Jul 11, 2016 - rh(Cu n h. ) = 0. δCu n,i h. Parallel Iterative Solver. Overview. Cu n,i h. = Cu n,i1 h. + αn,iδCu n,i h ... Gl := (ârh/âβ)|βl .... DG (LLF/SIP fluxes).
Parallel Iterative Solvers for Hybridized DG Methods with Application to Large-Eddy Simulation P. Fernandez, N.C. Nguyen and J. Peraire Aerospace Computational Design Laboratory Department of Aeronautics and Astronautics, MIT
Simulation of turbulent flows Simulating transitional and turbulent flows is a very challenging problem: -
Small numerical dissipation and dispersion required to capture transition to turbulence
-
Turbulent flows have multiple scales: Ratio largest-to-smallest length scales
Ratio largest-to-smallest temporal scales
tl = O(Re1/2 ) t⌘
l = O(Re3/4 ) ⌘
BL t n
le
u rb
Tu
on i t i ns a Tr
Instantaneous velocity field L B r ina
m La
Mean velocity field
Q-criterion isosurface colored by pressure NACA 65-(18)10 compressor cascade 3
Simulation of turbulent flows Computational approaches
Direct Numerical Simulation (DNS)
Treatment of turbulence
Time dependency
Computational cost
Accuracy
Resolve all scales
Unsteady
Prohibitive: Cost is O(Re11/4 ).
Accurate for all flows
Reynolds-Averaged Navier-Stokes equations (RANS)
Model all scales
Large-Eddy Simulation (LES)
Resolve large scales Model small scales
Affordable even with limited computing Inaccurate for transitional resources and complex and separated flows flows
Steady
Unsteady
4
Affordable only on large supercomputers and simple flows
Accurate for most flows
LES vs. RANS LES Instantaneous
RANS Time average
Mach number field of T106C LP turbine cascade at Re2s = 80,000 | M2s = 0.65 5
LES vs. RANS Suction side (transition & separation) Friction coe/cient
Pressure coe/cient 1
Time-averaged LES RANS
Time-averaged LES RANS
0.03
0.5
0.02
0.01
Cf
-Cp
0 0
-0.5 -0.01
-0.02
-1
-0.03 -1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0.1
0.2
0.3
0.4
0.5
0.6
x=c
x=c
Pressure side (no transition, no separation)
RANS fails whenever the flow transitions and separates 6
0.7
0.8
0.9
1
Goals of this research • Large-Eddy Simulation of flows of interest require time integration of spatial discretizations with 100M+ DOFs over 10k+ time steps
• High-level goal of this research: Enable accurate and efficient LES for industrial applications through convenient choices of -
Efficient and scalable parallel nonlinear solver: Newton-GMRES method
-
Physical models (SGS modeling, wall modeling, etc.): Under investigation
7
Goals of this research • Large-Eddy Simulation of flows of interest require time integration of spatial discretizations with 100M+ DOFs over 10k+ time steps
• High-level goal of this research: Enable accurate and efficient LES for industrial applications through convenient choices of -
Hybridized DG: Computational Efficiency Indicator of computational cost: Number of non-zero entries in Jacobian of global system ( NNZ ) Values of ↵NNZ (tetrahedral mesh)
NNZ = Np Nc2 ↵NNZ NNZ ⌘ Number of non-zeros Np ⌘ Number of mesh vertices Nc ⌘ Number of components of the PDE
