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C1.1 Inviscid flow over a bump2nd International Workshop on High-Order CFD Methods

D. C. Del Rey Fernández1, P.D. Boom1, and D. W. Zingg1, and J. E. Hicken2

1University of Toronto Institute of Aerospace Studies, Toronto, Ontario, M3H 5T6, Canada

2Rensselaer Polytechnic Institute, Troy, New York, 12180

May 27, 2013

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Diablo Flow Solver

Algorithm for solving the three-dimensional Euler/Navier-Stokes/RANS equations

Spalart-Allmaras 1-equation turbulence model1 (currently second-order with first-order convective terms)

Code implemented to solve the discrete equations, in parallel, using multi-block domaindecomposition on structured grids

Weak imposition of boundary conditions using simultaneous-approximation-terms (SATs)2

Only requires C0 continuity between blocksCommunication overhead, for parallel processing, remains the same regardless of the order of the discretization

1Michal Osusky and David W. Zingg. “A Parallel Newton-Krylov-Schur flow solver for the Reynolds-averaged Navier-Stokesequations”. In: AIAA Paper 2012-0442 (2012).

2Jason E. Hicken and David W. Zingg. “A parallel Newton-Krylov solver for the Euler equations discretized using simultaneousapproximation terms”. In: AIAA Journal 46.11 (2008), pp. 2773–2786.

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Spatial derivatives discretized using high-order summation-by-parts (SBP), centred,finite-difference operators

Mimetic of integration by partsDiscretely satisfies divergence theorem → conservativeEquipped with discrete norm and higher-order quadrature rule3

Amenable to the energy method → prove time stability for linearized NS equationsSuperconvergence of functionals if the discretization is dual-consistent and the solution is sufficiently smooth4

Second derivative can be approximated by application of the first derivative twice or minimum-width-stencil operator5

The nonlinear system of equations, resulting from the discretization, is solved using aninexact Newton-Krylov algorithm with a pseudo-transient time continuation startup phase

The linear system is solved with FGMRES and a Parallel Approximate-Schur preconditioner

General implementation for implicit and explicit multistep Runge-Kutta (MRK) methods

which specifically includes6:Linear multistep methods (Euler, BDF, Trapezoidal, . . .)Runge-Kutta methods (Explicit RK, SDIRK, ESDIRK, . . .)

Newton’s method is accelerated for implicit time-marching methods using:Lagrange polynomial extrapolation from step/stage solution valuesDelayed preconditioner updates for individual stages or stepsRelative tolerance termination of the nonlinear subiterations

3Jason E. Hicken and D. W. Zingg. “Summation-by-parts operators and high-order quadrature”. In: Journal of Computational andApplied Mathematics 237 (2013), pp. 111–125.

4J. E. Hicken and D. W. Zingg. “Superconvergent functional estimates from summation-by-parts finite-difference discretizations.” In:SIAM Journal on Scientific Computing 33.2 (2011), pp. 893–922.

5David C. Del Rey Fernández and D. W. Zingg. “High-Order Compact-Stencil Summation-By-Parts Operators for the SecondDerivative with Variable Coefficients”. In: ICCFD7-2803 (2012).

6Michal Osusky et al. “An efficient Newton-Krylov-Shur parallel solution algorithm for the steady and unsteady Navier-Stokesequations”. In: ICCFD7. 2012.

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Simulation parameters and characteristics

Mach number = 0.5

Grid, 5 block structured grid

Farfield distance to midpoint of bump 10

No dissipation for RHS computations

TauBench reference time is 9.5968 sec

Computations were performed on the General Purpose Cluster (GPC) supercomputer at the SciNet HPC Consortium. SciNet isfunded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario ResearchFund - Research Excellence; and the University of Toronto.

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Diablo Flow Solver Results Questions

10−3

10−2

10−1

10−9

10−8

10−7

10−6

10−5

10−4

10−3

1/√DOF

||E(E

ntr

opy)|| H

O(D(2,1,2))=1.9449

O(D(4,2,3))=2.9154

O(D(6,3,4))=4.7324

100

101

102

103

10−9

10−8

10−7

10−6

10−5

10−4

10−3

Work-units

||E(E

ntropy)|| H

D(2,1,2)

D(4,2,3)

D(6,3,4)

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Questions

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