# A New Gaskinetic Model to Analyze Background Flow Effects on Weak Gaseous Jet Flows from Electric Propulsion Devices

## Abstract

**:**

## 1. Introduction

## 2. Collisionless Jet Expanding into a Uniform Dilute Background Flow

#### 2.1. Derivations for the Mixture Density, Velocities, Temperature, and Pressure

#### 2.2. Validations and Discussions on the Derived Formulas

#### 2.3. Background Flow Effects on Electric Propulsion (EP) Devices’ Performance

## 3. New Methods to Recover the Flow Parameters with Limited Measurements

#### 3.1. Linearized Governing Equations for the Density and Number Fluxes

- Step 1:
- perform initial estimations on the seven parameters: ${n}_{0g}$, ${S}_{0g}$, ${n}_{bg}$, ${S}_{bg}$, ${\alpha}_{0g}$, ${T}_{0g}$ and ${T}_{bg}$;
- Step 2:
- select a total of ${N}_{1}$ measured densities ${n}_{m}$ at different points, $0\le {N}_{1}<6$;
- Step 3:
- select a total of ${N}_{2}$ measured number fluxes along the X-direction, ${\left(nU\right)}_{m}$, ${N}_{2}\ge 0$;
- Step 4:
- select a total of ${N}_{3}$ measured number fluxes along the Y-direction, ${\left(nV\right)}_{m}$, ${N}_{3}\ge 0$;Here ${N}_{1}+{N}_{2}+{N}_{3}\ge 7$. The properties can be from the same or different points, e.g., measured number density and fluxes at one point offer three relations.
- Step 5:
- for each selected point, compute the corresponding ${\theta}_{2}$ and ${\theta}_{1}$;
- Step 6:
- for each of the ${N}_{1}$ measured densities, use ${\theta}_{1}$ and ${\theta}_{2}$, and the five guessed parameters, ${n}_{0g}$, ${S}_{0g}$, ${n}_{bg}$, ${S}_{bg}$, and ${\alpha}_{0g}$, to compute the corresponding ${A}_{1}$–${A}_{5}$, and a total of ${N}_{1}$ guessed jet number number density ${n}_{jg}$ can be computed by using Equation (2). With the measured densities, ${n}_{m}$, a total of ${N}_{1}$ linear algebraic equations about ${\u03f5}_{1}$–${\u03f5}_{5}$, are properly constructed;
- Step 7:
- for each of the ${N}_{2}$ measured number fluxes along the X-direction, ${\left(nU\right)}_{m}$, by using the ${\theta}_{1}$ and ${\theta}_{2}$ for each point, and the seven guessed parameters, ${n}_{0g}$, ${S}_{0g}$, ${n}_{bg}$, ${S}_{bg}$, ${\alpha}_{0g}$, ${T}_{0g}$ and ${T}_{bg}$, a set of coefficients ${B}_{1}$–${B}_{7}$ can be computed, and a guessed jet number flux ${\left(nU\right)}_{jg}$ can also be computed by using Equation (3). A total of ${N}_{2}$ linear algebraic equations about ${\u03f5}_{1}$–${\u03f5}_{7}$ are properly constructed;
- Step 8:
- for each of the ${N}_{3}$ measured number fluxes along the Y-direction, ${\left(nV\right)}_{m}$, by using the ${\theta}_{1}$ and ${\theta}_{2}$, and the seven guessed parameters, ${n}_{0g}$, ${S}_{0g}$, ${n}_{bg}$, ${S}_{bg}$, ${\alpha}_{0g}$, ${T}_{0g}$ and ${T}_{bg}$, seven coefficients ${C}_{1}$–${C}_{7}$ can be computed. A guessed jet number fluxes along the Y-direction, ${\left(nV\right)}_{jg}$ can also be computed by using Equation (4). Together with the measured data ${\left(nV\right)}_{m}$, a total of ${N}_{3}$ linear algebraic equations about ${\u03f5}_{1}$–${\u03f5}_{7}$ are properly constructed;
- Step 9:
- solve the above seven linear algebraic equations, e.g., by using a Gaussian elimination method, and obtain ${\u03f5}_{1}$–${\u03f5}_{7}$;
- Step 10:
- if these seven small perturbation parameters are sufficiently small, then stop the iterations; the seven true parameters characterizing the whole flowfield are recovered;
- Step 11:
- if these seven small perturbation numbers are not small, then update the guessed values by multiplying with $1+{\u03f5}_{1}$, $1+{\u03f5}_{2}$, $1+{\u03f5}_{3}$, $1+{\u03f5}_{4}$, $1+{\u03f5}_{5}$, $1+{\u03f5}_{6}$ and $1+{\u03f5}_{7}$; go to Step 5, and repeat the above steps.

