## Case Studies using the GUI-HDMR Software

The GUI-HDMR software has been applied to a number of case studies in the field of environmental and combustion modelling:

### 3D street canyon model

In this case study the flow field in a street canyon in the city of York (UK) was investigated using the micro-scale computational fluid dynamics (CFD) model MISKAM. MISKAM consists of a three-dimensional non-hydrostatic flow model and an Eulerian dispersion model based on the Reynolds averaged Navier-Stokes equations. Four main input parameters have been addressed which are the roughness lengths (inflow, surface and wall) and the background wind direction. Various outputs have been selected, among them the wind components and the turbulent kinetic energy (TKE) at certain measuring points. A sample size of *N*=512 was sufficient in order to construct a very accurate metamodel based on a second order HDMR expansion and up to third order polynomials. It was shown that most of the first order effects are non-linear. For some of the output parameters, interactions could be identified, which were responsible for more than 5% of the overall output variance. Detailed information about the set up of the HDMR method and the results can be found in Ziehn and Tomlin (2008a). More discussions and physical insights are given in Benson et al. (2008).

### Reactive plume model

A combined Lagrangian stochastic model with a micro mixing sub-model was used to investigate a reactive plume of nitric oxide (NO) in a turbulent grid flow doped with ozone. A chemical reaction scheme (including photolysis) consisting of eight reactions between the species NO, O, O3 and NO2 has been implemented and the effects of uncertainties in temperature dependent rate parameters, photolysis rates, temperature and NO:NO2 ratio at the source on the predicted pollutant profiles were studied. In total 21 parameters were assumed to be uncertain. The output of interest was the maximum NO2 concentration at the plume centre. An accurate metamodel has been constructed using only *N*=1024 samples and a second order HDMR expansion using up to fifth order polynomials. No significant parameter interactions could be identified and the overall variance of the model output could therefore be explained by parameters acting independently. Results and more detailed information can be found in Ziehn et al. (2008b).

### Premixed methane flame model

The GUI-HDMR software was applied to a one dimensional low pressure premixed methane flame model used to investigate the influence of fuel sulphur and nitrogen on the emission of nitrogen oxides within the flame. A one-dimensional steady state reaction advection diffusion model has been used considering the uncertainties in rate constants and thermodynamic data. This led to a study of 176 input parameters (153 reaction rates and 23 enthalpies of formation). The final NO mole fraction within the burnt gas region was considered as the target model output. It was shown that screening methods can be generally avoided by applying a threshold to exclude unimportant component functions. A sample size of *N*=1024 and up to tenth order polynomials were sufficient in order to construct an accurate metamodel and to calculate variance based sensitivity indices up to second order. However, in one certain case, a combination of a screening approach and HDMR was shown to be computationally more efficient. Three different scenarios have been investigated where the uncertainty ranges for the 23 enthalpy of formation are changed according to different sources in the literature. The parameter importance ranking based on the first order sensitivity indices was shown to be critically dependent on the uncertainty ranges chosen. No significant parameter interactions were identified for this case. The whole case study with detailed information and results can be found in Ziehn and Tomlin (2008b).

### Cyclohexane model

This case study presents a global sensitivity analysis of simulations of low temperature isothermal cyclohexane oxidation under fuel rich conditions. A well mixed non steady state box model with highly complex chemistry and complex temporal dynamics has been applied. The analysis was used to investigate the important features of the oxidation process, as well a possible factors underlying qualitative discrepancies between simulations and experiments. The chemical scheme compromises 60 species and 238 reactions, where the input parameters representing the temperature independent part of the rate constants (A-factors) are considered to be uncertain, i.e. 238 parameters. The selected target outputs were the time for the maximum rate of reaction for O2 and the percentage of O2 remaining at the time for the maximum rate. It was revealed that the model had significant higher order effects and the calculation of the second order sensitivity indices would have required a large number of samples, especially in connection with the large input space dimension. Therefore, the Morris method was used in order to exclude unimportant parameters and as a result the HDMR method was applied using 33 potentially important parameters. Up to tenth order polynomials have been used and a threshold has been applied in order to exclude unimportant parameter interactions. For more detailed information about this case study please refer to Ziehn et al. (2008a).

**References**

Ziehn, T., Tomlin, A. S., 2008a. Global sensitivity analysis of a 3D street canyon model - Part I: The developemnt of high dimensional model representations. Atmospheric Environment 42, 1857-1873.

Benson, J., Ziehn, T., Dixon, N. S., Tomlin, A. S., 2008. Global sensitivity analysis of a 3D street canyon model - Part II: Application and physical insight using sensitivity analysis. Atmospheric Environment 42, 1874-1891.

Ziehn, T., Tomlin, A. S., 2008b. A global sensitivity study of sulphur chemistry in a premixed methane flame model using HDMR. International Journal of Chemical Kinetics 40, 742-753.

Ziehn, T., Hughes, K. J., Griffiths, J. F., Porter, R., Tomlin, A. S., 2008a. A global sensitivity study of cyclohexane oxidation under low temperature fuel rich conditions using HDMR methods. Combustion Theory and Modelling, under review.

Ziehn, T., Dixon, N. S., Tomlin, A. S., 2008b. The effects of parametric uncertainties in simulations of a reactive plume using a lagrangian stochastic model layout. Croatian Meteorological Journal 43, 37-41.