On the formation of an external cluster flow in a supersonic jet

. In this work, the influence of the cluster formation process on the gas dynamics of a supersonic flow is studied by the method of electron-beam visualization. Corrections to the model for describing the transverse dimensions of supersonic flows under conditions of developed condensation are presented. Possible causes and conditions for the formation of a secondary cocurrent flow are analyzed. The influence of background gas penetration into the clastered flow has been studied. The conditions for the occurrence of an anomalous afterglow of a clastered flow upon initiation by an electron discharge are studied.


Introduction
Studies of the nature of the outflow of gases from nozzles of various configurations have been carried out for many years [1][2][3][4][5][6].A range of verified theoretical and empirical models has been obtained that describe the processes occurring in gas flows and allow modeling and predicting the gas dynamics of the outflow with sufficient accuracy without the need for additional laboratory experiments.
In turn, a great potential for study is the direction associated with the formation of condensed particles (clusters), one of the sources of which is a supersonic flow.Clusters united by weak van der Waals forces have unique properties, which is reflected in fundamental research and applied applications in various technologies [7][8][9][10][11][12].
In the second half of the last century, O. Hagena and other authors [7][8] obtained an empirical model relating the average size of clusters formed in the flow 〈S〉 with the initial parameters of its outflow from the nozzle: )  , Г * =  0  0  eq   0 − , (1) where Г * is the dimensionless Hagen similarity parameter, deq = c•d*/tgα is the equivalent diameter of the nozzle, d* is the diameter of the critical section of the nozzle, α is the opening half-angle of the diffuser part of the nozzle, P0 and T0 are the stagnation pressure and temperature, a, b, k0, c, q, s, i = γ/(γ-1) are numerical constant coefficients determined empirically.According to studies [9], the proportion of condensate in the flow increases to a limiting value (~30%) with the development of condensation and remains constant with further growth of clusters in size.Proceeding from this, the main fraction of particles in a clustered flow are monomeric particles (atoms or molecules), as in flows without a condensed phase.
However, it should be noted that until recently, insufficient attention has been paid to the issues of the influence of condensation on the gas dynamics of flows.It was found in [6] that the development of condensation in the flow at certain gas-dynamic parameters of the outflow leads to the formation of a cocurrent flow external to the "traditional" jet. Figure 1 shows visualizations of argon outflows from supersonic nozzles under conditions of small (a) and developed (b) condensation.The resulting photographs have been edited in a form that makes it possible to represent on the general frame individual sections of the jets, which are distinguished by a significant difference in the intensity of the glow.When flowing from supersonic nozzles in the studied regimes, a spindle-shaped underexpanded jet with an Xshaped configuration is observed in the region of closure of lateral hanging shocks.The flow shape in Fig. 1, a can be called "traditional", because it is fixed in the works of many researchers [1][2][3][4][5], and is also characteristic of various conditions.At the same time, in E3S Web of Conferences 459, 01002 (2023) https://doi.org/10.1051/e3sconf/202345901002XXXIX Siberian Thermophysical Seminar the regime with developed condensation (Fig. 1, b), in addition to the "traditional" jet, an external flow is observed, which has a similar shape, but with much larger dimensions, a complex structure, and a weakly damped glow when the radiation is excited by an electron discharge.
It has been established that this flow is not observed behind large-diameter sonic nozzles (Fig. 1, a), at low stagnation pressures, and also in flows of weakly condensing gases [13].It was suggested that this flow is formed as a result of the ability of heavy clusters with high momentum to overcome the hanging shock waves of the "traditional" jet.This assumption was also confirmed in [14][15].This flow depends on the processes occurring in the initial jet and is generated by them, which is why in [6] it was called the "cluster wake".
In this paper, the influence of the condensation process on the gas dynamics of a supersonic flow is considered using electron-beam visualization, the process of background gas penetration into a clasted supersonic flow is studied, and possible causes and conditions for the formation of the detected secondary flow and its anomalous afterglow are analyzed.