#### 3.2. Test Cases and Discussions

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Expressions for Coefficients A_{i}, B_{i}, and C_{i}

## References

- Sanna, G.; Tomassetti, G. Introduction to Molecular Beams Gas Dynamics; Imperial College Press: London, UK, 2005. [Google Scholar]
- Maev, R.; Leshchynsky, V. Introduction to Low Pressure Gas Dynamic Spray; Wiley-Vch: Weinheim, Germany, 2008. [Google Scholar]
- Hastings, D.; Garrett, H. Spacecraft-Environment Interactions; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
- Simons, G.A. Effects of nozzle boundary layers on rocket exhaust plumes. AIAA J.
**1972**, 10, 1534–1535. [Google Scholar] [CrossRef] - Jian, H.; Chu, Y.; Cao, H.; Cao, Y.; He, X.; Xia, G. Three-dimensional IFE-PIC numerical simulation of background pressure’s effect on accelerator grid impingement current for ion optics. Vacuum
**2015**, 116, 130–138. [Google Scholar] [CrossRef] - Byers, C.C.; Dankanich, J.W. A review of facility effects on Hall effect thrusters. In Proceedings of the 31st International Electric Propulsion Conference, Ann Arbor, MI, USA, 20–24 September 2009. IEPC-2009-076.
- Gildea, S.R.; Sanchez, M.M.; Nakles, M.R.; Hargus, W.A. Experimentally characterizing the plume of a divergent cusped field thruster. In Proceedings of the 31st International Electric Propulsion Conference, Ann Arbor, MI, USA, 20–24 September 2009. AFRL-RZ-ED-TP-2009-316.
- Dankanich, J.W.; Swiatek, M.W.; Yim, T. A step towards electric propulsion testing standards: Pressure measurements and effective pumping speeds. In Proceedings of the48th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Altanta, GA, USA, 30 July–1 August 2012. AIAA paper 2012-3737.
- Jones, M.L. Results of large vacuum facility tests of an MPD arc thruster. AIAA J.
**1966**, 4, 1455–1456. [Google Scholar] [CrossRef] - Sovey, J.S.; Mantenieks, M.A. Performance and lifetime assessment of MPD arc thruster technology. In Proceedings of the 24th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Boston, MA, USA, 11–13 July 1988. AIAA Paper 88-3211.
- Nakles, M.R.; Hargus, W.A. Background pressure effects on ion velocity distribution within a medium-power Hall thruster. J. Propuls. Power
**2011**, 27, 737–743. [Google Scholar] [CrossRef] - Randolph, T.; Kim, V.; Kozubsky, K.; Zhurin, V.; Day, M. Facility effects on stationary plasma thruster testing. In Proceedings of the 23rd International Electric Propulsion Conference, Seattle, WA, USA, 13–16 September 1993. IEPC-93-93 844.
- Hofer, R.R.; Peterson, P.Y.; Gallimore, A.D. Characterizing vacuum facility back pressure effects on the performance of a Hall thruster. In Proceedings of the 27th International Electric Propulsion Conference, Pasadena, CA, USA, 15–19 October 2001. IEPC-01-045.
- Walker, M.L.R.; Gallimore, A.D.; Boyd, I.D.; Cai, C. Vacuum chamber pressure maps of a Hall thruster cold-flow expansion. J. Propuls. Power
**2004**, 20, 1127–1132. [Google Scholar] [CrossRef] - Rovey, J.L.; Walker, M.L.R.; Gallimore, A. Magnetically filtered Faraday probe for measuring the ion current density profile of a Hall thruster. Rev. Sci. Instrum.
**2006**, 77, 013503. [Google Scholar] [CrossRef] [Green Version] - Kamhawi, H.; Huang, W.; Haag, T.; Spektor, R. Investigation of the effects of facility background pressure on the performance and voltage-current characteristics of the high voltage Hall accelerator. In Proceedings of the 50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference 2014, Cleveland, OH, USA, 28–30 July 2014. AIAA Paper 2014-17063.
- Huang, W.; Kamhawi, H.; Hagg, W. Facility effect characterization test of NASA’s HERMes Hall thruster. In Proceedings of the 52nd AIAA/SAE/ASEE Jointed Propulsion Conference, Salt Lake City, UT, USA, 25–27 July 2016. AIAA paper 2016-4842.
- Li, X.; Zhang, T.; Jia, Y.; Chen, J. Numerical simulations of space background pressure effects on stability parameters for ion thruster grids. Vac. Cryog.
**2012**, 18, 71–76. Available online: http://www.cqvip.com/qk/95890x/201202/43092185.html (accessed on 1 December 2016). (In Chinese)[Google Scholar] - Yim, J.; Burt, J. Characterization of vacuum facility background gas through simulation and considerations for electric propulsion ground testing chambers. In Proceedings of the 51st AIAA/SAE/ASEE Joint Propulsion Conference, Orlando, FL, USA, 27–29 July 2015. AIAA paper 2015-3825.
- Bird, G. Molecular Gas Dynamics and the Direct Simulation of Gas Flows, 2nd ed.; Clarendon Press: Oxford, UK, 1994. [Google Scholar]
- Narasimha, R. Collisionless expansion of gases into vacuum. J. Fluid Mech.
**1962**, 12, 294–308. [Google Scholar] [CrossRef] - Liepmann, H.W. Gas kinetics and gasdynamics of orifice flow. J. Fluid Mech.
**1961**, 10, 65–79. [Google Scholar] [CrossRef] - Cai, C.; Zou, C. Gaskinetic solutions for high Knudsen number planar jet impingement flows. Commun. Comput. Phys.
**2013**, 14, 960–978. [Google Scholar] [CrossRef] - Cai, C.; Huang, X. High speed rarefied round jet impingement flows. AIAA J.
**2012**, 50, 2908–2911. [Google Scholar] [CrossRef] - Cai, C.; Khasawneh, K. Collisionless gas flows over a flat cryogenic pump plate. J. Vac. Sci. Technol. A
**2009**, 27, 601–610. [Google Scholar] [CrossRef] - Cai, C.; Boyd, I.D.; Sun, Q. Free molecular background flow in a vacuum chamber equipped with two-sided pumps. J. Vac. Sci. Technol. A
**2006**, 24, 9–19. [Google Scholar] [CrossRef] - Cai, C.; Boyd, I.D.; Sun, Q. Free molecular flows between two plates equipped with pumps. J. Thermophys. Heat Transf.
**2007**, 21, 95–104. [Google Scholar] [CrossRef] - Noller, H.G. Approximate calculation of expansion of gas from nozzles into high vacuum. J. Vac. Sci. Technol.
**1966**, 6, 202. [Google Scholar] [CrossRef] - Liu, H.; Cai, C.; Zou, C. An object-oriented implementation of the DSMC method. Comput. Fluids
**2012**, 57, 65–75. [Google Scholar] [CrossRef]

