High Resolution
Scheme
In
this study a simple and efficient method is developed for Euler equation and
Navier-Stokes equation
arising from the conservation law. A class of high order weighted
essentially non-oscillatory (WENO) schemes are applied to resolve the
complicated flow structure and shock wave. Classical WENO schemes are
computationally expensive in calculating the non-linear weight and smoothness
indicators, etc. We propose a block-structured adaptive mesh method together
with a modified hybrid-WENO scheme to reduce the cost. The reconstruction are only performed at non-smooth region.
Comparisons of WENO scheme with various smoothness indicators and different
Lax-Friedrich flux vector splitting methods are performed under the adaptive
mesh.
1-D Shock-Entropy Wave
Interaction Problem
Fig. 1
AMR computation of the Shu-Osher problem using the
WENO scheme.
2-D Double Mach
Reflection Problem
(a) Left, WENO_JS; right, WENO_JS_UD
(b) Left, WENO_Z; right, WENO_Z_UD
Fig. 2
Numerical results for the 2-D double Mach reflection problems with different
WENO schemes.
2-D Riemann Problem
(a) WENO_Z_GLF, WENO-Z-UD-GLF,
(b) WENO_Z_BLF, WENO-Z- UD-BLF.
Fig. 3
Numerical results for the 2-D Riemann problems with different WENO schemes.
2-D Shock Wave Boundary
Layer Interaction
Fig. 4
Contours of density at T=1 obtained on the adaptive mesh level 2-7, left:
Re=200; right: Re=1000 (blocks of the finest level are shown).
Fig. 5
Convergence test for density along the wall obtained
at T = 1, left: Re=200; right: Re=1000.
3-D Shock Wave Boundary
Layer Interaction
Fig. 6 Iso-surface of vortex structure (
) visualized by the Q-criterion at
different time.
Reference
C. Liu*, C. Hu, An Adaptive High Order WENO Solver for
Conservation Laws, submitted.
C. Liu, C. Hu, Block-Structured AMR method for High
Accurate Shock Wave Capturing Schemes, The 29th Computational Fluid
Dynamics Symposium (第29回数値流体力学シンポジウム),
Fukuoka, Japan, 2015.