205 research outputs found
Large eddy simulation of two-dimensional isotropic turbulence
Large eddy simulation (LES) of forced, homogeneous, isotropic,
two-dimensional (2D) turbulence in the energy transfer subrange is the subject
of this paper. A difficulty specific to this LES and its subgrid scale (SGS)
representation is in that the energy source resides in high wave number modes
excluded in simulations. Therefore, the SGS scheme in this case should assume
the function of the energy source. In addition, the controversial requirements
to ensure direct enstrophy transfer and inverse energy transfer make the
conventional scheme of positive and dissipative eddy viscosity inapplicable to
2D turbulence. It is shown that these requirements can be reconciled by
utilizing a two-parametric viscosity introduced by Kraichnan (1976) that
accounts for the energy and enstrophy exchange between the resolved and subgrid
scale modes in a way consistent with the dynamics of 2D turbulence; it is
negative on large scales, positive on small scales and complies with the basic
conservation laws for energy and enstrophy. Different implementations of the
two-parametric viscosity for LES of 2D turbulence were considered. It was found
that if kept constant, this viscosity results in unstable numerical scheme.
Therefore, another scheme was advanced in which the two-parametric viscosity
depends on the flow field. In addition, to extend simulations beyond the limits
imposed by the finiteness of computational domain, a large scale drag was
introduced. The resulting LES exhibited remarkable and fast convergence to the
solution obtained in the preceding direct numerical simulations (DNS) by
Chekhlov et al. (1994) while the flow parameters were in good agreement with
their DNS counterparts. Also, good agreement with the Kolmogorov theory was
found. This LES could be continued virtually indefinitely. Then, a simplifiedComment: 34 pages plain tex + 18 postscript figures separately, uses auxilary
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An embedded boundary integral solver for the stokes equations
We present a new method for the solution of the Stokes equations. Our goal is to develop a robust and scalable methodology for two and three dimensional, moving-boundary, flow simulations. Our method is based on Anita Mayo\u27s method for the Poisson\u27s equation: âThe Fast Solution of Poisson\u27s and the Biharmonic Equations on Irregular Regionsâ, SIAM J. Num. Anal., 21 (1984), pp. 285â 299. We embed the domain in a rectangular domain, for which fast solvers are available, and we impose the boundary conditions as interface (jump) conditions on the velocities and tractions. We use an indirect boundary integral formulation for the homogeneous Stokes equations to compute the jumps. The resulting integral equations are discretized by Nystrom\u27s method. The rectangular domain problem is discretized by finite elements for a velocity-pressure formulation with equal order interpolation bilinear elements (Q1-Q1). Stabilization is used to circumvent the inf-sup condition for the pressure space. For the integral equations, fast matrix vector multiplications are achieved via a NlogN algorithm based on a block representation of the discrete integral operator, combined with (kernel independent) singular value decomposition to sparsify low-rank blocks. Our code is built on top of PETSc, an MPI based parallel linear algebra library. The regular grid solver is a Krylov method (Conjugate Residuals) combined with an optimal two-level Schwartz-preconditioner. For the integral equation we use GMRES. We have tested our algorithm on several numerical examples and we have observed optimal convergence rates
A scalar auxiliary variable unfitted FEM for the surface Cahn-Hilliard equation
The paper studies a scalar auxiliary variable (SAV) method to solve the
Cahn-Hilliard equation with degenerate mobility posed on a smooth closed
surface {\Gamma}. The SAV formulation is combined with adaptive time stepping
and a geometrically unfitted trace finite element method (TraceFEM), which
embeds {\Gamma} in R3. The stability is proven to hold in an appropriate sense
for both first- and second-order in time variants of the method. The
performance of our SAV method is illustrated through a series of numerical
experiments, which include systematic comparison with a stabilized
semi-explicit method.Comment: 23 pages, 12 figure
Des avancĂ©es dans la rĂ©duction de modĂšle de type PGD pour les EDPs dâordre Ă©levĂ©, le traitement des gĂ©omĂ©tries complexes et la rĂ©solution des Ă©quations de Navier-Stokes instationnaires
