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ARTICLE

An Improved Unsplit and Convolutional Perfectly Matched Layer Absorbing Technique for the Navier-Stokes Equations Using Cut-Off Frequency Shift

Roland Martin¹, Carlos Couder-Castaneda¹

1 Université de Pau et des Pays de l’Adour, CNRS and INRIA Magique-3D. Laboratoire de Modélisation et Imagerie en Géosciences UMR CNRS 5212, Avenue de l’Université, 64013 Pau cedex, France. E-mail: roland.martin@univ-pau.fr

Computer Modeling in Engineering & Sciences 2010, 63(1), 47-78. https://doi.org/10.3970/cmes.2010.063.047

Abstract

We develop an unsplit convolutional perfectly matched layer (CPML) technique to absorb efficiently compressible viscous flows and their related supersonic or subsonic regimes at the outer boundary of a distorted computational domain. More particularly subsonic outgoing flows or subsonic wall-boundary layers close to the PML are well absorbed, which is difficult to obtain without creating numerical instabilities over long time periods. This new PML (CPML) introduces the calculation of auxiliary memory variables at each time step and allows an unsplit formulation of the PML. Damping functions involving a high shift in the frequency domain allow a much better absorption of the flow than for cases with no shift or low frequency shifts. The CPML has demonstrated its convenience because the time evolution of damping mechanisms do not need to be split and only the space derivatives of fluxes and primitive variables (velocities and temperature) need to be stored at each time step, reducing by this mean the number of computational arrays used in the numerical code. The results obtained show that CPML can absorb efficiently the out-going subsonic and supersonic fluxes at the outlet condition with very few reflections propagating back into the main domain.
The Navier-Stokes equations are applied in an extremely wide variety of industrial processes and geophysical flow simulations. As an example of interest for the industry, CPML is applied to the particular case of a critical air ejector-diffuser simulation in which the flow propagates along a converging-diverging tube, one main goal being for instance to obtain an efficient tool to model numerically different diffuser designs. In this context we investigate the impact of the PML on an unsteady flow submitted to supersonic expansion at the end of the ejector-diffuser while it remains subsonic in the wall-boundary layer. The numerical integration of the whole system of equations introduces a two-step predictor-corrector time-stepping scheme and a finite difference spatial discretization using a curvilinear coordinates transformation that is adapted to the ejector geometry. In this distorted mesh, the spatial finite difference scheme involves a backward-forward discretization and the CPML is able to deal with the distorted mesh in the direction parallel to the base of the PML layer.

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