Project

# Efficient uncertainty quantification of complex flows

Goal: Uncertainty quantification and sensitivity analysis of stochastic CFD problems using efficient methods.

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## Project log

In the current paper nondeterministic computational fluid dynamics (CFD) computations of three-dimensional (3D), developing, and statistically steady turbulent flow through an asymmetric diffuser with moderate adverse pressure gradient are presented. The inflow condition is assumed to be uncertain. The inlet streamwise velocity is supposed to be a stochastic process and described by the Karhunen-Loève (KL) expansion. In addition, the inlet turbulence intensity and turbulent length scale are assumed to be uncertain. The nonintrusive polynomial chaos (NIPC) expansion is used to propagate the inflow uncertainties in the flow field. The developed code is verified using a Monte Carlo (MC) simulation with 1000 Latin Hypercube samples on a planar asymmetric diffuser. A very good agreement is observed between the results of MC and polynomial chaos expansion methods. The verified uncertainty quantification method is then applied to stochastic developing turbulent flow through a 3D asymmetric diffuser. It was observed that the eigenvalues of covariance kernel rapidly decay due to the large correlation lengths and thus a few terms in the truncated KL expansion are used to describe the stochastic inlet velocity. For the KL expansion, the mean and the standard deviation are set to those measured experimentally. The uncertain inlet condition has a significant influence on the numerical results of velocity and turbulence fields specially in the developing region before the shear layers meet. It is concluded that one of the reasons for discrepancies between experimental and deterministic CFD results is the uncertainty in inflow condition. A sensitivity analysis is also performed using the Sobol' indices and contribution of each uncertain parameter on outputs variance is presented.
Most engineering problems contain a large number of input random variables, and thus their polynomial chaos expansion (PCE) suffers from the curse of dimensionality. This issue can be tackled if the polynomial chaos representation is sparse. In the present paper a novel methodology is presented based on combination of $$\ell _1$$-minimization and multifidelity methods. The proposed method employ the $$\ell _1$$-minimization method to recover important coefficients of PCE using low-fidelity computations. The developed method is applied on a stochastic CFD problem and the results are presented. The transonic RAE2822 airfoil with combined operational and geometrical uncertainties is considered as a test case to examine the performance of the proposed methodology. It is shown that the new method can reproduce accurate results with much lower computational cost than the classical full Polynomial Choas (PC), and $$\ell _1$$-minimization methods. It is observed that the present method is almost 15–20 times faster than the full PC method and 3–4 times faster than the classical $$\ell _1$$-minimization method.
Uncertainties are present in most engineering applications such as turbomachines. The performance of turbomachinery can be highly affected by the presence of uncertainties and these effects should be considered in the design procedure. The present study is the first work to quantify the impact of various uncertainties in a hydraulic machine. The paper is concerned with the combined effects of geometrical and operational uncertainties on the flow field and performance of a low specific-speed centrifugal pump. The uncertainty analysis is performed for three different flow rates, i.e., Q/Qd=0.825,1.0 and 1.1175. The volumetric flow rate, rotational speed and blade geometry of the pump are assumed to be stochastic with uniform Probability Distribution Functions (PDFs). The randomness in the blade geometry, due to the manufacturing tolerances, is imposed through two Karhunen–Loève (KL) expansions. The eigenvalues of covariance kernel of KL expansions rapidly decay due to the large correlation lengths and thus a few terms are used in the truncated KL expansion to describe the blade geometrical uncertainties. The uncertainties are propagated in the flow field and performance of the pump using the Non-Intrusive Polynomial Chaos (NIPC) method. The respective effects of assumed uncertain variables on the quantities of interest are assessed using Sobol’ indices. The uncertain operational and geometrical conditions have significant influences on the flow field and performance of the pump. It is observed that while variation of the pump head coefficient under operational and geometrical uncertainties is significant, the pump efficiency shows a robust behavior under assumed uncertain conditions.
The Polynomial Chaos Expansion (PCE) methodology is widely used for uncertainty quantification of stochastic problems. The computational cost of PCE increases exponentially with the number of input uncertain variables (known as curse of dimensionality). Therefore, use of PCE for uncertainty quantification of industrial applications with large number of uncertain variables is challenging. In this paper, a novel methodology is presented for efficient uncertainty quantification of stochastic problems with large number of input random variables. The proposed method is based on PCE with combination of ℓ1-minimization and multifidelity methods. The developed method employs the ℓ1-minimization method to recover important coefficients of PCE using low-fidelity computations. The low-fidelity evaluations should be accurate enough to capture the physical trends well. After that the multifidelity PCE method is utilized to correct a subset of recovered coefficients using high-fidelity computations. A threshold parameter is defined in order to select the subset of recovered coefficients to be corrected. Two challenging analytical and CFD test cases namely, the Ackley function and the transonic RAE2822 airfoil with combined operational and geometrical uncertainties are considered to examine the performance of the methodology. It is shown that the proposed method can reproduce accurate results with much lower computational cost than the classical full Polynomial Chaos (PC), and ℓ1-minimization methods. It is observed that for the considered examples, the present method can achieve comparable accuracy with respect to the full PC and the ℓ1-minimization methods with significantly lower number of samples.
The stochastic computations of a NASA gas turbine vane are conducted to investigate the effects of the operational uncertainties on the flow and heat transfer characteristics of the NASA C3X blade. The blade contains ten internal cooling channels to decrease the temperature. In order to minimize the analysis error the full conjugate heat transfer methodology has been employed to simulate the behavior of external hot gas flows, internal cooling air passages and the solid blade simultaneously. The v2 − f turbulence model is used and it is shown the simulation results have relatively good agreement with the experimental data. The inlet total pressure, total temperature, turbulence intensity, turbulent length scale and the outlet static pressure are assumed to be stochastic with uniform probability distribution functions. The effects of these uncertainties on the pressure, heat transfer coefficient and temperature of the blade outer surface at the midspan are studied. The polynomial chaos method with polynomials order p = 2 is used to quantify the uncertainties using the quasi random Sobol’ sampling. It is shown that the experimental data has the close agreement with the simulation data considering the uncertainties.
Most engineering problems contain a large number of input random variables, and thus their polynomial chaos expansion (PCE) suffers from the curse of dimensionality. This issue can be tackled if the polynomial chaos representation is sparse. In the present paper a novel methodology is presented based on combination of $\ell_1$-minimization and multifidelity methods. The proposed method employ the $\ell_1$-minimization method to recover important coefficients of PCE using low-fidelity computations. The developed method is applied on a stochastic CFD problem and the results are presented. The transonic RAE2822 airfoil with combined operational and geometrical uncertainties is considered as a test case to examine the performance of the proposed methodology. It is shown that the new method can reproduce accurate results with much lower computational cost than the classical full Polynomial Choas (PC), and $\ell_1$-minimization methods. It is observed that the present method is almost 15-20 times faster than the full PC method and 3-4 times faster than the classical $\ell_1$-minimization method.