The above section demonstrates how to determine the combined flowfield properties from background and jet flows, which are described by the seven parameters, ${n}_{0}$, ${S}_{0}$, ${T}_{0}$, ${n}_{b}$, ${S}_{b}$, ${T}_{b}$, and ${\alpha}_{0}$. The reverse problem is more practical and important: from limited experimental measurements, is it possible to recover the seven parameters? If DSMC simulations are adopted for this parameter recover work, then a large number of different parameter combinations must be assumed, and the simulations shall be performed in trial-and-error, just like shooting in dark, to find the combinations with the best match between the simulated results and the measurements. The procedure must be very lengthy because there is no information about towards which direction shall the parameters be tuned. This section aims to address this question with a new approach.

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

Because there are seven parameters describing the rarefied background and jet flows, only seven relations or equations are needed, with a minimum of seven experimental measurements.

The critical step in this new method is to linearize Equations (

7)–(

9), for the number density and number fluxes. The process starts with good initial estimations on the seven parameters, denoted as

${n}_{0g}$,

${S}_{0g}$,

${n}_{bg}$,

${S}_{bg}$,

${\alpha}_{0g}$,

${T}_{0g}$ and

${T}_{bg}$, where the subscript

$\u201c{g}^{\u2033}$ represents a guessed or estimated value. These guessed values are assumed to have small deviations from the corresponding true values, and a set of small perturbation parameters,

${\u03f5}_{i}$, are introduced:

These seven guessed values will update by multiplying $1+{\u03f5}_{i}$ after each iteration; hence, if these parameters ${\u03f5}_{i}$ continue to decrease, then the guessed values approach to the true parameter values.

Equations (

7)–(

9) and (

20) lead to the following linearized expressions:

where

${A}_{i}$,

${B}_{i}$ and

${C}_{i}$ are expressions with current estimated properties, and their specific expressions are included in the

Appendix; the right hand side terms

${n}_{m}(X,Y)$,

$\left({n}_{m}{U}_{m}\right)(X,Y)$ and

$\left({n}_{m}{V}_{m}\right)(X,Y)$ represent measured number density, number fluxes along the

X- and

Y-directions;

${\left({n}_{m}\right)}_{g}(X,Y)$,

${\left({n}_{m}{U}_{m}\right)}_{g}(X,Y)$ and

${\left({n}_{m}{V}_{m}\right)}_{g}(X,Y)$ represent the corresponding density and number flux values computed by using Equations (

7)–(

9) with the seven estimated parameters.

The expression for the mixture temperature is based on those expressions for the mixture number density and number fluxes, it is unnecessary to introduce a new perturbation parameter which shall be a combination of the other seven small perturbation parameters. It is inconvenient to use the temperature result for perturbations because the expressions are rather complex.

To determine the seven parameters can be achieved with different approaches. For example, by measuring the number densities, and number fluxes along the X- and Y- directions, at several points. The coefficients ${A}_{i}$, ${B}_{i}$, and ${C}_{i}$ have different values at each point, due to the different geometry factors, i.e., ${\theta}_{1}$ and ${\theta}_{2}$, and the local flowfield properties.

The new methods take the following steps to recover the seven parameters.

- 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.

The above process is fast, because in the end, it only requires to solve seven linear algebraic equations, and the whole process may require seconds. Most time in the above procedure is spent on computing coefficients

${A}_{i}$,

${B}_{i}$, and

${C}_{i}$. This method requires as fewer as seven measurements, and the whole flowfield can be further computed by using the seven recovered parameters for the jet and background flows. The three recovered parameters for the jet can be used to compute the pure jet flow (in a true vacuum condition) very conveniently by adopting Equations (

2)–(

5). It is also possible to estimate the background flow effects on the thruster by using the four recovered parameters for the background flows and Equations (

13)–(

19). For example, the amount of background flow may entrain or enter the thruster exit, if the exit area is available; and the extra force that the background flow may create on the devices, if the devices’ surface geometry information is available.

