School of somethingFACULTY 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)

Overview

1. Motivation.

2. Case study.

3. Model setup.

4. Sensitivity analysis.

5. Results.

6. Conclusions.

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.

• 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

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.

Model input parameters

• Surface roughness length z0, 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.

0

0lnz

zzuu

u – horizontal wind velocity u* - friction velocityz – distance from surfacez0 – surface roughness lengthκ – Von Karman Constant

Input parameter ranges

Input parameter range

surface roughness length 0.5-50cm

building roughness length 0.5-10cm

inflow roughness length 5-50cm

background wind direction (θ) θ±10°

• Uniform input parameter distributions.

• Ranges chosen based on model limitations and modellers experience.

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.

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.

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.

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.

Comparison of model results and experimental field results

G3 TKE/Um2. 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.

Mean TKE model results for θ = 90±10°

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

Measurement point sensitivity analysis results – G3 TKE at θ=90±10°

Sensitivity method

Pearson correlations

Spearman Ranked correlations

HDMR first order

r r2 rsp rsp2 Si

surface roughness

-0.5578 0.3112 -0.5690 0.3238 0.4258

building roughness

-0.3091 0.0955 -0.2803 0.0786 0.1154

inflow roughness

0.5131 0.2632 0.5542 0.3071 0.2533

wind direction (θ)

0.3689 0.1361 0.3643 0.1327 0.1610

total 0.8060 0.8422 0.9555

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°.

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°.

Measurement point sensitivity analysis results – G3 U at θ=90±10°

G3 U Pearson correlationsSpearman Ranked

correlationsHDMR first

order

  r r2 rsp rsp2  Si

surface roughness -0.4924 0.2424 -0.4917 0.2417 0.2833

building roughness -0.1846 0.0341 -0.1968 0.0387 0.0491

inflow roughness -0.1747 0.0305 -0.1651 0.0273 0.0359

wind direction (θ) -0.7610 0.5791 -0.7570 0.5731 0.6438

total 0.8861 0.8808 1.0121

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°.

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°.

Measurement point sensitivity analysis results – G3 TKE at θ=180±10°

Pearson correlations Spearman Ranked Correlations

HDMR first order

r r2 rsp rsp2 Si

surface roughness -0.0140 0.0002 -0.0113 0.0001 0.0005

building roughness -0.0247 0.0006 0.0059 0.0000 0.0009

inflow roughness 0.4159 0.1730 0.4495 0.2020 0.1520

wind direction 0.7525 0.5662 0.7320 0.5359 0.7433

total 0.7400 0.7380 0.8967

Parameter interaction HDMR second order Sij

surface roughness & wind direction 0.0003

surface roughness & building roughness 0.0014

surface roughness & inflow roughness 0.0003

building roughness & wind direction 0.0205

building roughness & inflow roughness 0.0057

wind direction & inflow roughness 0.0365

total 0.0647

Cross section of TKE sensitivity at θ=90±10°

Surface roughness length Building roughness length

Inflow roughness length Background wind direction θ

Cross section of U sensitivity at θ=90±10°

Surface roughness length Building roughness length

Inflow roughness length Background wind direction θ

Sensitivity across all wind angles

Relative sensitivity at (a) G3 and (b) G4 of un-normalised TKE (m2s-2) to all input parameters across all background wind angles.х - surface roughness

length,o - building surface roughness length, ●–inflow roughness length, * -

θ

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.

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.