236 research outputs found
Application of the Hughes-LIU algorithm to the 2-dimensional heat equation
An implicit explicit algorithm for the solution of transient problems in structural dynamics is described. The method involved dividing the finite elements into implicit and explicit groups while automatically satisfying the conditions. This algorithm is applied to the solution of the linear, transient, two dimensional heat equation subject to an initial condition derived from the soluton of a steady state problem over an L-shaped region made up of a good conductor and an insulating material. Using the IIT/PRIME computer with virtual memory, a FORTRAN computer program code was developed to make accuracy, stability, and cost comparisons among the fully explicit Euler, the Hughes-Liu, and the fully implicit Crank-Nicholson algorithms. The Hughes-Liu claim that the explicit group governs the stability of the entire region while maintaining the unconditional stability of the implicit group is illustrated
Experiments with a Malkus-Lorenz water wheel: Chaos and Synchronization
We describe a simple experimental implementation of the Malkus-Lorenz water
wheel. We demonstrate that both chaotic and periodic behavior is found as wheel
parameters are changed in agreement with predictions from the Lorenz model. We
furthermore show that when the measured angular velocity of our water wheel is
used as an input signal to a computer model implementing the Lorenz equations,
high quality chaos synchronization of the model and the water wheel is
achieved. This indicates that the Lorenz equations provide a good description
of the water wheel dynamics.Comment: 12 pages, 7 figures. The following article has been accepted by the
American Journal of Physics. After it is published, it will be found at
http://scitation.aip.org/ajp
Waves attractors in rotating fluids: a paradigm for ill-posed Cauchy problems
In the limit of low viscosity, we show that the amplitude of the modes of
oscillation of a rotating fluid, namely inertial modes, concentrate along an
attractor formed by a periodic orbit of characteristics of the underlying
hyperbolic Poincar\'e equation. The dynamics of characteristics is used to
elaborate a scenario for the asymptotic behaviour of the eigenmodes and
eigenspectrum in the physically relevant r\'egime of very low viscosities which
are out of reach numerically. This problem offers a canonical ill-posed Cauchy
problem which has applications in other fields.Comment: 4 pages, 5 fi
Principle of Maximum Entropy Applied to Rayleigh-B\'enard Convection
A statistical-mechanical investigation is performed on Rayleigh-B\'enard
convection of a dilute classical gas starting from the Boltzmann equation. We
first present a microscopic derivation of basic hydrodynamic equations and an
expression of entropy appropriate for the convection. This includes an
alternative justification for the Oberbeck-Boussinesq approximation. We then
calculate entropy change through the convective transition choosing mechanical
quantities as independent variables. Above the critical Rayleigh number, the
system is found to evolve from the heat-conducting uniform state towards the
convective roll state with monotonic increase of entropy on the average. Thus,
the principle of maximum entropy proposed for nonequilibrium steady states in a
preceding paper is indeed obeyed in this prototype example. The principle also
provides a natural explanation for the enhancement of the Nusselt number in
convection.Comment: 13 pages, 4 figures; typos corrected; Eq. (66a) corrected to remove a
double counting for ; Figs. 1-4 replace
Tidal instability in a rotating and differentially heated ellipsoidal shell
The stability of a rotating flow in a triaxial ellipsoidal shell with an
imposed temperature difference between inner and outer boundaries is studied
numerically. We demonstrate that (i) a stable temperature field encourages the
tidal instability, (ii) the tidal instability can grow on a convective flow,
which confirms its relevance to geo- and astrophysical contexts and (iii) its
growth rate decreases when the intensity of convection increases. Simple
scaling laws characterizing the evolution of the heat flux based on a
competition between viscous and thermal boundary layers are derived
analytically and verified numerically. Our results confirm that thermal and
tidal effects have to be simultaneously taken into account when studying
geophysical and astrophysical flows
Magnetic field induced by elliiptical instability in a rotating tidally distorded sphere
It is usually believed that the geo-dynamo of the Earth or more generally of
other planets, is created by the convective fluid motions inside their molten
cores. An alternative to this thermal or compositional convection can however
be found in the inertial waves resonances generated by the eventual precession
of these planets or by the possible tidal distorsions of their liquid cores. We
will review in this paper some of our experimental works devoted to the
elliptical instability and present some new results when the experimental fluid
is a liquid metal. We show in particular that an imposed magnetic field is
distorted by the spin- over mode generated by the elliptical instability. In
our experiment, the field is weak (20 Gauss) and the Lorenz force is negligible
compared to the inertial forces, therefore the magnetic field does not modify
the fluid flow and the pure hydrodynamics growth rates of the instability are
recovered through magnetic measurements
Tilt-over mode in a precessing triaxial ellipsoid
The tilt-over mode in a precessing triaxial ellipsoid is studied
theoretically and numerically. Inviscid and viscous analytical models
previously developed for the spheroidal geometry by Poincar\'e [Bull. Astr. 27,
321 (1910)] and Busse [J. Fluid Mech., 33, 739 (1968)] are extended to this
more complex geometry, which corresponds to a tidally deformed spinning
astrophysical body. As confirmed by three-dimensional numerical simulations,
the proposed analytical model provides an accurate description of the
stationary flow in an arbitrary triaxial ellipsoid, until the appearance at
more vigorous forcing of time dependent flows driven by tidal and/or
precessional instabilities.Comment: http://link.aip.org/link/doi/10.1063/1.350435
A systematic numerical study of the tidal instability in a rotating triaxial ellipsoid
The full non-linear evolution of the tidal instability is studied numerically
in an ellipsoidal fluid domain relevant for planetary cores applications. Our
numerical model, based on a finite element method, is first validated by
reproducing some known analytical results. This model is then used to address
open questions that were up to now inaccessible using theoretical and
experimental approaches. Growth rates and mode selection of the instability are
systematically studied as a function of the aspect ratio of the ellipsoid and
as a function of the inclination of the rotation axis compared to the
deformation plane. We also quantify the saturation amplitude of the flow driven
by the instability and calculate the viscous dissipation that it causes. This
tidal dissipation can be of major importance for some geophysical situations
and we thus derive general scaling laws which are applied to typical planetary
cores
Bifurcations in annular electroconvection with an imposed shear
We report an experimental study of the primary bifurcation in
electrically-driven convection in a freely suspended film. A weakly conducting,
submicron thick smectic liquid crystal film was supported by concentric
circular electrodes. It electroconvected when a sufficiently large voltage
was applied between its inner and outer edges. The film could sustain rapid
flows and yet remain strictly two-dimensional. By rotation of the inner
electrode, a circular Couette shear could be independently imposed. The control
parameters were a dimensionless number , analogous to the Rayleigh
number, which is and the Reynolds number of the
azimuthal shear flow. The geometrical and material properties of the film were
characterized by the radius ratio , and a Prandtl-like number . Using measurements of current-voltage characteristics of a large number of
films, we examined the onset of electroconvection over a broad range of
, and . We compared this data quantitatively to
the results of linear stability theory. This could be done with essentially no
adjustable parameters. The current-voltage data above onset were then used to
infer the amplitude of electroconvection in the weakly nonlinear regime by
fitting them to a steady-state amplitude equation of the Landau form. We show
how the primary bifurcation can be tuned between supercritical and subcritical
by changing and .Comment: 17 pages, 12 figures. Submitted to Phys. Rev. E. Minor changes after
refereeing. See also http://mobydick.physics.utoronto.c
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