6 The Investigation of the Evolution of Cluster Beam …
83
of the inner region of the nozzle (Fig. 6.4). The growth of a thick boundary layer
inside the supersonic part of the micro-nozzle sharply reduces the actual degree of
expansion of the nozzle. This leads to the decrease in the expansion of the flow in the
nozzle and increase in temperature in comparison with the adiabatic inviscid flow.
For given parameters (Table 6.3), the jet flowing from the micro-nozzle is under
expanded, continues to expand, and accelerate in the open space behind the nozzle.
Immediately after the nozzle exit, a standing shock wave arises, which removes excessive over-expansion of the flow in the jet and propagates downstream of the gas. In
the region x = 0.023–0.025 m, a hanging shock wave falls on the axis of symmetry
and is reflected from it in an irregular manner. Due to the large rarefaction of the
flow and the insufficiently detailed grid, the structure of irregular reflection with a
triple configuration of shock waves is strongly smeared in space and is observed in
Figs. 6.7 and 6.9 in the form of a zone of temperature increase and a decrease in the
Mach number.
In the calculations, a monotonic increase in the mass fraction of the liquid fraction
along the nozzle axis is shown (Figs. 6.5 and 6.6). In this case, behind the zone of
irregular reflection of the standing shock wave from the axis of symmetry in the
region of the new acceleration (expansion) of the flow, the condensation intensity
increases (Figs. 6.5 and 6.6). A strong shock wave arises in front of the skimmer, the
front of which is located at x = 0.029 m, beyond which the temperature rises sharply
(Fig. 6.9). The flow becomes subsonic. In this case, drops of liquid argon behind the
front of this shock wave evaporate almost completely (Fig. 6.10) and almost complete
denucleation occurs. The mass fraction of liquid droplets, the average radius of the
droplet, and the mass concentration of clusters in the flow decrease to almost zero.
Subsequently, new nucleation and condensation growth of argon clusters take place
in the skimmer.
6.5 Conclusions
The mathematical model of gas-dynamic flows with the phase transformations
(condensation and evaporation) is developed. The system of the Navier–Stokes equations is used to describe the flow parameters, and the system of moment equations is
used to describe the parameters of a two-phase medium. A numerical algorithm for
solving the general system of equations is constructed on the basis of the Godunov
scheme with the approximation AUSM+ [38] for solving the Riemann problem.
The developed numerical model was optimized and adapted for the case of pure
argon condensation in a nozzle based on the Hagena’s semi-empirical theory [29,
30]. In numerical experiments, certain values of the parameters of the condensation
model are determined. For example, the values of the coefficient of accommodation
and the nucleation correction factor multiplier are determined.
The processes of argon condensation–evaporation in the micro-system-jetskimmer system for generating cluster beams are studied. The fields of the flow
83
of the inner region of the nozzle (Fig. 6.4). The growth of a thick boundary layer
inside the supersonic part of the micro-nozzle sharply reduces the actual degree of
expansion of the nozzle. This leads to the decrease in the expansion of the flow in the
nozzle and increase in temperature in comparison with the adiabatic inviscid flow.
For given parameters (Table 6.3), the jet flowing from the micro-nozzle is under
expanded, continues to expand, and accelerate in the open space behind the nozzle.
Immediately after the nozzle exit, a standing shock wave arises, which removes excessive over-expansion of the flow in the jet and propagates downstream of the gas. In
the region x = 0.023–0.025 m, a hanging shock wave falls on the axis of symmetry
and is reflected from it in an irregular manner. Due to the large rarefaction of the
flow and the insufficiently detailed grid, the structure of irregular reflection with a
triple configuration of shock waves is strongly smeared in space and is observed in
Figs. 6.7 and 6.9 in the form of a zone of temperature increase and a decrease in the
Mach number.
In the calculations, a monotonic increase in the mass fraction of the liquid fraction
along the nozzle axis is shown (Figs. 6.5 and 6.6). In this case, behind the zone of
irregular reflection of the standing shock wave from the axis of symmetry in the
region of the new acceleration (expansion) of the flow, the condensation intensity
increases (Figs. 6.5 and 6.6). A strong shock wave arises in front of the skimmer, the
front of which is located at x = 0.029 m, beyond which the temperature rises sharply
(Fig. 6.9). The flow becomes subsonic. In this case, drops of liquid argon behind the
front of this shock wave evaporate almost completely (Fig. 6.10) and almost complete
denucleation occurs. The mass fraction of liquid droplets, the average radius of the
droplet, and the mass concentration of clusters in the flow decrease to almost zero.
Subsequently, new nucleation and condensation growth of argon clusters take place
in the skimmer.
6.5 Conclusions
The mathematical model of gas-dynamic flows with the phase transformations
(condensation and evaporation) is developed. The system of the Navier–Stokes equations is used to describe the flow parameters, and the system of moment equations is
used to describe the parameters of a two-phase medium. A numerical algorithm for
solving the general system of equations is constructed on the basis of the Godunov
scheme with the approximation AUSM+ [38] for solving the Riemann problem.
The developed numerical model was optimized and adapted for the case of pure
argon condensation in a nozzle based on the Hagena’s semi-empirical theory [29,
30]. In numerical experiments, certain values of the parameters of the condensation
model are determined. For example, the values of the coefficient of accommodation
and the nucleation correction factor multiplier are determined.
The processes of argon condensation–evaporation in the micro-system-jetskimmer system for generating cluster beams are studied. The fields of the flow