Antonio Ghidoni

Very high-order accurate discontinuous Galerkin computation of transonic turbulent flows on aeronautical configurations

This chapter presents high-order DG solutions of the RANS and k-ω turbulence model equations for transonic flows around aeronautical configurations. A directional shock-capturing term, proportional to the inviscid residual, is employed to control oscillations around shocks. Implicit time integration is applied to the fully coupled RANS and k-ω equations. Several high-order DG results of 2D and 3D transonic turbulent test cases proposed within the ADIGMA project demonstrate the capability of the method.

High-order discontinuous Galerkin discretization of transonic turbulent flows

This paper describes an approach for computing transonic turbulent flows by using a high-order discontinuous Galerkin (DG) method. The method solves the RANS and k-w turbulence model equations on hybrid grids consisting of arbitrary sets of elements (triangles and quadrangles in 2D, tetrahedrons, prisms, pyramids and hexahedrons in 3D) with possibly curved faces. Oscillations of high-order solutions around shocks are controlled by means of a viscous-like term, explicitly added to the discretized equations, which acts to damp the growth of higher-order components of the solution. For steady state solutions of turbulent flow computations the DG discretized equations are integrated in time by using the backward Euler method. The code is fully parallelized and relies on METIS for grid partitioning and on PETSc for linear algebra. Numerical results of some test cases proposed within the EU ADIGMA project will demonstrate the capabilities of the method. Copyright

Up to sixth-order accurate A-stable implicit schemes applied to the Discontinuous Galerkin discretized Navier–Stokes equations

Abstract In this paper a high-order implicit multi-step method, known in the literature as Two Implicit Advanced Step-point (TIAS) method, has been implemented in a high-order Discontinuous Galerkin (DG) solver for the unsteady Euler and Navier–Stokes equations. Application of the absolute stability condition to this class of multi-step multi-stage time discretization methods allowed to determine formulae coefficients which ensure A-stability up to order 6. The stability properties of such schemes have been verified by considering linear model problems. The dispersion and dissipation errors introduced by TIAS method have been investigated by looking at the analytical solution of the oscillation equation. The performance of the high-order accurate, both in space and time, TIAS-DG scheme has been evaluated by computing three test cases: an isentropic convecting vortex under two different testing conditions and a laminar vortex shedding behind a circular cylinder. To illustrate the effectiveness and the advantages of the proposed high-order time discretization, the results of the fourth- and sixth-order accurate TIAS schemes have been compared with the results obtained using the standard second-order accurate Backward Differentiation Formula, BDF2, and the five stage fourth-order accurate Strong Stability Preserving Runge–Kutta scheme, SSPRK4.

Optimal Runge–Kutta smoothers for the p-multigrid discontinuous Galerkin solution of the 1D Euler equations

This work presents a family of original Runge–Kutta methods specifically designed to be effective relaxation schemes in the numerical solution of the steady state solution of purely advective problems with a high-order accurate discontinuous Galerkin space discretization and a p-multigrid solution algorithm. The design criterion for the construction of the Runge–Kutta methods here developed is different form the one traditionally used to derive optimal Runge–Kutta smoothers for the h-multigrid algorithm, which are designed in order to provide a uniform damping of the error modes in the high-frequency range only. The method here proposed is instead designed in order to provide a variable amount of damping of the error modes over the entire frequency spectrum. The performance of the proposed schemes is assessed in the solution of the steady state quasi one-dimensional Euler equations for two test cases of increasing difficulty. Some preliminary results showing the performance for multidimensional applications are also presented.

Time Integration in the Discontinuous Galerkin Code MIGALE - Unsteady Problems

This chapter presents recent developments of a high-order Discontinuous Galerkin (DG) method to deal with unsteady simulation of turbulent flows by using high-order implicit time integration schemes. The approaches considered during the IDIHOM project were the Implicit Large Eddy Simulation (ILES), where no explicit subgrid-scale (SGS) model is included and the DG discretization itself acts like a SGS model, and two hybrid approaches between Reynolds-averaged Navier- Stokes (RANS) and Large Eddy Simulation (LES) models, namely the Spalart-Allmaras Detached Eddy Simulation (SA-DES) and the eXtra- Large Eddy Simulation (X-LES). Accurate time integration is based on high-order linearly implicit Rosenbrock-type Runge-Kutta schemes, implemented in the DG code MIGALE up to sixth-order accuracy. Several high-order DG results of both incompressible and compressible 3D turbulent test cases proposed within the IDIHOM project demonstrate the capability of the method.

Simulation of the Transitional Flow in a Low Pressure Gas Turbine Cascade With a High-Order Discontinuous Galerkin Method

In the last decade, discontinuous Galerkin (DG) methods have been the subject of extensive research efforts because of their excellent performance in the high-order accurate discretization of advection-diffusion problems on general unstructured grids, and are nowadays finding use in several different applications. In this paper, the potential offered by a high-order accurate DG space discretization method with implicit time integration for the solution of the Reynolds-averaged Navier–Stokes equations coupled with the k-ω turbulence model is investigated in the numerical simulation of the turbulent flow through the well-known T106A turbine cascade. The numerical results demonstrate that, by exploiting high order accurate DG schemes, it is possible to compute accurate simulations of this flow on very coarse grids, with both the high-Reynolds and low-Reynolds number versions of the k-ω turbulence model.

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 Sorensen (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).

Investigation of Near-Wall Grid Spacing Effect in High-Order Discontinuous Galerkin RANS Computations of Turbomachinery Flows

In the last decade, Discontinuous Galerkin (DG) methods have been the subject of extensive research effort because of their excellent performance in the high-order accurate discretization of advection-diffusion problems on general unstructured grids, and are nowadays finding use in several different applications. In this paper, the potential offered by a high-order accurate DG space discretization method with implicit time integration for the solution of the Reynolds-averaged Navier-Stokes equations coupled with the \(\boldsymbol{k}\)-\(\boldsymbol{\omega }\) turbulence model is investigated in the numerical simulation of the turbulent flow through the well known T106A turbine cascade. The numerical results demonstrate that, by exploiting high order accurate DG schemes, it is possible to compute accurate simulations of this flow on grids with few elements.

High-Order Discontinuous Galerkin Solution of Unsteady Flows by Using an Advanced Implicit Method

The aim of this paper is to investigate and evaluate a multi-stage and multi-step method that is an evolution of the more common Backward Differentiation Formulae (BDF). This new class of formulae, called Two Implicit Advanced Step-point (TIAS), has been applied to a high-order Discontinuous Galerkin (DG) discretization of the Navier-Stokes equations, coupling the high temporal accuracy gained by the TIAS scheme with the high space accuracy of the DG method. The performance of the DG-TIAS scheme has been evaluated by means of two test cases: an inviscid isentropic convecting vortex and a laminar vortex shedding behind a circular cylinder. The advantages of the high-order time discretization are illustrated comparing the sixth-order accurate TIAS scheme with the second-order accurate BDF scheme using the same spatial discretization.

Robust and Efficient Implementation of Very High-Order Discontinuous Galerkin Methods in CFD

Discontinuous Galerkin (DG) methods are a very powerful numerical techniques, that offer high degree of robustness, accuracy and flexibility, nowadays necessary for the solution of complex fluid flows. The drawback is the relatively high computational cost and storage requirement. This work will focus on two approaches which can be adopted to enhance the computational efficiency of this class of methods: (i) a DG discretization based upon co-located tensor product basis functions, and (ii) a p-multigrid solution strategy. The effectiveness of the proposed approaches has been demonstrated by computing 3D inviscid and turbulent test cases.