Euler equations of Gas Dynamics
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Euler equations of Euler equations of Gas Dynamics Gas Dynamics Hailiang Liu Hailiang Liu Math517@ISU Math517@ISU October 15, 2004 October 15, 2004
description
Euler equations of Gas Dynamics. Hailiang Liu Math517@ISU October 15, 2004. Simulated Gas Flow. Nonlinear Conservation Laws: of great practical importance since they govern a variety of physical phenomena that appear in fluid mechanics, - PowerPoint PPT Presentation
Transcript of Euler equations of Gas Dynamics
Euler equations of Gas Euler equations of Gas Dynamics Dynamics
Hailiang LiuHailiang Liu
Math517@ISUMath517@ISU
October 15, 2004October 15, 2004
Simulated Gas FlowSimulated Gas Flow
0)( ufu xt
vmvEp
pEv
pv
m
f
E
mu
)5.0)(1(
)(
22
Where u has components of density, momentum and energy.
Nonlinear Conservation Laws: of great practical importance since they govern a variety of physical phenomena that appear in fluid mechanics, astrophysics, groundwater flow, meterology, semiconductors, and reactive flows
Euler equation: Modeling inviscid non-heat conducting ( ideal polytropic) gases
Eigen-structureEigen-structure
cpc
pccpccR
cppcR
cvvcv
iii
ii
ttss
ti
i
1-
1 vs, v,
1-
1 vs,
invariants
Riemann 0
)0,,(),0,()0,,(
321
Here c^2=\gamma p/\rho=dh/dp, and c is called the sound speed.
Numerical Test by SodNumerical Test by Sod
Riemann problem proposed by Sod (1978)
Tr
Tl
r
l
u
u
xu
xuxu
)25.0,0,125.0(
)5.2,0,1(
0
0)0,(
Numerical Test by LaxNumerical Test by LaxRiemann problem proposed by Lax (1974)
Tr
Tl uu )4275.1,0,5.0()928.8,311.0,445.0(
Woodward-Collela’s testWoodward-Collela’s test
• Woodward-Collela proposed this test in 1984
)100,0,1()01.0,0,1()100,0,1(),,(
.19.0
9.01.0
1.00
)0,(
,,
rml
r
m
l
pm
xu
xu
xu
xu
The shock InteractionThe shock Interaction
A model of the interaction of a sinusoidal disturbance and a shock wave due to the initial conditions
8.0),1,0),5sin(2.01(
8.0),629369.2,333333.10,857143.3()0,)(,,(
xx
xxpv
A density wave will emerge behind the shock discontinuity, its fine structure makes the current problem a suitable test case for high order methods.
Test I--requiredTest I--required
• The first order central non-staggered LxF scheme (1954)
• The staggered NT scheme (4.2), (4.4a), (4.5).• The non-staggered NT second order accurate
central scheme, (2.18), (4.4a), (4.5).• Ref: Paper by Nessyahu and Tadmor(1990).
Test II--optionalTest II--optional
• The first order accurate Godunov-type scheme of Roe (1983)
• The ENO second order accurate scheme (1988)
