Chapter 8 Conclusions
8.2 Recommended Future Studies
There is still has something to make the PDSC complete. The diagram of recommended future studies is shown in Fig. 2.4 and described in the following:
1. To develop hybrid mesh code in PDSC. This can be a hybrid code with structured/unstructured and tetrahedral/hexahedral mesh system, which can save the cell number and reduce the time of tracking particle;
2. To couple with other numerical solves which can extend its capability of simulate complex flows. Combining with computational fluid dynamics (CFD) solver can solve the flow has continuum flow region. Combining with particle-in-cell (PIC) method can simulate flows with plasma. Incorporating with particle flux method (PFM ) can simulate inviscid flow.
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Appendix A Derivation of the Probability of Dissociation/Exchange
How to derive the reaction probability of the particle method is the most important issue to process the chemical reaction. For the dissociation and exchange reactions, the total collision energy model (TCE) is used in M ONACO. To make sure the MONACO uses the TCE model as mentions in Bird’s book [11], simple derivation in the following shows the dissociation probability of MONACO is the same as the Eq. (6.10) in Bird’s book.
From \PHYS\col_model.c directory
ETA[iclass]=refcxs*2.0/sqrt(PI)*pow((2.0*GASCONST*Trefvhs)/reducedm[iclass],om
From \PHYS\chem.c directory
ZETArot[iclass] = species[ispec].DOFrot + species[jspec].DOFrot=ζr,1 +ζr,2; ZETAvib[iclass] = species[ispec].DOFvib + species[jspec].DOFvib=ζv,1+ζv,2; ZETAc[iclass] = DOFrel + ZETArot[iclass] +ZETAvib[iclass]
=2(2−omega)+ζr,1+ζr,2ζv,1+ζv,2=2(5 2−ω12)+ζr,1+ζr,2 +ζv,1+ζv,2
1 12 12 REAphi b REAphi
b− +ω − + −ω +ζ − = +ζ + −
r0 = REAphi2[iclass][ireac]+1.0=(5 2−ω12)+ζ
r1 = REAphi1[iclass][ireac]+1.0=b+ζ +3 2−REAphi3 r2 = ZETAvib[iclass]/2.0=(ζv,1 +ζv,2) 2
r3 = r2 + REAphi3[iclass][ireac]= (ζv,1+ζv,2) 2+REAphi3
r4 = REAphi1[iclass][ireac]-REAphi2[iclass][ireac]+REAphi3[iclass][ireac]=b+ω12 −1
For Dissociation Reaction
REAbeta[iclass][ireac] = mc_gamma(r0)/mc_gamma(r1)
*mc_gamma(r2)/mc_gamma(r3)
P[ireac] = REAbeta[iclass][ireac]*pow(Ediff,REAphi1[iclass][ireac])
*pow(Ecoll,-REAphi2[iclass][ireac]) By comparing with Eq. (6.10) of the Bird’s book.
AB AB
So, the probability of dissociation uses the Eq. (6.10) of the Bird’s book. The probability of exchange also uses the same equation expect the constants of the Arrhenis equation.
Appendix B Derivation of the Probability of Recombination
A three-body model for recombination reaction is proposed by Professor Boyd.
The third body is used to provide the energy to process recombination reaction. The detail of this method is described in Section 6.1.2. The following paragraph is the equation of reaction probability.
From \PHYS\col_model.c directory ETA[iclass] From \PHYS\chem.c directory
REAphi1[iclass][ireac]=REAb[iclass][ireac]-0.5+omega=b−12+omega =χ r0 = 7.0/2.0+species[ireac].DOFrot/2.0-omega=7 2+ζ1 2−omega
r1 = r0 + REAphi1[iclass][ireac]= 72+ζ1 2−omega+χ
Precom=*nobj*volinv * REAbeta[iclass][kspec]
*pow((Ecoll+E3),REAphi1[iclass][kspec])
=
By comparing with Eq. (13) (Recombination probability of binary collision) of the paper of Boyd, Phys.Fluids A 4(1), 1992, pp.178-185.
