Andrea Crivellini

An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations

Discontinuous Galerkin (DG) methods have proved to be well suited for the construction of robust high-order numerical schemes on unstructured and possibly nonconforming grids for a variety of problems. Their application to the incompressible Navier-Stokes (INS) equations has also been recently considered, although the subject is far from being fully explored. In this work, we propose a new approach for the DG numerical solution of the INS equations written in conservation form. The inviscid numerical fluxes both in the continuity and in the momentum equation are computed using the values of velocity and pressure provided by the (exact) solution of the Riemann problem associated with a local artificial compressibility perturbation of the equations. Unlike in most of the existing methods, artificial compressibility is here introduced only at the interface flux level, therefore resulting in a consistent discretization of the INS equations irrespectively of the amount of artificial compressibility introduced. The discretization of the viscous term follows the well-established DG scheme named BR2. The performance and the accuracy of the method are demonstrated by computing the Kovasznay flow and the two-dimensional lid-driven cavity flow for a wide range of Reynolds numbers and for various degrees of polynomial approximation.

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

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.

Assessment of a sponge layer as a non-reflective boundary treatment with highly accurate gust–airfoil interaction results

ABSTRACT This paper deals with the numerical performance of a sponge layer as a non-reflective boundary condition. This technique is well known and widely adopted, but only recently have the reasons for a sponge failure been recognised, in analysis by Mani. For multidimensional problems, the ineffectiveness of the method is due to the self-reflections of the sponge occurring when it interacts with an oblique acoustic wave. Based on his theoretical investigations, Mani gives some useful guidelines for implementing effective sponge layers. However, in our opinion, some practical indications are still missing from the current literature. Here, an extensive numerical study of the performance of this technique is presented. Moreover, we analyse a reduced sponge implementation characterised by undamped partial differential equations for the velocity components. The main aim of this paper relies on the determination of the minimal width of the layer, as well as of the corresponding strength, required to obtain a reflection error of no more than a few per cent of that observed when solving the same problem on the same grid, but without employing the sponge layer term. For this purpose, a test case of computational aeroacoustics, the single airfoil gust response problem, has been addressed in several configurations. As a direct consequence of our investigation, we present a well documented and highly validated reference solution for the far-field acoustic intensity, a result that is not well established in the literature. Lastly, the proof of the accuracy of an algorithm for coupling sub-domains solved by the linear and non-liner Euler governing equations is given. This result is here exploited to adopt a linear-based sponge layer even in a non-linear computation.

Assessment of a high-order accurate Discontinuous Galerkin method for turbomachinery flows

ABSTRACT In this work the capabilities of a high-order Discontinuous Galerkin (DG) method applied to the computation of turbomachinery flows are investigated. The Reynolds averaged Navier–Stokes equations coupled with the two equations k-ω turbulence model are solved to predict the flow features, either in a fixed or rotating reference frame, to simulate the fluid flow around bodies that operate under an imposed steady rotation. To ensure, by design, the positivity of all thermodynamic variables at a discrete level, a set of primitive variables based on pressure and temperature logarithms is used. The flow fields through the MTU T106A low-pressure turbine cascade and the NASA Rotor 37 axial compressor have been computed up to fourth-order of accuracy and compared to the experimental and numerical data available in the literature.

Second derivative time integration methods for discontinuous Galerkin solutions of unsteady compressible flows

