43,656 research outputs found
A numerical method for junctions in networks of shallow-water channels
There is growing interest in developing mathematical models and appropriate
numerical methods for problems involving networks formed by, essentially,
one-dimensional (1D) domains joined by junctions. Examples include hyperbolic
equations in networks of gas tubes, water channels and vessel networks for
blood and lymph in the human circulatory system. A key point in designing
numerical methods for such applications is the treatment of junctions, i.e.
points at which two or more 1D domains converge and where the flow exhibits
multidimensional behaviour. This paper focuses on the design of methods for
networks of water channels. Our methods adopt the finite volume approach to
make full use of the two-dimensional shallow water equations on the true
physical domain, locally at junctions, while solving the usual one-dimensional
shallow water equations away from the junctions. In addition to mass
conservation, our methods enforce conservation of momentum at junctions; the
latter seems to be the missing element in methods currently available. Apart
from simplicity and robustness, the salient feature of the proposed methods is
their ability to successfully deal with transcritical and supercritical flows
at junctions, a property not enjoyed by existing published methodologies.
Systematic assessment of the proposed methods for a variety of flow
configurations is carried out. The methods are directly applicable to other
systems, provided the multidimensional versions of the 1D equations are
available
Diffusion-Based Coarse Graining in Hybrid Continuum-Discrete Solvers: Theoretical Formulation and A Priori Tests
Coarse graining is an important ingredient in many multi-scale
continuum-discrete solvers such as CFD--DEM (computational fluid
dynamics--discrete element method) solvers for dense particle-laden flows.
Although CFD--DEM solvers have become a mature technique that is widely used in
multiphase flow research and industrial flow simulations, a flexible and
easy-to-implement coarse graining algorithm that can work with CFD solvers of
arbitrary meshes is still lacking. In this work, we proposed a new coarse
graining algorithm for continuum--discrete solvers for dense particle-laden
flows based on solving a transient diffusion equation. Via theoretical analysis
we demonstrated that the proposed method is equivalent to the statistical
kernel method with a Gaussian kernel, but the current method is much more
straightforward to implement in CFD--DEM solvers. \textit{A priori} numerical
tests were performed to obtain the solid volume fraction fields based on given
particle distributions, the results obtained by using the proposed algorithm
were compared with those from other coarse graining methods in the literature
(e.g., the particle centroid method, the divided particle volume method, and
the two-grid formulation). The numerical tests demonstrated that the proposed
coarse graining procedure based on solving diffusion equations is theoretically
sound, easy to implement and parallelize in general CFD solvers, and has
improved mesh-convergence characteristics compared with existing coarse
graining methods. The diffusion-based coarse graining method has been
implemented into a CFD--DEM solver, the results of which are presented in a
separate work (R. Sun and H. Xiao, Diffusion-based coarse graining in hybrid
continuum-discrete solvers: Application in CFD-DEM solvers for particle laden
flows)
Hydrodynamics of Suspensions of Passive and Active Rigid Particles: A Rigid Multiblob Approach
We develop a rigid multiblob method for numerically solving the mobility
problem for suspensions of passive and active rigid particles of complex shape
in Stokes flow in unconfined, partially confined, and fully confined
geometries. As in a number of existing methods, we discretize rigid bodies
using a collection of minimally-resolved spherical blobs constrained to move as
a rigid body, to arrive at a potentially large linear system of equations for
the unknown Lagrange multipliers and rigid-body motions. Here we develop a
block-diagonal preconditioner for this linear system and show that a standard
Krylov solver converges in a modest number of iterations that is essentially
independent of the number of particles. For unbounded suspensions and
suspensions sedimented against a single no-slip boundary, we rely on existing
analytical expressions for the Rotne-Prager tensor combined with a fast
multipole method or a direct summation on a Graphical Processing Unit to obtain
an simple yet efficient and scalable implementation. For fully confined
domains, such as periodic suspensions or suspensions confined in slit and
square channels, we extend a recently-developed rigid-body immersed boundary
method to suspensions of freely-moving passive or active rigid particles at
zero Reynolds number. We demonstrate that the iterative solver for the coupled
fluid and rigid body equations converges in a bounded number of iterations
regardless of the system size. We optimize a number of parameters in the
iterative solvers and apply our method to a variety of benchmark problems to
carefully assess the accuracy of the rigid multiblob approach as a function of
the resolution. We also model the dynamics of colloidal particles studied in
recent experiments, such as passive boomerangs in a slit channel, as well as a
pair of non-Brownian active nanorods sedimented against a wall.Comment: Under revision in CAMCOS, Nov 201
Finite volume approach for the instationary Cosserat rod model describing the spinning of viscous jets
The spinning of slender viscous jets can be described asymptotically by
one-dimensional models that consist of systems of partial and ordinary
differential equations. Whereas the well-established string models possess only
solutions for certain choices of parameters and set-ups, the more sophisticated
rod model that can be considered as -regularized string is generally
applicable. But containing the slenderness ratio explicitely in the
equations complicates the numerical treatment. In this paper we present the
first instationary simulations of a rod in a rotational spinning process for
arbitrary parameter ranges with free and fixed jet end, for which the hitherto
investigations longed. So we close an existing gap in literature. The numerics
is based on a finite volume approach with mixed central, up- and down-winded
differences, the time integration is performed by stiff accurate Radau methods
A flow-pattern map for phase separation using the Navier-Stokes Cahn-Hilliard model
We use the Navier-Stokes-Cahn-Hilliard model equations to simulate phase
separation with flow. We study coarsening - the growth of extended domains
wherein the binary mixture phase separates into its component parts. The
coarsening is characterized by two competing effects: flow, and the
Cahn-Hilliard diffusion term, which drives the phase separation. Based on
extensive two-dimensional direct numerical simulations, we construct a
flow-pattern map outlining the relative strength of these effects in different
parts of the parameter space. The map reveals large regions of parameter space
where a standard theory applies, and where the domains grow algebraically in
time. However, there are significant parts of the parameter space where the
standard theory does not apply. In one region, corresponding to low values of
viscosity and diffusion, the coarsening is accelerated compared to the standard
theory. Previous studies involving Stokes flow report on this phenomenon; we
complete the picture by demonstrating that this anomalous regime occurs not
only for Stokes flow, but also, for flows dominated by inertia. In a second
region, corresponding to arbitrary viscosities and high Cahn-Hilliard
diffusion, the diffusion overwhelms the hydrodynamics altogether, and the
latter can effectively be ignored, in contrast to the prediction of the
standard scaling theory. Based on further high-resolution simulations in three
dimensions, we find that broadly speaking, the above description holds there
also, although the formation of the anomalous domains in the
low-viscosity-low-diffusion part of the parameter space is delayed in three
dimensions compared to two.Comment: 17 pages, 13 figure
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