Construction of the vortex-surface field from tomographic particle image velocimetry data of flow past a vortex generator

Abstract

We extend the vortex-surface field (VSF), a Lagrangian-based flow diagnostic method, to experimental data of the tomographic particle image velocimetry (Tomo-PIV). The boundary-constraint method is applied to construct the VSF from the instantaneous Tomo-PIV velocity field in the wake flow of a ramp vortex generator (VG) at a moderate Reynolds number. Under finite experimental noises, the VSF construction has satisfactory errors, showing the applicability of the VSF to visualize Tomo-PIV data. From a Lagrangian viewpoint, the VSF is used to elucidate the formation and evolution of coherent structures in the VG wake. The initially planar vortex surfaces consisting of undisturbed vortex lines in the laminar boundary layer are first lifted as the flow past over the VG. Subsequently, the bulge-like outer vortex surfaces in the near wake of VG generate a strong shear layer, and the near-wall inner vortex surface downstream to VG is lifted by the streamwise vortices formed from the lateral VG edges. Further downstream, the outer vortex surfaces break up into arch- or hairpin-like structures due to the Kelvin–Helmholtz instability. The geometric deformation of vortex surfaces is quantified by conditional means of the VSF gradient.

Appendix A Validation of the VSF method with synthetic data

We assess the robustness and performance of the VSF evolution using synthetic experimental data of a Taylor–Green (TG) flow (Brachet et al. 1983) with $Re = 1/ \nu =400$. The constant-density viscous flow is governed by the incompressible Navier–Stokes (NS) equations. The DNS for solving the NS equations was carried out in a periodic box with sides $(2 \pi )^3$ on uniform grid points $256^3$ with a standard pseudo-spectral method. The simulation details can be found in Yang and Pullin (2011).

The initial flow field is $\varvec{u} = (\sin x \cos y \cos z, -\cos x \sin y \cos z, 0)$ and the initial exact VSF is $\phi _v = (\cos 2x-\cos 2y)\cos z$ at $t=0$. A 5% level Gaussian-distributed random noise is imposed to the DNS velocity field of each time step as synthetic experimental data. The original DNS fields and the synthetic fields are referred to as “DNS” and “synthetic” below, respectively. The median filter and the DFS method (Wang et al. 2016) were applied to smooth and correct the synthetic velocity field at every time step, respectively. The corresponding field is referred to as “filtered” below. The VSFs are calculated from a time series of DNS, synthetic, and filtered datasets using the two-time method (Yang and Pullin 2011), and their evolutions are compared to assess effects of the noise, smoothing, and correction on the VSF solution.

Figure 14 plots time evolutions of the VSF isosurface of $\phi _v = 0.2$ at $t=0$, 2, and 4 calculated from the DNS, synthetic, and filtered data. The VSFs are successfully constructed for all datasets to track coherent structure in the TG flow. The evolutions of large-scale structures are consistent. The VSF isosurface for synthetic data is less smooth than that for DNS data at $t=0$ and $t=2$ and is more dissipated (Han and Yang 2022) at $t=4$ due to the imposed noises. After filtering and DFS correction, the quality of VSF solutions is significantly improved, as the VSF isosurfaces for the filtered data are close to those for the DNS data.

Fig. 14

Evolution of the isosurface of $\phi _v=0.2$ color-coded by $|\varvec{\omega }|$ in the TG flow for different datasets

Figure 15 plots evolutions of the volume-averaged VSF deviation $\langle |\lambda _{\omega } |\rangle$ for three types of datasets. The averaged VSF deviation less than $3\%$ is very small for DNS data; it is large around $30\%$ for synthetic data; it is reduced to $6\%$ for filtered data. Thus, the filtering and the DFS method can improve the VSF results calculated from 3D experimental data.

Fig. 15

Evolution of the volume-averaged VSF deviation in the TG flow for different datasets