542 research outputs found
Angular Momentum preserving cell-centered Lagrangian and Eulerian schemes on arbitrary grids
We address the conservation of angular momentum for cell-centered discretization of compressible fluid dynamics on general grids. We concentrate on the Lagrangian step which is also sufficient for Eulerian discretization using Lagrange+Remap. Starting from the conservative equation of the angular momentum, we show that a standard Riemann solver (a nodal one in our case) can easily be extended to update the new variable. This new variable allows to reconstruct all solid displacements in a cell, and is analogous to a partial Discontinuous Galerkin (DG) discretization. We detail the coupling with a second- order Muscl extension. All numerical tests show the important enhancement of accuracy for rotation problems, and the reduction of mesh imprint for implosion problems. The generalization to axi-symmetric case is detailed
High-Order Unstructured Lagrangian One-Step WENO Finite Volume Schemes for Non-Conservative Hyperbolic Systems: Applications to Compressible Multi-Phase Flows
In this article we present the first better than second order accurate
unstructured Lagrangian-type one-step WENO finite volume scheme for the
solution of hyperbolic partial differential equations with non-conservative
products. The method achieves high order of accuracy in space together with
essentially non-oscillatory behavior using a nonlinear WENO reconstruction
operator on unstructured triangular meshes. High order accuracy in time is
obtained via a local Lagrangian space-time Galerkin predictor method that
evolves the spatial reconstruction polynomials in time within each element. The
final one-step finite volume scheme is derived by integration over a moving
space-time control volume, where the non-conservative products are treated by a
path-conservative approach that defines the jump terms on the element
boundaries. The entire method is formulated as an Arbitrary-Lagrangian-Eulerian
(ALE) method, where the mesh velocity can be chosen independently of the fluid
velocity.
The new scheme is applied to the full seven-equation Baer-Nunziato model of
compressible multi-phase flows in two space dimensions. The use of a Lagrangian
approach allows an excellent resolution of the solid contact and the resolution
of jumps in the volume fraction. The high order of accuracy of the scheme in
space and time is confirmed via a numerical convergence study. Finally, the
proposed method is also applied to a reduced version of the compressible
Baer-Nunziato model for the simulation of free surface water waves in moving
domains. In particular, the phenomenon of sloshing is studied in a moving water
tank and comparisons with experimental data are provided
BETHE-Hydro: An Arbitrary Lagrangian-Eulerian Multi-dimensional Hydrodynamics Code for Astrophysical Simulations
In this paper, we describe a new hydrodynamics code for 1D and 2D
astrophysical simulations, BETHE-hydro, that uses time-dependent, arbitrary,
unstructured grids. The core of the hydrodynamics algorithm is an arbitrary
Lagrangian-Eulerian (ALE) approach, in which the gradient and divergence
operators are made compatible using the support-operator method. We present 1D
and 2D gravity solvers that are finite differenced using the support-operator
technique, and the resulting system of linear equations are solved using the
tridiagonal method for 1D simulations and an iterative multigrid-preconditioned
conjugate-gradient method for 2D simulations. Rotational terms are included for
2D calculations using cylindrical coordinates. We document an incompatibility
between a subcell pressure algorithm to suppress hourglass motions and the
subcell remapping algorithm and present a modified subcell pressure scheme that
avoids this problem. Strengths of this code include a straightforward
structure, enabling simple inclusion of additional physics packages, the
ability to use a general equation of state, and most importantly, the ability
to solve self-gravitating hydrodynamic flows on time-dependent, arbitrary
grids. In what follows, we describe in detail the numerical techniques employed
and, with a large suite of tests, demonstrate that BETHE-hydro finds accurate
solutions with 2-order convergence.Comment: 51 pages in emulateapj, including 25 figures, replace with version
accepted to ApJS, corrected typos and included minor referee's comment
Simulating water-entry/exit problems using Eulerian-Lagrangian and fully-Eulerian fictitious domain methods within the open-source IBAMR library
In this paper we employ two implementations of the fictitious domain (FD)
method to simulate water-entry and water-exit problems and demonstrate their
ability to simulate practical marine engineering problems. In FD methods, the
fluid momentum equation is extended within the solid domain using an additional
body force that constrains the structure velocity to be that of a rigid body.
