The present analysis covers a wide range of the involved flow paramters. More specifically, the study was carried out for moderate () and large values () of the aspect ratio , for three pressure ratios corresponding to small (), moderate (), and large pressure drops (), while the rarefaction parameter varies from 0 up to 10, representing gas flows from free molecular regime up to early slip regime. In addition, three indicative values of the dimensionless channel length, namely, , and two values of the temperature ratio , namely, , were considered.
4.1. Isothermal Case
Table 3 shows the hard-sphere values of the dimensionless mass flow rate
for various combinations of
,
,
, and
. As expected, in all cases, the increase in the pressure ratio
(or decrease in the pressure drop), which is the driving force of the studied gas flow, causes a significant decrease in the mass flow rate. Depending on the level of the gas rarefaction, as the pressure ratio
increases from 0.1 to 0.8, the dimensionless mass flow rate
can be reduced by 49–78%. As observed, the geometry parameters
and
have a significant effect on
. In general, the mass flow rate increases either by decreasing the dimensionless length
or by increasing the aspect ratio
. This behavior is justified by the fact that by increasing the dimensionless length or decreasing the aspect ratio, the wall effects become more pronounced, enhancing the wall resistance to the induced flow. The effect of the geometry parameters
and
on
remains significant for all examined pressure ratio values. In quantitative terms, the increase in the aspect ratio
from 2 to 5 (2.5 times) causes an increase in
by about 1.07–1.09, 1.5–1.9, and 1.7–2.3 times in the case of
, 5 and 10, respectively. By comparing the present results for the plane geometry with those in [
24], which considers conical pipes, we can conclude that for similar pressure ratio conditions, a similar effect of the aspect ratio
on
is observed. This means that when fast engineering calculations are required, the mass flow rate through diverging conical pipes for pressure ratios different than zero (which are not considered here since the plane geometry is studied) may be estimated, as a rough approximation, by using the available data in literature for straight tubes and modifying them considering the effect of the aspect ratio
on
reported in this work for the plane geometry. However, when accuracy is important, the modeling of the flow through diverging conical pipes is required and suggested.
The data for
, given in
Table 3, confirm the existence of a shallow Knudsen minimum for the case of small values of the aspect ratio
, as well as for all values of the pressure ratio
. However, as the channel becomes more open (higher aspect ratio
), no Knudsen minimum is observed even for the longest considered channels with
10. It is noted that, also in [
18], the authors observed the Knudsen minimum for the same values of
and
assuming
. However, for the shorter channels with
and for all considered values of
, following the behavior previously reported in [
28] for the straight channels, the dimensionless mass flow rate
always increases monotonically with the rarefaction parameter
.
In order to reveal the quantitative differences between the straight channel flows and diverging channel flows in
Figure 2, the ratio
is plotted as a function of
, for
, and
as well as for two values of the pressure ratio
, namely,
. The data for straight channels (
) were taken from [
28], where an extensive database of the dimensionless mass flow for gas flows through straight channels is provided. As shown in
Figure 2, under the same gas rarefaction conditions, the dimensionless mass flow rate of the diverging channel flows is higher than that of the corresponding straight channel flows. It was found that this increase in the mass flow rate in the transition regime is about 1.4–1.7 times for the case of short channels with
and becomes even larger and about 1.6–3.6 times for the longer channels with
.
Table 4 presents a comparison between the numerical values for the dimensionless mass flow rate
obtained from the approximate kinetic approach (see
Section 3.2) and those obtained from the complete 4D kinetic solution. In columns four, six, and eight of
Table 4, the relative difference
of the dimensionless mass flow rates defined as
for various values of
,
,
, and
are reported. As can be seen, the results obtained from the approximate kinetic approach differ significantly from those obtained from the complete 4D kinetic solution. As expected, the relative difference increases as the dimensionless length
decreases or the aspect ratio
increases. It is noted that, by either decreasing
or increasing
, we expect the contribution of the end-effect phenomena to become more pronounced. It is also observed that the increase in the pressure ratio (decrease in pressure drop) leads to a decrease in the relative difference
, which is expected if someone considers that, in this case, the local pressure gradient decreases too. However, the relative difference
remains large in all examined values of
,
,
, and
, with the approximate approach always overestimating the mass flow rate values. It should be mentioned that in [
24], large deviations between the complete solution and the approximate kinetic approach were reported for conical diverging flows into vacuum. A significant improvement in the description of the approximate kinetic approach is expected if someone considers the end-effect phenomena at the channel ends. The end-effect correction can be introduced into the approximate kinetic approach by elongating the length of the channel according to the local gas rarefaction conditions at the channel ends, i.e.,
. The correction lengths as function of the gas rarefaction parameter are available in the literature [
68] in the case of gas flows through a straight channel. Given the fact that, to the best of the authors’ knowledge, the corresponding end-effect correction lengths for diverging channels are not available in the literature, the presented errors in columns five, seven, and nine of
Table 4 are based on the end-effect length values for straight channels. In general, we expect further improvement in the predictions of the approximate kinetic approach after the introduction of the exact end-effect correction values for diverging channels, but this requires performing a dedicated analysis which is beyond the scope of the present study. As can be seen, by introducing the end-effect corrections of the straight channels the results of the approximate kinetic approach are improved significantly. We can conclude that in the case of diverging channel flows, the approximate kinetic approach coupled with the end-effect correction approach for straight channels can be applied to predict the mass flow rate within 15%, for small aspect ratio
and
, as long as
, while this error drops to 10% for longer diverging channels with
and
. Further improvement of the approximate kinetic approach is expected in the case of longer diverging channels with
, with its validity range being extended even for larger aspect ratios
when
.
