On improvement of PIV image interrogation near stationary interfaces

Abstract

In this paper the problem posed by interfaces when present in PIV measurements is addressed. Different image pre-processing, processing and post-processing methodologies with the intention to minimize the interface effects are discussed and assessed using Monte Carlo simulations. Image treatment prior to the correlation process is shown to be incapable of fully removing the effects of the intensity pedestal across the object edge. The inherent assumption of periodicity in the signal causes the FFT-based correlation technique to perform the worst when the correlation window contains a signal truncation. Instead, an extended version of the masking technique introduced by Ronneberger et al. (Proceedings of the 9th international symposium on applications of laser techniques to fluid mechanics, Lisbon, 1998) is able to minimize the interface-correlation, resolving only the particle displacement peak. Once the displacement vector is obtained, the geometric center of the interrogation area is not the correct placement. Instead, the centre of mass position allows an unbiased representation of the wall flow (Usera et al. in Proceedings of the 12th international symposium on applications of laser techniques to fluid mechanics, Lisbon, 2004). The aforementioned concepts have been implemented in an adaptive interrogation methodology (Theunissen et al. in Meas Sci Technol 18:275–287, 2007) where additionally non-isotropic resolution and re-orientation of the correlation windows is applied near the interface, maximizing the wall-normal spatial resolution. The increase in resolution and robustness are demonstrated by application to a set of experimental images of a flat-plate, subsonic, turbulent boundary layer and a hypersonic flow over a double compression ramp.

Appendix: Symmetric-mask-exclusion direct cross-correlation by means of FFT

Similar to the digital mask methodology proposed by Gui et al. (2003), the normalized correlation coefficient is defined as

$$ \phi \left({m,n} \right)=\frac{\sum\nolimits_{i,j=1}^{k\cdot W_S} {F\left( {i,j} \right)\cdot m_b \left({i+m,j+n} \right)\cdot \left({I_{\rm b} \left( {i+m,j+n} \right)-{\overline{I_{\rm b}}} \left({m,n} \right)} \right)}}{\sqrt {\sum\nolimits_{i,j=1}^{k\cdot W_S} {m_{\rm b}^2 \left({i+m,j+n} \right)\cdot \left({I_{\rm b} \left({i+m,j+n} \right)-{\overline{I_{\rm b}}} \left({m,n} \right)} \right)}^2}} $$

(10)

where

$$ F\left({i,j} \right)=\frac{m_{\rm a} \left({i,j} \right)\cdot \left({I_{\rm a} \left({i,j} \right)-{\overline{I_a}}} \right)}{\sqrt {\sum\nolimits_{i,j=1}^{k\cdot W_S} {m_{\rm a}^2 \left({i,j} \right)\cdot \left( {I_{\rm a} \left({i,j} \right)-{\overline{I_{\rm a}}}} \right)^2}}} $$

(11)

To separate the seeded flow from the object area, binary masking arrays ‘m _a’ and ‘m _b’ are introduced for the first and second recording respectively (Fig. 19e,f). After expanding the multiplicative operation in the nominator of Eq. 10, each of the individual terms involves a cross-correlation, which can be performed by means of fast Fourier transforms. However, while ‘F’ needs to be computed only once before the correlation operation, both mean intensity ‘${\overline{I_{\rm b}}}$’ and mask ‘m _b’ require recalculation for each offset (m,n). The latter necessitates a direct approach for the computation of ‘ϕ’, rendering the presented scheme computationally intensive.

Fig. 19

a Image with selected interrogation area of size ‘W _S ’, b second snapshot with extended search area ‘k·W _S ’, c extended and padded interrogation area ‘I _a’, d selected search area ‘I _b’, e mask ‘m _a’ (white =1), f mask ‘m _b’, g mask ‘W’ used in calculation of mean intensity and covariance

Following the FFT-based free shape correlation (Ronneberger et al. 1998), the interrogation area in the first image is extended and padded with zeros to equal the size ‘k · W _S ’ of the larger search area in the second partial image (Fig. 19a–d). This zero padding operation can be automatically taken into account by the binary mask ‘m _a’. When measurement points are located on a structured grid factor ‘k’ is set to 2. With every iteration the disparity between the deformed images will converge to zero, eventually allowing values of ‘k’ approaching unity. In case of window rotation and non-isotropic sizing the dimensions of the enlarged search area are given by ‘k _η · W _Sη’ and ‘k _ξ · W _Sξ’ respectively in wall-tangent and normal direction, where

$$ k_{\eta ,\xi} =1+\frac{\min \left({W_{S\eta} ,W_{S\xi}} \right)}{2\cdot W_{S\eta ,\xi}} $$

(12)

To negate the need of repetitive computation of ‘${\overline {I_{\rm b}}}(m,n)$’ and ‘m _b(m,n)’, the introduction of a third binary mask ‘W’ of similar size as the search area is proposed containing unity values inside the interrogation area and zero otherwise as depicted in Fig. 19g. Consequently, the mean operator can be translated into a correlation involving ‘W’, ‘m _b’ and ‘I _b ’ (Eq. 13). Calculation of ‘${\overline{I_b}} (m,n)$’ in Eq. 13 is hence reduced to a one-time correlation operation by means of 2 Fourier transforms. Hereafter determination of ‘${\overline{I_b}}$’ at (m,n) becomes a mere lookup action.

$$ {\overline{I_{\rm b}}} \left({m,n} \right)=\frac{\sum\nolimits_{i,j=1}^{k\cdot W_S } {m_{\rm b} \left({i+m,j+n} \right)\cdot I_{\rm b} \left({i+m,j+n} \right)} }{\sum\nolimits_{i,j=1}^{k\cdot W_S} {m_{\rm b} \left({i+m,j+n} \right)}}=\left[ {\frac{W\ast \left({m_{\rm b} I_{\rm b}} \right)}{W\ast m_{\rm b}}} \right]_{\left({m,n} \right)} $$

(13)

Concisely, the direct cross-correlation function can be expressed as a series of FFT operations (Eq. 14), which drastically reduces the computational effort compared to the direct spatial computation.

$$ \phi =\frac{F\ast \left({m_{\rm b} I_{\rm b}} \right)-{\overline{I_{\rm b}}} \left({F\ast m_{\rm b}} \right)}{\sqrt {\left({W\ast m_{\rm b} I_{\rm b}^2} \right)-{\overline{I_{\rm b}}} ^2\left({W\ast m_{\rm b}} \right)}}\quad \hbox{where}\; {\overline{I_{\rm b}}} =\frac{W\ast m_{\rm b} I_{\rm b}}{W\ast m_{\rm b}} $$

(14)