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Uncertainty and Blades

Name Affiliation Phone Number Email Address
Francesco Motomoli Imperial College London f.montomoli@imperial.ac.uk
Richard Ahlfeld Imperial College London ahlfeld.richard@googlemail.com
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Oil & Gas

Formula 1


The NACA0012 wing tip airfoil is a well-studied test case that has been used in many different applications. It is used for example to evaluate the propagation of front wing vortices in F1 and it has been used in turbomachinery compressors. The overall objective of this case is to quantify the impact of angle misalignment of the airfoil and the impact on drag and lift given datum uncertainty in form of non-Askey scheme PDFs and/or histograms.

This test case is challenging for two reasons: tip vortex representation and modelling of histograms and non-Askey scheme PDFs. The simulation of the tip vortex is highly affected by the numerical scheme used and it has been used to test and quantify the dissipation of the numerical models.

The data of this test case, geometry and input distributions can be given.

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The UQ&M objective of this test case is to increase the confidence level in the simulation considering the impact of installation variations.

2.1 Process Inputs

The uncertain inputs are given as histograms, as obtained in case of optical measurements. The input distributions are uncorrelated and are characterized by skewed fat tailed inputs. In this case PDF fitting cannot be applied because it introduces a bias/error in the output.

2.2 Propagation

The forward propagation of uncertainty is carried out using a CFD simulation and does not require any feedback mechanism, inversion or re-iteration. The absence of singularities in the flow region and a low number of random input parameters suggests the use of a Non-Intrusive Polynomial Chaos expansion method for uncertainty quantification. These methods have limited suitability to resolve discontinuities and suffer from what is referred to as ‘curse of dimensionality’. This means that they become increasingly difficult to apply for higher number of inputs. For the given case with only three uncertain parameters and without discontinuities, however, they are extremely efficient and their high efficiency will allow performing the required UQ analysis with computationally intensive CFD methods within a feasible time frame.

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What makes this test case challenging, is that small variations in the vortex reconstruction can completely change the histograms of the drag and lift of the airfoil. In other words: the underlying design space is highly non-linear with respect to small disalignment errors. The figure above shows an under-sampled oscillating response surface. A higher polynomial order than shown would be required to avoid large approximation errors and evaluate the suitability of the automatically based on measurement data created polynomial response surface. The use of higher order polynomial orders is very problematic, however, if it is based on limited measurement data. The reason is that the mapping of the polynomial coefficients from the moments of measurement data becomes highly ill-posed for high orders. Consequently, the difficulty of this problem is that it lies in the intersecting area of scarce input data and highly non-linear response surfaces

2.3 Interpretation and Communication of Results

The results of the performed uncertainty quantification should complement the currently available research on of wingtip vortices over lifting surface with an accurate understanding of how installation errors can affect the lift and drag coefficient. Ideally, the study should help to identify regions, for example for the pressure coefficient distribution, in which numerical solutions deviate from experimental results due to so far unconsidered installation errors. To this end, a detailed comparison between near field and far field vortices with various RANS and LES models and experimental results is required.


  • The proposed test case has been investigated before in depth at the department of Aeronautical Engineering at Imperial College. Not only various RANS, but also an implicit LES solution obtained with the in house software Nectar++ are available for numerical investigation. The numerical investigation of wingtip vortices is, however, a field of on-going research, where no single model can accurately describe the near field and far field properties of the create vortices with high reliability. A high numerical understanding is consequently crucial to identify the subtle differences between various methods and their uncertainty.
  • Uncertainty quantification based on the moments of optical measurement data for skewed and fat-tailed data has also been developed recently at Imperial. However, the ill-posedness of the moment problem has so far not been sufficiently investigated for higher order problems with oscillatory or highly non-linear response surfaces. From the current state it appears that further numerical precondition or bigger amounts of input data are required to solve this problem.


Chow, J. S., Zilliac, G. G., & Bradshaw, P. (1997). Mean and Turbulence Measurements in the Near Field of a Wingtip Vortex. AIAA Journal, 35(10), 1561–1567. http://doi.org/10.2514/2.1

Lombard, J.-E. W., Moxey, D., Sherwin, S. J., Hoessler, J. F. A., Dhandapani, S., & Taylor, M. J. (2015). Implicit Large-Eddy Simulation of a Wingtip Vortex. AIAA Journal, 1–13. http://doi.org/10.2514/1.J054181

Ahlfeld, R., Belkouchi, B., & Montomoli, F. (2016). SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. Under Review in Journal of Computational Physics, 1–29

Lucor, D., & Karniadakis, G. E. (2004). Adaptive generalized polynomial chaos for nonlinear random oscillators. SIAM Journal on Scientific Computing, 26(2), 720–735.