（様式 20）成果報告書

TSUBAME 共同利用 令和３年度 学術利用 成果報告書

利用課題名 液体金属流れ CFD 手法の開発及び核融合研究への応用
英文：High-order flux reconstruction method for the hyperbolic formulation of the incompressible
Navier-Stokes equations
on unstructured grids
利用課題責任者
胡 長洪
所属
九州大学応用力学研究所 新エネルギー力学部門
https://www.tj.kyushu-u.ac.jp/
邦文抄録（300 字程度）
英文抄録（100 words 程度)
A high-order Flux reconstruction implementation of the hyperbolic formulation for theincompressible Navier-Stokes equation is presented. The governing equations employ Chorin’s
classical artificial compressibility (AC) formulation cast in hyperbolic form. Instead of splitting thesecond-order conservation law into two equations, one for the solution and another for the gradient,
the Navier-Stokes equation is cast into a first-order hyperbolic system of equations. Including thegradients in the AC iterative process results in a significant improvement in accuracy for thepressure, velocity, and its gradients. Furthermore, this treatment allows for taking larger timesteps since the hyperbolic formulation eliminates the restriction due to diffusion.

Keywords: Hyperbolic method, Flux reconstruction, Unstructured grid, Incompressible NavierStokes equations, Artificial compressibility method背景と目的High
ordermethods,
suchasflux
reconstruction (FR) method, can achieve highorder spatial accuracy on complicated geometries
using compact stencils that only involve
immediate face neighbors. Furthermore, highorder methods, such as the flux-reconstruction
and discontinuous Galerkin methods havecompact stencils which renders them particularly
suitable for computation on modern hardware
such as general purpose graphical processing
units (GPGPUs). A challenge that arises whensolving the Navier-Stokes equations using theclassic formulation is the severe time-step
restriction in diffusion dominated problems.
Even in advection dominated problems (i.e.,high Reynolds number flows), localized highdiffusion areas, either due to a high turbulent
eddy viscosity or an artificially introduced
stabilization viscosity, can have an adverse effecton stability, especially if such regions overlappedwith highly refined mesh zones. To address thislimitation in the context of the flux
reconstruction method for incompressible flows, a
hyperbolic method [1] for solving the advectiondiffusion method is implemented in the flux
reconstruction code PyFR[2]. Using hyperbolic
diffusion, the restriction on stability fromdiffusion can be significantly alleviated.
Additionally, the hyperbolic method was found toincrease the accuracy and order of accuracy ofvariables and their gradients. This can be adesirable feature in applications that require
accurate gradients.
概要
This research aims to develop next generation
CFD techniques for solving incompressible, freesurface flows. Such phenomena are important ina variety of fields and applications such as
flooding and tsunami simulations, applications in
naval and marine engineering, and nuclear
fusion applications (flow of liquid metal as
plasma-facing material). This development isbased on the high-order Flux Reconstruction
method, which allows obtaining more accurate
results while utilizing modern hardware more
efficiently when compared to conventional CFD
techniques. The Tsubame super-computer wasused to carry carry out simulations for testingand validation of GPU-accelerated, high-order
CFD
codeforsolving
freesurface
incompressible flows. The simulations aimed totest the accuracy and computational efficiency
and scaling of the code for problems involving alarge number of degrees of freedom.

報告書結果および考察
Numericaltests
ofthe
hyperbolic
incompressible
flux
reconstruction
implementation showed significant reductions in
the absolute error of the field variables and thegradient of the velocity have been demonstrated .
Additionally, it has been shown that equal ordersof accuracy can be obtained for both the field
variables and velocity gradients. Numericalresults suggest that the improvement in theorder of accuracy of the velocity gradients lead toa matching improvement in the order of accuracyof pressure. Analysis shows that the time-steprequirements are significantly relaxed whenusing the hyperbolic solver.
This leads to a considerable speed-up of
convergence especially for diffusion dominatedproblems where the parabolic restriction can bequite severe.
The strong scaling performance of thedeveloped solver has been shown to be superior
to the existing INS-FR solver due to the extra
communication required for the computation of
the viscous fluxes in the latter.

Figure 1: Convergence of the residual of thepressure for laminar flow past a sphere at
Re=100 using the classic incompressibile NS(blue) and hyperbolic method (red)Figure 1 shows the significant reduction in thenumber of iterations required by the hyperbolic
solver to reach convergence, when compared tothe classical incompressible solver.

Figure 2: Strong scaling performance for thedeveloped solver on multiple GPUs inTSUBAME
Figure 2 shows the result of a strong scaling
study carried out on TSUBAME, demonstrating
the better scaaling performance of the hyperbolic
flux reconstruction method.

まとめ、今後の課題
In this report, progress on the development of ahigh-order hyperbolic method for incompressibloe
flows for use in high-accuracy and large-scale
simulations has been summarized. Satisfactory
results of benchmark tests were obtained. Future
work includes optimization of model parameters
and reducing the memory foot-print andcomputational cost of the solver.

References[1] H. Nishikawa, Y. Liu, Hyperbolic NavierStokes Method for HighReynolds-Number
Boundary Layer Flows.
[2] Witherden FD, Farrington AM, Vincent PE.
PyFR: An open source framework for solving
advection–diffusion type problems on streamingarchitectures using the flux reconstruction approach.
Computer Physics Communications. 2014 Nov
1;185(11):3028-40.