A Single Formulation for Uncertainty Propagation in Turbomachinery: SAMBA PC

Abstract

This work newly proposes an uncertainty quantification (UQ) method named sparse approximation of moment-based arbitrary polynomial chaos (SAMBA PC) 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 probability density functions (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 nonintrusive polynomial chaos (NIPC) methods, the number of required simulations grows exponentially with increasing numbers of input uncertainties: the so-called “curse of dimensionality.” 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 computational fluid dynamics (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. It 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 ten input random variables are required). It is shown in this paper 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 preserve 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 (LPT) blade-based on H2 profile. The relative importance of manufacturing errors in different location of the blade is analyzed.

A hybrid approach combining DNS and RANS simulations to quantify uncertainties in turbulence modelling

“…The first step is to describe the design space uncertainties in a probabilistic framework. For example, model parameters are replaced with random variables (Loeven, Witteven and Bijl [4], Ahlfeld and Montomoli [12]), or a domain region is defined as a random field (Dow and Wang [13], Doostan, Geraci and Iaccarino [14]). In the second step, the model is run repeatedly for random samples drawn from the input probability distributions.…”

Data-driven uncertainty quantification for Formula 1: Diffuser, wing tip and front wing variations

“…The validation and demonstration of SAMBA was already performed in Ahlfeld and Montomoli 12 and is repeated here with respect to diffusers, which are common parts of race cars. The main objective when simulating a diffuser is to investigate the down-force on the car or the negation of lift.…”

“…The errors resulting from PDF fitting were first shown by Oladyshkin and Nowak 11 for geological problems and then later by Ahlfeld and Montomoli 12 for turbomachinery problems. Both authors propose a new solution to the problem, and that is the use of an efficient expansion based on pure measurement data that do not require distribution fitting: the data-driven Polynomial Chaos Expansion, also called arbitrary Polynomial Chaos.…”

Effects of Inlet Incidence Perturbations on Compressor Cascade Performance using Adaptive Sparse Grid Collocation

“…Compared with the defects of long cycles and high costs exposed in the early experimental method, MCS can perform UQ of aerodynamics in a quick and accurate manner. With the increasing complexity of aerodynamic shapes and uncertain inducements, sensitivity analysis methods (Putko et al 2002;Luo and Liu 2018) such as the method of moment and the probabilistic models (Xiu and Karniadakis 2003;Loeven et al 2007;Hosder et al 2010;Loeven and Bijl 2010;Liu et al 2014;Panizza et al 2014;Seshadri et al 2015;Wunsch et al 2015;Ahlfeld and Montomoli 2017;Wang and Zou 2019;Xia et al 2019a) have been developed rapidly in the recent decade, further improving the efficiency of MCS. By using the method of moment, Putko et al (2002) determined the uncertainty propagations of geometric and flow variations separately in the quasi-one-dimensional flow.…”

“…Among those methods available for establishing the probabilistic models, non-intrusive polynomial chaos (NIPC) (Xiu and Karniadakis 2003) and nonintrusive probabilistic collocation (NIPRC) (Loeven et al 2007) (Loeven et al 2007;Hosder et al 2010;Panizza et al 2014;Seshadri et al 2015;Ahlfeld and Montomoli 2017;Wang and Zou 2019). In NIPRC, collocation points are chosen according to Gauss quadrature nodes, and the probability distribution of the solution is constructed with Lagrange interpolation.…”