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A hybrid Eulerian-Lagrangian flow solver  [PDF]
Artur Palha,Lento Manickathan,Carlos Simao Ferreira,Gerard van Bussel
Physics , 2015,
Abstract: Currently, Eulerian flow solvers are very efficient in accurately resolving flow structures near solid boundaries. On the other hand, they tend to be diffusive and to dampen high-intensity vortical structures after a short distance away from solid boundaries. The use of high order methods and fine grids, although alleviating this problem, gives rise to large systems of equations that are expensive to solve. Lagrangian solvers, as the regularized vortex particle method, have shown to eliminate (in practice) the diffusion in the wake. As a drawback, the modelling of solid boundaries is less accurate, more complex and costly than with Eulerian solvers (due to the isotropy of its computational elements). Given the drawbacks and advantages of both Eulerian and Lagrangian solvers the combination of both methods, giving rise to a hybrid solver, is advantageous. The main idea behind the hybrid solver presented is the following. In a region close to solid boundaries the flow is solved with an Eulerian solver, where the full Navier-Stokes equations are solved (possibly with an arbitrary turbulence model or DNS, the limitations being the computational power and the physical properties of the flow), outside of that region the flow is solved with a vortex particle method. In this work we present this hybrid scheme and verify it numerically on known 2D benchmark cases: dipole flow, flow around a cylinder and flow around a stalled airfoil. The success in modelling these flow conditions presents this hybrid approach as a promising alternative, bridging the gap between highly resolved and computationally intensive Eulerian CFD simulations and fast but less resolved Lagrangian simulations.
The Lagrangian chemistry and transport model ATLAS: simulation and validation of stratospheric chemistry and ozone loss in the winter 1999/2000
I. Wohltmann, R. Lehmann,M. Rex
Geoscientific Model Development (GMD) & Discussions (GMDD) , 2010, DOI: 10.5194/gmd-3-585-2010
Abstract: ATLAS is a new global Lagrangian Chemistry and Transport Model (CTM), which includes a stratospheric chemistry scheme with 46 active species, 171 reactions, heterogeneous chemistry on polar stratospheric clouds and a Lagrangian denitrification module. Lagrangian (trajectory-based) models have several important advantages over conventional Eulerian models, including the absence of spurious numerical diffusion, efficient code parallelization and no limitation of the largest time step by the Courant-Friedrichs-Lewy criterion. This work describes and validates the stratospheric chemistry scheme of the model. Stratospheric chemistry is simulated with ATLAS for the Arctic winter 1999/2000, with a focus on polar ozone depletion and denitrification. The simulations are used to validate the chemistry module in comparison with measurements of the SOLVE/THESEO 2000 campaign. A Lagrangian denitrification module, which is based on the simulation of the nucleation, sedimentation and growth of a large number of polar stratospheric cloud particles, is used to model the substantial denitrification that occured in this winter.
The Lagrangian chemistry and transport model ATLAS: simulation and validation of stratospheric chemistry and ozone loss in the winter 1999/2000
I. Wohltmann,R. Lehmann,M. Rex
Geoscientific Model Development Discussions , 2010, DOI: 10.5194/gmdd-3-769-2010
Abstract: ATLAS is a new global Lagrangian Chemistry and Transport Model (CTM), which includes a stratospheric chemistry scheme with 46 active species, 171 reactions, heterogeneous chemistry on polar stratospheric clouds and a Lagrangian denitrification module. Lagrangian (trajectory-based) models have several important advantages over conventional Eulerian models, including the absence of spurious numerical diffusion, efficient code parallelization and no limitation of the largest time step by the Courant-Friedrichs-Lewy criterion. This work describes and validates the stratospheric chemistry scheme of the model. Stratospheric chemistry is simulated with ATLAS for the Arctic winter 1999/2000, with a focus on polar ozone depletion and denitrification. The simulations are used to validate the chemistry module in comparison with measurements of the SOLVE/THESEO 2000 campaign. A Lagrangian denitrification module, which is based on the simulation of the nucleation, sedimentation and growth of a large number of polar stratospheric cloud particles, is used to model the substantial denitrification that occured in this winter.
