Statistical Error Estimation Methods for Engineering-Relevant Quantities From Scale-Resolving Simulations

Abstract

Scale-resolving simulations, such as large eddy simulations, have become affordable tools to investigate the flow in turbomachinery components. The resulting time-resolved flow field is typically analyzed using first- and second-order statistical moments. However, two sources of uncertainty are present when recording statistical moments from scale-resolving simulations: the influence of initial transients and statistical errors due to the finite number of samples. In this paper, both are systematically analyzed for several quantities of engineering interest using time series from a long-time large eddy simulation of the low-pressure turbine cascade T106C. A set of statistical tools to either remove or quantify these sources of uncertainty is assessed. First, the Marginal Standard Error Rule is used to detect the end of the initial transient. The method is validated for integral and local quantities and guidelines on how to handle spatially varying initial transients are formulated. With the initial transient reliably removed, the statistical error is estimated based on standard error relations considering correlations in the time series. The resulting confidence intervals are carefully verified for quantities of engineering interest utilizing cumulative and simple moving averages. Furthermore, the influence of periodic content from large scale vortex shedding on the error estimation is studied. Based on the confidence intervals, the required averaging interval to reduce the statistical uncertainty to a specific level is indicated for each considered quantity.

Turbulence anisotropy analysis at the middle section of a highly loaded 3D linear turbine cascade using Large Eddy Simulation

“…with an axial chord length l ax ¼ 0:06 m and an approximate axial velocity at the domain outlet u ax,out % 77 m=s. The start of the sampling period was chosen by evaluating several convergence criteria, such as acting forces on the blade surface, probe data at several locations, and analyzing the truncated mean-squared error, as described in Bergmann et al (2022). After the initial transient is washed out, the statistics are accumulated for 40 convective times.…”

A Numerical Test Rig for Turbomachinery Flows Based on Large Eddy Simulations With a High-Order Discontinuous Galerkin Scheme—Part III: Secondary Flow Effects

“…4 against experimental [19] and numerical [14,15,20] available on sampling error. In the following, we will show 68% 345 confidence intervals for all our LES runs [22]. In this figure, they 346 are within the line thickness for this rapidly converging quantity.…”

“…These effects have been reviewed in the lit-4 erature for both compressors [1,2] and turbines [3] using predomi-5 nantly experimental methods or, due to the computational expense, [4][5][6][7][8][9][10]. 20 For example, Cui et al [4] presented an extensive analysis of 21 both midspan and end wall flow physics for the T106A LPT with 22 parallel end walls at a Reynolds number of 160,000 using an incompressible second order accurate, unstructured Finite Volume (FV) 24 method. The flow physics at midspan were illuminated using span-25 wise periodic simulations with and without incoming wakes at 26 a reduced frequency of 0.68 based on chord length and trailing 27 edge (TE) free stream velocity.…”

“…It then marks the end of the initial transient as the time where the confidence interval becomes minimal. A detailed description this method applied to the LES of an LPT can be found in [22].…”

Comparison of Techniques for the Estimation of Flow Parameters of Fan Inflow Turbulence from Noisy Hot-Wire Data

“…This operating condition produces a laminar separation in the aft section of the suction side, as seen in Figure 4. This simulation was performed and reported by Bergmann et al in [39] and cordially made available by Christian Morsbach from the Numeric Methods department of the DLR in Cologne for the purpose of this current study. The blade chord length is c = 93 mm and the stagger angle is equal to 30.7 degrees.The data analyzed in this work were extracted from the axial slices marked with black dots, labeled "Wake cut", only the slice on the right-hand side of this figure.…”