χ
Table 1. 1 Comparison of some well-known DSM C codes.
a Dynamic Domain Decomposition
b Variable Time-Step Scheme
c Quantum Vibration M odel
d Graphic User Interface
Simulator Coordinate S ystem
Grid S ystem
Parallel
Computing DDD a VTS b Chemistry QVM c GUI d
Visual DS MC Program
2-D/Axis./3-D
Structured Sub-cells Adaptive
No No Yes Yes Yes Yes
DAC 2-D/Axis./3-D
Unstructured 2-level embedded
Cartesian
Yes Yes - Yes - -
MONACO 2-D/Axis./3-D Unstructured
Sub-cell Yes M anual Yes Yes Yes No
S MILE 2-D/Axis./3-D Rectangular
Adaptive Yes Yes Yes Yes Yes Yes
PDSC 2-D/3-D Unstructured
Adaptive Yes Yes Yes Yes Yes Yes
Table 3. 1 The complete listing of physical and VHS parameters of a hypersonic flow over a cylinder.
N2 gas Kn∞=0.025 n∞=5.1775E19 (#/m3) U∞= 1823.149 (m/s) Ma∞=20 ω=0.74 mref= 4.65E-26 (Kg) dref=4.17E-10 (m) Tref=273 (K) Tw=291.6 (K) T∞=20 (K) To=1620 (K)
Zr∞=21 T*=79.8
Table 3. 2 The cell numbers of adaptive mesh with or without cell quality control.
Level 0 1 2 3 4
cell quality control
- No Yes No Yes No Yes No Yes cell no. 7,025 13,916 13,916 33,737 33,773 67,060 67,021 75,305 75,099 (Knc)m in 0.066 0.104 0.104 0.176 0.171 0.300 0.289 0.314 0.321
Table 3. 3 The complete listing of physical and VHS parameters of a hypersonic flow over a 15o-compression ramp.
N2 gas Kn=λ∞/Xc=2E-4 ReL=1.04E5 ρ∞=5.221E-4 (kg/m3) p∞=12.79 (Pa) Ma∞=14.36 U∞= 2652.1 (m/s) ω=0.75 mref= 4.65E-26 (Kg) dref=4.17E-10 (m) Tref=273 (K) Tw=294.4 (K)
T∞=84.83 (K) Zr=5
Table 3. 4 The cell numbers of adaptive mesh with or without cell quality control.
Level 0 1 2
cell quality control
- No Yes No Yes
cell no. 15,063 30,219 30,219 83,398 83,776
Table 3. 5 The complete listing of physical and VHS parameters of a hypersonic flow over a sphere.
N2 gas Kn=λ∞/D=1.035E-1 ReL=30 n∞=9.77e20 (#/m3) p∞=12.79 (Pa) Ma∞=4.2 U∞= 697.022 (m/s) ω=0.74 mref= 4.65E-26(Kg) dref=4.17E-10 (m) Tref=273 (K) Tw=300 (K)
To=300 (K) T∞=66.25 (K) Zr=5
Table 3. 6 The cell numbers of adaptive mesh with or without cell quality control.
Level 0 1 2
cell quality control
- No Yes No Yes
cell no. 5,353 22,510 22,510 151,732 164,276
Table 4. 1 Comparisons of the parallel machines at NCHC.
System Hardware Configuration
IBM -SP2 IBM -SM P
CPU
1 per node P2SC –160 -M Hz
(64 cpu)
4 per node Power3-II 375-M Hz
(128 cpu) M emory 256 -M B (per node) 4 -GB (per node)
L1 Cache (per CPU) 128-KB 32KB/64KB
Table 4. 2 Number of cells and particles for three different problem sizes for the driven cavity flow
Problem size Small M edium Large
Cell numbers 11,250 45,000 180,000
Particle numbers 225,000 900,000 3,600,000
Table 6. 1 The constants of the rate coefficients in chem.dat file.
