Ensemble-Corrected RANS (EC-RANS): Online Data Assimilation for Turbulence Model Correction

Mathematical Foundation and Single-Case Validation

Abstract

This paper presents the mathematical foundation and single-case validation of an Ensemble-Corrected RANS (EC-RANS) framework, in which the RANS Reynolds stress field is sequentially corrected through LETKF assimilation of high-fidelity observation data, with divergence-freeness enforced analytically via Helmholtz decomposition. The framework couples four modular blocks: (A) a k-ω SST RANS baseline with stress-ratio indexing, (B) spatially adaptive LES patches governed by the synchronisation length scale, (C) LETKF-SCV correction with Helmholtz-decomposed forcing that preserves analytical divergence-freeness, and (D) ResDMD temporal propagation with formal Koopman convergence guarantees. The architecture contains a single calibrated parameter (ρ_faktor); remaining design choices are fixed by asymptotic scaling or standard practice. The present paper validates Blocks A and C on a 1D canonical flow (channel, Re_τ = 590) and a 3D separated flow (periodic hill, Re = 5600); Blocks B and D are formulated but not exercised in this work. This deliberate scope restriction allows mathematical rigour to be established before engineering claims are made. Formal proofs of three mathematical properties are provided: existence and uniqueness of the Helmholtz potentials (Lax-Milgram), per-cycle contraction of the coupled RANS-LETKF operator with constant q < 1, and convergence of the ResDMD operator for systems supporting a physical measure. Calibration against DNS of fully developed channel flow at Re_τ = 590 yields RMSE(U)/U⁺_c = 0.397%, a 78% improvement over the k-ω SST baseline, with less than 7% sensitivity to all fixed design thresholds in one-dimensional tests. Three-dimensional validation on the periodic hill at Re = 5600 (ESI OpenFOAM v2412, 2.56×10⁶ cells, M = 32 LETKF ensemble members, N = 2 independent replications with fresh random seeds) demonstrates monotone convergence to within 4.7% of DNS (Krank et al. 2018) for the primary realisation (Run E, seed = 42) over 12 assimilation cycles when the GRF perturbation amplitude σ = 0.30 is combined with multiplicative inflation ρ = 1.50; an independent-seed replication (Run E2, seed = 123) yields a converged RMSE of 5.7%, giving an N = 2 sample standard deviation of 0.007 on RMSE(U), reflecting the inherent RANS-baseline variability across decomposition seeds (see Supplementary S2.5). A controlled three-point σ-sweep (σ ∈ {0.05, 0.20, 0.30}) identifies a regime-specific failure mode at σ = 0.05 in which ensemble spread collapses and analysis error diverges; this failure was reproducibly documented across two independent runs. Projected performance against the NASA wall-mounted hump and the Ahmed body remains a literature-consistent scaling estimate awaiting three-dimensional validation and may differ substantially from actual computed values. The primary contribution is the mathematical framework itself — combining formal convergence proofs, a single calibrated parameter, per-cell confidence volumes, and a modular sensor interface for digital-twin integration — rather than a claim of validated universal accuracy. Three-dimensional implementation and community validation are identified as essential next steps.

Reproducibility package: https://github.com/pederwahlberg/ECRANS (Apache 2.0).