Experimental equipment
The work was carried out on the gas-dynamic stand LEMPUS-2 of the Department of Applied Physics of the Novosibirsk State University.A detailed description of the stand is given in [16].The LEMPUS-2 stand is a vacuum chamber equipped with a set of vacuum evacuation means: fore-vacuum oil-free, high-vacuum turbomolecular and helium cryogenic pumps that provide compensation for the gas flow through the nozzle when the pressure in the background space is selected in the range of 10 -4 -10 2 Pa.Inside the main vacuum volume there is a pre-chamber with a nozzle of the selected configuration.The generated outflow is visualized using a high-voltage, well-focused electron beam [17].Registration of initiated radiation is performed using the selected diagnostic system installed behind the quartz optical window of the vacuum chamber.The variants of the used optical paths are shown in Fig. 2.
For spectral analysis of the emission products, the radiation is collected with a quartz lens and a waveguide at the entrance slit of the Ocean Optics USB4000 spectral instrument.The localization of the recorded area of the studied gas object, due to the size of the spectrometer slit, the cross section of the waveguide and the diameter of the focused electron beam, taking into account the optical image magnification factor, does not exceed 1 mm 3 .The presence of a coordinate mechanism for moving the nozzle block and the supersonic jet flowing through the nozzle, makes it possible to shift the localization of measurements to different areas of the gas object.The coordinate mechanism of the lens makes it possible to additionally shift the localization of measurements relative to the electron beam within the size of the optical window (Fig. 2, a).The possibility of independent movement of not only the gaseous object, but also the recording optical system relative to the stationary electron beam makes it possible to carry out measurements in various modes of changing parameters.The replacement of the spectral complex (Fig. 2, a) with a Nikon D7200 camera or a PCO Panda 4.2 bi UV UV camera (Fig. 2, b) makes it possible to register visualized outflow patterns.The coordinate mechanism of the nozzle pre-chamber makes it possible to fix the point of intersection of the optical axis y and the axis of the gaseous object x (Zjl = 0) to eliminate optical distortion of the recorded image along the z axis (Fig. 1,b).Observation on the frames recorded by the camera of structural elements of the installation of known sizes allows, taking into account the optical distortion of objects, to register the sizes of gas flows.This technique is described in detail in [18].
The ambient gas pressure P∞ in the vacuum chamber, measured by Agilent Technologies CDG-500 and Helix Technology Micro-Ion Plus gauges, varies under control in the range of 10 -4 -10 2 Pa with an error of < 1%.The pressure in the nozzle pre-chamber P0 (stagnation pressure) is set in the range of 3-1000 kPa and is measured using a membrane absolute pressure sensor Siemens Sitrans P7MF1564 (measurement error 0.3%).To measure the stagnation temperatures T0 and gas in the background space T∞, the stand is equipped with thermometers (measurement error ~ 0.1%) installed on the nozzle pre-chamber and on the wall of the experimental stand, respectively.Measurements of the electron beam current with an error of < 2% are made at two points: at the exit of the electron gun (set electron current) and at the collector located at the bottom of the vacuum chamber (current of electrons passing through the gas object).
The additional leakage system installed in the expansion chamber makes it possible to change the composition of the atmosphere in the vacuum chamber by adding a jet of any other type of gas to the residual working gas.Control over the percentage of gases in the background space is carried out using the ExtorrXT-300M quadrupole analyzer (0 -300 amu, measurement error < 1-2%) with its own pumping system.

Influence of the condensation process on the dimensions of the supersonic flow
The technical equipment of the LEMPUS-2 gasdynamic stand [16] made it possible to carry out series of measurements of the radius of a supersonic argon jet from a supersonic conical nozzle (d* = 0.24 mm, α = 12.3°, diffuser part length L = 3.0 mm, nozzle outlet diameter da = 1.55 mm) when one of the initial outflow parameters (P∞, P0, T0) is varied and the others are kept constant.The expiration parameters used in these series are presented in Table 1.Fig. 3 shows visualizations of outflows with variations in ambient pressure P∞ (a-b), stagnation pressure P0 (cd) and stagnation temperature T0 (e-f).
The work [1] based on [2][3] proposed a model to describe the diameter of the maximum jet cross section: where rm is the radius of the jet in the maximum section, ra is the radius of the outlet section of the nozzle, k* is a constant numerical coefficient, N = P0 / P∞ is the degree of expansion of the jet in pressure,  ̅ is the degree of expansion of the nozzle in area, θ+ is the characteristic angle jet expansion.However, it should be noted that this model was obtained under conditions of gas outflow in the absence of condensation.Measurement of the radius of a "traditional" clustered jet in various flow modes (Table 1) using the method described in [18] made it possible to establish [19] that during the development of the condensation process in the flow, the coefficient k*, which is constant in the absence of condensation or in the absence of its change (for example, with variation of P∞), increases with an increase in pressure P0 and a decrease in temperature T0, hence, with an increase in the size of clusters and the proportion of condensate.Fig. 4 shows the change in the coefficient k* as a function of the average cluster size 〈S〉 (1) obtained for various supersonic conical nozzles.The obtained experimental points form a dependence that increases from the initial value and has a saturation It should be noted that the data sample in this study is small, due to the set of supersonic nozzles used, and the validity of the model has been proven only in a limited range of stagnation temperatures, geometric dimensions and shapes of supersonic nozzles, and only for one gas, argon, which is why the proposed correction cannot be considered completely verified.At the same time, it can be asserted with a sufficient degree of certainty that, in the presence of condensation, the model [1] can be used to describe the sizes of the formed flows only with the introduction of the proposed correction.

