[Paper Review] Ensemble-variational assimilation of statistical data in large eddy simulation

Vincent Mons, Yifan Du+1 more|arXiv (Cornell University)|Sep 25, 2021

Fluid Dynamics and Turbulent Flows82 references42 citations

TL;DR

This paper introduces an ensemble-variational data assimilation (EnVar) framework to calibrate Smagorinsky subgrid models in LES using reference statistics from DNS, producing data-assimilated LES (DA-LES) that better reproduce mean and second-order turbulence statistics.

ABSTRACT

A non-intrusive data assimilation methodology is developed to improve the statistical predictions of large-eddy simulations (LES). The ensemble-variational (EnVar) approach aims to minimize a cost function that is defined as the discrepancy between LES predictions and reference statistics from experiments or, in the present demonstration, independent direct numerical simulations (DNS). This methodology is applied to adjust the Smagorinsky subgrid model and obtain data assimilated LES (DA-LES) which accurately estimate the statistics of turbulent channel flow. To separately control the mean and fluctuations of the modeled subgrid tensor, and ultimately the first- and second-order flow statistics, two types of model corrections are considered. The first one optimizes the wall-normal profile of the Smagorinsky coefficient, while the second one introduces an adjustable steady forcing in the momentum equations to independently act on the mean flow. Using these two elements, the data assimilation procedure can satisfactorily modify the subgrid model and accurately recover reference flow statistics. The retrieved subgrid model significantly outperforms more elaborate baseline models such as dynamic and mixed models, in a posteriori testing. The robustness of the present data assimilation methodology is assessed by changing the Reynolds number and considering grid resolutions that are away from usual recommendations. Taking advantage of the stochastic formulation of EnVar, the developed framework also provides the uncertainty of the retrieved model.

Motivation & Objective

• Motivate data assimilation to enhance LES fidelity by leveraging statistical observations from DNS/experiments.
• Develop an EnVar-based method to calibrate subgrid models, primarily the Smagorinsky coefficient profile and an optional mean forcing.
• Assess how well DA-LES reproduces mean flows and Reynolds stresses in turbulent channel flow.
• Compare the EnVar-calibrated Smagorinsky model against dynamic and mixed subgrid models.
• Evaluate robustness to Reynolds number and grid resolution; quantify uncertainty in the retrieved model.

Proposed method

• Formulate a control vector gamma consisting of Cs(y) and optionally a steady forcing sigma(y) in the LES equations.
• Define observations m as mean flow and/or subgrid tensor statistics derived from DNS or experiments.
• Minimize a Bayesian-type cost function J combining prior information on gamma (B) and data misfit (R) via EnVar.
• Represent the control vector in an ensemble-based subspace using a POD projection to obtain a tractable optimization problem.
• Compute H, the LES-predicted observations minus references, for multiple ensemble realizations and perform POD-based reduction (E, V, H_POD).
• Obtain assimilated gamma by solving a linearized, reduced-cost optimization (w^a) and reconstruct gamma^a = gamma^f + E w^a; iterate if desired to refine.

Experimental results

Research questions

• RQ1 Can EnVar assimilate limited statistical observations to accurately recover mean and second-order statistics in LES?
• RQ2 Does adjusting the wall-normal Separable Smagorinsky profile Cs(y) together with a mean forcing sigma(y) enable independent control of mean and fluctuating subgrid dissipation?
• RQ3 How does DA-LES performance compare to dynamic and scale-similarity/mixed subgrid models in terms of a posteriori statistics?
• RQ4 Is the EnVar-based approach robust to Reynolds number changes and grid resolutions beyond standard recommendations?
• RQ5 What uncertainty quantification does EnVar provide for the retrieved subgrid-model corrections?

Key findings

• DA-LES with Cs(y) and/or sigma(y) can modify the subgrid model to match reference statistics of turbulent channel flow.
• The retrieved subgrid model through EnVar can outperform dynamic and mixed models in a posteriori testing.
• Simultaneous use of Cs(y) and sigma(y) provides efficient correction for both mean flow and second-order statistics.
• The framework demonstrates robustness to varying Reynolds numbers and grid resolutions beyond standard guidelines.
• EnVar offers a probabilistic view by providing uncertainty characterization of the retrieved model.