ترغب بنشر مسار تعليمي؟ اضغط هنا

Convergence of a Vector Penalty Projection Scheme for the Navier-Stokes Equations with moving body

81   0   0.0 ( 0 )
 نشر من قبل Adrien Doradoux
 تاريخ النشر 2016
  مجال البحث
والبحث باللغة English
 تأليف Vincent Bruneau




اسأل ChatGPT حول البحث

In this paper, we analyse a Vector Penalty Projection Scheme (see [1]) to treat the displacement of a moving body in incompressible viscous flows in the case where the interaction of the fluid on the body can be neglected. The presence of the obstacle inside the computational domain is treated with a penalization method introducing a parameter $eta$. We show the stability of the scheme and that the pressure and velocity converge towards a limit when the penalty parameter $epsilon$, which induces a small divergence and the time step $delta$t tend to zero with a proportionality constraint $epsilon$ = $lambda$$delta$t. Finally, when $eta$ goes to 0, we show that the problem admits a weak limit which is a weak solution of the Navier-Stokes equations with no-sleep condition on the solid boundary. R{e}sum{e} Dans ce travail nous analysons un sch{e}ma de projection vectorielle (voir [1]) pour traiter le d{e}placement dun corps solide dans un fluide visqueux incompressible dans le cas o` u linteraction du fluide sur le solide est n{e}gligeable. La pr{e}sence de lobstacle dans le domaine solide est mod{e}lis{e}e par une m{e}thode de p{e}nalisation. Nous montrons la stabilit{e} du sch{e}ma et la convergence des variables vitesse-pression vers une limite quand le param etre $epsilon$ qui assure une faible divergence et le pas de temps $delta$t tendent vers 0 avec une contrainte de proportionalit{e} $epsilon$ = $lambda$$delta$t. Finalement nous montrons que leprob{`i} eme converge au sens faible vers une solution des equations de Navier-Stokes avec une condition aux limites de non glissement sur lafront{`i} ere immerg{e}e quand le param etre de p{e}nalisation $eta$ tend vers 0.



قيم البحث

اقرأ أيضاً

We present a projection-based framework for solving a thermodynamically-consistent Cahn-Hilliard Navier-Stokes system that models two-phase flows. In this work we extend the fully implicit method presented in Khanwale et al. [{it A fully-coupled fram ework for solving Cahn-Hilliard Navier-Stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes.}, arXiv:2009.06628 (2020)], to a block iterative hybrid method. We use a projection-based semi-implicit time discretization for the Navier-Stokes and a fully-implicit time discretization for the Cahn-Hilliard equation. We use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) formulation. Pressure is decoupled using a projection step, which results in two linear positive semi-definite systems for velocity and pressure, instead of the saddle point system of a pressure-stabilized method. All the linear systems are solved using an efficient and scalable algebraic multigrid (AMG) method. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. The overall approach allows the use of relatively large time steps with much faster time-to-solve. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise and Rayleigh-Taylor instability.
The paper extends a stabilized fictitious domain finite element method initially developed for the Stokes problem to the incompressible Navier-Stokes equations coupled with a moving solid. This method presents the advantage to predict an optimal appr oximation of the normal stress tensor at the interface. The dynamics of the solid is governed by the Newtons laws and the interface between the fluid and the structure is materialized by a level-set which cuts the elements of the mesh. An algorithm is proposed in order to treat the time evolution of the geometry and numerical results are presented on a classical benchmark of the motion of a disk falling in a channel.
This paper is based on a formulation of the Navier-Stokes equations developed by P. Constantin and the first author (texttt{arxiv:math.PR/0511067}, to appear), where the velocity field of a viscous incompressible fluid is written as the expected valu e of a stochastic process. In this paper, we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with $frac{1}{N}$ times the sum over these $N$ copies. (We remark that our formulation requires one to keep track of $N$ stochastic flows of diffeomorphisms, and not just the motion of $N$ particles.) We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space $holderspace{1}{alpha}$ which consists of differentiable functions whose first derivative is $alpha$ Holder continuous (see Section ref{sGexist} for the precise definition). Further, we show that as $N to infty$ the system converges to the solution of Navier-Stokes equations on any finite interval $[0,T]$. However for fixed $N$, we prove that this system retains roughly $O(frac{1}{N})$ times its original energy as $t to infty$. Hence the limit $N to infty$ and $Tto infty$ do not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as $t to infty$ explicitly.
In two dimensions, we propose and analyze an a posteriori error estimator for finite element approximations of the stationary Navier Stokes equations with singular sources on Lipschitz, but not necessarily convex, polygonal domains. Under a smallness assumption on the continuous and discrete solutions, we prove that the devised error estimator is reliable and locally efficient. We illustrate the theory with numerical examples.
134 - Hajer Bahouri 2013
We prove a weak stability result for the three-dimensional homogeneous incompressible Navier-Stokes system. More precisely, we investigate the following problem : if a sequence $(u_{0, n})_{nin N}$ of initial data, bounded in some scaling invariant s pace, converges weakly to an initial data $u_0$ which generates a global regular solution, does $u_{0, n}$ generate a global regular solution ? A positive answer in general to this question would imply global regularity for any data, through the following examples $u_{0,n} = n vf_0(ncdot)$ or $u_{0,n} = vf_0(cdot-x_n)$ with $|x_n|to infty$. We therefore introduce a new concept of weak convergence (rescaled weak convergence) under which we are able to give a positive answer. The proof relies on profile decompositions in anisotropic spaces and their propagation by the Navier-Stokes equations.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا