Variation method based on the interpolation for navier-stokes solutions for transient uncompressible flow

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Authors:

Chengxi Liu, College of Mathematics and Information Sciences, Neijiang Normal University, Neijiang, China

Abstract:

Purpose. Many studies have been devoted to using variational multiscale (VMS) methods to solve the incompressible flows. The analysis differs when applying the so-called first or second fluctuation operator. On the other hand, VMS methods are used to solve unsteady incompressible flows. Error estimates dependent on the reduced Reynolds number are obtained. On the other hand, the error estimates not dependent on the Reynolds number have already been obtained by using SD and CIP methods. Thus, we desire to obtain the same or similar results by using VMS methods.

Methodology. We propose a fully discrete stabilized method for the unsteady NSEs at high Reynolds number. We use Crank-Nicolson difference in time and use the SV elements in space to preserve the incompressibility. The convective effects are stabilized by adding a new projection-based VMS term. The stability and convergence of the approximation solution are proved. The error estimates hold irrespective of the Reynolds number, and hence also for the incompressible Euler equations, provided the exact solution is smooth.

Findings. We prove the stability and convergence of the approximation solution. The error estimates hold irrespective of the Reynolds number, and hence also for the incompressible Euler equations, provided the exact solution is smooth. This method has good stability. It preserves the incompressibility and it has error estimates not dependent on the viscosity.

Originality. In this paper, we propose a new fully discrete VMS method using SV elements for the unsteady Navier-Stokes at high Reynolds number. Incompressibility is preserved by using Scott-Vogelius elements and convective effects are stabilized by adding a new projection-based variational multiscale (VMS) term.

Practical value. Numerical experiments demonstrate that our method is very effective for incompressible flows at high Reynolds number. They also confirm that our method preserves the incompressibility strongly.

References/Список літератури

1. Liu, C., Newman, J. C. and Anderson, W. K., 2015. Petrov–Galerkin Overset Grid Scheme for the Navi er–Stokes Equations with Moving Domains. Aiaa Journal, Vol. 53, No. 11, pp. 1–16.

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8. Chen, G., Feng, M. F. and He, Y. N., 2013. Finite difference streamline diffusion method using nonconforming space for incompressible time-dependent Navier-Stokes equations. Applied Mathematics & Mechanics, Vol. 34, No. 9, pp. 1083–1096.

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Date 2016-11-15 Filesize 1.57 MB Download 319

Tags: Unsteady Navier-Stokes equations at high Reynolds numberScott-Vogelius elementsincompressibilityconvective effectsCrank-Nicolson differencevaritational multiscale method

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