Evaluation of aero-optical distortion effects in PIV

Abstract

Aero-optical distortion effects on the accuracy of particle image velocimetry (PIV) are investigated. When the illuminated particles are observed through a medium that is optically inhomogeneous due to flow compressibility, the resulting particle image pattern is subjected to deformation and blur. In relation to PIV two forms of error can be identified: position error and velocity error. In this paper a model is presented that describes these errors and particle image blur in relation to the refractive index field of the flow. In the case of 2D flows the model equations can be simplified and, furthermore, the background oriented schlieren technique (BOS) can be applied as a means to assess and correct for the optical error in PIV. The model describing the optical distortion is validated by both computer simulation and real experiments of 2D flows. Two flow features are considered: one with optical distortion normal to the velocity (shear layer) and one with optical distortion in the direction of the flow (expansion fan). Both simulation and experiments demonstrate that the major source for the velocity error is the second derivative of the refractive index in the direction of the velocity vector. The aero-optical distortion effect is less critical for shearing interfaces in comparison with compression/expansion fronts, the most critical case being represented by shock waves. Based on the results from the simulated experiments, it is concluded that for the 2D flow case the BOS method allows a measurement of the mean velocity error in PIV and can reduce it to a large extent.

Appendix

1.1 Scaling rules for high-speed industrial experimental facilities

Equation 2 indicates that the optical displacement vector, hence the error in the position of a particle, increases with W² (Z_D and integration length S being proportional to W). Therefore, the problem of optical distortion increases rapidly with the linear dimension (L) of the wind tunnel test section. Note that going from a small-scale research wind tunnel (L~0.1 m) to industrial scale facilities (L~1 m) commonly involves a scale factor for L², hence W², on the order of 100. However, the model dimensions and the flow features associated with the inviscid flow behaviour generally scale linearly with the wind tunnel size as well. As a consequence, the field of view is also linearly increased. Viscous flow features are expected to scale less than linear, with L^0.5 for laminar flow and approximately L^0.8 for turbulent flow (White 1991). Therefore it is expected that ∇n decreases with L^−0.5 or L^−0.8 (for a prescribed free stream density level) and that ∇²n hence decreases with L⁻¹ or L^−1.6 for laminar and turbulent flow, respectively. In conclusion, it is estimated that in dimensional units the position error (Eq. 2) scales at most as L^1.5, or in pixel units as L^0.5. The direct velocity error (Eq. 6) scales at most as L in dimensional units.

At optical interfaces, i.e. shock waves, the refractive index is discontinuous (its derivatives vary strongly with ${\ifmmode\expandafter\vec\else\expandafter\vecabove\fi{x}}$) and therefore Eqs. 2 and 6 no longer hold. In the vicinity of the interface, strong particle image blur is expected accompanied by a significant velocity error. The extent for the particle image blur scales as W, hence L, as is shown by Eq. 10. In pixel units the enlargement is independent of L.