In addition to the effect of the diverging channel geometry on the mass flow rate, we are also interested in examining the corresponding effect on the distribution of macroscopic quantities of practical interest.
Figure 3 illustrates the axial distributions of the dimensionless pressure
, Mach number
, and dimensionless temperature
along the symmetry axis (
) for
and
, as well as for two indicative values of the gas rarefaction parameter
. Also, the corresponding distributions of straight channel flows (
) are plotted for comparison purposes. The data for the straight channel flows have been extracted from [
28], where the DSMC method was applied. The reason the curves for
are not smooth is attributed to statistical noise inherent in the DSMC method. As it is seen, the axial distributions for
differ not only quantitatively but also qualitatively from that for
. With the increase in the aspect ratio
, the flow starts to obtain the characteristics of the slit flow [
7,
69], namely, a rapid increase in the axial velocity as the flow enters the channel followed by a rapid decrease after some distance from the channel inlet. The decrease in the pressure at the inlet of the channel becomes more rapid as the aspect ratio
increases and is enhanced significantly as the gas rarefaction level decreases (increase in
), where the viscous effects become more important. As the flow enters the channel it is always accelerated, with this acceleration being more pronounced for divergent channels due to the weakening of the lateral wall effect on the flow. To maintain the amount of gas passing through any plane perpendicular to the line of symmetry of the channel, the velocity in the case of channels of constant cross-section increases monotonically along the channel, while in diverging channels, it slows down within the channel to balance the increase in height of the channel and the corresponding decrease in number density. A significant influence of the diverging flow geometry on the temperature profile is also observed. More specifically, a noticeable temperature drop is observed in the regions with a strong increase in velocity and, vice versa, the temperature rises with the decrease in the velocity, as a consequence of conservation of energy.
In
Figure 4 and
Figure 5, we extend the study of the effect of the diverging channel geometry on the distributions of macroscopic quantities, focusing on the variation in the channel length and the pressure ratio. Number density distributions are shown in
Figure 4, while the corresponding axial velocity distributions are given in
Figure 5. The distributions are plotted for two indicative values of
and three values of
, covering the cases of short and long channels, as well as the cases of large, moderate, and small pressure ratios. As can be seen in
Figure 4, the behavior of the number density (and the pressure) presents the same qualitative behavior for all values of
and
. The effect of the aspect ratio
on the number density distributions is enhanced at larger pressure drops (smaller values of
), and longer channels. As observed in
Figure 5, the velocity distributions for
show qualitatively similar behavior for all examined pressure ratios. For the long channel case with
, where the diverging geometry plays a more significant role, we observe not only quantitative differences but also significant qualitative ones among the different pressure ratios. Similarly to the short channel case with
, for moderate and large values of
, the flow always accelerates at the inlet, reaching its maximum value after some small distance from the inlet, and then decreases monotonically until the outlet of the channel. However, for small values of the pressure ratio
and aspect ratio
, the flow through diverging channels maintains the well-known qualitative behavior of gas flows through straight channels, namely, the velocity increases monotonically inside the channel. This behavior is explained by the fact that for
and
, the change in the number density (the number density at the outlet is 10 times less compared to the inlet) between the inlet and the outlet of the channel is larger compared to the change in the channel area (the outlet area is two times larger compared to the inlet area), while for smaller values of the pressure ratio
, the increase in the area compensates for the change in the pressure drop. Likewise, the weaker change in the velocity inside the channel for
compared to
is explained. To obtain a better understanding of the Mach number variation inside the channel for various combinations of
,
,
and
, the average Mach number values at different sections of the channel are given in
Table 5. The values of the Mach number remain well less than 0.3 in all examined cases for
. This is indirect evidence that the compressibility effects are insignificant for this low-pressure drop. While, as the values of the Mach number indicate, they are expected to play a more significant role as the pressure drop increases with a simultaneous increase in the gas rarefaction parameter
.
