Approximation of the hydrostatic Navier–Stokes system for density stratified flows by a multilayer model: Kinetic interpretation and numerical solution

Audusse, Emmanuel; Bristeau, Marie-Odile; Pelanti, Marica; Sainte-Marie, Jacques

doi:10.1016/j.jcp.2011.01.042

Cited by 49 publications

(107 citation statements)

References 42 publications

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“…The vertical velocity can be computed by postprocessing, taking into account the incompressibility of the fluid. The model has been extended by Audusse et al in [7] to variable density flows, and by Sainte-Marie in [12] to non-hydrostatic flows.…”

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confidence: 99%

A Multilayer Method for the Hydrostatic Navier-Stokes Equations: A Particular Weak Solution

2013

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In this work we present a mutilayer approach to the solution of non-stationnary 3D Navier-Stokes equations. We use piecewise smooth weak solutions. We approximate the velocity by a piecewise constant (in z) horizontal velocity and a linear (in z) vertical velocity in each layer, possibly discontinuous across layer interfaces. The multilayer approach is deduced by using the variational formulation and by considering a reduced family of test functions. The procedure naturally provides the mass and momentum interfaces conditions. The mass and momentum conservation across interfaces is formulated via normal flux jump conditions. The jump conditions associated to momentum conservation are formulated by means of an approximation of the vertical derivative of the velocity that appears in the stress tensor. We approximate the multilayer model for hydrostatic pressure, by using a PVM finite volume scheme and we present some numerical tests that show the main advantages of the model: it improves the approximation of the vertical velocity, provides good predictions for viscous effects and simulates re-circulations behind solid obstacles.

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confidence: 99%

A Multilayer Method for the Hydrostatic Navier-Stokes Equations: A Particular Weak Solution

2013

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“…The computations are straightforward., we only detail the way to write the shallow water model with two velocities (1) from the bilayer model (3). Since h 1 = h 2 in (4), the definition of the water depth h in (5) leads to h 1 = h 2 = h/2.…”

Section: Relations With Other Shallow Water Type Modelsmentioning

confidence: 99%

“…the free surface remains a monovaluated function in space. Several generalizations of the model have been proposed for variable density flow [3], for hydrostatic Navier-Stokes equations [5] and for non hydrostatic flows [10]. In this context, the solution of the Riemann problem (1)-(2) appears as an important step in a better understanding of this now widely used family of free surface fluid models.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the Riemann Problem for a Shallow Water Model with Two Velocities

Aguillon¹,

Audusse²,

Godlewski³

et al. 2018

SIAM J. Math. Anal.

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Some shallow water type models describing the vertical profile of the horizontal velocity with several degrees of freedom have been recently proposed. The question addressed in the current work is the hyperbolicity of a shallow water model with two velocities. The model is written in a nonconservative form and the analysis of its eigenstructure shows the possibility that two eigenvalues coincide. A definition of the nonconservative product is given which enables us to analyse the resonance and coalescence of waves. Eventually, we prove the well-posedness of the two dimensional Riemann problem with initial condition constant by half-plane.

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“…Improvement 55 in the flow field resolution for these environmental flows has generally arisen from (i) using 3 higher-order polynomials [23], which is essentially the strategy applied to thin film flows in [17] and by co-workers, or (ii) using a multi-layer approach, e.g. [24,25,26,27]. …”

Section: Introductionmentioning

confidence: 99%

“…When there is significant mixing between 70 layers, or if recirculations are present, an alternative approach with layer boundaries located independent of the flow field (for example layers uniformly distributed through depth) and inter-layer fluxes accounted for by matching conditions between the flows in each layer is needed. This approach has been adopted by, for example, Audusse and co-workers [29,26,27] …”

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confidence: 99%

A multi-layer integral model for locally-heated thin film flow

Kay¹,

Hibberd

Power

2017

Journal of Computational Physics

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Based on an approach used to model environmental flows such as rivers and estuaries, we develop a new multi-layered model for thin liquid film flow on a locally-heated inclined plane. The film is segmented into layers of equal thickness with the velocity and temperature of each governed by a momentum and energy equation integrated across each layer individually. Matching conditions applied between the layers ensure the continuity of down-plane velocity, temperature, stress and heat flux. Variation in surface tension of the liquid with temperature is considered so that local heating induces a surface shear stress which leads to variation in the film height profile (the Marangoni effect). Moderate inertia and heat convection effects are also included.In the absence of Marangoni effects, when the film height is uniform, we test the accuracy of the model by comparing it against a solution of the full heat equation using finite differences. The multi-layer model offers significant improvements over that of a single layer. Notably, with a sufficient number of layers, the solution does not exhibit local regions of negative temperature often predicted using a single-layer model.With Marangoni effects included the film height varies however we find heat convection can mitigate this variation by reducing the surface temperature gradient and hence the surface shear stress. Numerical results corresponding to the flow of water on a vertical plane show that very thin films are dominated by the Marangoni shear stress which can be sufficiently strong to overcome gravity leading to a recirculation in the velocity field. This effect reduces with increasing film thickness and the recirculation eventually disappears. In this case heating is confined entirely to the interior of the film leading to a uniform height profile.

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Approximation of the hydrostatic Navier–Stokes system for density stratified flows by a multilayer model: Kinetic interpretation and numerical solution

Cited by 49 publications

References 42 publications

A Multilayer Method for the Hydrostatic Navier-Stokes Equations: A Particular Weak Solution

A Multilayer Method for the Hydrostatic Navier-Stokes Equations: A Particular Weak Solution

Analysis of the Riemann Problem for a Shallow Water Model with Two Velocities

A multi-layer integral model for locally-heated thin film flow

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