Formulation and analysis of a diffusion-velocity particle model for transport-dispersion equations

Mycek, Paul; Pinon, Grégory; Germain, Grégory; Rivoalen, Elie

doi:10.1007/s40314-014-0200-5

Cited by 14 publications

(11 citation statements)

References 45 publications

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“…An alternative numerical approach for diffusion is possible in the Lagrangian Vortex framework: the Diffusion Velocity Method (DVM), initially proposed by Ogami and Akamatsu [23] and recently analysed by Mycek et al [24] . This last study also offers some perspective on the three-dimensional treatment of diffusion with LES using a DVM approach [24] .…”

Section: Lagrangian Vortex Methods and Treatment Of Diffusionmentioning

confidence: 99%

“…Additionally the method presented here can function together with both of the most common treatments of diffusion in Lagrangian Vortex methods, namely the Particle Strength Exchange (PSE) [20][21][22] and the Diffusion Velocity Method (DVM) [23] . These methods can integrate turbulent diffusion models, such as Large Eddy Simulation [2][3][4]24] , to better represent all turbulent length scales.…”

Section: Introductionmentioning

confidence: 99%

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A stochastic method to account for the ambient turbulence in Lagrangian Vortex computations

Bex

Carlier

Fur

et al. 2020

Applied Mathematical Modelling

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This paper describes a detailed implementation of the Synthetic Eddy Method (SEM) initially presented in Jarrin et al. (2006) applied to the Lagrangian Vortex simulation. While the treatment of turbulent diffusion is already extensively covered in scientific literature, this is one of the first attempts to represent ambient turbulence in a fully Lagrangian framework. This implementation is well suited to the integration of PSE (Particle Strength Exchange) or DVM (Diffusion Velocity Method), often used to account for molecular and turbulent diffusion in Lagrangian simulations. The adaptation and implementation of the SEM into a Lagrangian method using the PSE diffusion model is presented, and the turbulent velocity fields produced by this method are then analysed. In this adaptation, SEM turbulent structures are simply advected, without stretching or diffusion of their own, over the flow domain. This implementation proves its ability to produce turbulent velocity fields in accordance with any desired turbulent flow parameters. As the SEM is a purely mathematical and stochastic model, turbulent spectra and turbulent length scales are also investigated. With the addition of variation in the turbulent structures sizes, a satisfying representation of turbulent spectra is recovered, and a linear relation is obtained between the turbulent structures sizes and the Taylor macroscale. Lastly, the model is applied to the computation of a tidal turbine wake for different ambient turbulence levels, demonstrating the ability of this new implementation to emulate experimentally observed tendencies.

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Section: Lagrangian Vortex Methods and Treatment Of Diffusionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A stochastic method to account for the ambient turbulence in Lagrangian Vortex computations

Bex

Carlier

Fur

et al. 2020

Applied Mathematical Modelling

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“…Recently, Lucchesi et al [25] exploited the regularized particle representation in (5) to construct different methods to approximate the fractional Laplacian (4) and the fractional diffusion flux (2). These included various variants of the so-called particle strength exchange (PSE) algorithms [39][40][41][42][43], a diffusion-velocity method [44], as well as a methodology based on direct (fractional) differentiation [36] of the particle representation in (5). Whereas the various approaches explored in [25] all led to consistent approximations, the resulting schemes exhibited different properties.…”

Section: Particle Approximationmentioning

confidence: 99%

Hierarchical matrix approximations for space-fractional diffusion equations

Boukaram

Lucchesi

Turkiyyah

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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Space fractional diffusion models generally lead to dense discrete matrix operators, which lead to substantial computational challenges when the system size becomes large. For a state of size N , full representation of a fractional diffusion matrix would require O(N 2) memory storage requirement, with a similar estimate for matrix-vector products. In this work, we present H 2 matrix representation and algorithms that are amenable to efficient implementation on GPUs, and that can reduce the cost of storing these operators to O(N) asymptotically. Matrix-vector multiplications can be performed in asymptotically linear time as well. Performance of the algorithms is assessed in light of 2D simulations of space fractional diffusion equation with constant diffusivity. Attention is focused on smooth particle approximation of the governing equations, which lead to discrete operators involving explicit radial kernels. The algorithms are first tested using the fundamental solution of the unforced space fractional diffusion equation in an unbounded domain, and then for the steady, forced, fractional diffusion equation in a bounded domain. Both matrix-inverse and pseudotransient solution approaches are considered in the latter case. Our experiments show that the construction of the fractional diffusion matrix, the matrix-vector multiplication, and the generation of an approximate inverse pre-conditioner all perform very well on a single GPU on 2D problems with N in the range 10 5-10 6. In addition, the tests also showed that, for the entire range of parameters and fractional orders considered, results obtained using the H 2 approximations were in close agreement with results obtained using dense operators, and exhibited the same spatial order of convergence. Overall, the present experiences showed that the H 2 matrix framework promises to provide practical means to handle large-scale space fractional diffusion models in several space dimensions, at a computational cost that is asymptotically similar to the cost of handling classical diffusion equations.

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“…The account of diffusion in Vortex Method can be treated in several manners. Apart from the pioneering method of random walk proposed by Chorin [44], mainly two methods revealed to be sufficiently accurate and efficient, namely the Diffusion Velocity Method (DVM) [45][46][47][48] and the Particle Strength Exchange (PSE) method [16][17][18]. The latter is currently the most commonly used.…”

Section: Appendix a Particles Emissionmentioning

confidence: 99%

Vortex kinematics around a submerged plate under water waves. Part II: Numerical computations

Pinon¹,

Perret²,

Cao

et al. 2017

European Journal of Mechanics - B/Fluids

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This paper presents numerical computations of the flow generated by a horizontal plate immersed in a regular wave field, and associated loads acting on the plate. This numerical work is the continuation of the Poupardin et al. (2012) experimental study. This numerical study is original in the way that the vortical aspects of the flow are not neglected. Therefore, a 2D Lagrangian Vortex method is used as a numerical scheme. These methods are particularly well suited for the computation of unsteady and highly rotational flows in an open domain. The velocity field is decomposed into rotational and potential components, using the Helmholtz decomposition. The rotational part of the velocity is calculated by the Biot-Savart equation using vortex particles. The plate is modelled by a distribution of normal dipoles and the wave field is taken into account by means of a Stokes formulation, which completes the potential part of the velocity. Firstly, the numerical code is validated by means of comparison with the experimental results of Poupardin et al. (2012). In particular, the complex vortical activity and the mean flow velocity field are well reproduced and physically analysed. Secondly, forces acting on the plate are analysed on a wide range of parameters by varying the plate immersions and lengths. In the end, a new scaling is found for the lift forces acting on the plate based on the modified Stokes velocity (i.e. the bottom Stokes velocity for a water depth equals to the plate immersion) and the square of the plate length.

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Formulation and analysis of a diffusion-velocity particle model for transport-dispersion equations

Cited by 14 publications

References 45 publications

A stochastic method to account for the ambient turbulence in Lagrangian Vortex computations

A stochastic method to account for the ambient turbulence in Lagrangian Vortex computations

Hierarchical matrix approximations for space-fractional diffusion equations

Vortex kinematics around a submerged plate under water waves. Part II: Numerical computations

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