* Rate constant exp( ) exp( )
kT aT E
aT T
k= b −θ = b − a
Reactions a * b * E * a
1. N2 +N2 →N +N+N2 6.170E-09 -1.6 1.561E-18
2. N2+N →N+N+N 1.850E-08 -1.6 1.561E-18
3. N +N+N2 →N2 +N2 5.691E-40 -1.6 0
4. N +N+N→N2 +N 1.706E-39 -1.6 0
5. O2 +O2 →O+O+O2 4.580E-11 -1.0 8.197E-19
6. O2 +O→O+O+O 1.375E-10 -1.0 8.197E-19
7. O+O+O2 →O2+O2 6.305E-44 -0.5 0.0
8. O+O+O→O2+O 1.905E-43 -0.5 0.0
9. NO+N2 →N+O+ N2 3.830E-13 -0.5 1.043E-18
10. NO+O2 → N+O+O2 3.830E-13 -0.5 1.043E-18
11. NO+NO→ N+O+NO 3.830E-13 -0.5 1.043E-18
12. NO+N →N +O+N 7.660E-13 -0.5 1.043E-18
13. NO+O→ N+O+O 7.660E-13 -0.5 1.043E-18
14. N +O+N2 →NO+ N2 1.583E-43 -0.5 0.0
15. N +O+O2 →NO+O2 1.583E-43 -0.5 0.0
16. N +O+NO→NO+NO 1.583E-43 -0.5 0.0
17. N+O+N→NO+N 3.180E-43 -0.5 0.0
18. N +O+O→NO+O 3.180E-43 -0.5 0.0
19. N2+O→NO+N 5.300E-17 0.10 5.177E-19
20. NO+N →N2 +O 2.020E-17 0.10 0.0
21. NO+O→O2+N 3.600E-22 1.29 2.719E-19
22. O2+N →NO+O 5.200E-22 1.29 4.970E-20
Table 6. 2 Comparison of the reaction probability and rate constant of N2 +N2 →N +N+N2 reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical) 5000 2.008310E+09 1.000000E+01 4.979312E-09 0.290278E-08 1.910565E-24 0.111380E-23 7000 1.005983E+09 1.030000E+02 1.023875E-06 0.999210E-06 4.273378E-22 0.417043E-21 9000 9.993514E+08 2.472600E+04 2.475205E-05 0.227658E-04 1.1000727E-20 0.101180E-19 11000 9.992670E+08 1.687040E+05 1.688278E-04 0.154345E-03 7.889357E-20 0.721257E-19 13000 9.813145E+08 5.963470E+05 6.077022E-04 0.551246E-03 2.960917E-19 0.268585E-18
Table 6. 3 Comparison of the reaction probability and rate constant of N2+N →N+N+N reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical) 5000 6.722381E+08 2.000000E+00 2.975137E-09 0.409370E-08 2.427084E-24 0.333959E-23 7000 6.790187E+08 8.070000E+02 1.188480E-06 0.140915E-05 1.054633E-21 0.125045E-20 9000 5.994706E+08 1.897000E+04 3.164459E-05 0.321058E-04 2.990167E-20 0.303375E-19 11000 5.729034E+08 1.262960E+05 2.204490E-04 0.217667E-03 2.190240E-19 0.216260E-18 13000 5.446063E+08 4.427480E+05 8.129689E-04 0.777403E-03 8.421626E-19 0.805319E-18
Table 6. 4 Comparison of the reaction probability and rate constant of N +N+N2 →N2+N2 reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical) 5000 2.653486E+08 6.000000E+01 2.261177E-07 0.235230E-06 6.602036E-46 0.686809E-45 7000 2.659365E+08 2.900000E+01 1.090486E-07 0.116441E-06 3.754451E-46 0.400896E-45 9000 2.625208E+08 2.000000E+01 7.618443e-08 0.678132E-07 3.012677E-46 0.268164E-45 11000 2.573439E+08 1.200000E+01 4.663021E-08 0.437044E-07 2.075406E-46 0.194518E-45 13000 5.382297 E+08 2.600000E+01 4.830651e-08 0.301891E-07 2.382503E-46 0.148895E-45
Table 6. 5 Comparison of the reaction probability and rate constant of N +N+N → N2 +N reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical) 3000 1.058814E+09 1.606000E+03 1.516791E-06 0.141031E-05 2.214313E-45 0.205886E-44 4000 1.064403E+09 7.730000E+02 7.262284E-07 0.698113E-06 1.250171E-45 0.120177E-44 5000 1.050878E+09 4.980000E+02 4.738893E-07 0.406569E-06 9.369865E-46 0.803879E-45 6000 1.031560E+09 3.370000E+02 3.266960E-07 0.262026E-06 7.2702517E-46 0.583110E-45 7000 1.011896E+09 2.590000E+02 2.559552E-07 0.180997E-06 6.311923E-46 0.446344E-45
