Our investigation raises an important question that is of relevance to the wider turbomachinery community: how do we estimate the spatial average of a flow quantity given finite (and sparse) measurements? This paper seeks to advance efforts to answer this question rigorously. In this paper, we develop a regularized multivariate linear regression framework for studying engine temperature measurements. As part of this investigation, we study the temperature measurements obtained from the same axial plane across five different engines yielding a total of 82 data-sets. The five different engines have similar architectures and therefore similar temperature spatial harmonics are expected. Our problem is to estimate the spatial field in engine temperature given a few measurements obtained from thermocouples positioned on a set of rakes. Our motivation for doing so is to understand key engine temperature modes that cannot be captured in a rig or in computational simulations, as the cause of these modes may not be replicated in these simpler environments. To this end, we develop a multivariate linear least squares model with Tikhonov regularization to estimate the 2D temperature spatial field. Our model uses a Fourier expansion in the circumferential direction and a quadratic polynomial expansion in the radial direction. One important component of our modeling framework is the selection of model parameters, i.e. the harmonics in the circumferential direction. A training-testing paradigm is proposed and applied to quantify the harmonics.
In this second part of our two-part paper, we provide a detailed, frequentist framework for propagating uncertainties within our multivariate linear least squares model. This permits us to quantify the impact of uncertainties in thermodynamic measurements-arising from calibrations and the data acquisition system-and the correlations therein, along with uncertainties in probe positions. We show how the former has a much larger effect (relatively) than uncertainties in probe placement.We use this non-deterministic framework to demonstrate why the well-worn metric for assessing spatial sampling uncertainty falls short of providing an accurate characterization of the effect of a few spatial measurements. In other words, it does not accurately describe the uncertainty associated with sampling a non-uniform pattern with a few circumferentially scattered rakes. To this end, we argue that our data-centric framework can offer a more rigorous characterization of this uncertainty. Our paper proposes two new uncertainty metrics: one for characterizing spatial sampling uncertainty and another for capturing the impact of measurement imprecision in individual probes. These metrics are rigorously derived in our paper and their ease in computation permits them to be widely adopted by the turbomachinery community for carrying out uncertainty assessments.
Aeroengine performance is determined by temperature and pressure profiles along various axial stations within an engine. Given limited sensor measurements, we require a statistically principled approach for inferring these profiles. In this paper we detail a Bayesian methodology for interpolating the spatial temperature or pressure profile at axial stations within an aeroengine. The profile at any given axial station is represented as a spatial Gaussian random field on an annulus, with circumferential variations modelled using a Fourier basis and radial variations modelled with a squared exponential kernel. This Gaussian random field is extended to ingest data from multiple axial measurement planes, with the aim of transferring information across the planes. To facilitate this type of transfer learning, a novel planar covariance kernel is proposed. In the scenario where frequencies comprising the temperature field are unknown, we utilise a sparsity-promoting prior on the frequencies to encourage sparse representations. This easily extends to cases with multiple engine planes whilst accommodating frequency variations between the planes. The main quantity of interest, the spatial area average is readily obtained in closed form. We term this the Bayesian area average and demonstrate how this metric offers far more representative averages than a sector area average---a widely used area averaging approach. Furthermore, the Bayesian area average naturally decomposes the posterior uncertainty into terms characterising insufficient sampling and sensor measurement error respectively. This too provides a significant improvement over prior standard deviation based uncertainty breakdowns.
This paper introduces the Bayesian mass average and details its computation. Owing to the complexity of flow in an engine and the limited instrumentation and the precision of the sensor apparatus used, it is difficult to rigorously calculate mass averages. Building upon related work, this paper views any thermodynamic quantity's spatial variation at an axial plane in an engine (or a rig) as a Gaussian random field. In cases where the mass flow rate is constant in the circumferential direction but can be expressed via a polynomial or spline radially, this paper presents an analytical calculation of the Bayesian mass average. In cases where the mass flow rate itself can be expressed as a Gaussian random field, a sampling procedure is presented to calculate the Bayesian mass average. Examples of the calculation of the Bayesian mass average for temperature are presented, including with a real engine case study where velocity profiles are inferred from stagnation pressure measurements.
For an accurate performance assessment of a multi-stage compressor, the circumferentially non-uniform flow at the compressor exit needs to be understood and sampled in a way that minimizes uncertainties. To quantify the effect of the measurement rake positions in the exit duct on compressor performance a combined computational and experimental approach is used on a modern 4-stage compressor. The computational analysis is based on unsteady calculations of a 180-degree sector of the test compressor and experimental verification is provided by comparing to area-traverse data downstream of the outlet guide vanes. It is shown that the exit measurement rakes are subject to circumferential flow variations caused primarily by the combined effect of the potential field of the struts housed within the exit duct and the wakes originating from the outlet guide vanes. A circumferential camber pattern, applied to the outlet guide vanes, designed to shield the upstream compressor blade rows against the potential field of the exit struts, is found to reduce the amplitude of the circumferential variation in stagnation pressure and shift its circumferential phase. Recognizing that a smaller numerical model, consisting only of the last rotor, the outlet guide vanes and the exit struts, is sufficient to capture the relevant flow mechanisms, the circumferential variations in stagnation pressure and temperature at the rake position are quantified as a function of the exit capacity. The stagnation pressure and temperature uncertainty within a +/-2 deg circumferential range around the nominal rake position is found to be up to 2.25 times larger than the change of the nominal values over an 87.1–106.0% variation of the exit capacity. Three options to position the rakes to reduce the uncertainty in compressor efficiency are presented — moving the rake downstream as well as leaning and verniering the rakes over the outlet guide vane pitch. Moving the rake from the leading edge to the trailing edge plane of the exit struts reduced the efficiency uncertainty by 2.6%, while leaning and verniering the rakes reduced the efficiency uncertainty by 0.2% and 0.7% respectively. The knowledge gained from the large-scale, detailed CFD predictions can used to support future measurement campaigns.
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