Influence of the condensation process on the background gas penetration into the flow
The work [20] presented a detailed study of diffusion processes in the mixing zone of a low-density supersonic jet.However, no attention was paid to the influence of all features of the condensation process on this process, since the effect of the formation of a "cluster wake" was discovered much later.
The technique proposed in [20] was used to analyze the background gas penetration into the supersonic flow.According to this technique, an admixture of another gas is added through the leak into the background gas, which consists of residual particles of the jet gas that have not been evacuated from the vacuum volume.The pressure of the mixture of gases in the medium surrounding the jet is defined as the sum of the partial pressures of the residual gas of the jet and the mixed gas of the impurity.During the experiment, the ratio of the mixture components in the background medium is maintained constant.
Nitrogen, which has a close scattering cross section, was chosen as a reference.In this case, it was assumed that the difference in the masses of the working and impurity gases does not have a significant effect on the collisional mechanism of penetration.To register the radiation of a nitrogen impurity in an argon flow, the most intense transitions of a neutral molecule N2 C 3 Пu → B 3 Пg (0-0, 337.1 nm) and ion N2 + B 2 ∑u + → X 2 ∑g + (0-0, 391.4 nm) was chosen.
To study the penetration process, an argon flow was formed with the formation of large clusters to create conditions for the formation of a "cluster wake", shown in Fig. 5.At distances Xnl = 39.4 and 108.4 mm, transverse profiles of changes in the nitrogen emission intensity were recorded, which, after excluding the contribution of the source gas to the emission, were converted into the values of the background gas penetration degree Rp, which are taken as 100% far beyond the flow in the background medium.The distances Xnl are chosen from considerations of the fundamental difference between the corresponding sections of the formed flow: Xnl = 39.4 mm corresponds to a section approximately in the middle of the first spindle-shaped "barrel", while Xnl = 108.4mm corresponds to a transverse section behind the first "barrel", in the area of distribution of the "cluster wake".A comparison of the transverse profiles of the degree of penetration Rp for the indicated sections is shown in Fig.
6 [21].Based on the comparison with the recorded visualization of the outflow (Fig. 5), profile (a) reaches the background value (point A) almost immediately after the localization of measurements goes beyond the boundary of the "traditional" jet.It should also be noted that the profile curve is smooth, has no kinks and features.In turn, profile (b) reaches the background value at point C, but there is an inflection at point B. Based on the comparison with the visualization, point B presumably demonstrates the exit of the localization of measurements beyond the border of the second "barrel" into the area of the "cluster wake", which also prevents penetration.In turn, point C, which indicates the achievement of the background value profile, is located immediately outside the visualized "cluster wake", which confirms this assumption.It should also be noted that in the region of the second "barrel", which is more rarefied in relation to the first one, the background gas is present in a certain fraction and on the jet axis, which is consistent with [20].Thus, we can conclude that the "cluster wake" is a gas-dynamic unit, and not an optical effect associated with the features of excitation of flow particles by an electron discharge.At the same time, these measurements confirm the fact that particles of the "cluster wake" collide with the background gas, which, similarly to the "traditional" jet, prevents it from penetrating into the flow, but with a different intensity.

