Abstract. 1 Introduction

Transcription

1 Prediction of species transport in urban canyons using an h-adaptive finite element approach David B. Carrington and Darrell W. Pepper Department of Mechanical Engineering, University of Nevada, Las Vegas Abstract A three-dimensional h-adaptive finite element model has been developed for predicting airflow and particulate dispersion within urban canyons. Petrov-Galerkin weighting for the advection terms, mass lumping and h- adaptation are used in the model. The code has been used for the solution of many environmental transport problems including calculation of atmospheric wind fields over the Nevada Test Site, Las Vegas Valley, and species transport for indoor environments. The use of refined grids reduces resource requirements, allowing larger problems to be run on small computer platforms. 1 Introduction Efforts are underway to develop a local, mesh-adaptive finite element model that will accurately calculate wind fields and particulate transport over complex terrain. Performance of the finite element model is greatly improved with the use of adaptive remeshing. Computational speed and resource requirements are reduced with the ability to locally refine the grid during solution of the transient, nonlinear equations of motion (e.g. Pepper Accurately predicting species concentration is important when assessing consequences resulting from atmospheric dispersion of hazardous material. A model must be able to accurately simulate the global flow field and yet, be

2 56 Development and Application of Computer Techniques to Environmental Studies capable of resolving fine scale phenomena around obstructions and complex terrain. The use of h- adaptive unstructured grids combined with the Petrov-Galerkin finite element method, approaches numerical "exactness". In addition, the method allows for both fine and coarse scale resolutions. Refinement of unstructured meshes increases resolution in areas of steep gradients, i.e. where changes are occurring rapidly over time. Refinement is accomplished by concentrating nodes in regions where most of the activity is occurring, and unrelfining in regions where the solution is smooth. Adaptive mesh techniques must be able to quickly refine or unrefine to a coarser grid if the solution process is to be effective for large-scale problems. This paper presents the use of an h-adaptive finite element algorithm to calculate 3-1) wind fields and particle transport over buildings and through urban canyons. The algorithm runs on Pentium PCs and SGI workstations. Large-scale problems are run on an SGI Origin 2000 located within the National Supercomputer Center for Energy and the Environment (NSCEE) at UNLV. 2 h-adaptation Two types of unstructured meshes are generally used in 3-D- tetrahedrals and hexahedrals. Embedding tetrahedrals into the computational domain is easily accomplished without creating nodes where integration is incomplete. Quadrilateral (2-D) and hexahedral elements leave "virtual" (or "hanging") nodes creating an element with at least one extra node on a face and mesh incompatiblility. Figure 1 shows the resulting mesh inconsistency that occurs from 3-D h-adaptation. Figure 1. Virtual nodes for 3-D hexahedral element. Hanging nodes are not automatically resolved in the integration process. However, this computational inconsistency can be easily handled by interpolation. The flux between the elements must remain as near to equal as the interpolation will create. Anything less is equivalent to a source term.

3 Development and Application of Computer Techniques to Environmental Studies 57 In an explicit scheme, correction of flux inconsistency is performed by interpolation at hanging nodes before advancing in time. The finite element's linear interpolation function provides an accurate method for this process. Higher order interpolation functions provide better determination of this value. In a finite difference or control volume formulation embedding of a mesh in this manner may create poor results since the best interpolation possible for this discreatization is an average. This is insufficient unless the location of the edge of the adaptation is in a location where the solution is smooth. The model used in this study employs linear interpolation functions, that is, finite element shape functions, to determine velocity, pressure and temperature at the hanging nodes. Holes in the adapted grid are filled to minimize the amount of time spent interpolating values at the hanging nodes and to provide higher accuracy (see Pepper et al [1]). 3 Governing Equations The mathematical relations which describe conservation of mass, fluid motion, and species transport for three- dimensional atmospheric flow can be written as (Pielke [2]) Conservation of Mass dp dpu dpv dpw _ dt dx dy dz Conservation of Momentum x-direction: dpu dpu* dpuv dpuw _ d ( du\ d( du] d( du] r H -r r 1 ; - >fy P"T" *h i T \ h 3 \^ dt ox ay az ctx\ dx) dy\ ay) az\ oz) y-direction: 2vco sin y/ - 2wco cos if/- dx dpu dpuv dpv <9pvw _ d (' dv\ d (, dv\ d (' dv\ dp_ (3) dy

4 58 Development and Application of Computer Techniques to Environmental Studies z-direction: dpw dpuw dpvw dpw d f. dw} d. dw\ d (, dw\ ~+~ +~~+~ = * +T dy\ **1T dy Jr * (4) * dz Species Transport j ducj ducj dvc; dwc; d ( dc;\ d ( dca +~a^+-5t'+~5r'+--^=3-r*-5r rir ** A r f dt ok o(y dz ok^ okj <9y^ <9yJ (5) where u, v, w are the constant latitude (x), constant longitude (y), and vertical (z) components of velocity, respectively. The terms associated with Coriolis acceleration are the angular velocity of the earth, w, and the latitude y. Species concentration is C and gravitational acceleration is g. Source or sink for the j* species is S that can include changes of state, chemical transformation, precipitation and sedimentation. In this model, the flow is approximated as an incompressible fluid. Horizontal mixing is approximated by (Smagorinsky [3]) du dv^\ ( dv dii dx dy) {dx dy (6) The vertical exchange coefficients in the surface layer (to about 100 m) for momentum and species transport are given by (Blackadar [4]) Pm T (7) The friction velocity w, given by 1, z - in ku Z (8)