4.2. Non-Isothermal Case—Effect of the Wall Temperature
In this subsection, we study the flow characteristics in the case that the wall temperature is not equal to the temperature of the gas in the reservoirs, i.e.,
. The influence of the wall temperature is investigated by assuming a large temperature ratio, namely,
. The reason for this choice is twofold: first, under such a large temperature difference, any potential effect of the temperature wall on the flow field is expected to be significant [
27], and second, this temperature ratio covers a wide range of the examined temperature differences in [
27], where the temperature wall effects were examined in the case of straight channels, allowing us to examine whether the observations made previously for the case of straight channel flows into vacuum (
) also hold true for diverging channel flows. The investigation of the wall temperature effect was performed for
, which can be considered as the closest one to that studied in [
27]. Given the well-known inability of the hard sphere model to describe strongly non-isothermal gas flows, due to the lack of an appropriate description of the viscosity variation with temperature, the inverse power law molecular model was applied for the non-isothermal cases. This requires the definition of the viscosity index
, with
, which depends on the working temperature range, and is gas-specific. In the present work, the value of
is considered, which corresponds to argon gas and reproduces the experimental argon viscosity data [
70] within 1.5% over the entire range of the temperature variation examined here. It is noted that, this value is representative also for other gases with similar viscosity index values (e.g., Krypton) [
2]. For consistency reasons, all the isothermal results presented in this subsection have been obtained using the inverse power law molecular model and the same value for
.
In
Table 6, the dimensionless mass flow rate data
obtained for
are compared with the corresponding isothermal ones for
. The comparison covers a wide range of all involved parameters, namely,
,
, and
. By comparing the data for
in
Table 6 and the corresponding hard-sphere ones in
Table 3, it is clearly seen that the effect of the molecular model on
in the transition regime is very weak. A similar weak dependence of the dimensionless mass flow rate on
in the case of isothermal gas flows was reported in [
5] for the case of straight channels. A strong effect of the wall temperature on
is observed at high values of
. Indeed, as observed in the case of straight channels [
27], the effect of the temperature wall is enhanced in low-rarefied atmospheres (large values of
), mainly due to the fact that the higher particle collision rate makes the energy transfer mechanism more effective. For the cases close to the slip flow regime with
, the impact of
on
is increased as the aspect ratio
decreases. However, at smaller values of
, the effect of the temperature ratio becomes weaker.
In
Figure 6, the temperature contours
are shown for
and
and for
,
and
. As can be seen, the temperature wall has a strong influence on temperature contours, which also depends strongly on the geometric characteristics of the channel. It is deduced that, for
, as the gas enters the channel, its dimensionless temperature increases rapidly and always remains above one (the equilibrium value in the reservoir), in contrast to the isothermal case, in which the dimensionless gas temperature is always below one. In addition, for the non-isothermal case, as the aspect ratio increases, the temperature of the gas inside the channel decreases but always remains above one. It is also evident that the increase in the gas rarefaction parameter
enhances the influence of the wall temperature, resulting in higher gas temperature values inside the channel. This can be considered as a consequence of the higher particle collision rate existing at larger values of
compared to the smaller ones, leading to an improvement in the energy transfer mechanism.
Next, in
Figure 7, the corresponding pressure
contours are shown. Inside the reservoirs and far away from the channel ends, the pressure remains uniform and equal to its equilibrium values. It is evident that the pressure for the non-isothermal flow case remains higher and is characterized by smoother changes compared to the isothermal flow case. In the case of
, as the flow passes the channel, the pressure decreases monotonically, while for
, this is the case only for large aspect ratios
. For smaller values of the aspect ratio (
), an overshoot in the pressure is observed at the entrance of the channel. The overshoot becomes more pronounced as both the channel length and the gas rarefaction increase. The overshoot is a consequence of the high inlet temperature that the gas acquires as it enters the channel in the case of non-isothermal flow and greatly exceeds the equilibrium temperature in the reservoirs. It is worthwhile mentioning that, for similar flow conditions, the overshoot in pressure was also pointed out in [
27] for gas flows through straight tubes.
In
Figure 8, for the same set of flow parameters as those in
Figure 6 and
Figure 7, the corresponding axial velocity
contours are presented. The velocity contours are overlaid by the velocity streamlines. The velocity contours under isothermal and non-isothermal conditions present the same qualitative picture. However, the velocity increases noticeably as the wall temperature increases too. The streamlines do not align with the
-axis, highlighting the 2D character of the gas velocity. This may partially explain the deviations mentioned in
Table 4 between the complete kinetic solution and the approximate kinetic solution. The approximate kinetic solution requires the velocity to be one-directional (in the
-direction) at each cross-section, and the flow is pure isothermal.