Error propagation dynamics of velocimetry-based pressure field calculations (2): on the error profile

Abstract

A recent study investigated the propagation of error in a velocimetry-based pressure (V-pressure) field reconstruction problem by directly analyzing the properties of the pressure Poisson equation (Pan et al 2016 Meas. Sci. Technol. 27 084012). In the present work, we extend these results by quantifying the effect of the error profile in the data field (shape/structure of the error in space) on the resultant error in the reconstructed pressure field. We first calculate the mode of the error in the data that maximizes error in the pressure field, which is the most dangerous error (called the worst error in the present work). This calculation of the worst error is equivalent to finding the principle mode of, for example, an Euler–Bernoulli beam problem in one-dimension and the Kirchhoff–Love plate in two-dimensions, thus connecting the V-pressure problem from experimental fluid mechanics to buckling elastic bodies from elastic mechanics. Taking advantage of this analogy, we then analyze how the error profile (e.g. spatial frequency of the error and the location of the most concentrated error) in the data field coupled with fundamental features of the flow domain (i.e. size, shape, and dimension of the domain, and the configuration of boundary conditions) significantly affects the error propagation from data to the reconstructed pressure. Our analytical results lend to practical applications in two ways. First, minimization of error propagation can be achieved by avoiding low frequency error profiles in data similar to the worst case scenarios and error concentrated at sensitive locations. Second, small amounts of the error in the data, if the error profile is similar to the worst error case, can cause significant error in the reconstructed pressure field; such a synthetic error can be used to benchmark V-pressure algorithms.