A Contribution to the Outflow Boundary Conditions for Navier-Stokes Time-Splitting Methods
We present in this paper a numerical scheme for incompressible Navier-Stokes equations with open boundary conditions, in the framework of the pressure and velocity correction schemes. In Poux et al. (J Comput Phys 230:4011–4027, 2011), the authors present
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Abstract We present in this paper a numerical scheme for incompressible Navier-Stokes equations with open boundary conditions, in the framework of the pressure and velocity correction schemes. In Poux et al. (J Comput Phys 230:4011– 4027, 2011), the authors presented an almost second-order accurate version of the open boundary condition with a pressure-correction scheme in finite volume framework. This paper proposes an extension of this method in spectral element method framework for both pressure- and velocity-correction schemes. A new way to enforce this type of boundary condition is proposed and provides a pressure and velocity convergence rate in space and time higher than with the present state of the art. We illustrate this result by computing some numerical tests.
1 Introduction A difficulty in obtaining the numerical solution of the incompressible Navier-Stokes equations, lies in the Stokes stage and specifically in the determination of the pressure field which will ensure a solenoidal velocity field. Several approaches are possible. We can for instance consider exact methods as the Uzawa [1] and augmented lagrangian [5] ones. In complex geometries or three dimensional methods, theses techniques are inappropriate since their computational time costs are very high. An alternative consists in decoupling the pressure from the velocity by means of a time splitting scheme. A large number of theoretical and numerical E. Ahusborde () University of Pau, LMAP UMR 5142 CNRS, PAU Cedex, France e-mail: [email protected] M. Azaïez S. Glockner A. Poux University of Bordeaux, IPB-I2M UMR 5295, Bordeaux, France e-mail: [email protected]; [email protected]; [email protected] M. Azaïez et al. (eds.), Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2012, Lecture Notes in Computational Science and Engineering 95, DOI 10.1007/978-3-319-01601-6__5, © Springer International Publishing Switzerland 2014
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studies have been published that discuss the accuracy and the stability properties of such approaches. The most popular methods are pressure-correction schemes. They were first introduced by Chorin-Temam [2,18], and improved by Goda (the standard incremental scheme) in [6], and later by Timmermans in [19] (the rotational incremental scheme). They require the solution of two sub-steps: the pressure is treated explicitly in the first one, and is corrected in the second one by projecting the predicted velocity onto an ad-hoc space. A less studied alternative technique known as the velocity-correction scheme, developed by Orszag et al. in [15], Karniadakis et al. in [11], Leriche et al. in [12] and more recently by Guermond et al. in [10], consists in switching the two sub-steps. In [17] and [7], the authors proved the reliability of such approaches from the stability and the convergence rate points of view. A series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in the r
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