10,056 research outputs found
A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales
In this work it is shown how the immersed boundary method of (Peskin2002) for
modeling flexible structures immersed in a fluid can be extended to include
thermal fluctuations. A stochastic numerical method is proposed which deals
with stiffness in the system of equations by handling systematically the
statistical contributions of the fastest dynamics of the fluid and immersed
structures over long time steps. An important feature of the numerical method
is that time steps can be taken in which the degrees of freedom of the fluid
are completely underresolved, partially resolved, or fully resolved while
retaining a good level of accuracy. Error estimates in each of these regimes
are given for the method. A number of theoretical and numerical checks are
furthermore performed to assess its physical fidelity. For a conservative
force, the method is found to simulate particles with the correct Boltzmann
equilibrium statistics. It is shown in three dimensions that the diffusion of
immersed particles simulated with the method has the correct scaling in the
physical parameters. The method is also shown to reproduce a well-known
hydrodynamic effect of a Brownian particle in which the velocity
autocorrelation function exhibits an algebraic tau^(-3/2) decay for long times.
A few preliminary results are presented for more complex systems which
demonstrate some potential application areas of the method.Comment: 52 pages, 11 figures, published in journal of computational physic
Systematic Stochastic Reduction of Inertial Fluid-Structure Interactions subject to Thermal Fluctuations
We present analysis for the reduction of an inertial description of
fluid-structure interactions subject to thermal fluctuations. We show how the
viscous coupling between the immersed structures and the fluid can be
simplified in the regime where this coupling becomes increasingly strong. Many
descriptions in fluid mechanics and in the formulation of computational methods
account for fluid-structure interactions through viscous drag terms to transfer
momentum from the fluid to immersed structures. In the inertial regime, this
coupling often introduces a prohibitively small time-scale into the temporal
dynamics of the fluid-structure system. This is further exacerbated in the
presence of thermal fluctuations. We discuss here a systematic reduction
technique for the full inertial equations to obtain a simplified description
where this coupling term is eliminated. This approach also accounts for the
effective stochastic equations for the fluid-structure dynamics. The analysis
is based on use of the Infinitesmal Generator of the SPDEs and a singular
perturbation analysis of the Backward Kolomogorov PDEs. We also discuss the
physical motivations and interpretation of the obtained reduced description of
the fluid-structure system. Working paper currently under revision. Please
report any comments or issues to [email protected]: 19 pages, 1 figure. arXiv admin note: substantial text overlap with
arXiv:1009.564
Spatially Adaptive Stochastic Methods for Fluid-Structure Interactions Subject to Thermal Fluctuations in Domains with Complex Geometries
We develop stochastic mixed finite element methods for spatially adaptive
simulations of fluid-structure interactions when subject to thermal
fluctuations. To account for thermal fluctuations, we introduce a discrete
fluctuation-dissipation balance condition to develop compatible stochastic
driving fields for our discretization. We perform analysis that shows our
condition is sufficient to ensure results consistent with statistical
mechanics. We show the Gibbs-Boltzmann distribution is invariant under the
stochastic dynamics of the semi-discretization. To generate efficiently the
required stochastic driving fields, we develop a Gibbs sampler based on
iterative methods and multigrid to generate fields with computational
complexity. Our stochastic methods provide an alternative to uniform
discretizations on periodic domains that rely on Fast Fourier Transforms. To
demonstrate in practice our stochastic computational methods, we investigate
within channel geometries having internal obstacles and no-slip walls how the
mobility/diffusivity of particles depends on location. Our methods extend the
applicability of fluctuating hydrodynamic approaches by allowing for spatially
adaptive resolution of the mechanics and for domains that have complex
geometries relevant in many applications
Inertial Coupling Method for particles in an incompressible fluctuating fluid
We develop an inertial coupling method for modeling the dynamics of
point-like 'blob' particles immersed in an incompressible fluid, generalizing
previous work for compressible fluids. The coupling consistently includes
excess (positive or negative) inertia of the particles relative to the
displaced fluid, and accounts for thermal fluctuations in the fluid momentum
equation. The coupling between the fluid and the blob is based on a no-slip
constraint equating the particle velocity with the local average of the fluid
velocity, and conserves momentum and energy. We demonstrate that the
formulation obeys a fluctuation-dissipation balance, owing to the
non-dissipative nature of the no-slip coupling. We develop a spatio-temporal
discretization that preserves, as best as possible, these properties of the
continuum formulation. In the spatial discretization, the local averaging and
spreading operations are accomplished using compact kernels commonly used in
immersed boundary methods. We find that the special properties of these kernels
make the discrete blob a particle with surprisingly physically-consistent
volume, mass, and hydrodynamic properties. We develop a second-order
semi-implicit temporal integrator that maintains discrete
fluctuation-dissipation balance, and is not limited in stability by viscosity.
