Instantaneous Pressure Determination from Unsteady Velocity Fields Using Adjoint-based Sequential Data Assimilation

Abstract

A sequential data assimilation (DA) method is developed for pressure determination of turbulent velocity fields measured by particle image velocimetry (PIV), based on the unsteady adjoint formulation. A forcing term F, which is optimized using the adjoint system, is added to the primary Navier–Stokes (N–S) equations to drive the assimilated flow toward the observations at each time step. Compared with the conventional unsteady adjoint method, which requires the forward integration of the primary system and the backward integration of the adjoint system, the present approach integrates the primary-adjoint system all the way forward, discarding the requirement of data storage at every time step, being less computationally resource-consuming, and saving space. The pressure determination method of integration from eight paths [J. O. Dabiri et al., “An algorithm to estimate unsteady and quasi-steady pressure fields from velocity field measurements,” J. Exp. Biol. 217, 331 (2014)] is also evaluated for comparison. Using synthetic PIV data of a turbulent jet as the observational data, the present DA method is able to determine the instantaneous pressure field precisely using the three-dimensional velocity fields, regardless of the observational noise. For the two-dimensional three-component (3C) or two-component (2C) velocity fields, which are not sufficient for pressure determination by the integration method due to the lack of off-plane derivatives, the present DA method is able to reproduce pressure fields whose statistics agree reasonably well with those of the referential results. The 3C and 2C velocity fields yield quite similar results, indicating the possibility of pressure determination from only planar-PIV measurements in turbulent flows. The tomography PIV measurements are also used as observational data, and a clear pressure pattern is obtained with the present DA method.