Brian C. Vermeire

ILES Using the Correction Procedure via Reconstruction Scheme

An implicit large eddy simulation (ILES) solver is presented using the novel correction procedure via reconstruction (CPR) scheme. This scheme allows for high-order accurate solutions of the unsteady Navier-Stokes equations on unstructured meshes containing curved boundary elements. Explicit and implicit temporal schemes are used, with the implicit method relying on a quasi Newton-GMRES approach with a third order singly diagonal implicit Runge-Kutta (SDIRK) scheme. Results for ILES of the Taylor-Green vortex problem show that the current high-order schemes can provide more accurate solutions at equivalent computational cost to the lower order approximations. Preliminary results for ow over an SD7003 airfoil at M = 0:2 and Re = 60;000 show the current implementation of the CPR scheme is capable of solving complex, unsteady, transitional and turbulent ows for aerospace applications.

On the utility of GPU accelerated high-order methods for unsteady flow simulations

First- and second-order accurate numerical methods, implemented for CPUs, underpin the majority of industrial CFD solvers. Whilst this technology has proven very successful at solving steady-state problems via a Reynolds Averaged NavierStokes approach, its utility for undertaking scale-resolving simulations of unsteady flows is less clear. High-order methods for unstructured grids and GPU accelerators have been proposed as an enabling technology for unsteady scale-resolving simulations of flow over complex geometries. In this study we systematically compare accuracy and cost of the high-order Flux Reconstruction solver PyFR running on GPUs and the industry-standard solver STAR-CCM+ running on CPUs when applied to a range of unsteady flow problems. Specifically, we perform comparisons of accuracy and cost for isentropic vortex advection (EV), decay of the TaylorGreen vortex (TGV), turbulent flow over a circular cylinder, and turbulent flow over an SD7003 aerofoil. We consider two configurations of STAR-CCM+: a second-order configuration, and a third-order configuration, where the latter was recommended by CD-adapco for more effective computation of unsteady flow problems. Results from both PyFR and STAR-CCM+ demonstrate that third-order schemes can be more accurate than second-order schemes for a given cost e.g. going from second- to third-order, the PyFR simulations of the EV and TGV achieve 75 and 3 error reduction respectively for the same or reduced cost, and STAR-CCM+ simulations of the cylinder recovered wake statistics significantly more accurately for only twice the cost. Moreover, advancing to higher-order schemes on GPUs with PyFR was found to offer even further accuracy vs. cost benefits relative to industry-standard tools.

Towards green aviation with python at petascale

Accurate simulation of unsteady turbulent flow is critical for improved design of greener aircraft that are quieter and more fuel-efficient. We demonstrate application of PyFR, a Python based computational fluid dynamics solver, to petascale simulation of such flow problems. Rationale behind algorithmic choices, which offer increased levels of accuracy and enable sustained computation at up to 58% of peak DP-FLOP/s on unstructured grids, will be discussed in the context of modern hardware. A range of software innovations will also be detailed, including use of runtime code generation, which enables PyFR to efficiently target multiple platforms, including heterogeneous systems, via a single implementation. Finally, results will be presented from a fullscale simulation of flow over a low-pressure turbine blade cascade, along with weak/strong scaling statistics from the Piz Daint and Titan supercomputers, and performance data demonstrating sustained computation at up to 13.7 DP-PFLOP/s.

Adaptive IMEX schemes for high-order unstructured methods

We present an adaptive implicit-explicit (IMEX) method for use with high-order unstructured schemes. The proposed method makes use of the Gerschgorin theorem to conservatively estimate the influence of each individual degree of freedom on the spectral radius of the discretization. This information is used to split the system into implicit and explicit regions, adapting to unsteady features in the flow. We dynamically repartition the domain to balance the number of implicit and explicit elements per core. As a consequence, we are able to achieve an even load balance for each implicit/explicit stage of the IMEX scheme. We investigate linear advection-diffusion, isentropic vortex advection, unsteady laminar flow over an SD7003 airfoil, and turbulent flow over a circular cylinder. Results show that the proposed method consistently yields a stable discretization, and maintains the theoretical order of accuracy of the high-order spatial schemes.

