulated by a newly developed CFD solver based on the semi-Lagrangian approach with a high-order backward forward sweep interpolation algorithm. Turbulence behaviour of the winds is simulated by a zero equation model. Parallel computing using the OpenMP library and a multi-grid solver are applied to speed up the convergence rate of solving the diffusion terms and Poisson equation. It shows that the new CFD solver can achieve a speedup of about 10 times with an improved accuracy than the conventional semi-Lagrangian method.

Keywords: Semi-Lagrangian method, Backward forward sweep interpolation, Urban airflows, zero equation turbulence model, Parallel computing.

INTRODUCTION Evaluation of wind aerodynamics in the vicinity of buildings has always attracted attentions due to the considerations of people’s comfort and at a large scale the overall outdoor living conditions. There existed many previous researches evaluating the wind effect for pedestrian safety and comfort by conducting: for example, the experimental studies using wind tunnels (Li et al., 1983; Yoshie et al., 2007; Yan et al, 1998), and the numerical simulation mostly using computational fluid dynamics (CFD) techniques (Blocken et al., 2009; Liu et al., 2016; Cheung et al., 2011). In comparison, CFD is a lower-cost solution to study urban airflow problems, for example, using some commercial CFD software, such as ANYSYS FLUENT, whereas on the other hand using these existing CFD tools is often time consuming when modelling airflows with large scales because of the numerical and mathematical constraints such as CFL constrains on the time steps and/or grid resolutions. It is typical in a matter of a couple of days to model urban airflows even for a small portion of a city.

Among the two main categories of CFD solutions, Eulerian and Lagrangian methods, the former solves airflows on fixed computational cells so its main disadvantage is the CFL constraint causing unstable simulation for large time steps and thus slow convergence (Hofmann, 2000). The problem worsens when solving large scale problems, such as urban airflows. In comparison, the Lagrangian approach considers the fluid as small particles, each of which moves along its characteristic curve. Although the CFL constraint may not apply to the Lagrangian method, the regeneration of the

solutions over the computational cells from the Lagrangian results makes the whole process quite complicated and computationally expensive. Therefore, both conventional Eulerian and Lagrangian methods seem to suffer from a certain levels of weaknesses when solving large scale fluid problems.

Researchers therefore sought for a mid-solution by combining the Eulerian and Lagrangian methods. In this approach, the so-called semi-Lagrangian method, the fluid is considered as a large number of particles which move along their characteristic curve on the fixed Eulerian cells (Mortezazadeh and Wang, 2017). Semi-Lagrangian method was first proposed by Courant et al. (1952) and recently has been used many fields including building airflows (Jin et al., 2013; Jin et al., 2015; Xue et al., 2016). For example, Jin et al. (2013) simulated the natural ventilation around buildings and compared the results with conventional CFD solvers, FLUENT, and they demonstrated the accuracy of the semi-Lagrangian method is acceptable. Xue et al. (2016) applied the semi-Lagrangian method to simulating mixing convection problem and their simulation showed a good agreement with the experimental work. Our literature review shows that the semi-Lagrangian method has been mainly used to simulate indoor airflows. In this paper, we will explore the capability of the semi-Lagrangian method to simulate urban airflows around buildings.

The conventional semi-Lagrangian method is often dependent on low-order interpolations and thus with low accuracy (Zuo et al., 2012; Zerroukat, 2010). Thus, various researchers proposed new methods to improve its accuracy (Mortezazadeh and Wang, 2017; Zuo et al., 2012; Zerroukat, 2010). Some of the proposed methods are too complicated to be easily applied to large scale problems. In the present work, a new high-order of semi-Lagrangian method (Mortezazadeh and Wang, 2017), the backward forward sweep interpolating algorithm, is used to simulate the urban airflow around four buildings in the downtown Montreal near Concordia University. The proposed method is easy to be implemented while providing high accurate results even on coarse grids when compared to the conventional semi-Lagrangian method. By a combination of the 3rd-order backward and the 3rd-order forward polynomial interpolation methods, a 4th-order interpolation accuracy can be achieved at a computing cost of the 3rd-order. To further speed up the calculation, the parallel OpenMP library is applied. OpenMP is a well-known application programming interface (API) for parallelizing a computing program through the Hyper-Threading Technology, with which we can assign independent work or mathematic operations to all accessible CPU threads. All the CFD simulations in the present work were completed on the system with 12 GB RAM and the Intel(R) Core(TM) i7-4790 CPU @ 3.60GHz.

