313 research outputs found
A semi-implicit hybrid finite volume / finite element scheme for all Mach number flows on staggered unstructured meshes
In this paper a new hybrid semi-implicit finite volume / finite element
(FV/FE) scheme is presented for the numerical solution of the compressible
Euler and Navier-Stokes equations at all Mach numbers on unstructured staggered
meshes in two and three space dimensions. The chosen grid arrangement consists
of a primal simplex mesh composed of triangles or tetrahedra, and an edge-based
/ face-based staggered dual mesh. The governing equations are discretized in
conservation form. The nonlinear convective terms of the equations, as well as
the viscous stress tensor and the heat flux, are discretized on the dual mesh
at the aid of an explicit local ADER finite volume scheme, while the implicit
pressure terms are discretized at the aid of a continuous
finite element method on the nodes of the primal mesh. In the zero Mach number
limit, the new scheme automatically reduces to the hybrid FV/FE approach
forwarded in \cite{BFTVC17} for the incompressible Navier-Stokes equations. As
such, the method is asymptotically consistent with the incompressible limit of
the governing equations and can therefore be applied to flows at all Mach
numbers. Due to the chosen semi-implicit discretization, the CFL restriction on
the time step is only based on the magnitude of the flow velocity and not on
the sound speed, hence the method is computationally efficient at low Mach
numbers. In the chosen discretization, the only unknown is the scalar pressure
field at the new time step. Furthermore, the resulting pressure system is
symmetric and positive definite and can therefore be very efficiently solved
with a matrix-free conjugate gradient method.
In order to assess the capabilities of the new scheme, we show computational
results for a large set of benchmark problems that range from the quasi
incompressible low Mach number regime to compressible flows with shock waves
An Arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations
We present a novel second-order semi-implicit hybrid finite volume / finite
element (FV/FE) scheme for the numerical solution of the incompressible and
weakly compressible Navier-Stokes equations on moving unstructured meshes using
an Arbitrary-Lagrangian-Eulerian (ALE) formulation. The scheme is based on a
suitable splitting of the governing PDE into subsystems and employs staggered
grids, where the pressure is defined on the primal simplex mesh, while the
velocity and the remaining flow quantities are defined on an edge-based
staggered dual mesh. The key idea of the scheme is to discretize the nonlinear
convective and viscous terms using an explicit FV scheme that employs the
space-time divergence form of the governing equations on moving space-time
control volumes. For the convective terms, an ALE extension of the Ducros flux
on moving meshes is introduced, which is kinetic energy preserving and stable
in the energy norm when adding suitable numerical dissipation terms. Finally,
the pressure equation of the Navier-Stokes system is solved on the new mesh
configuration using a continuous FE method, with Lagrange
elements.
The ALE hybrid FV/FE method is applied to several incompressible test
problems ranging from non-hydrostatic free surface flows over a rising bubble
to flows over an oscillating cylinder and an oscillating ellipse. Via the
simulation of a circular explosion problem on a moving mesh, we show that the
scheme applied to the weakly compressible Navier-Stokes equations is able to
capture weak shock waves, rarefactions and moving contact discontinuities. We
show that our method is particularly efficient for the simulation of weakly
compressible flows in the low Mach number limit, compared to a fully explicit
ALE schem
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