Electron-discharge-initiated radiation of a clustered flow
It was shown in [22] that the co-flow ("cluster wake") formed in clustered jets has a weakly fading glow at a large distance outside the exciting electron beam, which differs from the glow of a "traditional" jet.For a detailed study of the radiation processes of a clustered flow, the mode of argon outflow from a supersonic nozzle under conditions of developed condensation with the formation of a visible "cluster wake" was chosen, the visualization of which is shown in Fig. 7.When performing spectral measurements in the presented outflow, it was found that in the studied wavelength range (280-900 nm) the devices do not register the radiation of "cluster wake" particles in the region of direct excitation of the flow (on an electron beam, Xbl = 0 mm) at any distance from nozzle exit (Fig. 8,a).At the same time, this radiation was detected when the measurement area was shifted downstream by 5 mm or more (Fig. 8, b).This means that the radiation in the "cluster wake" region occurs with a time delay [23].However, it should be noted that in the region of radiation initiation (on an electron beam), a large contribution to the radiation is made by short-lived transitions with a high intensity, and therefore the radiation of the "cluster wake", when the profiles are normalized to the intensity maximum, can turn out to be an insignificant addition.Therefore, it cannot be unambiguously stated that there is no contribution from fast transitions in the radiation in the "cluster wake" region.
It was also found that the radiation in the "cluster wake" region is observed only on individual argon lines (Fig. 8, b), located mainly in the yellow-green part of the spectrum, which is consistent with the previously discovered effect of anomalous amplification of radiation initiated in clustered flows [24].To clarify the conditions for the appearance of luminescence in the "cluster wake", measurements were made of the average lifetime of flow particles in an excited state τ [21].The data obtained were divided into two groups with a noticeable difference in the values of τ.It turned out that excited particles emitting radiation at wavelengths at which the presence of a "cluster wake" is most noticeable (yellow-green region of the spectrum) in a clustered flow have significantly longer lifetimes in the excited state.
It was found in [22] that the glow effect of the "cluster wake" is not observed in the absence of background gas, which is why it is of interest to check the role of the composition of the background gas on the glow initiation.The results of registering the glow of the "cluster wake" during the outflow of argon into an atmosphere of various compositions are shown in Fig. 9. From the results obtained, it follows that when a clastated argon flow flows into an atmosphere of various composition, not only a change in the color and intensity of the "cluster wake" glow is observed (in the case of adding nitrogen or helium impurities to the background gas), but also its complete disappearance in the visible spectral range in the case of adding an oxygen impurity to the background gas.It is noteworthy that, due to the difference in the scattering efficiency of the flow particles, when the atmosphere of different composition flows out into the atmosphere, a change in the size of the formed flow is also observed with identical initial outflow parameters.It should also be noted that when studying the process of background gas penetration into a clustered flow outflowing into an atmosphere with an admixture of oxygen, the result was an identical distribution, demonstrating the resistance of the "cluster wake" to background gas penetration into the flow (Fig. 6).However, based on the visualization of the outflow of an argon flow into the atmosphere with the addition of an admixture of oxygen (Fig. 9, d), the emission of a "cluster wake" in this outflow is not observed at all, which may be due to the effect of collisional quenching of particle radiation by oxygen noted in [25].associated with the transfer of energy to non-radiative levels of the molecule O2.

