C1.2 Ringleb flow. 2nd International Workshop on High-Order CFD Methods. D. C. Del Rey Ferna ndez1, P.D. Boom1, and D. W. Zingg1, and J. E.

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1 C. Ringleb flow nd International Workshop on High-Order CFD Methods D. C. Del Rey Ferna ndez, P.D. Boom, and D. W. Zingg, and J. E. Hicken University of Toronto Institute of Aerospace Studies, Toronto, Ontario, M3H 5T6, Canada Rensselaer Polytechnic Institute, Troy, New York, 8 May 7, 3 /

2 Diablo Flow Solver Algorithm for solving the three-dimensional Euler/Navier-Stokes/RANS equations Spalart-Allmaras -equation turbulence model (currently second-order with first-order convective terms) Code implemented to solve the discrete equations, in parallel, using multi-block domain decomposition on structured grids Weak imposition of boundary conditions using simultaneous-approximation-terms (SATs) Only requires C continuity between blocks Communication overhead, for parallel processing, remains the same regardless of the order of the discretization Michal Osusky and David W. Zingg. A Parallel Newton-Krylov-Schur flow solver for the Reynolds-averaged Navier-Stokes equations. In: AIAA Paper -44 (). Jason E. Hicken and David W. Zingg. A parallel Newton-Krylov solver for the Euler equations discretized using simultaneous approximation terms. In: AIAA Journal 46. (8), pp /

3 Spatial derivatives discretized using high-order summation-by-parts (SBP), centred, finite-difference operators Mimetic of integration by parts Discretely satisfies divergence theorem conservative Equipped with discrete norm and higher-order quadrature rule3 Amenable to the energy method prove time stability for linearized NS equations Superconvergence 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 an inexact 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 values Delayed preconditioner updates for individual stages or steps Relative tolerance termination of the nonlinear subiterations 3 Jason E. Hicken and D. W. Zingg. Summation-by-parts operators and high-order quadrature. In: Journal of Computational and Applied Mathematics 37 (3), pp J. E. Hicken and D. W. Zingg. Superconvergent functional estimates from summation-by-parts finite-difference discretizations. In: SIAM Journal on Scientific Computing 33. (), pp David C. Del Rey Ferna ndez and D. W. Zingg. High-Order Compact-Stencil Summation-By-Parts Operators for the Second Derivative with Variable Coefficients. In: ICCFD7-83 (). 6 Michal Osusky et al. An efficient Newton-Krylov-Shur parallel solution algorithm for the steady and unsteady Navier-Stokes equations. In: ICCFD7.. 3/

4 Simulation parameters and characteristics Single block grid Characteristic boundary conditions for inflow/outflow boundaries, slip wall boundary condition for remaining boundaries TauBench reference time is sec 4/

5 Slip wall boundary conditions O(D(,,))=.6495 O()=.496 O()=3.366 E (E ntr opy) H E (E ntr opy) H 6 6 D(,,) / DO F 3 5/

6 Slip wall boundary conditions O(D(,,))=.3547 O()=.7559 O()=.4468 E (ρ) H E (ρ) H D(,,) / DO F 3 6/

7 Slip wall boundary conditions O(D(,,))=.7947 O()=.4 O()=.76 D(,,) E (ρu) H E (ρu) H / DO F 3 7/

8 Slip wall boundary conditions O(D(,,))=.654 O()=.9538 O()=.95 D(,,) E (ρv) H E (ρv) H / DO F 3 8/

9 Slip wall boundary conditions O(D(,,))=.364 O()=.668 O()=.4658 D(,,) E (e) H E (e) H / DO F 3 9/

10 Characteristic boundary conditions O(D(,,))=.864 O()=.79 O()=3.76 E (E ntr opy) H E (E ntr opy) H D(,,) / DO F /

11 Subsonic domain: slip wall boundary condition E (E ntr opy) H E (E ntr opy) H O(D(,,))=.48 9 O(D (4,,3))= D(,,).8 7 O()= / DO F. 3 /

12 Questions /