Application of Higher-Order Compact Finite Difference Schemes to Out of Plane Vorticity Measurements

Abstract

The accuracy of out-of-plane vorticity from in-plane experimental velocity measurements is investigated with particular application to Digital Particle Image Velocimetry (DPIV). Higher-order compact finite difference schemes are proposed as alternatives for the vorticity estimation. Simulations of known flow fields are used to quantify errors associated with the different methods of estimation. The effects of spatial sampling resolution with respect to the range of flow scales on the bias errors of the method accuracies are explored. In addition, estimation of the velocity measurement random error propagation into the vorticity measurement is presented. The higher-order compact schemes deliver improved accuracy compared with previously used methods, even when that domain is spatially undersampled or in the presence of strong velocity gradients. The compact schemes demonstrated less than 0.3% bias error throughout the core of the vortex, resolving flow structures as small as the Nyquist sampling frequency of the system, while the error in the conventional methods increased as the spatial sampling and the range of wave numbers present in the flow field was reduced. The bias error is an irrecoverable loss due to the truncation error of the method, and thus poses significant limitations to the system, whereas the random error propagation can be reduced for the higher-order schemes by applying a simple Gaussian smoothing to the flow field. Thus, the reduction in bias error is of greater importance to the accuracy of the system, particularly as modern global flow measurement technologies achieve higher spatial and temporal resolutions, as well as higher accuracies in velocity measurements. Overall, the compact schemes provide an improved approach for vorticity evaluation compared to conventional algorithms.