The workflow has eight steps, listed in Table 1 with the principal decision and the check attached to each. Figure 1 shows the sequence and the return path taken when a check fails. The subsections below take the steps in order, grouping those that are decided together.
Steps of the proposed workflow with the main decision and the quality check for each
| Step | Main decision | Quality check before proceeding |
|---|---|---|
| 1. Scales and volume | Volume size and position relative to the trailing edge | Volume spans at least one shedding wavelength |
| 2. Optics and seeding | Number of cameras, aperture, particle image density | Particles in focus through the depth; density near 0.05 per pixel |
| 3. Calibration | Target traverse and mapping function | Self-calibration residual below 0.1 pixel |
| 4. Acquisition | Phases sampled and repetitions per phase | Foil position recorded with every image pair |
| 5. Pre-processing | Background removal and foil mask | Uniform particle intensity; no foil reflections |
| 6. Reconstruction | Algorithm, iterations, voxel size | Intensity profile through depth shows clear volume edges |
| 7. Correlation | Interrogation volume and overlap | Few outliers; small residual divergence |
| 8. Vortex analysis | Identification criterion and threshold | Circulation consistent between neighbouring planes |
Sequence of the proposed tomographic PIV workflow for flapping-foil wakes, with the return path taken when a quality check fails
Planning starts from the flow and not from the equipment. The chord, the flapping frequency and the free-stream speed fix the Strouhal number and the streamwise spacing of the shed vortices; the trailing-edge excursion gives the initial wake width. The measurement volume should hold at least one full shedding wavelength in the stream direction and the whole span of the structures of interest, with the trailing edge either just inside the volume or just upstream of it. Where one volume cannot do both, overlapping volumes at successive downstream stations, each sampled at the same phases, are preferable to a single volume stretched too thin.
Volume thickness is the costly dimension, since light intensity falls and ghost intensity rises with it. Four cameras are the usual compromise. They are placed with a total angular aperture large enough to resolve depth, fitted with Scheimpflug adapters so that the focal plane lies along the mid-plane of the volume, and stopped down until particles remain in focus across the full thickness. For a four-camera system a particle image density of about 0.05 particles per pixel is the accepted working value (Elsinga et al., 2006; Scarano, 2013). The volume edges should be cut sharply with knife-edge slits, both to keep stray particles out of the lines of sight and to make the reconstructed depth profile a usable diagnostic later.
A target is traversed through the depth of the volume and a mapping function is fitted for each camera. This physical calibration is rarely accurate enough by itself, because small movements of the cameras, the tank walls or the target leave errors of a pixel or more. Volume self-calibration, introduced by Wieneke (2008), uses the particle images themselves: particles are triangulated, the residual disparity between cameras is mapped through the volume, and the mapping functions are corrected. Reducing the residual to below about 0.1 pixel is the accepted requirement for reconstruction. In a facility with a moving foil and carriage the self-calibration should be repeated whenever anything is disturbed and at intervals during a long campaign, using recordings at reduced seeding if necessary.
Low-repetition tomographic systems cannot follow a flapping cycle in time, so acquisition is locked to the phase of the motion. An encoder on the actuator triggers the laser and cameras at chosen phases, and the measured foil position is stored with each image pair. The number of phases should resolve the formation and release of each vortex, and the number of repetitions at each phase should be set by the convergence of phase-averaged vorticity, which is slower than that of velocity. Where a time-resolved system is available, the same workflow applies and tracking-based methods such as Shake-The-Box (Schanz et al., 2016) become an alternative to correlation.
Phase averaging deserves a deliberate choice of reference. Averaging at fixed phase of the foil motion is the natural default and separates the periodic wake from the residual fluctuation. Further downstream, where vortices wander from cycle to cycle, a fixed-phase average smears their cores and understates peak vorticity. Keeping the instantaneous fields allows a conditional average, aligned on the detected position of each vortex, to be formed afterwards and compared with the fixed-phase result. The difference between the two is itself informative, since it measures how repeatable the wake is at that station.
The foil itself is the main nuisance in the images. It reflects laser light, blocks lines of sight differently for each camera and shadows part of the volume. Matt black or fluorescent coatings reduce reflections at source. In pre-processing, a sliding minimum over each phase removes the stationary background, intensity is normalised between cameras, and a mask derived from the known foil geometry and the recorded position is projected into every camera image. Voxels that fewer than three cameras can see should be flagged and excluded, since reconstruction there is unreliable.
MART with around five iterations is the reference choice (Elsinga et al., 2006). Faster schemes that begin from a multiplicative first guess and update all voxels simultaneously were introduced by Atkinson and Soria (2009) and are now standard for large volumes. Where pairs of exposures are available, motion-tracking enhancement (Novara et al., 2010) uses the fact that real particles move coherently between exposures whereas ghosts do not, and so permits higher seeding. The voxel size is matched to the pixel size in physical units. The check on this step is the profile of reconstructed intensity through the depth: it should be flat inside the illuminated slab and drop sharply outside, and the ratio of the two levels is a direct indicator of ghost intensity.
Displacements are obtained by three-dimensional cross-correlation with iterative volume deformation, ending at an interrogation volume that contains several particles and overlaps its neighbours. Spurious vectors are detected with the normalised median test of Westerweel and Scarano (2005), extended to three dimensions. Two checks are particular to volumetric data. For an incompressible flow the measured divergence should vanish, so its spread is a measure of error in the velocity gradients. And ghost particles, which move with the average displacement of the particles that create them, bias gradients toward zero across the depth; a vortex core that looks weaker in the depth direction than in the others is a warning sign.
Vorticity magnitude alone does not separate vortices from shear layers, and the wake of a flapping foil contains both. The Q criterion and the λ2 criterion, both set out and compared by Jeong and Hussain (1995), are the usual choices, applied to phase-averaged fields with a threshold stated explicitly. From the identified structures the analysis extracts the quantities that connect to performance: the circulation of each loop, obtained by integrating vorticity over planes cut through it; the spacing and inclination of successive loops; and the momentum carried by the jet between them. Circulation computed on neighbouring planes through the same vortex tube should agree, which gives a final internal check. Pressure fields and loads can be derived from the velocity data by the methods reviewed by van Oudheusden (2013), given adequate temporal information.