Quantifying Truncation-Related Uncertainties in Unsteady Fluid Dynamics Reduced Order Models

Abstract

In this paper, we present a new method to quantify the uncertainty introduced by the drastic dimensionality reduction commonly practiced in the field of computational fluid dynamics, the ultimate goal being to simulate accurate priors for real-time data assimilation. Our key ingredient is a stochastic Navier-Stokes closure mechanism that arises by assuming random unresolved flow components. This decomposition is carried out through Galerkin projection with a Proper Orthogonal Decomposition (POD-Galerkin) basis. The residual velocity fields, model structure and evolution of coefficients of the reduced order's solutions are used to compute the resulting multiplicative and additive noise's correlations. The low computational cost of these consistent correlation estimators makes them applicable to the study of complex fluid flows. This stochastic POD-ROM is applied to 2D and 3D DNS wake flows at Reynolds 100 and 300, respectively, with Uncertainty Quantification (UQ) and forecasting outside the learning interval being the main focus. The proposed stochastic POD-ROM approach is shown to stabilize the unstable temporal coefficients and to maintain their variability under control, while exhibiting an impressively accurate predictive capability.

Real-time estimation and prediction of unsteady flows using reduced-order models coupled with few measurements

“…Here, we do consider this noise term. As in [29], we have implemented the POD-Galerkin of the Navier-Stokes model under location uncertainty (3). We obtain the following ROM:…”

“…where ( βk ) k are n independent one-dimensional white noises and, by convention, the remaining parameters are b 0 = 1, M 0 = 0 and ϕ 0 = v. Using the physical equations ( 3) and (4) and corresponding technical statistical estimators from stochastic calculus, the ROM coefficients l, f , c, and αR are determined from the resolved spatial modes ϕ i , the resolved temporal modes b i , and the POD residual velocity v ′ . Interested readers can refer to Appendix B or to [29] for more details. Aimed at applied mathematicians, [29] extensively rely on stochastic calculus notations whereas we have tried to make this paper and its appendices accessible to a broader audience.…”

“…Inspired from the theoretical work of [24], [22] has introduced that stochastic closure and [23] generalized it. It has led to huge improvements in uncertainty and model error quantification both in high-dimensional CFD [25,26,27] and reduced state spaces [28,29]. Nevertheless, no associated ensemble-based data assimilation algorithm has been proposed yet.…”

“…This paper will be the first on this path. Our new fast data assimilation algorithm for fluid flows includes both an efficient background state ensemble emulator [29] and a particle filter to correct this ensemble. This paper will be organized as follows: section 2 will recall the main aspects of POD-Galerkin ROM (POD-ROM), section 3 will present the key player of our algorithm: a randomized version of POD-Galerkin ROM, section 4 explains the data assimilation procedure, and finally, section 6 will showcase its potential through some of our numerical results.…”

Input-output analysis of the stochastic Navier-Stokes equations: application to turbulent channel flow

“…It starts from a stochastic version of the Navier-Stokes equations, originally introduced by Mémin (2014), which is based on the stochastic transport of conserved quantities. The formalism has been successfully employed to perform large eddy simulations (Chandramouli et al, 2018), geophysical flow modelling (Resseguier et al, 2017a,b,c;Chapron et al, 2018;Bauer et al, 2020a,b), near-wall flow modelling (Pinier et al, 2019), data assimilation Mémin, 2017, 2018;Chandramouli et al, 2020) and reduced-order modelling (Resseguier et al, 2017d(Resseguier et al, , 2021). An advantage of the approach is the formulation of closure by defining statistics of a stochastic unresolved time-decorrelated (with respect to the time scales of the resolved part) velocity field.…”