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Lagrangian vs. Eulerian: An Analysis of Two Solution Methods for Free-Surface Flows and Fluid Solid Interaction Problems
As a step towards addressing a scarcity of references on this topic, we compared the Eulerian and Lagrangian Computational Fluid Dynamics (CFD) approaches for the solution of free-surface and Fluid–Solid Interaction (FSI) problems. The Eulerian approach uses the Finite Element Method (FEM) to spatially discretize the Navier–Stokes equations. The free surface is handled via the volume-of-fluid (VOF) and the level-set (LS) equations; an Immersed Boundary Method (IBM) in conjunction with the Nitsche’s technique were applied to resolve the fluid–solid coupling. For the Lagrangian approach, the smoothed particle hydrodynamics (SPH) method is the meshless discretization technique of choice; no additional equations are needed to handle free-surface or FSI coupling. We compared the two approaches for a flow around cylinder. The dam break test was used to gauge the performance for free-surface flows. Lastly, the two approaches were compared on two FSI problems—one with a floating rigid body dropped into the fluid and one with an elastic gate interacting with the flow. We conclude with a discussion of the robustness, ease of model setup, and versatility of the two approaches. The Eulerian and Lagrangian solvers used in this study are open-source and available in the public domain.
Authors:
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Award ID(s):
Publication Date:
NSF-PAR ID:
10349476
Journal Name:
Fluids
Volume:
6
Issue:
12
Page Range or eLocation-ID:
460
ISSN:
2311-5521
5. We consider the Euler equations for the potential flow of an ideal incompressible fluid of infinite depth with a free surface in two-dimensional geometry. Both gravity and surface tension forces are taken into account. A time-dependent conformal mapping is used which maps the lower complex half-plane of the auxiliary complex variable $w$ into the fluid’s area, with the real line of $w$ mapped into the free fluid’s surface. We reformulate the exact Eulerian dynamics through a non-canonical non-local Hamiltonian structure for a pair of the Hamiltonian variables. These two variables are the imaginary part of the conformal map and the fluid’s velocity potential, both evaluated at the fluid’s free surface. The corresponding Poisson bracket is non-degenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. The new Hamiltonian structure is a generalization of the canonical Hamiltonian structure of Zakharov ( J. Appl. Mech. Tech. Phys. , vol. 9(2), 1968, pp. 190–194) which is valid only for solutions for which the natural surface parametrization is single-valued, i.e. each value of the horizontal coordinate corresponds only to a single point on the free surface. In contrast, the new non-canonical Hamiltonian equationsmore »