PDF modeling of near-wall turbulent flows: A New model, Weak second-order scheme and a numerical study in a Hybrid configuration  [PDF]
Sergio Chibbaro,Jean-Pierre Minier
Physics , 2010,
Abstract: In this work, we discuss some points relevant for stochastic modelling of one- and two-phase turbulent flows. In the framework of stochastic modelling, also referred to PDF approach, we propose a new Langevin model including all viscosity effects and thus that is consistent with viscous Navier-Stokes equations. In the second part of the work, we show how to develop a second-order unconditionally stable numerical scheme for the stochastic equations proposed. Accuracy and consistency of the numerical scheme is demonstrated analytically. In the last part of the work, we study the fluid flow in a channel flow with the proposed viscous method. A peculiar approach is chosen: the flow is solved with a Eulerian method and after with the Lagrangian model proposed which uses some of the Eulerian quantities. In this way attention is devoted to the issue of consistency in hybrid Eulerian/Lagrangian methods. It is shown that the coupling is important indeed and that to couple the Lagrangian model to an Eulerian one which is not consistent with the same turbulence physics leads to large errors. This part of the work complements a recent article [Chibbaro and Minier International Journal of Multiphase flows submitted (arXiv:0912.2045)]
On Adaptive Eulerian-Lagrangian Method for Linear Convection-Diffusion Problems  [PDF]
Xiaozhe Hu,Young-Ju Lee,Jinchao Xu,Chensong Zhang
Mathematics , 2012,
Abstract: In this paper, we consider the adaptive Eulerian--Lagrangian method (ELM) for linear convection-diffusion problems. Unlike the classical a posteriori error estimations, we estimate the temporal error along the characteristics and derive a new a posteriori error bound for ELM semi-discretization. With the help of this proposed error bound, we are able to show the optimal convergence rate of ELM for solutions with minimal regularity. Furthermore, by combining this error bound with a standard residual-type estimator for the spatial error, we obtain a posteriori error estimators for a fully discrete scheme. We present numerical tests to demonstrate the efficiency and robustness of our adaptive algorithm.
A Lagrangian model of air-mass photochemistry and mixing using a trajectory ensemble: the Cambridge Tropospheric Trajectory model of Chemistry And Transport (CiTTyCAT) version 4.2
T. A. M. Pugh, M. Cain, J. Methven, O. Wild, S. R. Arnold, E. Real, K. S. Law, K. M. Emmerson, S. M. Owen, J. A. Pyle, C. N. Hewitt,A. R. MacKenzie
Geoscientific Model Development (GMD) & Discussions (GMDD) , 2012, DOI: 10.5194/gmd-5-193-2012
Abstract: A Lagrangian model of photochemistry and mixing is described (CiTTyCAT, stemming from the Cambridge Tropospheric Trajectory model of Chemistry And Transport), which is suitable for transport and chemistry studies throughout the troposphere. Over the last five years, the model has been developed in parallel at several different institutions and here those developments have been incorporated into one "community" model and documented for the first time. The key photochemical developments include a new scheme for biogenic volatile organic compounds and updated emissions schemes. The key physical development is to evolve composition following an ensemble of trajectories within neighbouring air-masses, including a simple scheme for mixing between them via an evolving "background profile", both within the boundary layer and free troposphere. The model runs along trajectories pre-calculated using winds and temperature from meteorological analyses. In addition, boundary layer height and precipitation rates, output from the analysis model, are interpolated to trajectory points and used as inputs to the mixing and wet deposition schemes. The model is most suitable in regimes when the effects of small-scale turbulent mixing are slow relative to advection by the resolved winds so that coherent air-masses form with distinct composition and strong gradients between them. Such air-masses can persist for many days while stretching, folding and thinning. Lagrangian models offer a useful framework for picking apart the processes of air-mass evolution over inter-continental distances, without being hindered by the numerical diffusion inherent to global Eulerian models. The model, including different box and trajectory modes, is described and some output for each of the modes is presented for evaluation. The model is available for download from a Subversion-controlled repository by contacting the corresponding authors.
