Chapter 7 Applications and Exam ples
7.3 A Three-Dimensional Re-entry S phere Flow
The third application is a three dimensional sphere case, which is at 90 km altitude and simulated with 19 chemical reactions [89]. According to the verifications of a single cell, the order of recombination probability is much less than the dissociation and exchange reactions. Thus, recombination is negligible in this simulation. Only one in sixteenth physical domain is simulated because its axis-symmetric geometry. Related flow conditions are listed as follows [69] the altitude of this case is 90 km and the free-stream density is 3.43E-6 kg/m3; the initial mole fraction of oxygen molecule, nitrogen molecule and oxygen atoms are 0.209, 0.788 and 0.004, respectively; the free-stream velocity and temperature are 7,500 m/s and 188 K, respectively; the wall temperature is 350 K with full diffuse surface; the corresponding Knudsen number Kn and Reynolds number Re(based on the diameter of the sphere, which is 1.6 meter) are 0.01 and 3,243, respectively. The two-level adaptive cell number and particle number of
this simulation are about 670,000 and 3,000,000, respectively. The adaptive mesh refinement can easily capture the bow shock in the front of the sphere because it is a hypersonic flow, which is shown as Fig. 7.12.
Properties Contour
Figure 7.13(a)~(c) illustrate flow field contours of this simulation, which including normalized density, normalized overall temperature and M ach number on the X-Z plane, respectively. The normalized properties are based on the free-stream condition. As we can see, there is a very strong bow shock stands in the front of the sphere and the wake region exists behind the object. The normalized density and overall temperature are increased along the X-axis first because the bow shock effect, which the maximum value are about 120 and 60, respectively. Figure 7.14 represents the mole fractions for five species, which are N2, O2, NO, N and O, along the stagnation line. Nitrogen and oxygen molecules stay constant before the bow shock and then decreasing near the shock region because the temperature goes very high, which can process the dissociation reaction. Thus, the mole fractions of nitric oxide, nitrogen atom and oxygen atom are increased. Figure 7.15 shows the surface coefficients, including pressure, skin-friction and heat transfer coefficients, along the symmetric line of the surface of the sphere. This figure is plotted as a function of the circumferential angle (θ) measured clockwise from the stagnation point and the simulation data of Dogra et al. [69] is also plotted into this figure for comparison. The pressure (Fig. 7.15(a)) and heat transfer (Fig.
7.15(c)) coefficient decreased with increasing circumferential angle and the maximum value are at stagnation point θ=0o. The simulated result agrees very well with the reference [69]. The skin-friction coefficient increased then decreasing along the sphere surface and the maximum value is near θ=40o. As we can see the results of skin-friction and heat transfer coefficients are pretty scattering because the particle sampling is not enough. M ore particle number and longer time-steps can overcome this problem and obtain more accurate simulated results. Although the results are not exact the same, the results are acceptable by comparing with reference data.
7.4 Concluding Remarks
This chapter presents several three-dimensional applications, which are a near-continuum parallel twin-jet, the Apollo re-entry vehicle flow and a hypersonic re-entry sphere flow. All these cases are interesting and important but it is very
complicated and the computation is time-consuming. By using the present PDSC, it can efficiently simulate and obtain accurate results by using reasonable computing time, which demonstrates its capability.
Chapter 8 Conclusions
8.1 Summary
The direct simulation of M onte Carlo (DSM C) method is used to simulate gas flows under rarefied gas environment in these four decades. M esh resolution, huge computational cost and flow involving specific problems are three main drawbacks by using the DSM C method. To overcome these disadvantages, a general-purpose parallel DSM C code (PDSC) is presented and verified by comparing with experiment and references. The current PDSC has following features;
1. It is a three-dimensional DSM C code with unstructured tetrahedral mesh, which is easier and flexible to deal with the flow with complex and irregular geometry. A cell-by-cell particle tracking technique can easily and correctly track particle movements.
2. A general unstructured adaptive mesh refinement with variable time-step scheme is developed to save computational time and to obtain accurate results.
The particle number of a 3-D hypersonic sphere flow by using constant time-step scheme is 1.7 million, which can be reduce to 0.34 million particles by applying variable time-step scheme and the transient time is decreased to only 25%. The computational efficiency can be speeded up to 10 times and more accurate result can be obtained if simulation uses suitable mesh resolution and time-step.
3. Parallelization of a 3-D DSM C method is developed to save the computing time. The parallel D SM C code with dynamic domain decomposition is also implemented to speed up the computational efficiency. The dynamic domain decomposition function can alleviate the unbalancing loading between each processor. This method can save 30-100% for the driven cavity flow by using static domain decomposition method.
4. Conservative weighting scheme (CWS) for flows with trace species is incorporated in PDSC to obtain reasonable number of simulated particles. A quasi 2-D hypersonic cylinder flow is used to verify this function. The total particle number by using constant weighting scheme is up to 10 million, which can be reduced to 0.24 million if the conservative weighting scheme is
activated.
5. Chemical reaction is developed for hypersonic reactive flows. It is tested by several cases, which includes comparisons the reaction probability (Pr) and rate constant (kf) of each single reaction, degree of dissociation of pure species and the mole fraction of air (5 species) for a 2-D single cell. This function can correctly simulate dissociation, exchange and recombination reactions and the results agree with theoretical data.
Finally, several applications, which are a 3-D parallel underexpanded twin-jets, a hypersonic Apollo re-entry vehicle and a hypersonic re-entry sphere flow, are simulated to demonstrate potential and ability of the PDSC.
8.2 Recommended Future S tudies
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
No No Yes Yes Yes Yes