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Comptes Rendus Mathématique
Volume 354, n° 11
pages 1124-1131 (novembre 2016)
Doi : 10.1016/j.crma.2016.06.007
Received : 11 May 2016 ;  accepted : 27 June 2016
A kinematic vector penalty–projection method for incompressible flow with variable density
Une méthode cinématique de pénalité–projection vectorielle pour l'écoulement incompressible à densité variable
 

Fig. 1




Fig. 1 : 

Edge-based generalized MAC-type unstructured mesh. Topology of the 3-D primal mesh with vertices, edges, faces and an interface Σ: p ,ρ ,ϕ unknowns located at all vertices a or b and velocity components v t on each edge [a ,b ].


Fig. 2




Fig. 2 : 

Detection of an interface represented by a Lagrangian marker chain on the primal mesh. Left: topology of the primal mesh with vertices, edges, cells and the interface – Right: intersections of the connected marker chain cutting across some primal edges.


Fig. 3




Fig. 3 : 

Second-order space convergence rate for the local curvature of an ellipse with a =1 and b =0.75: error in L 2-norm versus number N of Lagrangian interface markers.


Fig. 4




Fig. 4 : 

Laplace uniform capillary pressure p c =400Pa in a disk droplet of radius R =2.510−3m for a constant surface tension σ =1N/m: the velocity field is zero in both cases with no parasite current. Left: unstructured mesh non-fitted to the interface-markers circle – Right: unstructured mesh fitted to the interface.


Fig. 5




Fig. 5 : 

Dynamics of the air bubble of diameter d =0.01m, mass density ρ g =1.1768kg/m3, dynamic viscosity μ g =1.8510−5 Pas in a vertical cavity of width l =410−2m and height h =10−1m. The bubble rises in the liquid steel with ρ l =104kg/m3, μ l =10−3Pas and σ =1.5N/m. Left: pressure field p [−9235,0]Pa (p =0 at the bottom left) at time t =0.05s – Center: vertical velocity field v z [−0.48,1.55]m/s and streamlines at t =0.05s – Right: Some bubble positions and shapes during time and vertical velocity field v z at final time t =0.2s.

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