A Single Formulation for Uncertainty Propagation in Turbomachinery: SAMBA PC

Abstract

This work newly proposes an uncertainty quantification method named SAMBA PC (Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos) that offers a single solution to many current problems in turbomachinery applications. At the moment every specific case is characterized by a variety of different input types such as histograms (from experimental data), normal PDFs (design rules) or fat tailed PDFs (for rare events). Thus, the application of UQ requires the adaptation of ad hoc methods for each individual case. A second problem is that parametric PDFs have to be determined for all inputs. This is difficult if only few samples are available. In gas turbines, however, the collection of statistical information is difficult, expensive and having scarce information is the norm. A third critical limitation is that if using Non-Intrusive Polynomial Chaos methods the number of required simulations grows exponentially with increasing numbers of input uncertainties: the so-called ‘curse of dimensionality’. In this work it is shown that the fitting of parametric PDFs to small data sets can lead to large bias and the direct use of the available data is more accurate. This is done by propagating uncertainty through several test functions and the CFD simulation of a diffuser, highlighting the impact of different PDF fittings on the output. From the results it is concluded that the direct propagation of the experimental data set is preferable to the fit of parametric distributions if data is scarce. Thus, the suggested method offers an alternative to the maximum entropy theorem to handle scarce data. SAMBA simplifies the mathematical procedure for many different input types by basing the polynomial expansion on moments. Its moment-based expansion automatically takes care of arbitrary combinations of different input data. SAMBA is also numerically efficient compared to other UQ implementations. The relationship between the number of random variables and number of simulation is linear (only 21 simulations for 10 input random variables are required). It is shown that SAMBA’s algorithm can propagate a high number of input distributions through a set of nonlinear analytic test functions. Doing this the code needs a very small number of simulations and preserves a 5% error margin. SAMBA’s flexibility to handle different forms of input distributions and a high number of input variables is shown on a low pressure turbine blade based on H2 profile. The relative importance of manufacturing errors in different location of the blade is analyzed.

Uncertainty analysis of global and local performance impact of inflow and geometric uncertainties using sparse grid-based non-intrusive polynomial chaos

“…33 However, the Askey scheme does not provide the polynomial basis function for truncated distribution. The basis function corresponding to the truncated Gaussian distribution can be obtained by moment-based arbitrary PC method 24,34 in this study. In practice, Equation (18) can be rewritten as the following truncated formwhere N p is the number of basis functions, and p is the order of polynomial chaos; truea⌢j are unknown coefficients, corresponding to ai1i2i3 in Equation (18); ψ j ( ξ ) are polynomial basis functions, corresponding to χ i (italicξi1, …, italicξin) in Equation (18), which satisfies the complete orthogonal condition in the L 2 space.…”

“…33 However, the Askey scheme does not provide the polynomial basis function for truncated distribution. The basis function corresponding to the truncated Gaussian distribution can be obtained by moment-based arbitrary PC method 24,34 in this study. In practice, Equation ( 18) can be rewritten as the following truncated form…”

“…Besides, more open literatures concerned on NIPC-based performance UQ for turbomachinery blades considering only the effects of either geometric 23,24 or inflow variations. 25,26 In addition to UQ, Global sensitivity analysis (GSA) is of great significance in quantifying the relative importance of each uncertain source.…”

“…Xia et al 22 used an adaptive NIPC to study the fluctuation of inlet pressure for a transonic compressor rotor, Rotor 67. Besides, more open literatures concerned on NIPC-based performance UQ for turbomachinery blades considering only the effects of either geometric 23,24 or inflow variations. 25,26…”

“…So far, Sobol’s analysis has been widely introduced to aerodynamic sensitivity analysis for wings considering only the random fluctuation of flows 18,19 and for turbine blades considering only the geometric variations. 24 However, few studies have involved aerodynamic sensitivity analysis considering the coexistence effect of the geometric and inflow variations of turbomachinery blades. Furthermore, the aforementioned stochastic problems of turbomachinery mostly have focused on UQ and GSA of the overall performance.…”