11,762 research outputs found
An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations
We investigate the long tim behavior of the following efficient second order
in time scheme for the 2D Navier-Stokes equation in a periodic box:
\frac{3\omega^{n+1}-4\omega^n+\omega^{n-1}}{2k} +
\nabla^\perp(2\psi^n-\psi^{n-1})\cdot\nabla(2\omega^n-\omega^{n-1}) -
\nu\Delta\omega^{n+1} = f^{n+1}, \quad -\Delta \psi^n = \om^n. The scheme is
a combination of a 2nd order in time backward-differentiation (BDF) and a
special explicit Adams-Bashforth treatment of the advection term. Therefore
only a linear constant coefficient Poisson type problem needs to be solved at
each time step. We prove uniform in time bounds on this scheme in \dL2,
\dH1 and provided that the time-step is sufficiently small.
These time uniform estimates further lead to the convergence of long time
statistics (stationary statistical properties) of the scheme to that of the NSE
itself at vanishing time-step. Fully discrete schemes with either Galerkin
Fourier or collocation Fourier spectral method are also discussed
Stability of Vortex Solutions to an Extended Navier-Stokes System
We study the long-time behavior an extended Navier-Stokes system in
where the incompressibility constraint is relaxed. This is one of several
"reduced models" of Grubb and Solonnikov '89 and was revisited recently (Liu,
Liu, Pego '07) in bounded domains in order to explain the fast convergence of
certain numerical schemes (Johnston, Liu '04). Our first result shows that if
the initial divergence of the fluid velocity is mean zero, then the Oseen
vortex is globally asymptotically stable. This is the same as the Gallay Wayne
'05 result for the standard Navier-Stokes equations. When the initial
divergence is not mean zero, we show that the analogue of the Oseen vortex
exists and is stable under small perturbations. For completeness, we also prove
global well-posedness of the system we study.Comment: 24 pages, 1 figure, updated to add authors' contact information and
to address referee's comment
Some experiences with the viscous-inviscid interaction approach
Methods for simulating compressible viscous flow using the viscid-inviscid interaction approach are described. The formulations presented range from the more familiar full-potential/boundary-layer interaction schemes to a method for coupling Euler/Navier-Stokes and boundary-layer algorithms. An effort is made to describe the advantages and disadvantages of each formulation. Sample results are presented which illustrate the applicability of the methods
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
The instanton method and its numerical implementation in fluid mechanics
A precise characterization of structures occurring in turbulent fluid flows
at high Reynolds numbers is one of the last open problems of classical physics.
In this review we discuss recent developments related to the application of
instanton methods to turbulence. Instantons are saddle point configurations of
the underlying path integrals. They are equivalent to minimizers of the related
Freidlin-Wentzell action and known to be able to characterize rare events in
such systems. While there is an impressive body of work concerning their
analytical description, this review focuses on the question on how to compute
these minimizers numerically. In a short introduction we present the relevant
mathematical and physical background before we discuss the stochastic Burgers
equation in detail. We present algorithms to compute instantons numerically by
an efficient solution of the corresponding Euler-Lagrange equations. A second
focus is the discussion of a recently developed numerical filtering technique
that allows to extract instantons from direct numerical simulations. In the
following we present modifications of the algorithms to make them efficient
when applied to two- or three-dimensional fluid dynamical problems. We
illustrate these ideas using the two-dimensional Burgers equation and the
three-dimensional Navier-Stokes equations
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