div v(x) = 0, n terr = 0, v terr = v t,

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1 Proeedings of the Ceh Japanese Seminar in Applied Mathematis 6 Ceh Tehnial University in Prague, September 4-7, 6 pp. 4 8 FLOW AND POLLUTION TRANSPORT IN THE STREET CANYON PETR BAUER, AND ZBYŇEK JAŇOUR Abstrat. Air pollution is one of the serious problems in almost all ountries, espeially those with high population density and large industrial enters. We develop a mathematial model based on Navier-Stokes equations for visous inompressible flow and diffusion-onvetion equation desribing pollution transport, and solve the model using finite element method (FEM). A simple algebrai turbulene model is inluded with the turbulent visosity saled for urban area problems. We present the reent numerial results of Navier-Stokes flow and pollution transport in the D street anyon. Key words. FEM, atmospheri boundary layer, Navier-Stokes equations. AMS subjet lassifiations. 35K6, 35K65, 65N6, 68U. Introdution. We use a D model of air flow and pollution transport on a polygonal domain whih represents a vertial ut through the street area. We onsider the ase of stationary Navier-Stokes flow and diffusion-onvetion equation for one type of pollutant. We solve the following system of equations on (, T ) : (t, ) + v() grad (t, ) = D (t, ) + f(t, ), t v() v() ν v() + grad p() = g(), div v() =, (, ) = (), n terr =, v terr = v t, where R is a bounded domain, (t, ) is the onentration of the pollutant, v() is the veloity, is the initial ondition for onentration, n is the unit outer normal and terr denotes the terrain. The term f(t, ) represents the pollution soure and g() is the eternal fore. On the surfae, we use the Neumann boundary ondition for onentration and the Dirihlet boundary ondition for veloity, whih are appropriate from the physial point of view; see [9]. We use either Dirihlet or Neumann boundary onditions on the other parts of the boundary, like in = i, v in = v i, n out=, v n out=.. Weak formulation and numerial solution of the problem. The Navier- () Stokes problem is weakly formulated as follows. Let V = ( W ()), X = (W () ()), H = {q L () : q d = }, w X: w = v Γ in the weak sense, w n ds = div u d = for Dirihlet boundary ondition. We denote u = v w Department of Mathematis, Faulty of Nulear Sienes and Physial Engineering, Ceh Tehnial University, Prague. Institute of Thermomehanis, Aademy of Sienes of the Ceh Republi, Prague. 4

2 Flow in the Street Canyon 5 Fig... Lumped regions Fig... Lagrange and Cruei-Raviart elements (( w, s)) = w i s i j j = ( w, s), b( u, v, s) = i,j= We seek v X and p H, suh that: v (u i j i,j= j s i u j v i s i (( v, s)) + b( v, v, s) (p, div s) = ( g, s) (( w, s)) s V, (q, div u) = (q, div w) q H. j ). The inde h denotes finite-dimensional subspaes V h V, X h X, H h H. The mied formulation in finite-dimensional ase stands: (( v h, s)) h + b h ( v h, v h, s h ) (p h, div h s) h = ( g h, s) h (( w h, s)) h, (q, div h u h ) h = (q, div h w h ) h s V h q H h. The nonlinear term b h ( u h, u h, s h ) must be omputed iteratively. Diret appliation of this approah results in osillations in the solution. We use the upwinding tehnique proposed by [6], based on dual triangulation. This approah eventually leads to the iterative sheme: (.) ( A(u k ) B B T ) ( u k p k ) = ( G(u k ) H For the Diffusion-Convetion Equation, we use impliit Rothe method [4] and the method of harateristis [5] whih separates diffusion from onvetion. This approah uses pre-omputed veloity field, and is therefore independent of the flow type used. Appliation of these methods leads to the following linear system with a positive-definite matri; see []. For i =,..., m: m αj k [(v j, v i ) + τd( v j, v i )] = (τf k + k ϕ k, v i ) τd( w, v i ). j= ).

3 6 P. Bauer and Z. Jaňour p Fig. 3.. Navier-Stokes flow - pressure Fig. 3.. Navier-Stokes flow - veloity Numerial Solution using the Finite Element Method. We have hosen the finite element method in order to treat different terrain shapes easily. We use the linear Lagrange elements for onentration, Cruei-Raviart (Fig..) elements for veloity and pieewise onstant elements for pressure. Current mesh struture allows storing of multiple meshes together with their hierarhi struture to support the effiient multigrid solver whih is being developed. We an treat two different types of boundary onditions on eah part of the boundary.

4 Flow in the Street Canyon (a) t=. (b) t= () t=.3 (d) t= (e) t=.5 (f) t=. Fig Time evolution of onentration 3. Results. We show the reent results of steady state Navier-Stokes flow in the street anyon with Reynolds number Re = using paraboli veloity profile on the inlet. Note that the pressure is determined up to a onstant; we have hosen ero as a mean value in this ase. The domain has unit sie, and the parameters are: g =, D =. A onstant soure of pollution is loated at the street level simulating

5 8 P. Bauer and Z. Jaňour ehaust gases from the traffi. We an observe the umulation of pollutant at the leeward side of the the street. This is in agreement with the eperimental results, and ours due to the irular flow in the anyon. Due to relatively low Reynolds number and isothermal model, only a small amount of pollutant esapes the anyon beause of onvetion. Aknowledgment. This work was arried out under the projet Jindřih-Nečas Center for Mathematial Modelling LC65 of the Ministry of Eduation of the Ceh Republi, and with the support of the European Community - Researh Infrastruture Ation under the FP6 Struturing the European Researh Area Programme and within the HPC-EUROPA projet (RII3-CT ). REFERENCES [] P. G. Ciarlet, The Finite-Element Method for Ellipti Problems, North-Holland, Amsterdam (978). [] F. Brei, and M. Fortin, Mied and Hybrid Finite-Element Methods, Springer Verlag, New York (99). [3] M. Feistauer, Mathematial Methods in Fluid Dynamis, Longman, New York (993). [4] J. Kačur, Method of Rothe in Evolution Equations, BSB B. G. Teubner Verlagsgesellshaft, Leipig (985). [5] J. Kačur, Solution of Degenerate Convetion-Diffusion Problems by the Method of Charateristis, SIAM Journal on Numerial Analysis, 39 (), [6] F. Shiewek, and L. Tobiska, An Optimal Order Error Estimate for Upwind Disretiation of the Navier-Stokes Equation, Numerial methods in partial differential equations, no. 4 (996), [7] M. Beneš, Z. Jaňour, and F. Rys, Numerial Solution of the Falkner-Skan Equation, proeedings on 3rd Seminar on Euler and Navier-Stokes Equations (Theory, Numerial Solution, Appliations), Institute of Thermomehanis, Prague, May (998), 3 4. [8] M. Beneš, and Z. Jaňour, A Numerial Solution of the Planetary-Boundary-Layer Equations, etended abstrat in Workshop on urban boundary layer parametrisations, COST Ation 75, Zurih, May (), EC, [9] M. Beneš, and R.F. Holub, Aerosol Wall Deposition in Enlosures Investigated by Means of a Stagnant Layer, Environment International,, Suppl. (996), [] P. Bauer, Mathematial Modelling and Numerial Simulation of Pollution Transport in the Atmospheri Boundary Layer, Proeedings on the Conferene Topial Problems of Fluid Mehanis, Praha (5), 7.