State observer-based data assimilation: a PID control-inspired observer in the pressure equation

Abstract

A novel state observer-based data-assimilation technique is described and then assessed with a numerical test case. This novel technique assimilates pressure data through a state observer that is constructed based on a proportional-integral-derivative control law and that acts on the pressure equation. The technique is assessed by comparing the performance to a standard simulation and a data-assimilated simulation using a previously-established technique, wherein a proportional observer is placed in the momentum equations. First, the mechanism through which measured pressure data is assimilated into simulations is described for both the previously-established and the novel technique. Next, the techniques are applied to a square cylinder in cross-flow at a Reynolds number of 100. A reference simulation is run on a dense mesh, and both standard and data-assimilated simulations are run on a much coarser mesh. The primary characteristic used to evaluate the techniques is their dynamic performance, in terms of how quickly vortex shedding is realized and how accurately the frequency of the vortex shedding is modeled on the coarser mesh. Both of the data assimilation techniques produce simulations with a much faster transition from initial conditions to vortex shedding than the standard simulation on the coarse mesh. The data-assimilated simulation using an observer in the pressure equations most accurately estimates the pressures and velocities at the probed locations and produces a frequency spectrum that most closely matches the reference simulation.

Calculation of the Pressure Field for Turbulent Flow around a Surface-Mounted Cube Using the SIMPLE Algorithm and PIV Data

“…The idea of integrating measurements into a CFD code is not new, since a methodology appertaining to the family of state observers has been proposed [21], where the SIMPLER algorithm uses partial experimental information as feedback for the reconstruction of the boundary conditions of turbulent flows. State observers can be defined as functions aiming to reconstruct the state of a system from an incomplete set of measurements [22]. The principal motive behind the integration of measurements is the necessity of reproducing the exact structure of real complex flows, especially in disciplines such as meteorology and feedback flow control [23].…”

Nudging-based data assimilation of the turbulent flow around a square cylinder

“…Incompressible flows have generally been considered in the previous studies discussed below, and, depending on the considered type of observation (namely velocity or pressure measurements), the feedback term may act on the momentum equations (Imagawa & Hayase 2010) and/or on the pressure Poisson equation (Neeteson & Rival 2020). The intensity of this feedback term is tuned through a single scalar, which thus allows us to adjust to which extent the simulation is driven towards the reference data.…”

“…Nudging may also be categorised as a special case of the proportional-integral-derivative (PID) controller that only includes the proportional contribution. The benefits of considering the temporal integral and derivative of the discrepancies between observations and predictions have been recently investigated by Neeteson & Rival (2020) and Saredi et al (2021).…”

“…Nudging was able to compensate for uncertainties in the initial condition (Imagawa & Hayase 2010) or numerical errors due to the use of a coarser grid in the second simulation (Hayase & Hayashi 1997), and thus successfully reproduced the reference solution. Neeteson & Rival (2020) also considered nudging and more general PID controllers to drive a coarse-grid simulation based on synthetic pressure data of the flow past a square cylinder at Re = 100. Suzuki & Hasegawa (2017) combined a nudging/state observer-like approach with linear stochastic estimation to reconstruct turbulent channel flows at friction Reynolds number Re τ = 100 from wall measurements, the same DNS set-up was used to both generate data and perform estimation.…”

“…In particular, nudging was used to recover a reference forced isotropic turbulent flow, assuming that the forcing of the reference flow was unknown. Various types of observations of the reference flow were considered among Eulerian observations, as in Hayase & Hayashi (1997), Imagawa & Hayase (2010) and Neeteson & Rival (2020), Lagrangian data or in spectral space. The effect of the observation density on the ability of nudging in recovering unmeasured scales was also investigated.…”