RESEARCH ARTICLE

Corrections for one- and two-point statistics measured with coarse-resolution particle image velocimetry

Antonio Segalini • Gabriele Bellani •

Gaetano Sardina • Luca Brandt • Evan A. Variano

Received: 10 September 2013 / Revised: 23 March 2014 / Accepted: 22 April 2014 Springer-Verlag Berlin Heidelberg 2014

Abstract A theoretical model to determine the effect of the size of the interrogation window in particle image velocimetry measurements of turbulent flows is presented.

The error introduced by the window size in two-point velocity statistics, including velocity autocovariance and structure functions, is derived for flows that are homogeneous within a 2D plane or 3D volume. This error model is more general than those previously discussed in the literature and provides a more direct method of correcting biases in experimental data. Within this model framework, simple polynomial approximations are proposed to provide a quick estimation of the effect of the averaging on these statistics. The error model and its polynomial approximation are validated using statistics of homogeneous isotropic turbulence obtained in a physical experiment and in a direct numerical simulation. The results demonstrate that the present formulation is able to correctly estimate the turbulence statistics, even in the case of strong smoothing due to a large interrogation window. We discuss how to use these results to correct experimental data and to aid the comparison of numerical results with laboratory data. 1 Introduction

Particle image velocimetry (PIV) is a measurement technique that allows the characterization of a velocity field in space and time by calculating the displacement of groups of tracer particles in ‘‘interrogation areas’’, which are discrete subregions of the measurement area (Raffel et al. 2001; Adrian 2005). From a theoretical point of view, the

PIV algorithm can be seen as a spatiotemporal filter (see

Westerweel 1997) of the velocity field, whose cutoff frequency and wavelength depend mainly on the interval between two subsequent images, Dt, and the size of the interrogation area, L, respectively. The first can be made very small thanks to double-cavity lasers, which can shoot two pulses at arbitrarily small intervals, and efficient subpixel interpolation schemes that can precisely resolve small displacements caused by small Dt values (Chen and Katz 2005; Nobach et al. 2005). The frequency resolution is also increasing due to improved cameras and algorithms (see

Scarano and Moore 2012). Spatial resolution of PIV has also been improved by new algorithms, e.g., those with iterative window offset and deformation (Westerweel et al. 1997; Scarano and Riethmuller 2000; Scarano 2002).

However, PIV spatial resolution is always limited by the fundamental trade-off between interrogation area size and signal strength. That is, reducing the size of the interrogation area reduces the number of tracer particles used in the velocity calculation, having a negative effect on the signal-to-noise ratio (see Westerweel 1997; Foucaut et al. 2004). This constraint leads to a minimum size of the interrogation area, given the practical limits of tracer particle density (Poelma et al. 2006; Ka¨hler et al. 2012), and limits the spatial resolution capability of PIV. According to the latest comparative tests, it is very difficult in practice to use an interrogation window smaller than 16 16 pixels (Stanislas et al. 2008). In physical space, this corresponds to a window size that ranges between 0:5 0:5 mm2 and 2 2 mm2 for common optical setups (e.g., camera

A. Segalini (&) G. Sardina L. Brandt

Linne´ FLOW Centre, KTH Mechanics, 100-44 Stockholm,

Sweden e-mail: segalini@mech.kth.se

G. Bellani E. A. Variano

Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720, USA 123

Exp Fluids (2014) 55:1739

DOI 10.1007/s00348-014-1739-z resolution 1,024 1,024 pixels and 50 50 mm2 or 100 100 mm2-wide image areas). These values can be significantly higher than the smallest turbulent scales at high-Reynolds number, and thus, the effect of the unresolved scales must be taken into account.

Typical PIV data analyses use the velocity fields to compute quantities such as the spatial distribution of turbulent kinetic energy (TKE), vorticity, dissipation rate and two-point correlations. It is well known that the estimation of such quantities is strongly affected by spatial resolution (Saarenrinne et al. 2001). Ad hoc correction schemes can account for insufficient resolution for some specific quantities, such as the velocity variance (Saarenrinne and Piirto 2000; Tanaka and Eaton 2007; Scharnowski et al. 2012).

One-dimensional filtering effects can occur in more traditional measurement techniques, for instance in hot-wire anemometry, due to the finite length of the sensor. This problem was first addressed by Dryden et al. (1937) and further investigated by Frenkiel (1949), Wyngaard (1968) and Segalini et al. (2011a) among others, which proposed several correction schemes for hot-wire measurements. In particular, Wyngaard (1968) provided an elegant analysis in

Fourier space of the effect of spatial resolution in single- or

X-wire measurements in isotropic turbulence.

Several papers address spatial resolution issues for PIV (see for example Scarano 2003; Lavoie et al. 2007; Giordano and Astarita 2009; Ka¨hler et al. 2012). In particular,

Lavoie et al. (2007) extended the methodology of Wyngaard (1968) to estimate 2D filtering effects in PIV data of grid turbulence, assuming a flow field that is statistically homogeneous and isotropic.

In this paper, similarly to the work of Lavoie et al. (2007), we derive a rigorous analytical model of the 2Dfiltering effects for flows that are homogeneous within a 2D plane, but we do so in physical space, rather than in wavenumber space. The advantage of this approach is that it relates filtering effects to physical quantities like the