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4th International Conference On Building Energy, Environment

The main purpose of the present work is to investigate the capability of the high-order semi-Lagrangian method to simulating urban airflow problems on coarse grids. To capture the turbulence behaviour of the flow, at a preliminary study phase, we added a simple turbulence model, the zero equation (Chen and Xu 1989), which may not be the perfect model for urban airflows but it would be enough to evaluate the performance of the new CFD solver. We do not investigate any physical phenomena about the wind velocity around the buildings and leave these topics for the future works, for example, adding large eddy simulation models.

The paper was structured as follows: first, the methodology of the solver is explained. Then, the accuracy and performance of the solver is investigated. For the validation, the cavity driven problem, the typical CFD benchmark case, is used. As a demonstration case, the airflow around the downtown Montreal is simulated, followed by the last section of conclusion and discussion.

CONSERVATION EQUATIONS In this section, we show the details of the high-order backward forward sweeping interpolation scheme. The following equations are the non-dimensional incompressible mass and momentum equations to be solved:

. 0U (1)

2(1 ( / ))( . ) tUU U p U f

t Re

(2)

where Eq. (1) is the mass conservation and Eq. (2) is the

momentum or Navier-Stokes equation.U , t , p , t ,

and

f represent velocity vector, time, pressure, turbulence

viscosity, reference viscosity, and body force terms,

respectively. Here, Re is the Reynolds number:

U LRe

(3)

where , U

, and L are the reference values of density,

velocity, and length, respectively. In the next step, by using the projection method and a fractional step method [7, 9, 12], Eqs. (1) and (2), can be written as follows:

2(1 ( / ))tUU f

t Re

(4)

( . ) 0U

Ut

(5)

2 1.U

t

(6)

Up

t

(7)

By implicitly solving Eq. (4), we will calculate the diffusion terms and body force by using a three-level V-cycle geometric multigrid solver (Mortezazadeh and Wang, 2016) and consequently an intermediate velocity field is obtained. In the

next step, by solving the advection terms, Eq. (5), the intermediate velocity field is updated by using the high-order semi-Lagrangian approach proposed by Mortezazadeh and Wang (2017). Then, the Poisson equation, Eq. (6), is solved in a similar way as Eq. (4) and the new pressure field is calculated. In the end, the new velocity field is estimated by using the new pressure field and also the intermediate velocity field (Eq. (7)).

SEMI-LAGRANGIAN METHOD In this section, the procedure of solving the advection term, Eq. (5), is introduced. In the computational domain, the difficulty of solving the momentum equation or Eq. (2) is because of the existence of the non-linear advection term. Based on the Lagrangian perspective, we can convert the nonlinear advection term to a linear form of derivative by changing the Eulerian derivative to the Lagrangian derivative. So, Eq. (5) can be written as follows:

0dU

dS (8)

where S Udt and it is the characteristic curve, along which

the fluid particles move. The first-order discretization form of the above equation is:

( ) ( )a dX XU U (9)

whereaX and

dX are the position of the arrival and departure

points, respectively. Note that the arrival points are always on the Eulerian cell centres while the departure points are not. So, to calculate the scalar value of the unknown variable at the departure points, we need to use an interpolation scheme. Figure 1 shows the schematic of the semi-Lagrangian approach with the proposed backward forward sweep interpolation scheme.

Figure 1. Schematic of 2D semi-Lagrangian method for 3rd-order backward forward sweep interpolation method

In Fig. 1, the value of the air velocity at the departure point is calculated with the 3rd-order backward polynomial interpolation method at one-time step and subsequently it is calculated by the 3rd-order forward polynomial interpolation method at the next time step. This process is repeated afterwards. The combination of the 3rd-order forward and backward methods, each separate 3rd-order truncation error is removed so a 4th-order accuracy is achieved. Note that for a two-dimensional problem, the 3rd-order interpolation scheme needs 9 neighbour Eulerian cells (27 cells in three-

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4th International Conference On Building Energy, Environment

dimensional problem) around the departure point, whereas in comparison, the direct 4th-order interpolation method will need 16 cells for 2-D problem and 64 cells for 3-D. So, the proposed method is significantly faster than the direct 4th-order scheme. In addition, using each of the 3rd-order interpolation methods alone without sweeping, such as only 3rd-order backward interpolation method, will create oscillation or dispersion errors near flow regions with sharp gradients. The combination of backward and forward methods removes these dispersion errors. For more details, please refer to the work by Mortezazadeh and Wang (2017).

ZERO EQUATION TURBULENCE MODEL To account for the turbulence behaviour of the airflow with few computing efforts, zero equation model developed by Chen and Xu (1998) is used by the present work as shown in Eq. 10.

2 2 2 20.03874 .t L u v w (10)

where L is the distance of the cell to the nearest wall, and u

, v , and w are the velocity components in the x, y, and z-

coordinates, respectively.