**Figure 2.**Velocity phases. (

**a**) Jet; (

**b**) Background flow. $\angle KJu=\angle M{U}_{b}u={\theta}_{1}$, $\angle IJu=\angle L{U}_{b}u={\theta}_{2}$.

**Figure 3.**Nondimensional density contours for the jet/background mixture, ${n}_{b}/{n}_{0}=0.2$, ${S}_{0}=2.0$, ${\alpha}_{0}={45}^{\circ}$, ${S}_{b}=1.0$, ${T}_{0}=200$ Kelvin, and ${T}_{b}=300$ Kelvin. Red: analytical, black: direct simulation Monte Carlo (DSMC).

**Figure 4.**Nondimensional U-velocity component contours for the jet/background mixture, ${n}_{b}/{n}_{0}=0.2$, ${S}_{0}=2.0$, ${\alpha}_{0}={45}^{\circ}$, ${S}_{b}=1.0$, ${T}_{0}=200$ Kelvin, and ${T}_{b}=300$ Kelvin. Red: analytical, black: DSMC.

**Figure 5.**Nondimensional V-velocity component contours for the jet/background mixture, ${n}_{b}/{n}_{0}=0.2$, ${S}_{0}=2.0$, ${\alpha}_{0}={45}^{\circ}$, ${S}_{b}=1.0$, ${T}_{0}=200$ Kelvin, and ${T}_{b}=300$ Kelvin. Red: analytical, black: DSMC.

**Figure 6.**Nondimensional translational temperature contours for the jet/background mixture, ${n}_{b}/{n}_{0}=0.2$, ${S}_{0}=2.0$, ${\alpha}_{0}={45}^{\circ}$, ${S}_{b}=1.0$, ${T}_{0}=200$ Kelvin, and ${T}_{b}=300$ Kelvin. Red: analytical, black: DSMC.

