Modified ( n + 1 ) D Laplacian for smooth pressure reconstruction based on time-resolved velocimetry (1): analysis and numerics

We analyze a smooth pressure solver based on the ‘modified Poisson equation’: ∇ 2 p + ξ 2 ∂ 2 p ∂ t 2 = f ( u ( t ) ) , where p is the pressure field, u ( t ) is the velocity field measured by time-resolved image velocimetry, and ξ 2 is a tunable parameter to control the solver’s diffusive behaviour in time. This modified Poisson equation aims at obtaining smooth pressure fields from potentially noisy image velocimetry measurements, and is a part of the current four-dimensional (4D) pressure solver (implemented in, for example, DaVis 10.2) by LaVision. This work focuses on investigating three aspects of the ‘modified Poisson equation’: smoothing effect, error propagation, and drift in time. We first provide rigorous analysis and validate that this solver can sufficiently smooth the computed pressure field by setting a large enough ξ 2 . However, a large value of ξ 2 may cause large errors in the reconstructed pressure fields. Then we introduce an upper bound on the error in the reconstructed pressure fields to quantify the error propagation dynamics. Finally, we discuss the potential drift due to the partitioning in time, which is an optional strategy used in LaVision’s current 4D pressure solver to reduce computational costs. Our analysis and validation not only show that careful choice of the parameters (e.g. ξ 2 ) is needed for smooth and accurate pressure field reconstruction but provide theoretical guidelines for parameter tuning when similar pressure solvers are used for time-resolved image velocimetry data.