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  • 1. The seventh International Conference on Urban Climate, 29 June - 3 July 2009, Yokohama, Japan ON THE EFFECTS OF COMPLEX URBAN GEOMETRIES ON MESOSCALE MODELLING Adil Rasheed*, Darren Robinson*, Chidambaram Narayanan**, Djamel Lakehal** *Solar Energy and Building Physics Lab., EPFL, Switzerland; **Ascomp GmbH, Zurich, Switzerland Abstract The effects of urban structures on the atmosphere are studied using mesoscale models having a resolution ranging from a few meters to a few kilometers. Because of such a coarse resolution urban parameterizations are required to simulate the effects of the subgrid scales (buildings and canyons). Mostly, these parameterizations are developed by assuming a city to be comprised of an array of cubes. In the present study we attempt to verify this basic assumption by comparing spatially averaged quantities such as the velocity, turbulent kinetic energy and radiative exchanges in an urban domain comprising different geometrical structures. The spatially averaged velocity fields are computed using the Immersed Boundary Technique (IB) whereas the radiant exchanges are computed using a Simplified Radiosity Algorithm (which accounts for sky anisotropy). Key words: Urban Parametrization, Urban Complexity, Mesoscale Modelling 1. INTRODUCTION Mesoscales models are used to understand the heat and momentum exchanges between the air flowing above a city and the urban elements from which the city is comprised. However, with a resolution of a few hundreds of meters or a few kilometers these models are not capable of explicitly resolving the heat and momentum exchanges. They are instead parameterized and simulated using an Urban Canopy Model. Several urban canopy models have been developed in recent years eg Kondo [1], Martilli [2]. However, all of these canopy models share the assumption that a city is made up of a regular array of cubes or infinitely long canopies. The inputs to these models which include street width, building width, building density and a statistical representation of the buildings’ heights, are generally obtained through quantitative field surveys (which are very slow and time consuming to perform) or qualitative estimates. But in performing this geometric abstraction there is no way to ensure that the total built surfaces and volumes of the simplified geometry match those of the actual city. In this paper we aim to test the central hypothesis that cities can be accurately represented by a regular array of cubes or canopies. For this we investigate the effects of complexity in urban geometry on the spatially averaged drag forces, velocities, turbulent kinetic energy and shortwave radiation exchange. For drag computation we have used an Immersed Boundary Technique while for computing the radiation we have used a Simplified Radiosity Algorithm 2. METHODOLOGY 2.1 Simplified Radiosity Algorithm: In order to implement the SRA (Simplified Radiosity Algorithm) for the calculation of longwave and shortwave exchange in the urban context, view factors from each surface to a descretized sky vault, from the sun to each surface and from each surface to other surfaces are required. These view factors are computed by processing renderings of the urban scene from the relevant view position (surface or solar normal). The view factors of the sky are used with the cosine of the angle of incidence between the surfaces and the sky patches to obtain the sky contribution matrix. This matrix is then simply multiplied by the radiance of each sky patch to obtain the irradiance incident on each surface. Similarly, the sun view factors are used to obtain the direct contribution matrix. Finally, the view factors between surfaces are combined with the relative angle between the surfaces to obtain the surface contribution matrix which allows for multiple diffuse reflections to be taken into account. Further details including results from extensive validation tests can be found in Robinson (2004). 2.2 Immersed Boundary Technique: To use a purely conventional CFD approach to simulate air flow around complex geometries encountered in a city it is normal to use an unstructured mesh. But unstructured grid generation is a time consuming process and the numerical methods used for solving the conservation equations with unstructured mesh suffer from stability and convergence issues. In contrast, a Cartesian mesh is very easy to generate and efficient numerical algorithms can be used with them thus reducing the computational time significantly. However, it is not straightforward to fit Cartesian grids to complex geometries and doing so invariably creates a great many redundant cells. As a compromise we use a solution which combines numerical stability on Cartesian grids with a procedure for fitting these grids to complex geometries. In this approach, which we call