A Lagrangian model of air-mass photochemistry and mixing using a trajectory ensemble: the Cambridge Tropospheric Trajectory model of Chemistry And Transport (CiTTyCAT) version 4.2  [PDF]
T. A. M. Pugh,M. Cain,J. Methven,O. Wild
Geoscientific Model Development Discussions , 2011, DOI: 10.5194/gmdd-4-2469-2011
Abstract: A Lagrangian model of photochemistry and mixing is described (CiTTyCAT, stemming from the Cambridge Tropospheric Trajectory model of Chemistry And Transport), which is suitable for transport and chemistry studies throughout the troposphere. Over the last five years, the model has been developed in parallel at several different institutions and here those developments have been incorporated into one "community" model and documented for the first time. The key photochemical developments include a new scheme for biogenic volatile organic compounds and updated emissions schemes. The key physical development is to evolve composition following an ensemble of trajectories within neighbouring air-masses, including a simple scheme for mixing between them via an evolving "background profile", both within the boundary layer and free troposphere. The model runs along trajectories pre-calculated using winds and temperature from meteorological analyses. In addition, boundary layer height and precipitation rates, output from the analysis model, are interpolated to trajectory points and used as inputs to the mixing and wet deposition schemes. The model is most suitable in regimes when the effects of small-scale turbulent mixing are slow relative to advection by the resolved winds so that coherent air-masses form with distinct composition and strong gradients between them. Such air-masses can persist for many days while stretching, folding and thinning. Lagrangian models offer a useful framework for picking apart the processes of air-mass evolution over inter-continental distances, without being hindered by the numerical diffusion inherent to global Eulerian models. The model, including different box and trajectory modes, is described and some output for each of the modes is presented for evaluation. The model is available for download from a Subversion-controlled repository by contacting the corresponding authors.
Transformation Properties of the Lagrangian and Eulerian Strain Tensors  [PDF]
Thomas B. Bahder
Physics , 2002,
Abstract: A coordinate independent derivation of the Eulerian and Lagrangian strain tensors of finite deformation theory is given based on the parallel propagator, the world function, and the displacement vector field as a three-point tensor. The derivation explicitly shows that the Eulerian and Lagrangian strain tensors are two-point tensors, each a function of both the spatial and material coordinates. The Eulerian strain is a two-point tensor that transforms as a second rank tensor under transformation of spatial coordinates and transforms as a scalar under transformation of the material coordinates. The Lagrangian strain is a two-point tensor that transforms as scalar under transformation of spatial coordinates and transforms as a second rank tensor under transformation of the material coordinates. These transformation properties are needed when transforming the strain tensors from one frame of reference to another moving frame.
Eulerian and Lagrangian propagators for the adhesion model (Burgers dynamics)  [PDF]
Francis Bernardeau,Patrick Valageas
Physics , 2009, DOI: 10.1103/PhysRevD.81.043516
Abstract: Motivated by theoretical studies of gravitational clustering in the Universe, we compute propagators (response functions) in the adhesion model. This model, which is able to reproduce the skeleton of the cosmic web and includes nonlinear effects in both Eulerian and Lagrangian frameworks, also corresponds to the Burgers equation of hydrodynamics. Focusing on the one-dimensional case with power-law initial conditions, we obtain exact results for Eulerian and Lagrangian propagators. We find that Eulerian propagators can be expressed in terms of the one-point velocity probability distribution and show a strong decay at late times and high wavenumbers, interpreted as a "sweeping effect" but not a genuine damping of small-scale structures. By contrast, Lagrangian propagators can be written in terms of the shock mass function -- which would correspond to the halo mass function in cosmology -- and saturate to a constant value at late times. Moreover, they show a power-law dependence on scale or wavenumber which depends on the initial power-spectrum index and is directly related to the low-mass tail of the shock mass function. These results strongly suggest that Lagrangian propagators are much more sensitive probes of nonlinear structures in the underlying density field and of relaxation processes than their Eulerian counterparts.
Lagrangian-Eulerian Methods for Uniqueness in Hydrodynamic Systems  [PDF]
Peter Constantin
Mathematics , 2014,
Abstract: We present a Lagrangian-Eulerian strategy for proving uniqueness and local existence of solutions in path spaces of limited smoothness for a class of incompressible hydrodynamic models including Oldroyd-B type complex fluid models and zero magnetic resistivity magneto-hydrodynamics equations.
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