Sparse data assimilation for under-resolved large-eddy simulations

TL;DR

This work addresses the challenge of obtaining accurate scale-resolving LES predictions on coarse meshes by introducing a sparse, variational data assimilation framework that injects a stationary divergence-free forcing into the LES momentum equation. Using time-averaged velocity observations and a discrete adjoint, the method optimizes the forcing to correct mean flow, improve vortex-shedding frequency (Strouhal number), and infer anisotropic Reynolds stresses; the approach is demonstrated on flow over periodic hills and flow around a square cylinder. A subsequent step reconstructs the anisotropic Reynolds stresses from the corrective force and replaces the stationary forcing by velocity-fluctuation nudging to preserve mean flow while adjusting fluctuations, leading to improved near-wake dynamics and stress components. The results show substantial improvements in mean velocity and TKE predictions, as well as in the fidelity of Reynolds-stress anisotropy, highlighting the potential of LES enhanced by stationary, sparse data assimilation for efficient, accurate turbulence simulations, with implications for higher Reynolds-number applications and meshfree methods. Possible future directions include optimized data-point placement, inclusion of TKE data via a transport equation, and assessment across different SGS models and geometries.

Abstract

The need for accurate and fast scale-resolving simulations of fluid flows, where turbulent dispersion is a crucial physical feature, is evident. Large-eddy simulations (LES) are computationally more affordable than direct numerical simulations, but their accuracy depends on sub-grid scale models and the quality of the computational mesh. In order to compensate related errors, a data assimilation approach for LES is devised in this work. The presented method is based on variational assimilation of sparse time-averaged velocity reference data. Working with the time-averaged LES momentum equation allows to employ a stationary discrete adjoint method. Therefore, a stationary corrective force in the unsteady LES momentum equation is iteratively updated within the gradient-based optimization framework in conjunction with the adjoint gradient. After data assimilation, corrected anisotropic Reynolds stresses are inferred from the stationary corrective force. Ultimately, this corrective force that acts on the mean velocity is replaced by a term that scales the velocity fluctuations through nudging of the corrected anisotropic Reynolds stresses. Efficacy of the proposed framework is demonstrated for turbulent flow over periodic hills and around a square cylinder. Coarse meshes are leveraged to further enhance the speed of the optimization procedure. Time- and spanwise-averaged velocity reference data from high-fidelity simulations is taken from the literature. Our results demonstrate that adjoint-based assimilation of averaged velocity enables the optimization of the mean flow, vortex shedding frequency (i.e., Strouhal number), and anisotropic Reynolds stresses. This highlights the superiority of scale-resolving simulations such as LES over simulations based on the (unsteady) Reynolds-averaged equations.

Figures (13)

• Figure 1: Flowchart of the optimization-based data assimilation procedure.
• Figure 1: Test function $f_g^\mathrm{test}$ for flow over periodic hills normalized with its initial value $f^{(0)}$.
• Figure 2: Simulation domain of the periodic hill setup. The mean flow is in $x$-direction. All length scales are normalized by the hill crest height $H$. The domain extends over a width of $4.5 \, H$ in $z$-direction, in which periodicity is assumed.
• Figure 2: Test function $f_g^\mathrm{test}$ for flow around a square cylinder normalized with its initial value $f^{(0)}$.
• Figure 3: Streamwise velocity component of the flow over periodic hills. The LES solver ran until the convergence criterion $C_U = 0.15$ was reached.