Furthermore, the temporal scheme requires only constant-coefficient Poisson and
Helmholtz linear solvers, enabling a very efficient and simple FFT-based
implementation on GPUs. We numerically investigate the performance of the
method on several standard test problems...Comment: Contains a number of corrections and an additional Figure 7 (and
associated discussion) relative to published versio
The Stokes-Einstein Relation at Moderate Schmidt Number
The Stokes-Einstein relation for the self-diffusion coefficient of a
spherical particle suspended in an incompressible fluid is an asymptotic result
in the limit of large Schmidt number, that is, when momentum diffuses much
faster than the particle. When the Schmidt number is moderate, which happens in
most particle methods for hydrodynamics, deviations from the Stokes-Einstein
prediction are expected. We study these corrections computationally using a
recently-developed minimally-resolved method for coupling particles to an
incompressible fluctuating fluid in both two and three dimensions. We find that
for moderate Schmidt numbers the diffusion coefficient is reduced relative to
the Stokes-Einstein prediction by an amount inversely proportional to the
Schmidt number in both two and three dimensions. We find, however, that the
Einstein formula is obeyed at all Schmidt numbers, consistent with linear
response theory. The numerical data is in good agreement with an approximate
self-consistent theory, which can be used to estimate finite-Schmidt number
corrections in a variety of methods. Our results indicate that the corrections
to the Stokes-Einstein formula come primarily from the fact that the particle
itself diffuses together with the momentum. Our study separates effects coming
from corrections to no-slip hydrodynamics from those of finite separation of
time scales, allowing for a better understanding of widely observed deviations
from the Stokes-Einstein prediction in particle methods such as molecular
dynamics.Comment: Submitte
The Stochastic Dynamics of Rectangular and V-shaped Atomic Force Microscope Cantilevers in a Viscous Fluid and Near a Solid Boundary
Using a thermodynamic approach based upon the fluctuation-dissipation theorem
we quantify the stochastic dynamics of rectangular and V-shaped microscale
cantilevers immersed in a viscous fluid. We show that the stochastic cantilever
dynamics as measured by the displacement of the cantilever tip or by the angle
of the cantilever tip are different. We trace this difference to contributions
from the higher modes of the cantilever. We find that contributions from the
higher modes are significant in the dynamics of the cantilever tip-angle. For
the V-shaped cantilever the resulting flow field is three-dimensional and
complex in contrast to what is found for a long and slender rectangular
cantilever. Despite this complexity the stochastic dynamics can be predicted
using a two-dimensional model with an appropriately chosen length scale. We
also quantify the increased fluid dissipation that results as a V-shaped
cantilever is brought near a solid planar boundary.Comment: 10 pages, 15 images, corrected equation (8
Stochastic Eulerian Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations
We present approaches for the study of fluid-structure interactions subject
to thermal fluctuations. A mixed mechanical description is utilized combining
Eulerian and Lagrangian reference frames. We establish general conditions for
operators coupling these descriptions. Stochastic driving fields for the
formalism are derived using principles from statistical mechanics. The
stochastic differential equations of the formalism are found to exhibit
significant stiffness in some physical regimes. To cope with this issue, we
derive reduced stochastic differential equations for several physical regimes.
We also present stochastic numerical methods for each regime to approximate the
fluid-structure dynamics and to generate efficiently the required stochastic
driving fields. To validate the methodology in each regime, we perform analysis
of the invariant probability distribution of the stochastic dynamics of the
fluid-structure formalism. We compare this analysis with results from
statistical mechanics. To further demonstrate the applicability of the
methodology, we perform computational studies for spherical particles having
translational and rotational degrees of freedom. We compare these studies with
results from fluid mechanics. The presented approach provides for
fluid-structure systems a set of rather general computational methods for
treating consistently structure mechanics, hydrodynamic coupling, and thermal
fluctuations.Comment: 24 pages, 3 figure
Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators
We present an estimator-based control design procedure for flow control,
using reduced-order models of the governing equations, linearized about a
possibly unstable steady state. The reduced models are obtained using an
approximate balanced truncation method that retains the most controllable and
observable modes of the system. The original method is valid only for stable
linear systems, and we present an extension to unstable linear systems. The
dynamics on the unstable subspace are represented by projecting the original
equations onto the global unstable eigenmodes, assumed to be small in number. A
snapshot-based algorithm is developed, using approximate balanced truncation,
for obtaining a reduced-order model of the dynamics on the stable subspace. The
proposed algorithm is used to study feedback control of 2-D flow over a flat
plate at a low Reynolds number and at large angles of attack, where the natural
flow is vortex shedding, though there also exists an unstable steady state. For
control design, we derive reduced-order models valid in the neighborhood of
this unstable steady state. The actuation is modeled as a localized body force
near the leading edge of the flat plate, and the sensors are two velocity
measurements in the near-wake of the plate. A reduced-order Kalman filter is
developed based on these models and is shown to accurately reconstruct the flow
field from the sensor measurements, and the resulting estimator-based control
is shown to stabilize the unstable steady state. For small perturbations of the
steady state, the model accurately predicts the response of the full
simulation. Furthermore, the resulting controller is even able to suppress the
stable periodic vortex shedding, where the nonlinear effects are strong, thus
implying a large domain of attraction of the stabilized steady state.Comment: 36 pages, 17 figure
The stochastic dynamics of micron and nanoscale elastic cantilevers in fluid: fluctuations from dissipation
The stochastic dynamics of micron and nanoscale cantilevers immersed in a
viscous fluid are quantified. Analytical results are presented for long slender
cantilevers driven by Brownian noise. The spectral density of the noise force
is not assumed to be white and the frequency dependence is determined from the
fluctuation-dissipation theorem. The analytical results are shown to be useful
for the micron scale cantilevers that are commonly used in atomic force
microscopy. A general thermodynamic approach is developed that is valid for
cantilevers of arbitrary geometry as well as for arrays of multiple cantilevers
whose stochastic motion is coupled through the fluid. It is shown that the
fluctuation-dissipation theorem permits the calculation of stochastic
quantities via straightforward deterministic methods. The thermodynamic
approach is used with deterministic finite element numerical simulations to
quantify the autocorrelation and noise spectrum of cantilever fluctuations for
a single micron scale cantilever and the cross-correlations and noise spectra
of fluctuations for an array of two experimentally motivated nanoscale
cantilevers as a function of cantilever separation. The results are used to
quantify the noise reduction possible using correlated measurements with two
closely spaced nanoscale cantilevers.Comment: Submitted to Nanotechnology April 26, 200
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