In this paper we investigate the possibility of using the high-order accurate A(α)-stable Second Derivative (SD) schemes proposed by Enright for the implicit time integration of the Discontinuous Galerkin (DG) space-discretized Navier–Stokes equations. These multistep schemes are A-stable up to fourth-order, but their use results in a system matrix difficult to compute. Furthermore, the evaluation of the nonlinear function is computationally very demanding. We propose here a Matrix-Free (MF) implementation of Enright schemes that allows to obtain a method without the costs of forming, storing and factorizing the system matrix, which is much less computationally expensive than its matrix-explicit counterpart, and which performs competitively with other implicit schemes, such as the Modified Extended Backward Differentiation Formulae (MEBDF). The algorithm makes use of the preconditioned GMRES algorithm for solving the linear system of equations. The preconditioner is based on the ILU(0) factorization of an approximated but computationally cheaper form of the system matrix, and it has been reused for several time steps to improve the efficiency of the MF Newton–Krylov solver. We additionally employ a polynomial extrapolation technique to compute an accurate initial guess to the implicit nonlinear system. The stability properties of SD schemes have been analyzed by solving a linear model problem. For the analysis on the Navier–Stokes equations, two-dimensional inviscid and viscous test cases, both with a known analytical solution, are solved to assess the accuracy properties of the proposed time integration method for nonlinear autonomous and non-autonomous systems, respectively. The performance of the SD algorithm is compared with the ones obtained by using an MF-MEBDF solver, in order to evaluate its effectiveness, identifying its limitations and suggesting possible further improvements.

Hybrid OPENMP/MPI parallelization of a high-order Discontinuous Galerkin CFD/CAA solver

This paper describes the implementation of an hybrid OpenMP/MPI parallelization strategy in a Discontinuous Galerkin solver used for DNS and LES or CAA computations, to fruitfully exploit the modern massively parallel HPC facilities. It is usually believed that the sheared memory view of OpenMP can easily increase the parallel efficiency of codes dealing with multi-core clusters. The idea consists of running calculations on those machines restricting as much as possible the use of the MPI library to the communications between nodes and exploiting the shared memory paradigm within a node. However, in practice, the achievement of a real parallel performance gain is not straightforward. Moreover, as far as DG solvers are concerned, almost nothing is reported in the current literature about the hybrid MPI/OpenMP implementation. In this work a colouring algorithm has been employed for OpenMP. The resulting hybrid strategy performs quite satisfactory, since generally it is more efficient of the pure MPI implementation. However, the performances are heavily dependent on hardware platforms, as well as on computational details such as the polynomial order of space discretization or the number of computational elements. Several scalability tests have been performed, resulting in the conclusion that the best performance can be achieved only with a proper choice of the number of MPI partition and OpenMP threads to be used within a single node. The reliability of the method was here assessed by solving the Taylor Green vortex problem at Reynolds numbers equal to 800 and 1600 and the Linear Euler acustic scattering from a rigid sphere.

MATRIX-FREE MODIFIED EXTENDED BACKWARD DIFFERENTIATION FORMULAE APPLIED TO THE DISCONTINUOUS GALERKIN SOLUTION OF COMPRESSIBLE UNSTEADY VISCOUS FLOWS

In this work a matrix-free modified extended backward differentiation time integration method has been implemented in a high-order discontinuous Galerkin solver for the unsteady Navier-Stokes equations. The resulting non-linear systems at each time step are solved iteratively using a preconditioned inexact Newton/Krylov method. In order to speed-up the solution process a frozen preconditioner formulation and a polynomial extrapolation technique for computing a better initial guess for the Newton iterations have been considered. Numerical results for compressible inviscid and viscous test cases show the effectiveness of the proposed numerical strategies and the performance advantages of the matrix-free method compared to its matrix-explicit counterpart for this class of implicit multi-stage time schemes. Furthermore, the influence of some physical (low Mach) and space discretization (stretched grid) aspects is examined to highlight pros and cons of the proposed time integration algorithm and its potential in solving non-stiff and stiff systems with respect to the widely used explicit Runge-Kutta schemes.

Development and validation of a model for the simulation of the air drying phenomena in pipelines

The main aim of this research work consists in the development and validation of a model, based on a numerical method, and its relevant simulation software, to solve the differential equations governing the drying process of gas pipelines. The knowledge of this phenomenon represents the key factor for performing effective pre-commissioning activities in petrochemical industry; in order to avoid hydrate formation and pipe corrosion, it is in fact necessary to achieve an effective removal of all water prior the introduction of hydrocarbon gas. This is normally achieved by performing a bulk dewatering operation, followed by a drying operation which efficiency is dependent on the air flow rate, air pressure and temperature. In this paper, the model implemented in a finite volume-based simulation software, as well as the considerations that have been made for its development, is presented. Its predictions are compared with available air drying field data of two existing gas pipeline systems.