Generalised least squares approach for estimation of the log-law parameters of turbulent boundary layers

Abstract

Uncertainty in estimating the log-law parameters is arguably the greatest obstacle to establishing definitive conclusions regarding their numerical values and universality. This challenge is exacerbated by the limited number of studies that provide thorough uncertainty analyses of experimental data and fitting procedures, and those that do often adopt different approaches, undermining direct comparisons. The present study applies the generalised least squares (GLS) principle to the log-law velocity profile to establish a standardised, comprehensive framework for quantifying uncertainty in the log-law parameters across datasets. GLS contrasts with ordinary least squares (OLS) and weighted least squares (WLS), which do not account for correlation in errors across measured quantities, as well as with alternative heuristic methods that independently sample primitive variables. Instead, it incorporates a full covariance matrix of the residuals, propagated from the uncertainties in the primitive variables and consistent with the experimental methods employed. The study presents a systematic analysis of the response of the log-law regression model using synthetic data, emulating measurements from a hot-wire anemometer mounted on a linear traverse. This analysis serves as a predictive tool for experimental design, identifying a priori the dominant sources of uncertainty in the log-law parameters and potential mitigation strategies. The study also provides new insights into the correlation between the log-law parameters and proposes a new fitting procedure that eliminates the need to prescribe the location and extent of the log region. The open-source Python implementation of the log-law regression model is available for download on GitHub at https://github.com/ma2ferreira/gls_loglaw.git.