From Particles to Pressure: PIV-based pressure reconstruction for base flows

Pressure reconstruction based on particle image velocimetry (PIV) refers to the determination of pressure data from pictures of small tracer particles added to a flow. The technique possesses a unique combination of beneficial characteristics in that it non-intrusively provides simultaneous pressure and velocity data in the flow field without the need for instrumentation or other preparation the model. The present research provides a structured overview of different approaches to PIV-based pressure reconstruction and characterises their (relative) performance, particularly when applied to a transonic base flow. The unsteady, large-scale behaviour of this flow constitutes is subject of active research in the context of launcher aerodynamics to which experimental pressure field data would make a valuable contribution. Two techniques are analysed in depth through theoretical analyses, a simulated experiment based on a numerical simulation and several wind tunnel experiments: pseudo-tracking for the determination of instantaneous pressure fields and the Reynolds-averaging approach for the determination of time-averaged pressure fields.

Various PIV-based methods for instantaneous pressure determination are capable of reconstructing the main features of instantaneous pressure fields, including methods that reconstruct pressure fields from a single velocity snapshot. Highly accurate pressure fields can be obtained by tracking individual particles in combination with advanced processing techniques. In view of this outcome, it is recommended to let the choice for a specific technique be guided by the desired accuracy, resolution and dimensionality of the pressure results, while taking taking into account practical considerations, in particular limitations in the capabilities of available measurement equipment and the complexity of the measurement system. Without such intent, the potential difficulties and complexity of data acquisition were demonstrated with the use of a 12-camera/2-laser PIV system.

For instantaneous pressure reconstruction through pseudo-tracking new insights were obtained on its spatio-temporal filtering behaviour and the propagation of velocity measurement errors. A cut-off peak-response is specified as a function of the temporal track length and spatial resolution. Novel approaches are suggested to determine suitable temporal track lengths on the basis of the variation in material acceleration with track length and on the basis of pressure power spectra. Such spectra were also used estimate the local error margin of reconstructed pressure values. For the implementation of pseudo-tracking, it is recommended to first construct tracks by a combination of a second-order integration method and linear interpolation, using an integration time step that is sufficiently small to meet the Courant–Friedrichs–Lewy condition. The material acceleration may subsequently be estimated from the tracks by means of least-square fitting of a first-order polynomial or central differencing depending on the type of input data.

When calculating mean pressure fields with the Reynolds-averaging approach, it is recommended to only include the terms that are associated with the mean flow and Reynolds stresses. The impact of neglecting spatial and temporal density variations may be estimated as the difference between pressure solutions calculated with and without density-gradient terms. After validation, the approach was employed to study the effects of an exhaust plume and nozzle length on transonic and supersonic axisymmetric base flows. Amongst others, the results showed that depending on the nozzle length the presence of a plume may cause a decrease in base pressure in the transonic flow cases and an increase in base pressure in the supersonic flow cases, indicating the effects of entrainment and displacement, respectively. The results furthermore highlight the need of considering during vehicle design, that a longer nozzle in which a plume expands further, not only corresponds to a lower exit pressure in the plume, but also to a different ambient pressure near the nozzle exit.