599 research outputs found
Second Order Fully Discrete Energy Stable Methods on Staggered Grids for Hydrodynamic Phase Field Models of Binary Viscous Fluids
We present second order, fully discrete, energy stable methods on spatially staggered grids for a hydrodynamic phase field model of binary viscous fluid mixtures in a confined geometry subject to both physical and periodic boundary conditions. We apply the energy quadratization strategy to develop a linear-implicit scheme. We then extend it to a decoupled, linear scheme by introducing an intermediate velocity term so that the phase variable, velocity field, and pressure can be solved sequentially. The two new, fully discrete linear schemes are then shown to be unconditionally energy stable, and the linear systems resulting from the schemes are proved uniquely solvable. Rates of convergence of the two linear schemes in both space and time are verified numerically. The decoupled scheme tends to introduce excessive dissipation compared to the coupled one. The coupled scheme is then used to simulate fluid drops of one fluid in the matrix of another fluid as well as mixing dynamics of binary polymeric, viscous solutions. The numerical results in mixing dynamics reveals the dramatic difference between the morphology in the simulations obtained using the two different boundary conditions (physical vs. periodic), demonstrating the importance of using proper boundary conditions in fluid dynamics simulations
A Second Order Fully-discrete Linear Energy Stable Scheme for a Binary Compressible Viscous Fluid Model
We present a linear, second order fully discrete numerical scheme on a
staggered grid for a thermodynamically consistent hydrodynamic phase field
model of binary compressible fluid flow mixtures derived from the generalized
Onsager Principle. The hydrodynamic model not only possesses the variational
structure, but also warrants the mass, linear momentum conservation as well as
energy dissipation. We first reformulate the model in an equivalent form using
the energy quadratization method and then discretize the reformulated model to
obtain a semi-discrete partial differential equation system using the
Crank-Nicolson method in time. The numerical scheme so derived preserves the
mass conservation and energy dissipation law at the semi-discrete level. Then,
we discretize the semi-discrete PDE system on a staggered grid in space to
arrive at a fully discrete scheme using the 2nd order finite difference method,
which respects a discrete energy dissipation law. We prove the unique
solvability of the linear system resulting from the fully discrete scheme. Mesh
refinements and two numerical examples on phase separation due to the spinodal
decomposition in two polymeric fluids and interface evolution in the gas-liquid
mixture are presented to show the convergence property and the usefulness of
the new scheme in applications
Entropy-Production-Rate-Preserving Algorithms for Thermodynamically Consistent Nonisothermal Models of Incompressible Binary Fluids
We derive a thermodynamically consistent, non-isothermal, hydrodynamic model
for incompressible binary fluids following the generalized Onsager principle
and Boussinesq approximation. This model preserves not only the volume of each
fluid phase but also the positive entropy production rate under
thermodynamically consistent boundary conditions. Guided by the thermodynamical
consistency of the model, a set of second order structure-preserving numerical
algorithms are devised to solve the governing partial differential equations
along with consistent boundary conditions in the model, which preserve the
entropy production rate as well as the volume of each fluid phase at the
discrete level. Several numerical simulations are carried out using an
efficient adaptive time-stepping strategy based on one of the
structure-preserving schemes to simulate the Rayleigh-B\'{e}nard convection in
the binary fluid and interfacial dynamics between two immiscible fluids under
competing effects of the temperature gradient, gravity, and interfacial forces.
Roll cell patterns and thermally induced mixing of binary fluids are observed
in a rectangular region with insulated lateral boundaries and vertical ones
with imposed temperature difference. Long time simulations of interfacial
dynamics are performed demonstrating robust results of new structure-preserving
schemes
Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier–Stokes–Cahn–Hilliard system:Primitive variable and projection-type schemes
In this paper we describe two fully mass conservative, energy stable, finite
difference methods on a staggered grid for the quasi-incompressible
Navier-Stokes-Cahn-Hilliard (q-NSCH) system governing a binary incompressible
fluid flow with variable density and viscosity. Both methods, namely the
primitive method (finite difference method in the primitive variable
formulation) and the projection method (finite difference method in a
projection-type formulation), are so designed that the mass of the binary fluid
is preserved, and the energy of the system equations is always non-increasing
in time at the fully discrete level. We also present an efficient, practical
nonlinear multigrid method - comprised of a standard FAS method for the
Cahn-Hilliard equation, and a method based on the Vanka-type smoothing strategy
for the Navier-Stokes equation - for solving these equations. We test the
scheme in the context of Capillary Waves, rising droplets and Rayleigh-Taylor
instability. Quantitative comparisons are made with existing analytical
solutions or previous numerical results that validate the accuracy of our
numerical schemes. Moreover, in all cases, mass of the single component and the
binary fluid was conserved up to 10 to -8 and energy decreases in time
Staggered Schemes for Fluctuating Hydrodynamics
We develop numerical schemes for solving the isothermal compressible and
incompressible equations of fluctuating hydrodynamics on a grid with staggered
momenta. We develop a second-order accurate spatial discretization of the
diffusive, advective and stochastic fluxes that satisfies a discrete
fluctuation-dissipation balance, and construct temporal discretizations that
are at least second-order accurate in time deterministically and in a weak
sense. Specifically, the methods reproduce the correct equilibrium covariances
of the fluctuating fields to third (compressible) and second (incompressible)
order in the time step, as we verify numerically. We apply our techniques to
model recent experimental measurements of giant fluctuations in diffusively
mixing fluids in a micro-gravity environment [A. Vailati et. al., Nature
Communications 2:290, 2011]. Numerical results for the static spectrum of
non-equilibrium concentration fluctuations are in excellent agreement between
the compressible and incompressible simulations, and in good agreement with
experimental results for all measured wavenumbers.Comment: Submitted. See also arXiv:0906.242
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