Table 6. 6 Comparison of the reaction probability and rate constant of O2 +O2 →O+O+O2 reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
3000 909944997 224 2.461687E-07 0.127783E-06 7.407696E-23 0.384524E-22
4000 812138012 14664 1.805605E-05 0.125894E-04 5.838560E-21 0.407089E-20
5000 902565292 211049 2.338324E-04 0.185648E-03 7.994987E-20 0.634750E-19
6000 876586710 1116620 1.273827E-03 0.107052E-02 4.558466E-19 0.383093E-18
7000 846974877 3550774 4.192302E-03 0.363173E-02 1.559184E-18 0.135070E-17
Table 6. 7 Comparison of the reaction probability and rate constant of O2 +O→O+O+O reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
3000 433257607 97 2.238853E-07 0.201857E-06 1.280387E-22 0.115441E-21
4000 437929163 9191 2.098741E-05 0.198873E-04 1.289762E-20 0.122216E-19
5000 429199928 139810 3.257456E-04 0.293266E-03 2.116688E-19 0.190564E-18
6000 417020964 790388 1.895320E-03 0.169109E-02 1.289010E-18 0.115011E-17
7000 402801642 2608729 6.476461E-03 0.573701E-02 4.577705E-18 0.405505E-17
Table 6. 8 Comparison of the reaction probability and rate constant of O+O+O2 →O2 +O2 reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
3000 186447756 148 7.937880E-07 0.643972E-06 1.419123E-045 0.115128E-44
4000 188588480 120 6.363061E-07 0.488072E-06 1.299858E-045 0.997041E-45
5000 185131710 91 4.915419E-07 0.383242E-06 1.143789E-045 0.891781E-45
6000 179556576 67 3.731414E-07 0.309784E-06 9.805790E-046 0.814081E-45
7000 173254226 51 2.943651E-07 0.256480E-06 8.650219E-046 0.753692E-45
Table 6. 9 Comparison of the reaction probability and rate constant of O+O+O→O2+O reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
3000 746437311 3657 4.899273E-06 0.450826E-05 3.781211E-45 0.347944E-44
4000 755242580 2791 3.695501E-06 0.341685E-05 3.259020E-45 0.301328E-44
5000 741177555 2148 2.898091E-06 0.268297E-05 2.911261E-45 0.269516E-44
6000 718982844 1703 2.368624E-06 0.216870E-05 2.687134E-45 0.246033E-44
7000 694873458 1335 1.921213E-06 0.179554E-05 2.437256E-45 0.227783E-44
Table 6. 10 Comparison of the reaction probability and rate constant of NO+N2 →N +O+N2 reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
5000 249626105 464 1.858780E-06 0.194162E-05 1.412561E-21 0.147551E-20
7000 250173393 28670 1.146005E-04 0.113297E-03 9.473233E-20 0.936544E-19
9000 247166816 271610 1.098893E-03 0.103368E-02 9.672819E-19 0.909877E-18
11000 242704757 1077996 4.441594E-03 0.409384E-02 4.110775E-18 0.378892E-17
13000 238343701 2719260 1.140899E-02 0.103942E-01 1.100956E-17 0.100303E-16
Table 6. 11 Comparison of the reaction probability and rate constant of NO+O2 → N+O+O2 reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
5000 235424229 628 2.667525E-06 0.205645E-05 1.913965E-21 0.147551E-20
7000 236615349 38244 1.616294E-04 0.119997E-03 1.261472E-19 0.936543E-19
9000 233134814 343420 1.473053E-03 0.109481E-02 1.224226E-18 0.909877E-18
11000 228967220 1315563 5.745639E-03 0.433596E-02 5.020758E-18 0.378892E-17
13000 224779133 3221065 1.432991E-02 0.110090E-01 1.305607E-17 0.100303E-16
Table 6. 12 Comparison of the reaction probability and rate constant of NO+NO→ N+O+NO reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
5000 493274083 1872 3.795050E-06 0.389306E-05 1.427100E-021 0.146396E-20
7000 494547920 115934 2.344242E-04 0.227166E-03 9.588958E-020 0.929208E-19
9000 488493032 1070455 2.191341E-03 0.207259E-02 9.544769E-019 0.902750E-18
11000 479800926 4259445 8.877525E-03 0.820839E-02 4.065693E-018 0.375925E-17
13000 470268366 10667725 2.268434E-02 0.208411E-01 1.083194E-017 0.995177E-17
Table 6. 13 Comparison of the reaction probability and rate constant of NO+N →N +O+N reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
5000 266760622 980 3.673706E-06 0.363018E-05 2.986411E-21 0.295103E-20
7000 270420558 59722 2.208486E-04 0.211827E-03 1.952863E-19 0.187309E-18
9000 264623436 548760 2.073739E-03 0.193263E-02 1.952618E-18 0.181975E-17
11000 259673553 2150813 8.282757E-03 0.765412E-02 8.200219E-18 0.757785E-17
13000 254694166 5391097 2.116694E-02 0.194338E-01 2.184979E-17 0.200607E-16
Table 6. 14 Comparison of the reaction probability and rate constant of NO+O→ N+O+O reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
5000 223358315 978 4.378614E-06 0.433327E-05 2.981911E-021 0.295103E-20
7000 224668576 58772 2.615942E-04 0.252853E-03 1.937843E-019 0.187309E-18
9000 221564005 552839 2.495166E-03 0.230694E-02 1.968229E-018 0.181975E-17
11000 217709070 2156316 9.904576E-03 0.913655E-02 8.214855E-018 0.757785E-17
13000 213956988 5394345 2.521229E-02 0.231977E-01 2.180288E-017 0.200607E-16
Table 6. 15 Comparison of the reaction probability and rate constant of N +O+N2 →NO+N2 reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
5000 99122344 40 4.035417E-07 0.304911E-06 2.963733E-045 0.223936E-44
7000 99329520 24 2.416200E-07 0.218537E-06 2.092509E-045 0.189261E-44
9000 98280965 17 1.729735E-07 0.167800E-06 1.720584E-045 0.166912E-44
11000 96350646 10 1.037876E-07 0.134867E-06 1.161857E-045 0.150978E-44
13000 94522664 14 1.481126E-07 0.111954E-06 1.837347E-045 0.138879E-44
Table 6. 16 Comparison of the reaction probability and rate constant of N +O+O2 →NO+O2 reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
5000 99223776 32 3.225034E-07 0.304911E-06 2.368562E-045 0.223936E-44
7000 99226239 25 2.519495E-07 0.218537E-06 2.181966E-045 0.189261E-44
9000 98226610 18 1.832497E-07 0.167800E-06 1.822802E-045 0.166912E-44
11000 96373596 15 1.556443E-07 0.134867E-06 1.742370E-045 0.150978E-44
13000 94694018 13 1.372843E-07 0.111954E-06 1.703021E-045 0.138879E-44
Table 6. 17 Comparison of the reaction probability and rate constant of N +O+NO→NO+NO reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
5000 99073342 37 3.734607E-07 0.304911E-06 2.742809E-045 0.223936E-44
7000 99355778 21 2.113616E-07 0.218537E-06 1.830461E-045 0.189261E-44
9000 98045441 24 2.447844E-07 0.167800E-06 2.434893E-045 0.166912E-44
11000 96465977 17 1.762279E-07 0.134867E-06 1.972795E-045 0.150978E-44
13000 94639816 15 1.584957E-07 0.111954E-06 1.966150E-045 0.138879E-44
Table 6. 18 Comparison of the reaction probability and rate constant of N +O+N →NO+N reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
5000 222368524 225 1.011834E-06 0.914497E-06 2.476983E-045 0.223870E-44
7000 223682461 170 7.600059E-07 0.655444E-06 2.193882E-045 0.189205E-44
9000 220237391 124 5.630288E-07 0.503272E-06 1.866755E-045 0.166863E-44
11000 216670853 122 5.630661E-07 0.404463E-06 2.101188E-045 0.150933E-44
13000 212511245 74 3.482169E-07 0.335746E-06 1.439954E-045 0.138838E-44
Table 6. 19 Comparison of the reaction probability and rate constant of N +O+O→NO+O reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
5000 222535385 201 9.032271E-07 0.914497E-06 2.211112E-045 0.223870E-44
7000 223472402 185 8.278427E-07 0.655444E-06 2.389704E-045 0.189205E-44
9000 220318540 123 5.582826E-07 0.503272E-06 1.851018E-045 0.166863E-44
11000 216464094 117 5.405054E-07 0.404463E-06 2.016999E-045 0.150933E-44
13000 212461834 65 3.059373E-07 0.335746E-06 1.265119E-045 0.138838E-44
Table 6. 20 Comparison of the reaction probability and rate constant of N2 +O→NO+N reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
5000 448696187 43293 9.648622E-05 0.100183E-03 6.597438E-20 0.685024E-19
7000 450814795 374796 8.313747E-04 0.812581E-03 6.183565E-19 0.604378E-18
9000 444335334 1205511 2.713066E-03 0.257461E-02 2.148765E-18 0.203911E-17
11000 436669803 2480978 5.681588E-03 0.533062E-02 4.731359E-18 0.443909E-17
13000 428279387 4039183 9.431187E-03 0.878537E-02 8.188798E-18 0.762805E-17
Table 6. 21 Comparison of the reaction probability and rate constant of NO+N →N2 +O reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
5000 532736242 33141438 6.220984E-02 0.582376E-01 5.057131E-17 0.473422E-16
7000 534468471 31479363 5.889845E-02 0.553712E-01 5.208119E-17 0.489622E-16
9000 528640226 29895721 5.655211E-02 0.533227E-01 5.324907E-17 0.502083E-16
11000 519066616 28442223 5.479494E-02 0.517416E-01 5.424891E-17 0.512260E-16
13000 509389746 27193371 5.338421E-02 0.504612E-01 5.510638E-17 0.520890E-16
Table 6. 22 Comparison of the reaction probability and rate constant of NO+O→O2 +N reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
5000 446187472 264810 5.934950E-04 0.607356E-03 4.041802E-19 0.413620E-18
7000 448940904 1222346 2.722732E-03 0.265693E-02 2.016951E-18 0.196820E-17
9000 442632876 2987210 6.748731E-03 0.644966E-02 5.323514E-18 0.508760E-17
11000 435020516 5452565 1.253404E-02 0.118315E-01 1.039573E-17 0.981302E-17
13000 426429436 8435851 1.978252E-02 0.185425E-01 1.710737E-17 0.160350E-16
Table 6. 23 Comparison of the reaction probability and rate constant of O2 +N →NO+O reaction
Temp. (oK)
Total collisions between reactants
Number of real reaction
P r
(Simulation)
P r
(Theoretical)
k f
(Simulation)
k f
(Theoretical)
5000 446187472 264810 5.934950E-04 0.607356E-03 4.041802E-19 0.413620E-18
7000 448940904 1222346 2.722732E-03 0.265693E-02 2.016951E-18 0.196820E-17
9000 442632876 2987210 6.748731E-03 0.644966E-02 5.323514E-18 0.508760E-17
11000 435020516 5452565 1.253404E-02 0.118315E-01 1.039573E-17 0.981302E-17
13000 426429436 8435851 1.978252E-02 0.185425E-01 1.710737E-17 0.160350E-16
Table 6. 24 M ole percentage of each species by using original chem.dat.