**Figure 7.**Nondimensional pressure contours for the jet/background mixture, ${n}_{b}/{n}_{0}=0.2$, ${S}_{0}=2.0$, ${\alpha}_{0}={45}^{\circ}$, ${S}_{b}=1.0$, ${T}_{0}=200$ Kelvin, and ${T}_{b}=300$ Kelvin. Red: analytical, black: DSMC.

**Figure 8.**Background flow effects on centerline number density profiles for the mixture, ${n}_{b}/{n}_{0}=0.2$, ${S}_{0}=2.0$, ${S}_{b}=1.0$, ${T}_{0}=200$ Kelvin, and ${T}_{b}=300$ Kelvin.

**Figure 9.**Background flow effects on centerline u-velocity component profiles for the mixture, ${n}_{b}/{n}_{0}=0.2$, ${S}_{0}=2.0$, ${S}_{b}=1.0$, ${T}_{0}=200$ Kelvin, and ${T}_{b}=300$ Kelvin.

**Figure 10.**Background flow effects on centerline temperature profiles for the mixture, ${n}_{b}/{n}_{0}=0.2$, ${S}_{0}=2.0$, ${S}_{b}=1.0$, ${T}_{0}=200$ Kelvin, and ${T}_{b}=300$ Kelvin.

**Figure 11.**Background flow effects on centerline pressure profiles for the mixture, ${n}_{b}/{n}_{0}=0.2$, ${S}_{0}=2.0$, ${S}_{b}=1.0$, ${T}_{0}=200$ Kelvin, and ${T}_{b}=300$ Kelvin.

**Figure 12.**Background flow effects on the local number flux (normalized by ${n}_{0}\sqrt{2R{T}_{0}}$, along $r/\left(2H\right)=4.$, ${n}_{b}/{n}_{0}=0.5$, ${S}_{0}=0.2$, ${S}_{b}=0.1$, ${T}_{0}=200$ Kelvin, and ${T}_{b}=300$ Kelvin.

**Figure 14.**Background flow effects on the total number flux rate across different radius (normalized by ${n}_{0}\sqrt{2R{T}_{0}}$), within angle span $-{75}^{\circ}<\theta <{75}^{\circ}$, ${n}_{b}/{n}_{0}=0.5$, ${S}_{0}=0.2$, ${S}_{b}=0.1$, ${T}_{0}=200$ Kelvin, and ${T}_{b}=300$ Kelvin.

**Figure 16.**Test case: mixture centerline number flux $nU/\left({n}_{0}\sqrt{2R{T}_{0}}\right)$ profile development history.

**Figure 17.**Test case: mixture centerline number flux $nV/\left({n}_{0}\sqrt{2R{T}_{0}}\right)$ profile development history.

**Figure 19.**Test case: mixture number flux $nU/\left({n}_{0}\sqrt{2R{T}_{0}}\right)$ profile development history, along $R=3$ m.

**Figure 20.**Test case: mixture number flux $nV/\left({n}_{0}\sqrt{2R{T}_{0}}\right)$ profile development history, along $R=3$ m.

**Table 1.**Test Parameters: ${n}_{0}=1.0$ (non-dimensionalized), ${S}_{0}=1.2$, ${n}_{b}/{n}_{0}=0.8$, ${S}_{b}=0.8$, ${\alpha}_{0}={45}^{\circ}$, ${T}_{0}=200$ Kelvin, ${T}_{b}=300$ Kelvin. Number densities and fluxes are normalized by ${n}_{0}$ and ${n}_{0}\sqrt{2R{T}_{0}}$.

Point | Angle (${}^{\circ}$) | x (m) | y (m) | ${\mathit{n}}_{\mathit{m}}$ | ${\left(\mathit{nU}\right)}_{\mathit{m}}$ | ${\left(\mathit{nV}\right)}_{\mathit{m}}$ |
---|---|---|---|---|---|---|

