On the Extraction of Pressure Fields From PIV Velocity Measurements in Turbines

Abstract

In this study, the pressure field for a water turbine is derived from particle image velocimetry (PIV) measurements. Measurements are performed in a recirculating water channel facility. The PIV measurements include calculating the tangential and axial forces applied to the turbine by solving the integral momentum equation around the airfoil. The results are compared with the forces obtained from the Blade Element Momentum theory (BEMT). Forces are calculated by using three different methods. In the first method, the instantaneous and mean pressure fields are obtained from PIV velocity fields by solving the Poisson equation. The developed method can be applied to rectangular fields of view with any complex internal boundary such an airfoil. The boundary conditions are obtained from the Navier-Stokes momentum equations. A central finite difference approximation is applied to obtain velocities spatial gradients up to the second order. A forward difference approximation is applied to obtain the unsteady term (acceleration). In the second method, the pressure at the boundaries is determined by spatial integration of the pressure gradient using forward finite differences along the boundaries. The forces in the blade can be obtained using this method without the need to resolve the pressure field inside of the domain. In the third method, applicable only to incompressible, inviscid, irrotational, and steady flow, the pressure is calculated using the Bernoulli equation. This approximated pressure is known to be accurate far from the airfoil and outside of the wake for steady flows. This value is compared to the pressure obtained from the Poisson equation. Additionally, that pressure is used to solve for the force from the integral momentum equation on the blade. From the three methods proposed to solve for pressure and forces from PIV measurements, the first one, which is solved by using the Poisson equation, provides the best match to the BEM theory calculations. In the proposed methods, the pressure is calculated from a known PIV velocity field. On the other hand, in computational fluid dynamic (CFD) studies, the velocity is unknown. Thus, the proposed experimental calculations should be useful in those CFD studies.