COMPARISON OF POLYNOMIAL CHAOS EXPANSION METHODS FOR UNCERTAINTY QUANTIFICATION IN CFD SIMULATIONS

Computational Fluid Dynamics (CFD) computer codes have proven to be a powerful tool in the analysis of all kinds of fluid systems. However, there is still a lack of practical methods for determining the uncertainty of their results, as most current techniques require performing too many simulations to be affordable in industrial-scale situations. One of the most promising methods for uncertainty quantification in computational fluid dynamics is Polynomial Chaos Expansion, a name that includes a variety of techniques, all based on the same mathematical background: projecting the system’s response into a basis of orthogonal polynomials. This paper discusses the main advantages and drawbacks of three of these techniques, namely random sampling, Gaussian quadrature and linear regression, in terms of reliability, ease of use and computational costs. All three techniques were applied to simulations of the turbulent mixing of two streams of water inside a Y-shaped channel, and the results compared with experimental data. Results show that, in this test case, quadrature method provides more reliable results than the other two techniques, with a lower computational cost. Due to its robustness and low number of simulations required, Polynomial Chaos Expansion via quadrature methods might be suitable for most industrial CFD simulations. INTRODUCTION Computational Fluid Dynamics (CFD) computer codes enable researchers and engineers to simulate fluid systems with great detail, and have become an important research and design tool in many fields. However, as CFD codes became more capable and popular, the question on how to determine the uncertainty in the simulations raised. The development of uncertainty quantification methods suitable for CFD simulations is an area of active research, since most current techniques require performing too many simulations to be affordable in industrial-scale situations. One recent attempt to alleviate this problem is the use of Polynomial Chaos Expansion (PCE) methods, which, if properly implemented, can produce accurate results from a limited number of simulations. NOMENCLATURE B [any] Orthogonal basis d [-] Normalized distance between experimental points and the uncertainty band F [any] Generic integral FOM [-] Figure of merit f ,g [any] Arbitrary functions I [-] Turbulence intensity m [-] Number of simulations / Sample size n [-] Number of uncertain parameters N [-] Number of terms in the PC expansion pi [any] Probability distribution function of the i-th parameter q [-] Order of the PC expansion ~r [m] Position vector R [any] System’s response U [m/s] Mean velocity u′ [m/s] RMS of velocity fluctuations x [any] Vector of uncertain parameters X [any] Generic region of integration xi [any] i-th uncertain parameter Special characters αi [-] i-th coefficient of the PC expansion δi j [-] Kronecker delta λ [-] Parameter controlling the shape of the inlet velocity profile μ [any] Mean system’s response ωi [any] Weights of Gaussian quadrature φi [any] i-th basis function σ [any] Variance of the system’s response The PCE foundational paper can be found in [1], while a complete review of the different PCE techniques, along with some applications to CFD simulations, is presented in [2], [3] and [4]. PCE methods can be broadly classified into two distinct categories: (i) intrusive, which require modification of the CFD code, and (ii) non-intrusive, which allow using CFD software as a ‘black box’ with no need for modifications. Since modification of the CFD code is often difficult, or even impossible in the case of commercial software, intrusive methods are rarely used in practice. For this reason, the present paper is limited to the non-intrusive techniques. 12th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics