Numerical Calculations of Unsteady Flowsin Time-Space Domain

t = 50,100,300

At t = 0,

Example: Benchmark Prob. of ICASE in CAA95 (Category 2-1)

P1: 1 GpaRho1: 10 kg/m3

P2: .1 MpaRho2: 1 kg/m3

Grid 3000Total step 800CFL .99

Solid line is exact solution and red circle one numerical results

Air shock tube case

2-D EULER EQUATIONS FOR REACTING FLOWS

where Q is the unknown vector, F and G are the flux vectors, and H is the source term:

HGFQ

yxt

,

)(

,

2

uZ

upE

uv

pu

u

Z

E

v

u

FQ

0

0

0

0

,

)(

2 HG

vZ

vpE

pv

uv

v

where E= e – (1-Z)q0 + (u2+v2)/2 is the total energy with e as

the internal energy and q as the heat release

)exp( TEZK and

Schlieren images at t = 11.65 for detonation with f=3.0, q0=50, E+=50

From original NASA code From updated code

Mass fraction contours at t = 11.65 for detonation with f=3.0, q0=50, E+=50

From original NASA code From updated code

Vorticity contours at t = 11.65 for detonation with f=3.0, q0=50, E+=50

From original NASA code From updated code

Figure A.7: Schlieren plots of density for ZND Test 4. Roe-HLL EF 3*-Hwith Wave Propagation Method, Van Albada-limiter, 40 Pts/L1/2, CRoeCFL = 0.95From the Ph D thesis(p. 232) of by Ralf Deiterding(2003)

Schlieren plots of density for ZND Test 4. 50 Pts/L1/2, CFL = 0.99