#### 3.2. Test Cases and Discussions

A test case is presented here to demonstrate the procedure. The jet and background flows are assumed as argon with the following parameters:

${n}_{b}/{n}_{0}=0.8$,

${S}_{0}=1.2$,

${S}_{b}=0.8$,

${\alpha}_{0}={45}^{\circ}$,

${T}_{0}=200$ Kelvin and

${T}_{b}=300$ Kelvin. The number density at the jet exit is

${n}_{0}=1\times {10}^{10}$ ${m}^{-3}$, and it is used as a reference to normalize the background density. Those seven numbers are selected at random for demonstration.

Table 1 shows data for density, number fluxes along the

X- and

Y-directions, at nine points with a radius of 3 m from the jet exit center, but with different relative angles. The jet exit height is 1 m. The values in

Table 1 are assumed as accurate measured data.

The following initial estimated parameters are assumed to start the recover procedure:

${n}_{0g}=0.6$,

${S}_{0g}=1.0$,

${n}_{bg}/{n}_{0g}=1.2$,

${S}_{bg}=1.0$,

${\alpha}_{0g}={25}^{\circ}$,

${T}_{0g}=250$ Kelvin, and

${T}_{bg}=250$ Kelvin. The following properties are taken from

Table 1 and assumed known: number densities at points A, C; number fluxes along the

X-direction at points D, E, H; and number fluxes along the

Y-directions at points H, I. These values form the right hand side terms for the seven equations: two from Equation (

21), three from Equation (

22), and two from Equation (

23).

Table 2 shows the development histories for the absolute values for the seven parameters, as shown, after fifteen iterations, the seven estimated parameters are sufficiently close to the true parameters.

Table 3 shows the parameter convergence histories for

${\u03f5}_{i}$, each of them is correction percentage to the values from the previous iteration. As illustrated, the differences among the guessed and true values are negligible after the fifteen iterations.

Figure 15,

Figure 16 and

Figure 17 show the corresponding developing profiles for the jet centerline number density and number fluxes along the

X- and

Y-directions. Those profiles are computed with updated parameters. These three figures indicate that after three rounds of iterations, the profiles are actually close to the true profiles. At farfield, the profiles merge into the background flows. As discussed in

Section 2, the background flow can distort the jet flowfield patterns significantly, or even create totally opposite trends.

Figure 18,

Figure 19 and

Figure 20 show the corresponding profiles for the number density and number fluxes along the arc

$R=3$ m, where the data from

Table 1 are presented as solid points. As shown, the convergence develops quite fast, the differences are minor among the results obtained by using the parameters obtained after four and ten iterations. It is reasonable to conclude that by using the parameters obtained at the fifteen iterations, the whole flowfield parameters recovered accurately.

Some discussions are offered at the end of this section.

First, it is interesting to compare how this model and the DSMC simulations can recover the seven parameters from very limited measurements. DSMC simulations must start with specific given parameters, it is reasonable to adopt five values for each of the seven parameters, e.g., ${n}_{0g1}<{n}_{0g2}<{n}_{0g3}<{n}_{0g4}<{n}_{0g5}$; ${S}_{0g1}<...<{S}_{0g5}$; ${n}_{bg1}<...<{n}_{bg5}$; ${S}_{bg1}<...<{S}_{bg5}$; ${\alpha}_{0g1}<...<{\alpha}_{0g5}$; ${T}_{0g1}<...<{T}_{0g5}$, and ${T}_{bg1}<...<{T}_{bg5}$. Then there will be ${5}^{7}$ = 78,125 sets of parameter combinations from which to obtain the optimal parameter set, if the DSMC simulations are used. The simulation process may require one or several years. However, if the method developed in this paper is adopted, it may only take 10 seconds, with the same computer. This fact illustrates the significance of this new model.