Conclusion
Let us analyze the reasons why the secondary flow ("cluster wake") visualized in this work was not detected earlier.To do this, consider the conditions under which the present research was carried out.
First, supersonic nozzles with a small diffuser angle were used.This means that the formed supersonic jets are substantially compressed compared to the jets behind the sonic nozzles.According to the calculations performed by P.A. Skovorodko [9], the active process of condensation proceeds even in the nozzle, in its diffuser part.
Secondly, despite the high parameters of the stagnation pressure in the nozzle pre-chamber, the pumping system of the experimental facility ensured a low level of background gas pressure in the medium surrounding the jet.Under conditions of strongly underexpanded jets, a flow of large clusters was formed, moving along the streamlines within the total angle not exceeding 60 degrees [9].The lateral boundaries of the primary supersonic spindle-shaped jet were quite wide and blurred, which created favorable conditions for heavy clusters to overcome them.
Thirdly, based on the totality of spectral measurements and visualizations, we can conclude that the composition of the atmosphere surrounding the outflowing flow has a direct effect on the emission of "cluster wake" particles, which indirectly confirms the assumption [22] about the stage-by-stage process of energy transfer in the flow from flying excited clusters to particles of the background environment.
The excitation rate of the background particle in this case can be proportional to the density of the background gas and clusters, as well as the probability of energy exchange (the presence of a suitable energy level for energy transfer).Since the glow intensity of the "cluster wake" weakly depends on the position of the electron beam, it can be assumed that the average lifetime of the excited state of the cluster is long enough for the cluster to overcome the distance to the glow region.Consequently, the background pressure P∞ was low enough for the formation of an underexpanded supersonic jet with diffuse side boundaries, but also high enough for the required jet compression and the presence of static particles in the background space for collisions.
Based on the combination of the above features, it can be assumed that under conditions of outflow into a highly rarefied medium (vacuum), clusters freely expanding from the primary flow would not encounter significant resistance from the surrounding background gas.The flow clusters excited by the electron beam would not undergo collisions with background particles and would not transfer energy to them, therefore, due to the presence of long-lived levels in the clusters, the radiation would be dispersed over a wide region of the surrounding medium.On the other hand, during the formation of a strongly underexpanded jet isolated from the background medium, the lateral shock waves would be well concentrated, the mixing zone of the primary supersonic jet would be limited to an extremely narrow local area, so that it would be extremely difficult for even large clusters to overcome them.It should be noted that only a combination of factors can lead to the development of condensation in the flow, and, consequently, to the formation of a "cluster wake" in it.
On the basis of the photometric and spectral series of measurements carried out, knowledge was obtained about the conditions for the formation and emission of a "cluster wake" formed when heavy clusters pass through lateral shock waves.The effect of broadening of the "traditional" jet and "cluster wake", associated with the clustering of particles in the flow, made it possible to present a correction model for the empirical model [1], which describes the transverse dimensions of flows formed behind supersonic nozzles.On the basis of the photometric measurements carried out at various initial parameters of the outflow, the conditions for observing the anomalous structure of the "cluster wake" were discussed.In spectral studies of the "cluster wake" afterglow, a process was discovered related to the mechanisms of interaction of clusters formed in a supersonic jet with the surrounding background, the processes initiating the radiation of the "cluster wake" were determined, the assumption of the participation of large clusters in the process was substantiated, and it was shown that this process is selective and occurs only under certain conditions.
The work was performed using the shared equipment center "Applied physics" of the NSU Physics Department with the financial support of Russian Science Foundation (project number 22-11-00080).

Fig. 2 .
Fig.2.Schematic diagram of the measuring section of the gasdynamic stand LEMPUS-2[16] for spectral measurements (a) and photo-visualization (b).1nozzle pre-chamber, 2supersonic gas flow, 3electron beam, 4quartz short-focus lens in an insulated box, 5waveguide, 6 -Ocean Optics USB4000 spectrometer, 7camera Nikon D7200 or PCO Panda 4.2 bi UV, ↔↕ are the available directions of movement, Xnb is the distance along the x axis between the nozzle exit and the electron beam axis z (nozzlebeam), Xbl is the distance along the x axis between the electron beam axis z and the optical axis y (beam-localization), Xnl = Xnb + Xbl is the distance along the x axis between the nozzle exit and the optical axis y, Zjl is the distance along the electron beam axis z between the x axis of the jet and the optical axis y (jetlocalization).

Fig. 4 .
Fig. 4. Change in the coefficient k* for the "traditional" jet (2) in the series of measurements for various supersonic nozzles depending on the average size of clusters in the flow 〈〉.
lower limit is determined by a constant coefficient in the absence of condensation[1][2][3], and the upper limit is probably determined by the limitation of the condensate fraction in the jet, as well as the limit of the efficiency of monomer background scattering by cluster particles as their mass ratio tends to infinity.The range of variation of the coefficient k* is determined in the range of cluster sizes achievable in this study, 330 < 〈S〉 < 6200 particles/cluster.Based on the data obtained, dependence (2) under conditions of developed condensation can be represented as: min = 2.05 is the value of the numerical coefficient k* in the absence of condensation (〈〉 = 1),  * max is the maximum value of the coefficient k* in the clustered flow at 〈S〉 → ∞; Δk* =  * max - * min ; q = 1650 is a constant numerical coefficient.

Fig. 6 .
Fig. 6.Comparison of the transverse profiles of the fraction of the penetrating background gas Rp recorded at distances Xnl = 39.4(a) and 108.4(b) mm from the nozzle exit.

Fig. 8 .
Fig. 8.Comparison of the transverse profiles of the normalized intensity of In radiation recorded in an argon flow (Fig. 7) with a shift in the localization of measurements downstream from the initiating electron beam: Xnb = 136.5 mm, Xbl = 0 (a) and 5 (b) mm.

Table 1 .
Parameters of used expansion series.