Immersed Boundaries, complex geometries are handled using a non body conformal Cartesian grid in which the fluid solid interface is represented by a surface grid, but the Cartesian volume is generated with no regard to this surface grid. Thus the solid boundary would cut through the Cartesian volume grid. But because the grid doesn’t conform to the solid boundary, incorporating the boundary conditions requires us to modify the equations in the vicinity of the boundary. Assuming that such a modification is possible (of course it is) the modified governing equations would then be discretized using a finite difference, finite volume or a finite element technique without resorting to coordinate transformation or complex discretization operators. When compared with unstructured grid methods, the Cartesian grid-based IB method retains the advantage of being amenable to powerful line-iterative techniques and geometric multigrid methods, which can also lead to lower per-grid-point operation counts. In the Urban Context this new algorithm, as against the conventional CFD approach, offers us the possibility of simulating large domains. In principle, constructing urban geometry with a 3-D modeling tool (from google earth
  • 2. The seventh International Conference on Urban Climate, 29 June - 3 July 2009, Yokohama, Japan data) is trivial and quick and such geometries can be readily converted into STL (stereo lithography file) format, which holds information about the solid-liquid interface (in our case the building-atmosphere interface). The file is then processed to determine the solid and fluid zones. The flow equation is then solved only in the fluid zone so that in the solid zone the flow equations are switched off. For the present study we have used TransAT [5] code as this is the only code which offers the option to use IB. More detail about the IB can be found in [4] 3. TEST SET-UP AND RESULTS For the study we have chosen a part of Basel with a dimension of 1000m by 750 meter. A good approximation of the real geometry is sketched and it is assumed that all the buildings have a height of 15m. The total built vertical and horizontal surfaces are presented in Table 1. A simplified representation of the domain is also created; having the same built surface areas and volumes. The simplified representation consists of 20 X 18 cubes each of dimension 20m X 20m X 15m aligned in a regular array with a spacing of 30m in the stream wise direction and 20m in the span wise direction. Both the complex and simplified geometries are shown in Figure 1 Table 1: Geometric characteristic of the built surfaces in the domain Horizontal built area (Roofs) 144000 m2 Building Height 15 m Vertical built ares (Walls) 432000m2 Total Built Volume 2160000 m3 Ground area (Ground) 606000 m2 3.1 Radiation Set up: For radiation computation the surface in both the representations are tessellated into smaller surfaces. In the complex representation the roofs, walls and ground surfaces are subdivided into 992, 2658 and 870 triangular surfaces while for the simplified representation there are 2160, 7520 and 2242 triangular surfaces. Figure 1: Surface tesselization for Complex and Simplified Geometry (STL files) Figure 2: Comparison of the amount of Shortwave Radiation absorbed (by roof, wall and ground) every hour by the complex and simplified geometry 3.2 Radiation Results: It is quite clear that all domains of the same size will have the same amount of solar radiation entering them. However, for mesoscale modeling, the correct calculation of the distribution of the radiation amongst the wall, roof and ground surfaces is very important as this will result in differential heating of the different types of surfaces and hence result in different buoyancy forces being transferred to the air flowing over an urban area. In Figure 2 we therefore present the distribution of shortwave amongst ground (G), wall (W)
  • 3. The seventh International Conference on Urban Climate, 29 June - 3 July 2009, Yokohama, Japan and roof (R) surfaces over the full domain for both simplified (S) and complex (C) geometries on 7th of January for which the sky was clear. From Figure 3 we made the following observations: 1. For roofs: Since the horizontal roof surface areas in both the representations are the same and all the buildings are of the same height (and hence there is no obstruction to the sky) we observe that the amount of radiation absorbed during the whole day is similar which is quite expected. 2. For Ground: In the particular case of the complex representation that we have chosen it is clear that the ground receives more shortwave radiation compared to the simplified case, as views to the sky and sun are relatively unobstructed. 