1. How to use CFD (RANS or LES) models for urban parameterizations – and the problem of averages Alberto Martilli CIEMAT Madrid, Spain Martilli, Exeter, 3-4 May 2007. Pieter Bruegel the Elder. The Tower of Babel. 1563. Oil on panel. Kunsthistorisches Museum, Vienna, Austria

2. Only features larger than the volume over which the average is performed can be resolved. Mesoscale model numerical resolution (few kilometers or (at best) several hundreds of meters in the horizontal) implies a spatial average over a volume comparable with the grid cell. Martilli, Exeter, 3-4 May 2007.

3. ‘LES’ approach (spatial filter) But.. Spatial averages of what? In mesoscale modelling literature I could find two approaches: ‘RANS’ approach (spatial average of non- random fields) Martilli, Exeter, 3-4 May 2007.

4. Atmosphere is turbulent. Non-deterministic behavior. The next state of the environment is partially but not fully determined by the previous state of the environment. For a variable in a turbulent flow , we define the probability to get the value  as f(  ) (probability density function). Then the mean is (from Pope, 2000) This mean fulfill the Reynolds assumptions and filters out the stochastic, random component leaving only the deterministic component. ‘RANS’ approach Parameterization of turbulent effects must be indipendent than resolution. Martilli, Exeter, 3-4 May 2007.

5. Mesoscale models computes, then, spatial averages of the mean variables There is a double averaging Martilli, Exeter, 3-4 May 2007.

6. Over homogeneous terrain turbulent structures can form randomly anywhere. The mean fields do not see horizontal variations. mean So, we can reasonably assume that: Martilli, Exeter, 3-4 May 2007.

7. Over heterogeneous terrain (e. g. cities), the mean (deterministic) fields can show structure at the scale of the heterogeneities. The grid resolution of the mesoscale model is bigger than the scale of heterogeneities. Martilli, Exeter, 3-4 May 2007. There can be deterministic features that are subgrid (not resolved, must be prameterized)

8. Some consequences of the double averaging. Every variable can be seen as the sum of three terms. is the spatial variation of the mean field within the spatial averaging scale is the departure from the mean (turbulent part, stochastic) Resolved stress Reynolds stress (turbulent) Dispersive stress (it can be present also in laminar flows) Mean deterministic structures smaller than the averaging spatial scale. Martilli, Exeter, 3-4 May 2007.

9. How to validate? mean Homogeneous terrains: Where T is the integral time scale of the turbulence (Garratt 1994). Point measurements can be used to validate the simulations Martilli, Exeter, 3-4 May 2007.

10. How to validate? Hetereogenous terrains Point measurements cannot be compared with model results. A spatial average is needed. How to perform it? or Very dense set of measurements (difficult). Use a CFD model (RANS or LES). Martilli, Exeter, 3-4 May 2007.

11. LES approach The only average performed is a spatial average over the grid cell and the time step (Pielke, 1984, Jacobson, 1999) This average filters out all the features that are smaller than the grid cell (no matter if they are turbulent or not): Martilli, Exeter, 3-4 May 2007.

12. Advantages : only one averaging process (no dispersive stress, etc.) Disadvantages: Reynolds assumption is not (strictly) valid (Galmarini and Thunis, 1999) When the grid cell size becomes close to the size of the most energetic eddies (hundreds of meters during daytime), turbulent stochastic motions start to be resolved (we are in Terra Incognita, Wyngaard, 2004). The model solution represents then, only one of the possible states of the atmosphere (how to use these results?? Time averages??). Martilli, Exeter, 3-4 May 2007. Parameterizations should be resolution dependent

13. In any case, spatial averages are also needed. As for the previous approach, CFD (LES) models can be used to perform such averages over hetereogenous terrain. Martilli, Exeter, 3-4 May 2007.

14. An example with a CFD-RANS code. FLUENT over a regular array of cubes (Santiago, Coceal, Martilli, Belcher, about to be submitted) Simulations for steady state based on Reynolds-Averaged Navier-Stokes equations (RANS) Turbulence model: k-  standard Governing equations solved by means of a collocated grid system using finite volume method Pressure-velocity coupling : SIMPLE Advection-differencing scheme: QUICK Martilli, Exeter, 3-4 May 2007.

15. Aim: use the CFD RANS simulations to perform a parameter study to 1)Test the hypotesis of Martilli and Santiago (2007) on the modified drag parameterization. 2)Derive values of drag coefficients (C Dmod )for different packing densities Martilli, Exeter, 3-4 May 2007. Introduce two new velocity scales from turbulent and dispersive kinetic energy and define the drag as where dke=dispersive kinetic energy

16. Validation against DNS simulations (Coceal et al. 2006) Periodic Boundary conditions at inflow-outflow Symmetric on lateral. Martilli, Exeter, 3-4 May 2007.

17. Comparison DNS-RANS Martilli, Exeter, 3-4 May 2007. (spatial averages)

18. Comparison DNS-RANS Martilli, Exeter, 3-4 May 2007. (spatial averages)

19. To perform a parameter study, RANS is a better tool. RANS simulations compare worse with measurements than DNS, (or even LES). But they are much faster in terms of CPU (around 100 times faster than LES). Martilli, Exeter, 3-4 May 2007.

20. Study for different f f =0.0625 f =0.11 f =0.16 f =0.25 f =0.33 f =0.44 sparse dense Martilli, Exeter, 3-4 May 2007.

21. where A = 1.0, B = 6.4, C = -29. and D = 28.. for Martilli, Exeter, 3-4 May 2007.

22. Conclusions/Summary The averaging technique chosen is important over hetereogenous terrain (model validation, interpretation of the results, paramterizations, etc.) In urban areas point measurements are not representatives of spatial averaged values (in particular in the urban canopy). CFD models can be used to obtian such spatial averages CFD-RANS are less precise than LES or DNS, but much faster. They can be used for parameter studies. Using CFD-RANS simulations, a paramterization for C dmod have been derived as a function of the packing density. Martilli, Exeter, 3-4 May 2007.

23. Thank you! Martilli, Exeter, 3-4 May 2007.

24. Pressure drag P1P1 P2P2 Sink of momentum How to parameterize the drag? Usually for vegetation canopy flows:  k building density Different configuration alligned array Martilli and Santiago, 2007, BLM Martilli, Exeter, 3-4 May 2007.

25. Estimation of C D C D (z) Alternative Introduce two new velocity scales from turbulent and dispersive kinetic energy and define where and dke=dispersive kinetic energy C dmod constant with height kinetic energy of the time averaged structures smaller than the grid cell Martilli, Exeter, 3-4 May 2007.

26. Spatial variability of vertical profiles. Martilli, Exeter, 3-4 May 2007. WIND E F IHABCDG