The main purpose of this work is to describe a simulation method for the use of aPGD-based Model reduction Method (MOR) for solving high order partial differentialequations. First, the PGD method is used for solving fourth order PDEs and thealgorithm is illustrated on a lid-driven cavity problem. Transformations of coordinatesfor changing the complex physical domain into the simple computational domain arealso studied, which lead to extend the spatial PGD method to complex geometrydomains. Some numerical examples for different kinds of domain are treated toillustrate the potentialities of this methodology.Finally, a PGD-based space-time separation is introduced to solve the unsteadyStokes or Navier-Stokes equations. This decomposition makes use of common tem-poral modes for both velocity and pressure, which lead to velocity spatial modessatisfying individually the incompressibility condition. The adaptation and imple-mentation of a PGD approach into a general purpose finite volume framework isdescribed and illustrated on several analytic and academic flow examples. A largereduction of the computational cost is observed on most of the treated examples.Lâobjectif principal de ce travail est de proposer une nouvelle approche de simulationbasĂ©e sur une MĂ©thode de rĂ©duction du modĂšle (MOR) utilisant une dĂ©compositionPGD. Dans ce travail, cette approche est dâabord utilisĂ©e pour rĂ©soudre des Ă©quationsaux dĂ©rivĂ©es partielles dâordre Ă©levĂ© avec un exemple numĂ©rique pour les Ă©quations auxdĂ©rivĂ©es partielles du quatriĂšme ordre sur le problĂšme de la cavitĂ© entraĂźnĂ©e. Ensuiteun changement de coordonnĂ©es pour transformer le domaine physique complexe enun domaine de calcul simple est Ă©tudiĂ©, ce qui conduit Ă Ă©tendre la mĂ©thode PGDau traitement de certaines gĂ©omĂ©tries complexes. Divers exemples numĂ©riques pourdiffĂ©rents types de domaines gĂ©omĂ©triques sont ainsi traitĂ©s avec lâapproche PGD.Enfin, une sĂ©paration espace-temps est proposĂ©e pour rĂ©soudre les Ă©quations deNavier-Stokes instationnaires Ă lâaide dâune approche PGD. Cette dĂ©compositionest basĂ©e sur le choix de modes temporels communs pour la vitesse et la pression,ce qui conduit Ă une dĂ©composition basĂ©e sur des modes spatiaux satisfaisant in-dividuellement la condition dâincompressibilitĂ©. Lâadaptation dâune formulationvolumes finis Ă cette dĂ©composition PGD est prĂ©sentĂ©e et validĂ©e sur de premiersexemples analytiques ou acadĂ©miques pour les Ă©quations de Stokes ou Navier-Stokesinstationnaires. Une importante rĂ©duction des temps calculs est observĂ©e sur lespremiers exemples traitĂ©s
Effect of high order interpolation in the stability and efficiency of the time-integration process in vorticity-velocity CFD algorithms
The numerical solution of the incompressible Navier-Stokes Equations offers an effective alternative to the experimental analysis of Fluid-Structure interaction i.e. dynamical coupling between a fluid and a solid which otherwise is very complex, time consuming and very expensive. To have a method which can accurately model these types of mechanical systems by numerical solutions becomes a great option, since these advantages are even more obvious when considering huge structures like bridges, high rise buildings, or even wind turbine blades with diameters as large as 200 meters. The modeling of such processes, however, involves complex multiphysics problems along with complex geometries. This thesis focuses on a novel vorticity-velocity formulation called the KLE to solve the incompressible Navier-stokes equations for such FSI problems. This scheme allows for the implementation of robust adaptive ODE time integration schemes and thus allows us to tackle the various multiphysics problems as separate modules.
The current algorithm for KLE employs a structured or unstructured mesh for spatial discretization and it allows the use of a self-adaptive or fixed time step ODE solver while dealing with unsteady problems. This research deals with the analysis of the effects of the Courant-Friedrichs-Lewy (CFL) condition for KLE when applied to unsteady Stokeâs problem. The objective is to conduct a numerical analysis for stability and, hence, for convergence. Our results confirmthat the time step ât is constrained by the CFL-like condition ât †const. hα, where h denotes the variable that represents spatial discretization
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