RESULTS First, a validation study for the new CFD solver is done to compare the accuracy of the proposed method on the coarse grids, compared to conventional semi-Lagrangian method. A 2D benchmark problem, the cavity flow, is used here with the

Reynolds number of 1000Re . Schematic of the cavity flow

problem and its boundary condition are shown in Figure 2.

Figure 2. Schematic of boundary condition and velocity stream

line for 2D cavity driven problem, 1000Re .

As mentioned before, the backward forward sweep interpolation scheme is with a 4th-order of accuracy possibly providing accurate results even on coarse grids. In this test case, the results are compared with an experiment work (Ghia et al. 1982). Figure 3 shows the performance of the proposed method in comparison with conventional semi-Lagrangian method.

Figure 3. u-velocity along a vertical line passed the center of

computational domain, 2D cavity driven problem, 1000Re .

The performance of the new high-order semi-Lagrangian method is shown in Table 1. The last column of the Table 1 shows the speedups of the proposed method in comparison with conventional semi-Lagrangian method in the context of similar accuracy. For getting the same accuracy for this particular test case, the conventional semi-Lagrangian

method will need a total of 90,000 cells ( 300 300 ), which is

9 times more than the proposed method. As a result, there is a speedup of around 9.32: the computational time of the proposed method is 44 [s] and the conventional semi-Lagrangian approach is around 410 [s]. It is expected that the speedup of the proposed method could be more significant for large-scale problems, such as urban airflows.

Table 1. Speedup comparison between CPU and Parallel

programming, 2D cavity driven problem, 1000Re

Number of cells Speedup (CPU/ Parallel programming)

10,000 1.27

1,000,000 2.63

4,000,000 4.44

For the same accuracy: Present work: 10,000

Conventional SL; 90,000 9.32

URBAN AIRFLOW SIMULATION In this section, we will use the proposed method to simulate airflow around four buildings of the Concordia Sir George Williams’s campus. Here, a set of fine grids are used near the buildings for enough resolution (See Figure 4). The geometries and computational domain are created by using the CFD0 Editor from the US NIST. In this case, the total number of cells are 1,452,000 and the Reynolds number is

100000Re . The dimensions of the domain is

24.5 17.8 4 . The height of the tallest building, the EV

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4th International Conference On Building Energy, Environment

building, is equal to 1, in the non-dimensional form. The boundary conditions are shown in Figure 4.

Figure 4. Schematic of the computational cells around the Concordia Sir George Williams campus.

The total computational time for this problem was around 15 minutes, while using conventional CFD solvers and commercial software, we need significantly more computational time, such as many hours for even fewer grids (Tseng et al. 2006; Hoof et al. 2010). Note that this computational time can be reduced further if we use the GPU computing, which is our future work. Figure 5 shows the airflows around the buildings in the horizontal surface close to the ground (x-y coordinates). In this figure the areas with high wind speed are shown. With the proposed wind direction, we can see the wind velocity between EV and MB buildings are higher than other areas and it could affect pedestrian comfort or even safety.

Figure 5. Schematic of the wind velocity vectors around the Concordia Sir George Williams’s campus on the x-y surface, (z=0.2)

Figure 6-1 shows the velocity contour in x-z plane between but parallel to the EV and MB buildings. Additionally, Figure 6-2 shows the airflow streamline around the LB and HB buildings. As mentioned before, the purpose of this work is not to investigate the flow separation and recirculation. The results, however, show the new solver with the zero equation model is able to capture these flow features, and provide information on pedestrian comfort. We believe the proposed solver is useful for the study of urban aerodynamics for large cities with a potential addition of other turbulence models, such as large eddy simulation models.

Figure 6. Airflow simulation (1) behind the LB and H buildings, (2) between MB and EV buildings, x-z surface.

CONCLUSION In the present work, we investigated the performance of a new high-order semi-Lagrangian method with zero equation turbulence model to simulate outdoor airflow and wind velocity near a downtown area. It shows that the proposed method and the new CFD solver are able to simulate large scale problems in a fast manner with a certain level of accuracy. The advantage of the proposed method in comparison with conventional semi-Lagrangian method is that the method is able to capture high accurate results even on coarse grids, which is an important feature for modelling large-scale problems, such as urban airflows. By using the high-order scheme and the OpenMP parallel computing platform, we are able to speed up the solver 10 times faster than the conventional semi-Lagrangian method. It can be therefore used for the study of large city pedestrian wind comfort. For the future work, we will add other turbulence models and GPU computing capabilities to further improve the accuracy and the computing speed of the new CFD program.

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