Using this formulation, a single set of equations is solved over the entire
computational domain. The constraint force is calculated in two distinct ways:
one using an Eulerian-Lagrangian framework of the immersed boundary (IB) method
and another using a fully-Eulerian approach of the Brinkman penalization (BP)
method. Both FSI strategies use the same multiphase flow algorithm that solves
the discrete incompressible Navier-Stokes system in conservative form. A
consistent transport scheme is employed to advect mass and momentum in the
domain, which ensures numerical stability of high density ratio multiphase
flows involved in practical marine engineering applications. Example cases of a
free falling wedge (straight and inclined) and cylinder are simulated, and the
numerical results are compared against benchmark cases in literature.Comment: The current paper builds on arXiv:1901.07892 and re-explains some
parts of it for the reader's convenienc
A conservative coupling algorithm between a compressible flow and a rigid body using an Embedded Boundary method
This paper deals with a new solid-fluid coupling algorithm between a rigid
body and an unsteady compressible fluid flow, using an Embedded Boundary
method. The coupling with a rigid body is a first step towards the coupling
with a Discrete Element method. The flow is computed using a Finite Volume
approach on a Cartesian grid. The expression of numerical fluxes does not
affect the general coupling algorithm and we use a one-step high-order scheme
proposed by Daru and Tenaud [Daru V,Tenaud C., J. Comput. Phys. 2004]. The
Embedded Boundary method is used to integrate the presence of a solid boundary
in the fluid. The coupling algorithm is totally explicit and ensures exact mass
conservation and a balance of momentum and energy between the fluid and the
solid. It is shown that the scheme preserves uniform movement of both fluid and
solid and introduces no numerical boundary roughness. The effciency of the
method is demonstrated on challenging one- and two-dimensional benchmarks
High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs
Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching
Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers
In this paper we use the genuinely multidimensional HLL Riemann solvers
recently developed by Balsara et al. to construct a new class of
computationally efficient high order Lagrangian ADER-WENO one-step ALE finite
volume schemes on unstructured triangular meshes. A nonlinear WENO
reconstruction operator allows the algorithm to achieve high order of accuracy
in space, while high order of accuracy in time is obtained by the use of an
ADER time-stepping technique based on a local space-time Galerkin predictor.
The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the
grid, considering the entire Voronoi neighborhood of each node and allows for
larger time steps than conventional one-dimensional Riemann solvers. The
results produced by the multidimensional Riemann solver are then used twice in
our one-step ALE algorithm: first, as a node solver that assigns a unique
velocity vector to each vertex, in order to preserve the continuity of the
computational mesh; second, as a building block for genuinely multidimensional
numerical flux evaluation that allows the scheme to run with larger time steps
compared to conventional finite volume schemes that use classical
one-dimensional Riemann solvers in normal direction. A rezoning step may be
necessary in order to overcome element overlapping or crossing-over. We apply
the method presented in this article to two systems of hyperbolic conservation
laws, namely the Euler equations of compressible gas dynamics and the equations
of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to
fourth order of accuracy in space and time have been carried out. Several
numerical test problems have been solved to validate the new approach
Multi-Dimensional, Compressible Viscous Flow on a Moving Voronoi Mesh
Numerous formulations of finite volume schemes for the Euler and
Navier-Stokes equations exist, but in the majority of cases they have been
developed for structured and stationary meshes. In many applications, more
flexible mesh geometries that can dynamically adjust to the problem at hand and
move with the flow in a (quasi) Lagrangian fashion would, however, be highly
desirable, as this can allow a significant reduction of advection errors and an
accurate realization of curved and moving boundary conditions. Here we describe
a novel formulation of viscous continuum hydrodynamics that solves the
equations of motion on a Voronoi mesh created by a set of mesh-generating
points. The points can move in an arbitrary manner, but the most natural motion
is that given by the fluid velocity itself, such that the mesh dynamically
adjusts to the flow. Owing to the mathematical properties of the Voronoi
tessellation, pathological mesh-twisting effects are avoided. Our
implementation considers the full Navier-Stokes equations and has been realized
in the AREPO code both in 2D and 3D. We propose a new approach to compute
accurate viscous fluxes for a dynamic Voronoi mesh, and use this to formulate a
finite volume solver of the Navier-Stokes equations. Through a number of test
problems, including circular Couette flow and flow past a cylindrical obstacle,
we show that our new scheme combines good accuracy with geometric flexibility,
and hence promises to be competitive with other highly refined Eulerian
methods. This will in particular allow astrophysical applications of the AREPO
code where physical viscosity is important, such as in the hot plasma in galaxy
clusters, or for viscous accretion disk models.Comment: 26 pages, 21 figures. Submitted to MNRA
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