A | −75 | 0.776 | −2.898 | 0.803 | 0.542 | 0.538 |

B | −45 | 2.122 | −2.121 | 0.844 | 0.586 | 0.498 |

C | −30 | 2.598 | −1.500 | 0.894 | 0.667 | 0.472 |

D | −15 | 2.898 | −0.776 | 0.944 | 0.763 | 0.485 |

E | 0.000 | 3.000 | 0.000 | 0.952 | 0.785 | 0.539 |

F | 15 | 2.898 | 0.776 | 0.899 | 0.691 | 0.576 |

G | 30 | 2.598 | 1.500 | 0.822 | 0.561 | 0.549 |

H | 45 | 2.122 | 2.121 | 0.771 | 0.497 | 0.497 |

I | 75 | 0.776 | 2.898 | 0.780 | 0.532 | 0.509 |

**Table 2.**Parameter convergence histories. True values: ${n}_{0}=1.0$ (non-dimensionalized), ${S}_{0}=1.2$, ${n}_{b}/{n}_{0}=0.8$, ${S}_{b}=0.8$, ${\alpha}_{0}={45}^{\circ}$, ${T}_{0}=200$ Kelvin, ${T}_{b}=300$ Kelvin.

Parameter Names | Initial Values | 1st Iteration | 2nd | 3rd | 4th | 8th | 15th |
---|---|---|---|---|---|---|---|

${n}_{0g}$ | 0.60 | 0.989 | 1.007 | 1.018 | 1.002 | 0.999 | 1.000 |

${S}_{0g}$ | 1.0 | 1.175 | 1.232 | 1.197 | 1.205 | 1.201 | 1.199 |

${n}_{bg}$ | 1.2 | 0.802 | 0.801 | 0.799 | 0.799 | 0.800 | 0.799 |

${S}_{bg}$ | 1.0 | 0.649 | 0.697 | 0.779 | 0.794 | 0.800 | 0.799 |

${\alpha}_{0g}$ | 25.0 | 37.57 | 46.59 | 44.99 | 44.99 | 44.99 | 45.00 |

${T}_{0g}$ | 250.0 | 220.8 | 127.1 | 221.5 | 178.9 | 195.3 | 200.4 |

${T}_{bg}$ | 250.0 | 346.7 | 381.5 | 306.2 | 303.9 | 299.9 | 300.0 |

Relative Errors | Initial Values | 1st Iteration | 2nd | 3rd | 4th | 8th | 15th |
---|---|---|---|---|---|---|---|

${\u03f5}_{1}$ | $-40.0\%$ | $+64.9\%$ | $+1.73\%$ | $+1.09\%$ | $-1.59\%$ | $+0.04\%$ | $+3.4\times {10}^{-5}$ |

${\u03f5}_{2}$ | $-16.7\%$ | $+17.5\%$ | $+4.90\%$ | $-2.86\%$ | $+0.69\%$ | $+0.16\%$ | $-1.5\times {10}^{-4}$ |

${\u03f5}_{3}$ | $+50.0\%$ | $-33.1\%$ | $-0.16\%$ | $-0.18\%$ | $+0.09\%$ | $3.3\times {10}^{-6}$ | $-1.8\times {10}^{-7}$ |

${\u03f5}_{4}$ | $+25.0\%$ | $-35.1\%$ | $+7.46\%$ | $+11.8\%$ | $+1.96\%$ | $0.14\%$ | $-1.0\times {10}^{-4}$ |

${\u03f5}_{5}$ | $+45.0\%$ | $+50.3\%$ | $+24.0\%$ | $-3.46\%$ | $+0.001\%$ | $-0.04\%$ | $+3.9\times {10}^{-5}$ |

${\u03f5}_{6}$ | $+40.0\%$ | $-11.7\%$ | $-42.4\%$ | $+74.3\%$ | $-19.3\%$ | $-5.41\%$ | $+5.0\times {10}^{-3}$ |

${\u03f5}_{7}$ | $-16.7\%$ | $+38.6\%$ | $+10.0\%$ | $-19.7\%$ | $-0.76\%$ | $-0.25\%$ | $+1.7\times {10}^{-4}$ |

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**MDPI and ACS Style**

Cai, C.
A New Gaskinetic Model to Analyze Background Flow Effects on Weak Gaseous Jet Flows from Electric Propulsion Devices. *Aerospace* **2017**, *4*, 5.
https://doi.org/10.3390/aerospace4010005

**AMA Style**

Cai C.
A New Gaskinetic Model to Analyze Background Flow Effects on Weak Gaseous Jet Flows from Electric Propulsion Devices. *Aerospace*. 2017; 4(1):5.
https://doi.org/10.3390/aerospace4010005

**Chicago/Turabian Style**

Cai, Chunpei.
2017. "A New Gaskinetic Model to Analyze Background Flow Effects on Weak Gaseous Jet Flows from Electric Propulsion Devices" *Aerospace* 4, no. 1: 5.
https://doi.org/10.3390/aerospace4010005