2,795 research outputs found
An adaptive preconditioner for steady incompressible flows
This paper describes an adaptive preconditioner for numerical continuation of
incompressible Navier--Stokes flows. The preconditioner maps the identity (no
preconditioner) to the Stokes preconditioner (preconditioning by Laplacian)
through a continuous parameter and is built on a first order Euler
time-discretization scheme. The preconditioner is tested onto two fluid
configurations: three-dimensional doubly diffusive convection and a reduced
model of shear flows. In the former case, Stokes preconditioning works but a
mixed preconditioner is preferred. In the latter case, the system of equation
is split and solved simultaneously using two different preconditioners, one of
which is parameter dependent. Due to the nature of these applications, this
preconditioner is expected to help a wide range of studies
Collective phase description of oscillatory convection
We formulate a theory for the collective phase description of oscillatory
convection in Hele-Shaw cells. It enables us to describe the dynamics of the
oscillatory convection by a single degree of freedom which we call the
collective phase. The theory can be considered as a phase reduction method for
limit-cycle solutions in infinite-dimensional dynamical systems, namely, stable
time-periodic solutions to partial differential equations, representing the
oscillatory convection. We derive the phase sensitivity function, which
quantifies the phase response of the oscillatory convection to weak
perturbations applied at each spatial point, and analyze the phase
synchronization between two weakly coupled Hele-Shaw cells exhibiting
oscillatory convection on the basis of the derived phase equations.Comment: 16 pages, 4 figures, to appear in Chao
Dynamic Stability of the 3D Axi-symmetric Navier-Stokes Equations with Swirl
In this paper, we study the dynamic stability of the 3D axisymmetric
Navier-Stokes Equations with swirl. To this purpose, we propose a new
one-dimensional (1D) model which approximates the Navier-Stokes equations along
the symmetry axis. An important property of this 1D model is that one can
construct from its solutions a family of exact solutions of the 3D
Navier-Stokes equations. The nonlinear structure of the 1D model has some very
interesting properties. On one hand, it can lead to tremendous dynamic growth
of the solution within a short time. On the other hand, it has a surprising
dynamic depletion mechanism that prevents the solution from blowing up in
finite time. By exploiting this special nonlinear structure, we prove the
global regularity of the 3D Navier-Stokes equations for a family of initial
data, whose solutions can lead to large dynamic growth, but yet have global
smooth solutions
Koopman analysis of the long-term evolution in a turbulent convection cell
We analyse the long-time evolution of the three-dimensional flow in a closed
cubic turbulent Rayleigh-B\'{e}nard convection cell via a Koopman eigenfunction
analysis. A data-driven basis derived from diffusion kernels known in machine
learning is employed here to represent a regularized generator of the unitary
Koopman group in the sense of a Galerkin approximation. The resulting Koopman
eigenfunctions can be grouped into subsets in accordance with the discrete
symmetries in a cubic box. In particular, a projection of the velocity field
onto the first group of eigenfunctions reveals the four stable large-scale
circulation (LSC) states in the convection cell. We recapture the preferential
circulation rolls in diagonal corners and the short-term switching through roll
states parallel to the side faces which have also been seen in other
simulations and experiments. The diagonal macroscopic flow states can last as
long as a thousand convective free-fall time units. In addition, we find that
specific pairs of Koopman eigenfunctions in the secondary subset obey enhanced
oscillatory fluctuations for particular stable diagonal states of the LSC. The
corresponding velocity field structures, such as corner vortices and swirls in
the midplane, are also discussed via spatiotemporal reconstructions.Comment: 32 pages, 9 figures, article in press at Journal of Fluid Mechanic
Order-of-magnitude speedup for steady states and traveling waves via Stokes preconditioning in Channelflow and Openpipeflow
Steady states and traveling waves play a fundamental role in understanding
hydrodynamic problems. Even when unstable, these states provide the
bifurcation-theoretic explanation for the origin of the observed states. In
turbulent wall-bounded shear flows, these states have been hypothesized to be
saddle points organizing the trajectories within a chaotic attractor. These
states must be computed with Newton's method or one of its generalizations,
since time-integration cannot converge to unstable equilibria. The bottleneck
is the solution of linear systems involving the Jacobian of the Navier-Stokes
or Boussinesq equations. Originally such computations were carried out by
constructing and directly inverting the Jacobian, but this is unfeasible for
the matrices arising from three-dimensional hydrodynamic configurations in
large domains. A popular method is to seek states that are invariant under
numerical time integration. Surprisingly, equilibria may also be found by
seeking flows that are invariant under a single very large Backwards-Euler
Forwards-Euler timestep. We show that this method, called Stokes
preconditioning, is 10 to 50 times faster at computing steady states in plane
Couette flow and traveling waves in pipe flow. Moreover, it can be carried out
using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes
to these popular spectral codes. We explain the convergence rate as a function
of the integration period and Reynolds number by computing the full spectra of
the operators corresponding to the Jacobians of both methods.Comment: in Computational Modelling of Bifurcations and Instabilities in Fluid
Dynamics, ed. Alexander Gelfgat (Springer, 2018
Phase description of oscillatory convection with a spatially translational mode
We formulate a theory for the phase description of oscillatory convection in
a cylindrical Hele-Shaw cell that is laterally periodic. This system possesses
spatial translational symmetry in the lateral direction owing to the
cylindrical shape as well as temporal translational symmetry. Oscillatory
convection in this system is described by a limit-torus solution that possesses
two phase modes; one is a spatial phase and the other is a temporal phase. The
spatial and temporal phases indicate the position and oscillation of the
convection, respectively. The theory developed in this paper can be considered
as a phase reduction method for limit-torus solutions in infinite-dimensional
dynamical systems, namely, limit-torus solutions to partial differential
equations representing oscillatory convection with a spatially translational
mode. We derive the phase sensitivity functions for spatial and temporal
phases; these functions quantify the phase responses of the oscillatory
convection to weak perturbations applied at each spatial point. Using the phase
sensitivity functions, we characterize the spatiotemporal phase responses of
oscillatory convection to weak spatial stimuli and analyze the spatiotemporal
phase synchronization between weakly coupled systems of oscillatory convection.Comment: 35 pages, 14 figures. Generalizes the phase description method
developed in arXiv:1110.112
Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach
Five types of blow-up patterns that can occur for the 4th-order semilinear
parabolic equation of reaction-diffusion type
u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1,
\quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For
the semilinear heat equation , various blow-up patterns
were under scrutiny since 1980s, while the case of higher-order diffusion was
studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure
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