1,343 research outputs found
A CutFEM method for two-phase flow problems
In this article, we present a cut finite element method for two-phase
Navier-Stokes flows. The main feature of the method is the formulation of a
unified continuous interior penalty stabilisation approach for, on the one
hand, stabilising advection and the pressure-velocity coupling and, on the
other hand, stabilising the cut region. The accuracy of the algorithm is
enhanced by the development of extended fictitious domains to guarantee a well
defined velocity from previous time steps in the current geometry. Finally, the
robustness of the moving-interface algorithm is further improved by the
introduction of a curvature smoothing technique that reduces spurious
velocities. The algorithm is shown to perform remarkably well for low capillary
number flows, and is a first step towards flexible and robust CutFEM algorithms
for the simulation of microfluidic devices
Numerical analysis of a stabilized finite element approximation for the three-field linearized viscoelastic fluid problem using arbitrary interpolations
The original publication is available at www.esaimm2an.org.In this paper we present the numerical analysis of a three-field stabilized finite element formulation recently proposed to approximate viscoelastic flows. The three-field viscoelastic fluid flow problem may suffer from two types of numerical instabilities: on the one hand we have the two inf-sup conditions related to the mixed nature problem and, on the other, the convective nature of the momentum and constitutive equations may produce global and local oscillations in the numerical approximation. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the decomposition into their finite element component and a subscale, which is approximated properly to yield a stable formulation. The analyzed problem corresponds to a linearized version of the Navier-Stokes/Oldroyd-B case where the advection velocity of the momentum equation and the non-linear terms in the constitutive equation are treated using a fixed point strategy for the velocity and the velocity gradient. The proposed method permits the resolution of the problem using arbitrary interpolations for all the unknowns. We describe some important ingredients related to the design of the formulation and present the results of its numerical analysis. It is shown that the formulation is stable and optimally convergent for small Weissenberg numbers, independently of the interpolation used.Peer ReviewedPostprint (author's final draft
Three-dimensional viscous steady streaming in a rectangular channel past a cylinder
We consider viscous steady streaming induced by oscillatory flow past a
cylinder between two plates, where the cylinder's axis is normal to the plates.
While this phenomenon was first studied in the 1930s, it has received renewed
interest recently for possible applications in particle manipulations and
non-Newtonian flows. The flow is driven at the ends of the channel by the
boundary condition which is a series solution of the oscillating flow problem
in a rectangular channel in the absence of a cylinder. We use a combination of
Fourier series and an asymptotic expansion to study the confinement effects for
steady-streaming. The Fourier series in time naturally simplifies to a finite
series. In contrast, it is necessary to truncate the Fourier series in z, which
is in the direction of the axis of the cylinder, to solve numerically. The
successive equations for the Fourier coefficients resulting from the asymptotic
expansion are then solved numerically using finite element methods. We use our
model to evaluate how steady streaming depends on the domain width and distance
from the cylinder to the outer walls, including the possible breaking of the
four-fold symmetry due to the domain shape. We utilize the tangential
steady-streaming velocity along the radial chord at an angle of pi/4 to analyze
our solutions over an extensive range of oscillating frequencies and multiple
levels in the z-direction. Finally, higher-order solutions are computed and an
asymptotic correction to steady streaming is included.Comment: 25 pages, 14 figure
Least squares based finite element formulations and their applications in fluid mechanics
In this research, least-squares based finite element formulations and their applications
in fluid mechanics are presented. Least-squares formulations offer several computational
and theoretical advantages for Newtonian as well as non-Newtonian fluid flows. Most
notably, these formulations circumvent the inf-sup condition of Ladyzhenskaya-Babuska-
Brezzi (LBB) such that the choice of approximating space is not subject to any compatibility
condition. Also, the resulting coefficient matrix is symmetric and positive-definite. It
has been observed that pressure and velocities are not strongly coupled in traditional leastsquares
based finite element formulations. Penalty based least-squares formulations that
fix the pressure-velocity coupling problem are proposed, implemented in a computational
scheme, and evaluated in this study. The continuity equation is treated as a constraint on
the velocity field and the constraint is enforced using the penalty method. These penalty
based formulations produce accurate results for even low penalty parameters (in the range
of 10-50 penalty parameter). A stress based least-squares formulation is also being proposed
to couple pressure and velocities. Stress components are introduced as independent
variables to make the system first order. The continuity equation is eliminated from the
system with suitable modifications. Least-squares formulations are also developed for viscoelastic
flows and moving boundary flows. All the formulations developed in this study
are tested using several benchmark problems. All of the finite element models developed
in this study performed well in all cases.
A method to exploit orthogonality of modal bases to avoid numerical integration and have a fast computation is also developed during this study. The entries of the coefficient
matrix are calculated analytically. The properties of Jacobi polynomials are used and most
of the entries of the coefficient matrix are recast so that they can be evaluated analytically
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