Probabilistic evaluation of streamline topologies for the detection of preferential flow configurations in PIV applications

Abstract

While PIV allows the extraction of instantaneous streamlines, the availability of typically thousands of patterns hampers a statistical understanding of the underlying flow topology. A particular application is oscillating gas jets, encountered in scientific and industrial studies dealing with mass transfer and flow control. Multiple methods, such as statistical moments (including mean, variance, skewness, and kurtosis), proper orthogonal decomposition (POD), and Hartigan’s dip test are common post-processing methodologies. However, as demonstrated in this paper, they do not provide sufficient in-depth information from a statistical perspective to attribute probabilities to potential flow topologies. In this work, a novel approach to describe probability distributions based on extracted streamline patterns is proposed. Based on the available streamline patterns, the convolution with adaptive Gaussian kernels reveals the probability density function of the flow topology. The proposed methodology, as well as more traditional post-processing approaches, are assessed on the basis of synthetic flow fields and experimental PIV data of oscillating impinging gas jets, demonstrating the added value of the streamline probability map in characterising the scrutinised flow.

Graphic abstract

Appendix: generating of synthetic flapping jets

Given a PIV recording of $n_x\times n_y$ pixels$^2$, the jet origin is positioned at $\left( 1,\frac{n_y}{2}\right)$. The jet centerline is assumed to follow a parabolic trajectory $y_c=ax_c^2+bx_c+c$. Imposing the centerline to be located at $y_c=y_p$ for $x_c=n_x$, with a horizontal tangent at the origin, the coefficients are given by $a=\frac{y_p-\frac{n_y}{2}}{\left( n_x-1\right) ^2}$, $b=-2a$, and $c=a+\frac{n_y}{2}$. The arc length $s(x_c)$ along the centerline can be calculated as $s(x_c)=\frac{1}{4a}\left( f(2ax_c+b)-f(b)\right)$ where $f(t)=t\sqrt{1+t^2}+\mathrm{log}\left( t+\sqrt{1+t^2}\right)$. The displacement along the centerline is prescribed as $V_c(x_c)=8\left( 1+s(x_c)\right) ^{-0.5}$, i.e., a maximum displacement of 8 pixels in the origin is imposed. As the jet will spread out, the jet width is defined as $\sigma (x_c)=\sigma _0\left( 1+\frac{5s(x_c)}{n_x}\right)$ with $\sigma _0=10 pixels$. For any pixel location $(x_o,y_o)$, the normal distance, $\eta$, to the jet centerline can be determined. Denoting the intersection point of the line, normal to the centerline and passing through $(x_o,y_o)$, and the centerline by $(x_i,y_i)$, the local, total velocity can be calculated as $V_{tot}(x_o,y_o)=V_c(x_i)\exp \left( -\eta ^2/2\sigma ^2(x_i)\right)$. Local velocity components are then obtained as $U(x_o,y_o)=V_{tot}(x_o,y_o) \mathrm{cos}(\theta )$ and $V(x_o,y_o)=V_{tot}(x_o,y_o)\alpha \mathrm{sin}(\theta )$ with $\mathrm{cos}(\theta )=\left( 1+(2ax_i+b)^2\right) ^{-0.5}$ and $\alpha =\frac{2ax_i+b}{|2ax_i+b|}$.