High-order discontinuous Galerkin computation of axisymmetric transonic flows in safety relief valves F. Bassi a , F. Cecchi b , N. Franchina a,⇑ , S. Rebay c , M. Savini a a Università degli Studi di Bergamo, Dipartimento di Ingegneria Industriale, viale Marconi 5 – 24044 Dalmine (BG), Italy b Tai Milano Spa, via E. Petrella 21 – 20124 Milano (MI), Italy c Università degli Studi di Brescia, Dipartimento di Ingegneria Meccanica e Industriale, via Branze 38 – 25123 Brescia (BS), Italy article info Article history: Received 7 April 2011 Received in revised form 24 May 2011 Accepted 25 May 2011 Available online 13 June 2011 Keywords: Transonic axisymmetric flow RANS equations Discontinuous Galerkin High-order accurate discretization Safety relief valves abstract This paper presents a discontinuous Galerkin (DG) discretization of the compressible RANS and k–x turbulence model equations for two-dimensional axisymmetric flows. The developed code has been applied to investigate the transonic flow in safety relief valves. This new DG implementation has evolved from the DG method presented in [1]. An ‘‘exact’’ Riemann solver is used to compute the interface numerical inviscid flux while the viscous flux discterization relies on the BRMPS scheme [2,3]. Control of oscillations of high-order solutions around shocks is obtained by means of a shock-capturing technique developed and assessed within the EU ADIGMA project [4]. The code has been applied to compute the flow in a spring loaded safety valve at several back pressures and different disk lifts. The predicted device flow capacity and the pressure inside its bonnet have been checked against experimental data. The CFD simulations allow to clarify the complex flow patterns occur- ring and to explain the measured trends. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction In this work the DG discretized RANS and k–x equations are used to investigate the fluid flow through spring loaded safety valves. Safety valves are direct-acting pressure relief devices widely used in industrial equipment dealing with fluids under pressure such as pneumatic systems, gas pressure vessels, and steam boilers. These devices are required to work reliably within narrow ranges of pressure by allowing the fluid to flow out of the enclosure when the pressure happens to exceed the design lim- it and shutting off the flow when the pressure gets back to normal. Gas flow through relief valves is in general quite complex due to highly non-uniform flow conditions, flow separation, embedded supersonic regions and shock waves. In addition, flow patterns change dramatically under different opening conditions and the discharge efficiency is strongly affected by small back-pressure changes. The design, sizing and testing of safety valves is therefore a challenging task, see [5]. Additional effects, like real gas behav- iour and small deviations from a perfect geometrical similarity, can further complicate matters and raise the cost of an accurate experimental characterization of a family of safety valves. In this context, CFD provides a systematic approach to a de- tailed analysis of the flow through safety valves. In addition, a more comprehensive picture of the complex flow features provided by the numerical investigation can be very helpful to develop new designs of such devices. The approach of using numerical simulations for the characterization of safety valves has already been pursued at a profit, as reported, e.g., in [6]. However, there is still room for improvement of physical models and numerical discretization methods of CFD codes. In recent years several high-order methods have been emerging as practical tools to go beyond the formal second-order accuracy of standard finite volume discretizations of PDEs on general unstruc- tured grids, (see e.g. [7–14]). The DG method, in particular, has been gaining popularity as one of the most promising approaches to the accurate and robust numerical solution of ever more com- plex physical models and has attracted great efforts of many re- search groups into its development. The DG discretization is based on polynomial approximations which are discontinuous between elements. Like continuous finite element methods, DG methods can increase the accuracy raising the degree of polynomial approximation. On the other hand, the discontinuous approximation between elements allows to use upwind discretizations of interface fluxes, like in high-resolution finite volume methods. The result is that DG methods are accurate, flexible and robust at the same time. They allow arbitrary unstruc- tured geometries and easy control of accuracy without compromis- ing simulation stability. The majority of DG spatial operators presented in the literature have very compact support, which is a useful feature for implicit time integration schemes and for the parallelization of DG codes. Finally, an accurate treatment of 0045-7930/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2011.05.015 ⇑ Corresponding author. E-mail address: nicoletta.franchina@unibg.it (N. Franchina). Computers & Fluids 49 (2011) 203–213 Contents lists available at ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid