2,801 research outputs found
Finite element techniques for the Navier-Stokes equations in the primitive variable formulation and the vorticity stream-function formulation
Finite element procedures for the Navier-Stokes equations in the primitive variable formulation and the vorticity stream-function formulation have been implemented. For both formulations, streamline-upwind/Petrov-Galerkin techniques are used for the discretization of the transport equations. The main problem associated with the vorticity stream-function formulation is the lack of boundary conditions for vorticity at solid surfaces. Here an implicit treatment of the vorticity at no-slip boundaries is incorporated in a predictor-multicorrector time integration scheme. For the primitive variable formulation, mixed finite-element approximations are used. A nine-node element and a four-node + bubble element have been implemented. The latter is shown to exhibit a checkerboard pressure mode and a numerical treatment for this spurious pressure mode is proposed. The two methods are compared from the points of view of simulating internal and external flows and the possibilities of extensions to three dimensions
Development of a fractional-step method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems
A fractional step method is developed for solving the time-dependent three-dimensional incompressible Navier-Stokes equations in generalized coordinate systems. The primitive variable formulation uses the pressure, defined at the center of the computational cell, and the volume fluxes across the faces of the cells as the dependent variables, instead of the Cartesian components of the velocity. This choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell. The governing equations are discretized by finite volumes using a staggered mesh system. The solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a Poisson equation. This procedure, combined with the consistent approximations of the geometric quantities, is done to satisfy the discretized mass conservation equation to machine accuracy, as well as to gain the favorable convergence properties of the Poisson solver. The momentum equations are solved by an approximate factorization method, and a novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two- and three-dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method. Good agreement is obtained in all cases
A low-cost parallel implementation of direct numerical simulation of wall turbulence
A numerical method for the direct numerical simulation of incompressible wall
turbulence in rectangular and cylindrical geometries is presented. The
distinctive feature resides in its design being targeted towards an efficient
distributed-memory parallel computing on commodity hardware. The adopted
discretization is spectral in the two homogeneous directions; fourth-order
accurate, compact finite-difference schemes over a variable-spacing mesh in the
wall-normal direction are key to our parallel implementation. The parallel
algorithm is designed in such a way as to minimize data exchange among the
computing machines, and in particular to avoid taking a global transpose of the
data during the pseudo-spectral evaluation of the non-linear terms. The
computing machines can then be connected to each other through low-cost network
devices. The code is optimized for memory requirements, which can moreover be
subdivided among the computing nodes. The layout of a simple, dedicated and
optimized computing system based on commodity hardware is described. The
performance of the numerical method on this computing system is evaluated and
compared with that of other codes described in the literature, as well as with
that of the same code implementing a commonly employed strategy for the
pseudo-spectral calculation.Comment: To be published in J. Comp. Physic
Numerical Methods for the Stochastic Landau-Lifshitz Navier-Stokes Equations
The Landau-Lifshitz Navier-Stokes (LLNS) equations incorporate thermal
fluctuations into macroscopic hydrodynamics by using stochastic fluxes. This
paper examines explicit Eulerian discretizations of the full LLNS equations.
Several CFD approaches are considered (including MacCormack's two-step
Lax-Wendroff scheme and the Piecewise Parabolic Method) and are found to give
good results (about 10% error) for the variances of momentum and energy
fluctuations. However, neither of these schemes accurately reproduces the
density fluctuations. We introduce a conservative centered scheme with a
third-order Runge-Kutta temporal integrator that does accurately produce
density fluctuations. A variety of numerical tests, including the random walk
of a standing shock wave, are considered and results from the stochastic LLNS
PDE solver are compared with theory, when available, and with molecular
simulations using a Direct Simulation Monte Carlo (DSMC) algorithm
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
On the Accuracy of Explicit Finite-Volume Schemes for Fluctuating Hydrodynamics
This paper describes the development and analysis of finite-volume methods for the LandauâLifshitz NavierâStokes (LLNS) equations and related stochastic partial differential equations in fluid dynamics. The LLNS equations incorporate thermal fluctuations into macroscopic hydrodynamics by the addition of white noise fluxes whose magnitudes are set by a fluctuation-dissipation relation. Originally derived for equilibrium fluctuations, the LLNS equations have also been shown to be accurate for nonequilibrium systems. Previous studies of numerical methods for the LLNS equations focused primarily on measuring variances and correlations computed at equilibrium and for selected nonequilibrium flows. In this paper, we introduce a more systematic approach based on studying discrete equilibrium structure factors for a broad class of explicit linear finite-volume schemes. This new approach provides a better characterization of the accuracy of a spatiotemporal discretization as a function of wavenumber and frequency, allowing us to distinguish between behavior at long wavelengths, where accuracy is a prime concern, and short wavelengths, where stability concerns are of greater importance. We use this analysis to develop a specialized third-order RungeâKutta scheme that minimizes the temporal integration error in the discrete structure factor at long wavelengths for the one-dimensional linearized LLNS equations.Together with a novel method for discretizing the stochastic stress tensor in dimension larger than one, our improved temporal integrator yields a scheme for the three-dimensional equations that satisfies a discrete fluctuation-dissipation balance for small time steps and is also sufficiently accurate even for time steps close to the stability limit
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