1. School of something FACULTY OF OTHER THE SENSITIVITY OF A 3D STREET CANYON CFD MODEL TO UNCERTAINTIES IN INPUT PARAMETERS James Benson*, Nick Dixon, Tilo Ziehn and Alison S. Tomlin prejfb@leeds.ac.uk Energy & Resources Research Institute (ERRI)

2. Overview 1. Motivation. 2. Case study. 3. Model setup. 4. Sensitivity analysis. 5. Results. 6. Conclusions.

3. Motivation CFD models increasingly used for prediction of air flow in urban areas. -Individual buildings resolved. -3D flow structures are predicted. Currently lack of information on computational fluid dynamic (CFD) model sensitivity/ uncertainty. Need to: -Determine effects of lack of knowledge of input parameters. -Improve confidence in air pollution models. -Provide information to help develop pollution modelling system. Require suitable sensitivity and uncertainty analysis techniques.

4. Gillygate, York, UK. Typical street canyon. H/W ≈0.8. Site of extensive experimental campaign. (Boddy et al. 2005). Experimental results allow comparison/ validation of CFD model. Case Study

5. Model Model is CFD k-ε turbulent flow model MISKAM v4.21 (Eichorn, 1996). Commonly used as an operation model (Lohmeyer et al., 2000). Uncertainties exist in input parameters including: - Background wind direction θ. -Surface and building roughness lengths. -Inflow surface roughness length (determines effect of upwind terrain on wind and turbulence profiles). Interested in effects on predicted flow (u, v, w and mean wind speed, U) and turbulence (Turbulent Kinetic Energy - TKE) in street canyon.

6. Model input parameters Surface roughness length z 0, used in log-law of the wind. -Inflow, buildings and surface roughness lengths. Background wind direction θ: -To show the effect of misspecification when comparing to experimental results. u – horizontal wind velocity u * - friction velocity z – distance from surface z 0 – surface roughness length κ – Von Karman Constant

7. Input parameter ranges Input parameterrange surface roughness length0.5-50cm building roughness length0.5-10cm inflow roughness length5-50cm background wind direction (θ)θ±10° Uniform input parameter distributions. Ranges chosen based on model limitations and modellers experience.

8. Model domain grid setup Non-equidistant grid. Resolution 89 (270m) x 124 (400m) x 28 (100m) points. Measurement points at: -G3 (183,211,5.5m), 2m from canyon wall. -G4 (171,211,5.3m), 1m from canyon wall.

9. Sensitivity Techniques Random Sampling Monte-Carlo (RS-MC) with regression analysis: -Pearson correlation coefficient. -Spearman ranked correlation coefficient. Random-Sampling High Dimensional Model Representation (RS-HDMR): -First order sensitivity indices. -Second order sensitivity indices. Cross sectional sensitivity analysis of model domain (y=211m). Comparison to experimental results.

10. Monte Carlo sampling sensitivity analysis 10000 runs at each wind angle for stable output means and variance. Random sampling. Input parameter limits and distributions defined. Samples generated for each parameter from above limits. Model run using input parameters from samples. 30 -40 minutes runtime for each run on 2GHz computer: Time taken for Monte-Carlo runs using single desktop PC = 625 days. Time taken on ‘Everest’ 30 processors of distributed memory computer for 10000 Monte-Carlo runs = 21 days.

11. HDMR Sensitivity Analysis Monte Carlo analysis requires large numbers of model runs which are often computationally prohibitive. HDMR is a more effective way of determining sensitivities for non-linear models. -Input parameter limits and distributions defined. -Quasi-random samples generated for each parameter from above limits. -Model run using input parameters from samples. -Model replacement constructed from the responses of the output to the inputs. Model replacement used to generate sensitivity indices at much reduced computational cost. Time taken on ‘Everest’ 30 processors for 1024 HDMR runs = 2.1 days.

12. Comparison of model results and experimental field results G3 TKE/U m 2. Black circles: experimental 15 minute averages, grey dots: RS-MC model results. The error bars on the experimental data are 1 standard deviation from the mean. х - coefficient of variation for the model results.

13. Mean TKE model results for θ = 90±10° Canyon cross-section of mean TKE and u, w wind vectors for θ=90±10°

14. Measurement point sensitivity analysis results – G3 TKE at θ=90±10° Sensitivity method Pearson correlations Spearman Ranked correlations HDMR first order rr2r2 r sp r sp 2 SiSi surface roughness -0.55780.3112-0.56900.32380.4258 building roughness -0.30910.0955-0.28030.07860.1154 inflow roughness 0.51310.26320.55420.30710.2533 wind direction (θ) 0.36890.13610.36430.13270.1610 total0.80600.84220.9555 Sensitivity of mean TKE at G3 to each parameter given by Pearson and Spearman Ranked Correlation coefficients and RS-HDMR first order sensitivity indices for θ=90±10°.

15. HDMR first order component function for G3 TKE at θ=90±10° Scatter plot (a) and RS-HDMR component function (b) for surface roughness length and un-normalised TKE at G3 for θ = 90±10°.

16. Measurement point sensitivity analysis results – G3 U at θ=90±10° G3 UPearson correlations Spearman Ranked correlations HDMR first order rr2r2 r sp r sp 2 S i surface roughness-0.49240.2424-0.49170.24170.2833 building roughness-0.18460.0341-0.19680.03870.0491 inflow roughness-0.17470.0305-0.16510.02730.0359 wind direction (θ)-0.76100.5791-0.75700.57310.6438 total0.88610.88081.0121 Sensitivity of mean wind speed (U) at G3 to each parameter given by Pearson and Spearman Ranked Correlation coefficients and RS-HDMR first order sensitivity indices for θ=90±10°.

17. HDMR first order component function for v at G3, θ=90±10° Scatter plot (a) and RS-HDMR component function (b) for θ and along canyon wind component v at G3 for θ = 90±10°.

18. Measurement point sensitivity analysis results – G3 TKE at θ=180±10° Pearson correlations Spearman Ranked Correlations HDMR first order rr2r2 r sp r sp 2 SiSi surface roughness-0.01400.0002-0.01130.00010.0005 building roughness-0.02470.00060.00590.00000.0009 inflow roughness0.41590.17300.44950.20200.1520 wind direction0.75250.56620.73200.53590.7433 total0.74000.73800.8967 Parameter interactionHDMR second order S ij surface roughness & wind direction0.0003 surface roughness & building roughness0.0014 surface roughness & inflow roughness0.0003 building roughness & wind direction0.0205 building roughness & inflow roughness0.0057 wind direction & inflow roughness0.0365 total0.0647

19. Cross section of TKE sensitivity at θ=90±10° Surface roughness lengthBuilding roughness length Inflow roughness lengthBackground wind direction θ

20. Cross section of U sensitivity at θ=90±10° Surface roughness lengthBuilding roughness length Inflow roughness lengthBackground wind direction θ

21. Sensitivity across all wind angles Relative sensitivity at (a) G3 and (b) G4 of un-normalised TKE (m 2 s -2 ) to all input parameters across all background wind angles.х - surface roughness length,o - building surface roughness length, ●–inflow roughness length, * - θ

22. Conclusions Overall uncertainty is small in comparison to model output means even with all possible parameter uncertainty included. Sensitivity is highly location dependant. Sensitivity is highly wind direction dependant. HDMR method provides more detailed sensitivity information including non-linear and second order effects with reduced computational expense.

23. Acknowledgements Thanks to ERSPC for the project funding and the EC for supporting this presentation at SAMO 2007. Also thanks to A. Tomlin, T. Ziehn and N. Dixon.