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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 through a stationary corrective forcing term in the LES momentum equation working with the time-averaged LES momentum equation allows to employ a stationary discrete adjoint method. Therefore, the stationary forcing in the unsteady LES momentum equation is iteratively updated within the gradient-based optimization framework in conjunction with the adjoint gradient. The implementation is based on a three-dimensional LES solver in OpenFOAM, which is extended to allow the computation of adjoint quantities and, ultimately, the adjoint gradient. 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 turbulent kinetic energy. This highlights the superiority of scale-resolving simulations such as LES over simulations based on the (unsteady) Reynolds-averaged equations.

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.

The multiple scattering of an ultrashort laser pulse by a turbid dispersive medium (namely a cloud of bubbles in water) is investigated by means of Monte Carlo simulations. The theory of Gouesbet and Gr\'ehan [Part. Part. Syst. Charact. 17 213-224 (2000)] is used to derive an energetic model of the scattering transient. The spreading and extinction of the pulse is decoupled from the transient of scattering to describe each phenomenon individually. The transient of scattering is modeled with the Lorenz-Mie Theory, thus also valid for a relative refractive index lower than one, contrary to the Debye series expansion which does not converge close to the critical angle. To this aim, the Scattering Impulse Response Function (SIRF) allows to detect the different modes of scattering transient in time and direction. The present approach is more generic and can simulate clouds of air bubbles in water. Two Monte-Carlo approaches are proposed. The first is a pure Monte Carlo approach where the delay due to the scattering is randomly drawn at each event, while the second is based on the transport of the whole scattering signal. They are both embedded in the Monte Carlo code Scatter3D [JOSA A 24, 2206-2219 (2007)]. Both models produce equivalent trends and are validated against published numerical results. They are applied to the multiple scattering of ultra short pulse by a cloud of bubble in water in the forward direction. The pulse spread due to the propagation in water is computed for a wide range of traveled distances and pulse durations, and the optimal pulse duration is given to minimize the pulse spread at a given distance. The main result is that the scattered photons exit the turbid medium earlier than the ballistic photons and produce a double peak related to the refraction in the bubble. This demonstrates the possibility to develop new diagnostics to characterize dynamic bubbly flows., Comment: 30 pages, 19 Figures

We present a new method for modeling tissue perfusion on the capillary scale. The microvasculature is represented by a network of one-dimensional vessel segments embedded in the extra-vascular space. Vascular and extra-vascular space exchange fluid over the vessel walls. This exchange is modeled by distributed sources using smooth kernel functions for the extra-vascular domain. It is shown that the proposed method may significantly improve the approximation of the exchange flux, in comparison with existing methods for mixed-dimension embedded problems. Furthermore, the method exhibits better convergence rates of the relevant quantities due to the increased regularity of the extra-vascular pressure solution. Numerical experiments with a vascular network from the rat cortex show that the error in the approximation of the exchange flux for coarse grid resolution may be decreased by a factor of $3$. This may open the way for computing on larger network domains, where a fine grid resolution cannot be achieved in practical simulations due to constraints in computational resources, for example in the context of uncertainty quantification.

Conservation laws in the form of elliptic and parabolic partial differential equations (PDEs) are fundamental to the modeling of many problems such as heat transfer and flow in porous media. Many of such PDEs are stochastic due to the presence of uncertainty in the conductivity field. Based on the relation between stochastic diffusion processes and PDEs, Monte Carlo (MC) methods are available to solve these PDEs. These methods are especially relevant for cases where we are interested in the solution in a small subset of the domain. The existing MC methods based on the stochastic formulation require restrictively small time steps for high variance conductivity fields. Moreover, in many applications the conductivity is piecewise constant and the existing methods are not readily applicable in these cases. Here we provide an algorithm to solve one-dimensional elliptic problems that bypasses these two limitations. The methodology is demonstrated using problems governed by deterministic and stochastic PDEs. It is shown that the method provides an efficient alternative to compute the statistical moments of the solution to a stochastic PDE at any point in the domain. A variance reduction scheme is proposed for applying the method for efficient mean calculations.

Monte Carlo (MC) simulations of transport in random porous networks indicate that for high variances of the log-normal permeability distribution, the transport of a passive tracer is non-Fickian. Here we model this non-Fickian dispersion in random porous networks using discrete temporal Markov models. We show that such temporal models capture the spreading behavior accurately. This is true despite the fact that the slow velocities are strongly correlated in time, and some studies have suggested that the persistence of low velocities would render the temporal Markovian model inapplicable. Compared to previously proposed temporal stochastic differential equations with case specific drift and diffusion terms, the models presented here require fewer modeling assumptions. Moreover, we show that discrete temporal Markov models can be used to represent dispersion in unstructured networks, which are widely used to model porous media. A new method is proposed to extend the state space of temporal Markov models to improve the model predictions in the presence of extremely low velocities in particle trajectories and extend the applicability of the model to higher temporal resolutions. Finally, it is shown that by combining multiple transitions, temporal models are more efficient for computing particle evolution compared to correlated CTRW with spatial increments that are equal to the lengths of the links in the network.