Modified (n + 1)D Laplacian for smooth pressure reconstruction based on time-resolved Velocimetry: from analysis to experiments

Smooth reconstruction of pressure fields from time-resolved image velocimetry involves two general strategies: 1) suppressing the noise of the measured velocity field and/or 2) applying some smoother to the pressure reconstruction. For example, the 4D pressure solver in DaVis 10, LaVision, employed the idea of the latter category by introducing the “modified Poisson equation”: ▽²p+ξ²∂²p/∂t²=f(u), where p is the pressure field, u is the velocity field measured by time-resolved image velocimetry. The tunable parameter ξ² is used to control the solver's smoothing behavior by adjusting the diffusivity in time. However, the selection of ξ² is not trivial until the features of the modified Laplacian are known. This work focuses on investigating the fundamental properties of the modified Poisson equation. We first provide rigorous analysis, with numerical and experimental validation, to prove that this solver can smooth the computed pressure by setting a large enough ξ², and the error in the reconstructed pressure is bounded. We also discuss the potential drift of pressure due to the partitioning in time, which is an optional strategy used in LaVision's current 4D solver to reduce computational costs, with theoretical analysis backed by numerical and experimental tests. Our analysis and validation not only show that careful choice of the parameters (e.g., ξ²) is needed for smooth and accurate pressure field reconstruction but provide guidelines for parameter tuning when similar time-resolved solvers are used.