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The URANS equations provide a computationally efficient tool to simulate unsteady turbulent flows for a wide range of applications. To account for the errors introduced by the turbulence closure model, recent works have adopted data assimilation (DA) to enhance their predictive capabilities. Recognizing the challenges posed by the computational cost of 4DVar DA for unsteady flows, we propose a 3DVar DA framework that incorporates a time-discrete Fourier transform of the URANS equations, facilitating the use of the stationary discrete adjoint method in Fourier space. Central to our methodology is the introduction of a corrective, divergence-free, and unsteady forcing term, derived from a Fourier series expansion, into the URANS equations. This term aims at mitigating discrepancies in the modeled divergence of Reynolds stresses, allowing for the tuning of stationary parameters across different Fourier modes. Our implementation is built upon an extended version of the coupled URANS solver in OpenFOAM, enhanced to compute adjoint variables and gradients. This design choice ensures straightforward applicability to various flow setups and solvers, eliminating the need for specialized harmonic solvers. A gradient-based optimizer is employed to minimize discrepancies between simulated results and sparse velocity reference data. The effectiveness of our approach is demonstrated through its application to flow around a two-dimensional circular cylinder at a Reynolds number of 3900. Our results highlight the method's ability to reconstruct mean flow accurately and improve the vortex shedding frequency through the assimilation of zeroth mode data. Additionally, the assimilation of first mode data further enhances the simulation's capability to capture low-frequency dynamics of the flow, and finally, it runs efficiently by leveraging a coarse mesh.

Data assimilation (DA) plays a crucial role in extracting valuable information from flow measurements in fluid dynamics problems. Often only time-averaged data is available, which poses challenges for DA in the context of unsteady flow problems. Recent works have shown promising results in optimizing Reynolds-averaged Navier-Stokes (RANS) simulations of stationary flows using sparse data through variational data assimilation, enabling the reconstruction of mean flow profiles. In this study we perform three-dimensional variational data assimilation of sparse time-averaged data into an unsteady RANS (URANS) simulation by means of a stationary divergence-free forcing term in the URANS equations. Efficiency and speed of our method are enhanced by employing coarse URANS simulations and leveraging the stationary discrete adjoint method for the time-averaged URANS equations. The data assimilation codes were developed in-house using OpenFOAM for the URANS simulations as well as for the solution of the adjoint problem, and Python for the gradient-based optimization. Our results demonstrate that data assimilation of sparse time-averaged velocity measurements not only enables accurate mean flow reconstruction, but also improves the flow dynamics, specifically the vortex shedding frequency. To validate the efficacy of our approach, we applied it to turbulent flows around cylinders of various shapes at Reynolds numbers ranging from 3000 to 22000. Our findings indicate that data points near the cylinder play a crucial role in improving the vortex shedding frequency, while additional data points further downstream are necessary to also reconstruct the time-averaged velocity field in the wake region.

The Reynolds-averaged Navier-Stokes (RANS) equations provide a computationally efficient method for solving fluid flow problems in engineering applications. However, the use of closure models to represent turbulence effects can reduce their accuracy. To address this issue, recent research has explored data-driven techniques such as data assimilation and machine learning. An efficient variational data assimilation (DA) approach is presented to enhance steady-state eddy viscosity based RANS simulations. To account for model deficiencies, a corrective force term is introduced in the momentum equation. In the case of only velocity reference data, this term can be represented by a potential field and is divergence-free. The DA implementation relies on the discrete adjoint method and approximations for efficient gradient evaluation. The implementation is based on a two-dimensional coupled RANS solver in OpenFOAM, which is extended to allow the computation of the adjoint velocity and pressure as well as the adjoint gradient. A gradient-based optimizer is used to minimize the difference between the simulation results and the reference data. To evaluate this approach, it is compared with alternative data assimilation methods for canonical stationary two-dimensional turbulent flow problems. For the data assimilation, sparsely distributed reference data from averaged high-fidelity simulation results are used. The results suggest that the proposed method achieves the optimization goal more efficiently compared to applying data assimilation for obtaining the eddy viscosity, or a field modifying the eddy viscosity, directly. The method works well for different reference data configurations and runs efficiently by leveraging coarse meshes.

We study vector-boson scattering of the physical, gauge-invariant states in a reduced standard-model setup on the lattice for various parameter sets. To this end, the phase shift in the scalar channel is determined using a L\"uscher-type analysis. The results can be readily interpreted in terms of the Higgs properties and a reunitarized Fr\"ohlich-Morchio-Strocchi analysis at Born level. The only deviation appears for a Higgs mass below the elastic threshold, where we find a negative scattering length indicative of the bound-state nature of the physical scalar degree of freedom. We assess the possible implications for an experimental detection of the effect., Comment: 21 pages, 8 figures, 9 tables; v2: minor corrections to text, small modification to Figs. 3 - 5, version to appear in PRD

Simulating hypersonic flow around a space vehicle is challenging because of the multiscale and nonequilibrium nature inherent in these flows. To effectively deal with such flows, a novel particle particle hybrid scheme combining the stochastic particle Bhatnagar-Gross-Krook (BGK) method with Direct Simulation Monte Carlo (DSMC) was developed recently, but only for monatomic gases [Fei et. al., J. Comput. Phys. 2021]. Here this work is extended to the particle-particle hybrid method for polyatomic gases. In the near continuum regime, employing the Ellipsoidal Statistical BGK model proposed by Y. Dauvois, et. al. [Eur. J. Mech. B Fluids, 2021] with discrete levels of vibrational energy, the stochastic particle BGK method is first established following the idea of the unified stochastic particle BGK (USPBGK) scheme. In the fluid limit, it has been proven to be of second order temporal and spatial accuracy. Then, the USPBGK scheme with rotational and vibrational energies is combined with DSMC to construct a hybrid method for polyatomic gases. The present hybrid scheme is validated with numerical tests of homogenous relaxation, 1D shock structure and 2D hypersonic flows past a wedge and a cylinder. Compared to traditional stochastic particle methods, the proposed hybrid method can achieve higher accuracy at a much lower computational cost. Therefore, it is a more efficient tool to study multiscale hypersonic flows.