5 Development and Application of Computer Techniques to Environmental Studies 59 is assumed constant in surface layer where k is the von Karman constant. The Monin length L is defined as L = -0 ul J gke* where mean vertical temperature is 0 and 0* is the convective temperature. Businger-Dyer relations (Businger [5]) approximate p^ by p _ LI j = for L>0 (9) At the top of the surface layer, a transition or Ekinan layer is defined. In this layer, Coriolis effects are present. A vertical exchange coefficient for this layer is constant and is given by k=ku^s (Blackadar [4]). The standard weak formulation of the Galerkin weighted residual method is used to discretize the atmospheric equations of motion and species transport. The matrix equivalent forms of the governing equations are written as (Pepper and Brueckner [6]) (10) (11) where V is the velocity vector (m/s), CT is the gradient operator, and C. is species concentration (gm/nf) of the j* component. The matrix coefficients (denoted by [ ]) and column vectors ( { }) are integral relations based on the value of the interpolation function and its derivatives with respect to x, y, and z. The dot above the variable refers to time dependence. For example, the trial approximation for velocity is expressed where N. is the shape function. An explicit Euler scheme is used to advance the solution in time. Mass lumping is used for the time dependent term. Reduced integration is used (when elements are not overly distorted) in the solution process. The

6 60 Development and Application of Computer Techniques to Environmental Studies microscale domain in this study allows for the elimination of the Coriolis terms. In order to reduce numerical dispersion, a Petrov-Galerkin technique is used for the advection terms (Heinrich [7]). The technique applies a selective weight function to the advection terms, i.e., (13) where h, is element size, a = coth 0/2-2/0 with p = h^\v\/2k^, and is an effective diffusion in the direction of the local velocity vector. This weighting introduces artificial diffusion into the numerical scheme that acts along the local streamline. The use of Petrov-Galerkin weighting selectively eliminates the dispersive computational noise associated with steep gradient resolution. When combined with mesh adaptation, steep gradients are accurately resolved as the solution progresses without requiring extensive remeshing of the domain. 4 Results Velocity vectors are shown in Figure 2 around several obstructions after adaptation in a layer close to the surface. The vertical nature of the flow around the obstructions is shown in Figure 3. The adapted grid shown in Figure 4 demonstrates the ability to resolve the flow around the buildings and in the boundary layer using the algorithm. For point sources, Lagrangian particles are used to depict the transport of contaminant. These particles can be sized to specific radii and for density to account for deposition. The tra ectories for the Lagrangian particles are either interpolated using the finite element approximation functions or global interpolation functions. The particle trajectories can then be calculated in the global coordinate system. Distance of travel is determined by the time increment. This increment is calculated by limiting particle advection to be no further than its neighboring element. In this manner, accuracy of the trajectory is assured. Currently, particulate diffusivities are determined using Guassian probability density.

7 Development and Application of Computer Techniques to Environmental Studies 61 Figure 2. Velocity vectors in a lower layer. Figure 3. Velocity vectors in a vertical plane.

8 62 Development and Application of Computer Techniques to Environmental Studies Figure 4. Adapted mesh - resolution of flow field in areas of steep gradients. Figure 5. Transport and deposition of particles.

9 Development and Application of Computer Techniques to Environmental Studies 63 The distance a particle travels s dependent on its aerodynamic diameter and the effectiveness of gravitational setting. As the flow slows and swirls in the proximity of buildings or natural obstructions, concentrations increase. Deposition of the entrained particles occurs by impact with surfaces or by settling. Figure 5 demonstrates entrained particles being transported and deposited. 5 Conclusions An h-adaptive, finite element model has been developed for predicting species transport and dispersion within urban complexes. The h-adapting algorithm coupled with thefiniteelement method produces accurate solutions for environmental flow and transport problems. The 3-13 model developed in this study runs on enhanced personal computers, but is more suitable for running on workstations and larger computers to take advantage of memory architecture. A parallel version of the 3-Dalgorithm has been run on a SGI Origin Significant performance improvements for large-scale, three-dimensional atmospheric flows over complex terrain require execution on a parallel computer and the employment of optimized matrix solution schemes. The inclusion of p-adaptation with the h-adapting algorithm is being developed. References [1] Pepper, D.W., Carrington, D.B., and Lombardo, J.M., A Parallel h-adaptive Finite Element Model for Atmospheric Transport Prediction, Advances in Engineering Software, 29, pp , [2] Pielke, R.A., MesoscaleMeterologicalModeling, Academic Press, San Diego, pp , [3] Smagorinsky, L, Manabe, S., and Holloway, Jr., J.L., Numerical Results for a Nine-Level General Circulation Model of the Atmosphere, Monthly Weather Review, 93, pp , [4] Blackadar, A.K., Turbulence and Diffusion in the Atmosphere, Springer-Verlag, Berlin Heidelberg, pp , 1997.

10 64 Development and Application of Computer Techniques to Environmental Studies [5] Businger, J.A., Wyngaard, J.C., Izumi, Y., and Bradley, E.F., Flux-Profile Relationships in the Atmospheric Surface Layer, Journal of Atmospheric Science, 2%,w ,1971. [6] Pepper, D.W. and Brueckner, P.P., A Finite Element Model for Calculating 3-D Windfields Over Irregular Terrain, Measurement and Modeling of Environmental Flows, eds. Sherif S.A., Stock, D.E., Michaelides, E.E., Davis,LK, Celik, L, Khalighi, B., Kumar, R., FED-Vol. 143/HTD-VoL 232, American Society of Mechanical Engineers, New York, pp ,1992. [7] Yu, C.C. and Heinrich, J.C., Petrov-Galerkin Methods for the Time-Dependent Convective Transport Equation, International Journal of Numerical Methods in Engineering, 23, pp , 1986.