Implementation of an explicit algebraic Reynolds stress model in an implicit very high-order discontinuous Galerkin solver

In this work we present the main features of an implicit implementation of the explicit algebraic Reynolds stress model (EARSM) of Wallin and Johansson (J Fluid Mech 403:89–132, 2000) in the high-order Discontinuous Galerkin (DG) solver named MIGALE (Bassi et al. (2011) Discontinuous Galerkin for turbulent flows. In: Wang ZJ (ed) Adaptive high-order methods in computational fluid dynamics. Volume 2 of Advances in computational fluid dynamics. World Scientific). Explicit Algebraic Reynolds stress models replace the linear Boussinesq hypothesis by an algebraic approximation of the anisotropy transport equations, resulting in a non-linear constitutive relation for the Reynolds stress tensor in terms of mean flow strain-rate and rate-of-rotation tensors. The EARSM model has been implemented in the existing k-ω model of the DG code MIGALE without any recalibration of the constants and a basic assessment and validation of its near-near wall behaviour has been done on a turbulent flat plate test case (Slater et al. (2000) The NPARC verification and validation archive. ASME Paper 2000-FED-11233, ASME). Consistently with the mean-flow equations, the turbulence model equations have been discretized to a high-order spatial accuracy on hybrid type elements by using hierarchical and orthonormal polynomial basis functions, local to each element and defined in the physical space. Such discretization preserves its accuracy also for highly-stretched elements with curved boundaries as those used within turbulent boundaries layers. For steady-state computations, the time integration of the fully coupled system of governing equations is performed implicitly by means of the linearized backward Euler method where the Jacobian is derived analytically and a pseudo-transient continuation strategy is employed (Bassi et al. (2010) Very high-order accurate discontinuous Galerkin computation of transonic turbulent flows on aeronautical configurations. In: Norbert Kroll, Heribert Bieler, Herman Deconinck, Vincent Couaillier, Harmen van der Ven, and Kaare Sørensen (eds) ADIGMA – A European initiative on the development of adaptive higher-order variational methods for aerospace applications. Volume 113 of Notes on numerical fluid mechanics and multidisciplinary design. Springer, Berlin/Heidelberg, pp 25–38). The capabilities of the present version of the code will be demonstrated by computing an external aerodynamic problem proposed within the EU-funded project IDIHOM (Project IDIHOM (2012) Industrialisation of high-order methods a top-down approach).