Technical report | The Estimation of Uncertainty in Theoretical Corrections to Unsteady Pressure Measurements through Tubes

Abstract

A general method used to quantify uncertainty in theoretical corrections to unsteady pressure measurements through tubes is documented in this report. This method is based on a wellvalidated theoretical model, which produces a transfer function that may be used for correcting unsteady pressure measurements through an Nt number of tubes and Nv number of volumes. The uncertainty estimation methods employed are in accordance with AIAA Standards. A software tool, implementing a synthesis of the theoretical model and AIAA uncertainty estimation methodology, is also developed in this work. Aided by two separate case studies, it is found that by failing to account for uncertainty in the transfer function that is used to correct unsteady pressure data, the overall uncertainty in measured unsteady pressure may be misrepresented, depending on the spectral content of the measurements or tube configuration.

Executive Summary

Measuring unsteady static pressure—that is, static pressure that varies rapidly with time—through tubes while not optimal, is sometimes necessary in fluid-dynamics testing. It is well known that unsteady pressure waves are modulated in amplitude and phase when travelling through tubes and, unless corrections are applied, the measurements will not represent the true quantities one expects.

A Linear Time-Invariant (LTI) transfer-function model was developed by Bergh & Tijdeman [1] to correct unsteady pressure amplitude and phase modulations through an Nt number of tubes and Nv number of volumes. It has previously been demonstrated that transfer-function corrections based on the theory in [1] agree excellently with experimental data, but are also sensitive to tube dimensions and mean ambient conditions. This sensitivity has led many to experimentally obtain transfer-function corrections for unsteady pressure measurements, rather than rely on theoretical models. While experimental corrections are advantageous in some situations, they are not always feasible for reasons of cost, time, or tubing configuration. A general method for estimating uncertainty in the theoretical transfer function is reported here, which is based on the theory in [1] and is in accordance with AIAA Standards for uncertainty estimation [2]. Specifically, this methodology may be used to quantify the effect of uncertainty in the transfer function, due to uncertainties in tube dimensions and mean ambient conditions, on the overall uncertainty of unsteady pressure measurements. A software tool using this methodology is also developed and utilised in this work.

Two case studies presented here reveal that transfer-function uncertainty is highly sensitive to individual uncertainties in tube dimensions and moderately sensitive to mean ambient conditions. It is also found that by failing to account for uncertainty in the transfer-function parameters, the overall uncertainty in measured unsteady pressure may be misrepresented, depending on the spectral content of the measurements or tube configuration. Therefore, if measurements of unsteady pressure must be conducted through tubes, and theoretical transferfunction corrections are to be used, it is recommended that the methodology in this report be employed a priori to testing, so that the transfer-function uncertainty is known—or at least appreciated.