On the properties of energy stable flux reconstruction schemes for implicit large eddy simulation

We begin by investigating the stability, order of accuracy, and dispersion and dissipation characteristics of the extended range of energy stable flux reconstruction (E-ESFR) schemes in the context of implicit large eddy simulation (ILES). We proceed to demonstrate that subsets of the E-ESFR schemes are more stable than collocation nodal discontinuous Galerkin methods recovered with the flux reconstruction approach (FRDG) for marginally-resolved ILES simulations of the TaylorGreen vortex. These schemes are shown to have reduced dissipation and dispersion errors relative to FRDG schemes of the same polynomial degree and, simultaneously, have increased CourantFriedrichsLewy (CFL) limits. Finally, we simulate turbulent flow over an SD7003 aerofoil using two of the most stable E-ESFR schemes identified by the aforementioned TaylorGreen vortex experiments. Results demonstrate that subsets of E-ESFR schemes appear more stable than the commonly used FRDG method, have increased CFL limits, and are suitable for ILES of complex turbulent flows on unstructured grids.

A p-adaptive LCP formulation for the compressible Navier-Stokes equations

This paper presents a polynomial-adaptive lifting collocation penalty (LCP) formulation for the compressible Navier-Stokes equations. The LCP formulation is a high-order nodal scheme in differential form. This format, although computationally efficient, complicates the treatment of non-uniform polynomial approximations. In Cagnone and Nadarajah (2012) [9], we proposed to circumvent this difficulty by employing specially designed elements inserted at the interface where the interpolation degree varies. In the present study we examine the applicability of this approach to the discretization of the Navier-Stokes equations, with focus put on the treatment of the viscous fluxes. The stability of the scheme is analyzed with the scalar diffusion equation and the merits of the approach are demonstrated with various p-adaptive simulations.

PyFR: Next-Generation High-Order Computational Fluid Dynamics on Many-Core Hardware (Invited)

High-order numerical methods for unstructured grids combine the superior accuracy of high-order spectral or finite difference methods with the geometric flexibility of low-order finite volume or finite element schemes. The Flux Reconstruction (FR) approach unifies various high-order schemes for unstructured grids within a single framework. Additionally, the FR approach exhibits a significant degree of element locality, and is thus able to run efficiently on modern many-core hardware platforms, such as Graphical Processing Units (GPUs). The aforementioned properties of FR mean it offers a promising route to performing affordable, and hence industrially relevant, scale-resolving simulations of hitherto intractable unsteady flows within the vicinity of real-world engineering geometries. Here we present PyFR, an open-source Python based framework for solving advection-diffusion type problems using the FR approach. The framework is designed to solve a range of governing systems on mixed unstructured grids containing various element types. It is also designed to target a range of hardware platforms via use of a custom Mako-derived domain specific language. Specifically, the current release of PyFR is able to solve the compressible Euler and Navier-Stokes equations on grids of quadrilateral and triangular elements in two dimensions, and hexahedral, tetrahedral, prismatic, and pyramidal elements in three dimensions, targeting clusters of multi-core CPUs, NVIDIA GPUs (K20, K40 etc.), AMD GPUs (S10000, W9100 etc.), and heterogeneous mixtures thereof. Results will be presented for various benchmark and ‘real-world’ flow problems. PyFR is freely available under an open-source 3-Clause New-Style BSD license (www.pyfr.org).

A Polynomial Adaptive LCP Scheme for Viscous Compressible Flows

This paper presents a polynomial-adaptive liftingcollocation-penalty (LCP) formulation for the compressible Navier-Stokes equations. The problem of non-conforming polynomial approximations is dealt with an interface element approach. Emphasis is put on the treatment of diffusive fluxes, and p-adaptive viscous compressible flow simulations are performed.