Secondly, this work presents linearized density, and number fluxes along the

X- and

Y-directions. Is it feasible to derive expressions for the number fluxes along a radial direction? The answer is affirmative. If the angle is assumed as

λ, then Equations (

22) and (

23) can be combined linearly as:

Correspondingly, measurements of the number flux along the radial directions are required to form linear equations.

Thirdly, is it feasible to derive a linearized equation for the temperature from Equation (

10) or even one equation for pressure? The answer is affirmative. However, the process is lengthier and final results are more complex. The effort will be more demanding, and is not one concern of this feasibility study. This paper aims to demonstrate the method with simple examples.

Fourthly, what if there are more measured data points? They can help further develop more reasonable models. It is reasonable to assume there are at least two groups of highly rarefied background flows in a vacuum chamber. A group of molecules are from the chamber walls with a room temperature, and the other groups of molecules are from pumps with a low temperature. An advanced but similar model can be developed with four additional parameters for a pump,

${n}_{p}$,

${S}_{p}$,

${T}_{p}$, and

${\alpha}_{p}$, where subscript “

$p$” represents pump related property. There will be a total of 7 + 4 = 11 parameters involved in this new model. The new equations corresponding to Equations (

7)–(

9) shall include extra four new parameters. Each of the linearized Equations (

21)–(

23) shall include four extra coefficients. The expressions will be more complex, the effort will be more demanding; however, the procedures are the same as those described in this paper.

Fifthly, these approaches and results are superior than results obtained by using the Buckingham PI theorem. The PI theorem can only determine possible numbers of non-dimensional parameters, and possible parameter combinations. However, the PI theorem can not obtain detailed formulas with all factors explicitly embedded, like Equations (

7)–(

9).

Sixthly, these new approaches can conveniently explain the reasons for differences among measurements, even for the situation with the same nozzle exit parameters but in different vacuum chambers. Due to the geometry differences (e.g., distances and relative angles from the pumps) among ground vacuum chambers, or within the same chamber but different plume firing directions, the background parameters (e.g., the relative angle ${\alpha}_{0}$ ) may vary. The measurements may be quite different because they are actually for the mixture of jet and background flows, not only for the jet. With these new approaches presented in this paper, the recovered jet parameters $({n}_{0},{U}_{0},{T}_{0})$ shall be almost identically or quite close. It is also feasible to determine some special chamber characters for a ground vacuum chamber by performing extensive experimental tests and analysis, or even develop general guidelines applicable to many ground vacuum chambers.

Probably the most important question is, can the approaches applicable to more realistic problems, e.g., dilute neutral gas with collisions, or even dilute plasma flows? For the first problem to determine the mixed flowfield with given parameters, simple DSMC or Particle-In-Cell simulations can be performed, or the analytical results from this study can be used for crude approximations. For the second problem to recover the flow parameters from limited measurements, the same recovery procedure presented in this paper, can be applied first by assuming the flow is collisionless neutral or plasma flows. A set of 7 parameters can be recovered. The true parameters must be within the neighborhood of these recovered parameters in the parameter space. This is because the jet and background flows are assumed dilute and weak. A few more particle simulations, either DSMC (for collisional neutral gas), or DSMC+PIC (Particle-In-Cell) (for dilute plasma flows), with parameters from the neighborhood region can lead to the most close parameter set. It is feasible to offset these seven recovered parameters, e.g., each by $\pm 10\%$, then there are ${2}^{7}=128$ sets of parameters, with which DSMC or DSMC/PIC simulations can be performed and they may yield more accurate results. By doing this, the simulation amount can reduce by a factor of $78125/128=610$ times.

In this end, it shall be emphasized that the methods from this work do not intend to replace DSMC and any Computational Fluid Dynamics (CFD) simulations, instead, they can help simulations by providing a proper starting point, and significantly reducing the amount of work by narrowing down the parameter ranges. Similarly, this work does not compare the differences among existing experimental methods, nor intends to propose a new experimental method. Instead, the methods from this paper can help analyze the measurements, and perform proper predictions which may be very difficult to measure.