Temp.
(oK)
N2 (Original)
N2 (Fitting)
O2 (Original)
O2 (Fitting)
NO (Original)
NO (Fitting)
N (Original)
N (Fitting)
O (Original)
O (Fitting)
2000 80 80 20 20 0 0 0 0 0 0
3000 78.73293 79.30586 19.04047 19.49159 0.5480151 0.2640975 0.0005548 5.9726E-5 1.678009 0.9383817 4000 58.51967 58.68676 0.2123696 0.2220255 0.4692463 0.938484 1.096412 0.5560535 39.70229 39.59669 6000 25.41407 36.97584 0.0039259 0.0063654 0.1099764 0.2671703 38.56771 25.1045 35.90432 37.64613 8000 0.24792 0.2827485 0.0001338 3.5631E-5 0.0020865 0.0023249 70.51454 70.89766 29.23532 28.81723 10000 0.0164276 0.0224934 2.6668E-5 4.4456E-6 0.0005067 0.0005867 71.36262 71.67037 28.62042 28.30656 12000 0.0039995 0.0045668 9.9989E-6 6.6664E-6 0.0002100 0.0001900 71.63901 71.78274 28.35678 28.2125 14000 0.0016150 0.0016265 5.0041E-6 6.6657E-6 9.4986E-5 5.9992E-5 71.94937 72.16114 28.04892 27.83717 16000 0.0006602 0.0006750 0 1.6665E-6 2.0007E-5 4.5000E-5 72.29124 72.49075 27.70809 27.50853
Figure 1. 1 Effective limits of major approximations in the DSM C method [10, 11].
: vertices of the graph : cell faces
: cut edges
Figure 1. 2 Sketch of graph and mesh.
move all molecules
enter new molecules
print out the data NO
NO reset
sampling data YES
YES sort (index) molecules
reach steady flow?
sufficient sampling?
start
set initial state and read system data
collide molecules
sample flow field
stop
Figure 2. 1 The flowchart of the standard DSM C method.
Figure 2. 2 Schematic diagram of the PDSC.
(a)
(b)
Figure 2. 3 M uST Visual Preprocessor for numerical simulations [73].
(a)
(b)
Figure 2. 4 Important features of the PDSC (a) future PDSC; (b) planned study of PDSC.
L
1L
2Cell 1 Cell 2
N
1W
11
u
1
φ
∆ t
N
2W
22 2
φ
∆ t
2
21 ,
1 mv mv
φ =
Figure 3. 1 Sketch of the concept of variable-time-step scheme.
move all molecules
enter new molecules
print out the data NO
NO reset
sampling data YES
YES sort (index) molecules
NO
YES reach steady flow?
adapt the mesh?
sufficient sampling?
start
set initial state and read system data
collide molecules
sample flow field
mesh refinement
stop
Figure 3. 2 The flowchart of the DSM C method with mesh adaptation.
Initial grid
Initial grid Initial grid Initial grid
1 hanging node (isotropic)
2 hanging node (isotropic)
3 hanging node (isotropic)
4 hanging node (isotropic)
hanging node removed (anisotropic)
hanging nodes removed (anisotropic)
hanging nodes removed (isotropic)
hanging nodes removed (isotropic)
isotropic ( 1st stage)
isotropic ( 1st stage)