Application of discrete mechanics model to jump conditions in two-phase flows

2021

Physics of Fluids

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The proposal for a new formulation of the Navier-Stokes equations is based on a Helmholtz-Hodge decomposition where all the terms corresponding to the physical phenomena are written as the sum of a divergence-free term and another curl-free term. These transformations are founded on the bases of discrete mechanics, an alternative approach to the mechanics of continuous media, where conservation of the acceleration on a segment replaces that of the momentum on a volume. The equation of motion thus becomes a law of conservation of total mechanical energy per volume unit where the conservation of mass is no longer necessarily an additional law. The new formulation of the Navier-Stokes equations recovers the properties of the discrete approach without altering those of its initial form; the solutions of the classical form are also those of the proposed formulation. Writing inertial terms in two components resulting from the Helmholtz-Hodge decomposition gives the equation of motion new properties when differential operators are applied to it directly.

Section: Introductionsupporting

confidence: 57%

On a reformulation of Navier–Stokes equations based on Helmholtz–Hodge decomposition

2021

Physics of Fluids

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“…The numerical methodology implemented to simulate the various flows or FSI problems is very close to the mimetic methods initiated by Shaskhov [12] for the diffusion equation; since then, they have been applied successfully to different fields of physics, mechanics, electromagnetism, etc. This method is formally very close to the ideas retained for the discrete physical model itself; the connection between the two has been highlighted recently [11]. This methodology is of the ready-to-use type; indeed, no discretization is necessary to transform the discrete vector equation into an algebraic system of rank n e .…”

Section: Numerical Methodologymentioning

confidence: 77%

“…The discrete formulation already presented elsewhere [8,9,11] is summarized to give its main characteristics. It is based on established principles: (i) the principle of equivalence between gravitational and inertial effects; (ii) the Galilean principle of velocity relativity; (iii) the equivalence between energy and mass resulting from special relativity; and (iv) the Helmholtz-Hodge decomposition of a vector into one divergence-free component and another curl-free.…”

Section: One-dimensional Frameworkmentioning

confidence: 99%

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A Monolithic Approach of Fluid–Structure Interaction by Discrete Mechanics

Vincent

2021

Fluids

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The unification of the laws of fluid and solid mechanics is achieved on the basis of the concepts of discrete mechanics and the principles of equivalence and relativity, but also the Helmholtz–Hodge decomposition where a vector is written as the sum of divergence-free and curl-free components. The derived equation of motion translates the conservation of acceleration over a segment, that of the intrinsic acceleration of the material medium and the sum of the accelerations applied to it. The scalar and vector potentials of the acceleration, which are the compression and shear energies, give the discrete equation of motion the role of conservation law for total mechanical energy. Velocity and displacement are obtained using an incremental time process from acceleration. After a description of the main stages of the derivation of the equation of motion, unique for the fluid and the solid, the cases of couplings in simple shear and uniaxial compression of two media, fluid and solid, make it possible to show the role of discrete operators and to find the theoretical results. The application of the formulation is then extended to a classical validation case in fluid–structure interaction.

“…The article [7] refers mainly to numerical simulations performed with the discrete model for two-phase flows. The existence of a local reference frame where accelerations and velocities are expressed facilitates the treatment of jump conditions at interfaces.…”

Section: Originality Of the Workmentioning

confidence: 99%

An alternative to the concept of continuous medium

2021

Acta Mech

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Discrete mechanics proposes an alternative formulation of the equations of mechanics where the Navier-Stokes and Navier-Lamé equations become approximations of the equation of discrete motion. It unifies the fields of fluid and solid mechanics by extending the fields of application of these equations to all space and time scales. This article presents the essential differences induced by the abandonment of the notion of continuous medium and global frame of reference. The results of the mechanics of continuous medium validated by fluid and solid observations are not questioned. The concept of continuous medium is not invalidated, the discrete formulation proposed simply widens the spectrum of the applications of the classical equations.The discrete equation of motion introduces several important modifications, in particular the fundamental law of the dynamics on an element of volume becomes a law of conservation of the accelerations on an edge. The acceleration considered as an absolute quantity is written as a sum of two components, one soledoidal the other irrotational according to a local orthogonal Helmholtz-Hodge decomposition. The mass is abandoned and replaced by the compression and rotation energies represented by the scalar and vectorial potentials of the acceleration. The equation of motion and all the physical parameters are expressed only with two fundamental units, those of length and time. The essential differences between the two approaches are listed and some of them are discussed in depth. This is particularly the case with the known paradoxes of the Navier-Stokes equation or the importance of inertia for the Navier-Lamé equation.