3. For Walls: The simplified representation receives more shortwave radiation than its counterpart, due to an increased reflected contribution and an increased south facing surface area. Thus, for the particular day the walls in the simplified representation will be much hotter than in the complex one. The opposite will be true for the ground surfaces. This will result in very different surface temperatures of walls and ground and may influence the energy exchange in a significant way. CFD simulation set up: For the CFD simulation the geometrical representation is the same as for the radiative calculations. However, the domain has been extended on the inlet side by 500m and at the outlet side by 1000m. The spanwise boundaries are placed 125m away from the outermost building. The domain is discretized into a mesh of 175 X 175 X 40 cells (stream wise X span wise X vertical) using TransATmesh [4] (Figure 3). An inlet boundary condition with 1m/s velocity in the streamwise direction is imposed on the left side of the domain and an outlet boundary condition is applied at the right end. For the bottom side of the domain a wall boundary condition is specified and for the rest of the surfaces symmetry boundary conditions are imposed. The turbulence model used in this simulation is the standard k-e model, while the convective scheme used for density and velocity is the HYBRID. A preconditioned (multigrid) GMRES pressure solver is used for solving for the pressure field. The flow is solved in a steady state with convergence criteria of E-4 for velocities and kinetic energy and E-03 for dissipation. Figure 3: Mesh for CFD simulation of flow over complex and simplified geometry Figure 4: Velocity field at a height of 5.6m above the ground level (a)Complex Geometry (b) Simplified Geometry
  • 4. The seventh International Conference on Urban Climate, 29 June - 3 July 2009, Yokohama, Japan (a) (b) Figure 5: Comparison between the spatially averaged (a) velocity profile (b) Turbulent kinetic energy profile in complex and simplified geometries Table 2: Space averaged drag (Fx, Fy, Fz), shear (Sx, Sy,Sz) and total forces Fx Fy Sx Sy Fx+Sx Fy+Sy Complex 8207 157 985 2.5 9194 160 Simple 4931 -46 754 1.3 5685 -45 CFD results: Figure 4 gives the velocity field for both geometric representations at a height of 5.6m above the ground surface. The more complex (real) representation is characterized by the formation of large vortices formed in the inter-building spaces while in the simplified representation vortices are formed on the leeward side of the cubes. These are small and well isolated from each other. Moreover, the simplified case offers ample opportunity for the fluid to “short circuit” so that the fluid motion remains unidirectional in stream wise canopies. On the other hand the flow tends to be deflected (in the span wise direction) in the more complex representation due to the random orientation of buildings. This is evident from the magnitude of the spatially averaged drag forces presented in the Table 2. Large vortices formed in the complex scenario also results in larger values of turbulent kinetic energy (Figure 5(b)). However, from the spatially averaged profiles of the stream wise velocity (Figure 5(a)) it is observed that the profiles are almost the same. 4. CONCLUSION From the work done it appears to be clear that urban geometric complexity plays a major role in the distribution of radiant energy which might lead to differential heating of surfaces and hence to differences in the total energy exchange with the surrounding air. A future course of study will be to study the impact on the energy exchange itself. It should be noted here that we have not yet analyzed the long wave radiation distribution which being a function of surface temperature may be more influenced by the geometric complexity. Although the drag forces are affected by the complexity of urban geometry, its effect on the spatially averaged velocity is negligible. However, the mechanical behavior of the flow does need further investigation. But it is encouraging that if the complexity indeed doesn’t significantly affect the spatially averaged velocity profiles then a method to identify an equivalent geometry for a real complex urban scene may be based on radiation balances alone. 5. REFERENCES [1] Kondo, H., Genchi, Y., Kikegawa, Y., Ohashi, Y.,Yoshikado, H., and Komiyama, H. (2005). Development of a multi-layer urban canopy model for the analysis of energy consumption in a big city: Structure of the urban canopy model and its basic performance. Boundary-Layer Meteorology, 116:395–421. [2] Martilli, A., (2001) Development of an Urban Turbulence Parametrisation for Mesoscale Atmospheric Models, PhD Thesis, EPFL [3] Robinson, D., Stone, A., 2004, Solar radiation modelling in the urban context, Solar Energy, 77(3), p295-309 [4] Mittal, R. and Iaccarino, G., 2005, Immersed Boundary Methods, Annual Review Fluid Mechanics, 37:23961 [5] http://www.ascomp.ch/

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