In this section, the effect of the mixing chamber contraction angle (
φ) on the mixing layer development pattern is carefully explored based on the evaluation parameters presented in
Section 2.2. Furthermore, the fundamental relationship between
φ, the supersonic mixing layer evolution pattern and the performance of the ejector is further revealed.
3.1. Variation of Supersonic Mixing Layer Boundary
The static pressure at the center of the primary flow (Pp) is about 21.3 kPa at the nozzle exit section. In all operating conditions, the state parameters of the primary flow will not change with φ. When φ increases from 0° to 6°, the static pressure of the secondary flow (Ps) at the nozzle exit section gradually increases. Additionally, the static pressure ratio Ps/Pp increases from about 0.75 to 0.89. Therefore, the under-expanded primary flow dictates that the secondary flow boundary will gradually develop up to the wall of the mixing chamber.
As shown in
Figure 8, the position of the secondary flow covered point x
t decreases from 909.1 mm to 552 mm when
φ gradually expands. Compared with
φ = 2°, the non-mixed length
l is reduced by 22.12% for
φ = 6°. The secondary flow boundary is more curved and closer to the axial direction at a mixing chamber contraction angle of 2°, 4° or 6°. Thus, the secondary flow boundary is slightly compressed. There are two main reasons for this phenomenon. First, the increased contraction angle reduces the distance from the primary flow nozzle outlet to the wall of the mixing chamber. Then, the secondary flow boundary is more likely to develop to the wall of the mixing chamber, as in
Figure 8a. Second, a larger mixing chamber contraction angle will bring higher secondary flow static pressure at the nozzle exit section (
Figure 8b). The mixing layer is subject to stronger compression at this time.
Figure 9 shows the mass fraction distribution of the primary and secondary flows, and it can be more clearly visualized that the non-mixed length l decreases with the increase in
φ.
Different to the secondary flow, the boundary of the primary flow is distinct in its development.
Figure 10a shows the development of the primary flow boundary of the mixing layer for different mixing chamber contraction angles. At the beginning of the mixing of the primary and secondary streams, the higher static pressure of the primary stream makes it develop outward first. The development of the primary flow boundary along the radial direction is maximum when the contraction angle of the mixing chamber
φ = 0°. With a gradually increasing
φ, the initial development of the primary flow boundary along the radial direction is severely inhibited (
Figure 10, region A). Additionally, when the mixing layer develops inside the secondary throat, the primary flow boundary tends to smooth out. At this time, the primary flow boundary shows fluctuations due to the penetration of the complicated wave structure.
Figure 10b depicts the relationship between the secondary stream mass flow rate (
ms) and the non-mixed length
l. It can be observed that
ms gradually increases when the non-mixed length
l grows. Thus,
ms has a positive correlation with the non-mixed length
l.
The primary flow boundary developments at
φ = 0° are shown in
Figure 11. Numerical Schlieren image and pressure distribution comparisons are appended to
Figure 11 for better interpretation of the primary flow boundary fluctuations. In the graph, RS
1-RS
6 are oblique shock waves in the primary flow, and TS
1-TS
4 are oblique shock waves within the mixing layer. Obviously, the shock wave strength RS
1 > RS
2 > TS
1 > TS
2 > RS
3 > RS
4 > TS
3 > TS
4 > RS
5 > RS
6. The following analysis explains the fluctuating state exhibited by the development of the primary flow boundary.
Before the oblique shock wave RS1, the primary flow passes through a series of expansion waves, decreasing its static pressure. A sudden increase in static pressure is observed after the shock wave RS1. Therefore, during section AB, the static pressure within the mixing layer is relatively lower, enabling the primary flow boundary to develop toward the wall. Due to the energy and mass transfer between the primary and secondary flows, more high-energy primary flow enters the mixing layer. Additionally, the static pressure in the mixing layer keeps increasing. At spot B, the static pressure of the fluid is equal on both sides of the primary flow boundary. The static pressure is lower on the primary flow side, which turns it into a compressed state (section BC).
For the oblique shock waves RS2 and TS1, the intensity of RS2 is stronger. Additionally, the primary velocity is greater than that in the mixing layer. A larger static pressure value increase is observed in the primary flow after the oblique shock wave RS2. Again, the primary flow boundary develops toward the wall in section CD. For the oblique shock waves TS2 and RS3, the intensity of TS2 is higher, and its static pressure increment to the mixing layer is much greater. In section DE, the primary flow boundary is again compressed.
For the reasons above, in sections EF and FG, the magnitude of the static pressure increment after the fluid passes through the oblique shock wave determines whether the primary flow boundary develops outward or becomes compressed.
3.2. Growth and Pressurization Performance of the Mixing Layer
The thickness and pressure variations of the mixed layer reflect the mass and energy transfer pattern between the primary and secondary flows. Therefore, it is worthwhile to analyze the thickness of the mixing layer and the pressurization performance.
Figure 12 represents the development of the mixing layer thickness (
σ) along the range for different
φ. In the early stages of mixing layer development, i.e., in the range of the non-mixed length l, the mixing layer thickness (
σ) increases linearly. Additionally, it grows in a faster linear fashion as
φ increases. The result is that a large mixing chamber contraction angle promotes the growth of the mixing layer. Noticeably, the convective Mach number also varies in a small range when
φ is changed from 0° to 6°, i.e.,
Mc = 1.2~1.1. Ka A et al. [
23] also found the quasilinear growth pattern of the mixing layer thickness at convective Mach number
Mc = 1.4. The convective Mach number is a significant dimensionless parameter for characterizing the compressibility of a fluid and is defined as
where Δ
U is the velocity difference of two streams across the mixing layer;
a1 and
a2 are the speed of sound for both sides of the mixing layer of the primary and secondary flows.
The mixing layer grows linearly, followed by the fluctuating states of slow growth. Fluctuations are more dramatic as the thickness of the mixing layer grows to a greater
φ. From region A, we can see that the thickness of the mixing layer gradually increases as the non-mixed length
l decreases. It shows that
l is negatively correlated with the mixing layer thickness
δ. The interaction of the complicated wave structure with the mixing layer is the main reason for its thickness fluctuation (a detailed explanation is elaborated in
Section 3.1). Compared with
φ = 0°, when
φ = 2°, 4°, 6°, a new oblique shock wave will be generated to render the structure of the original wave system more complicated (
Figure 13).
The value of
P0δ/
P0s reflects the pressurization enhancement effect of the primary flow to a certain extent. As in
Figure 14, the evolution of the pressurization enhancement (
P0δ/
P0s) along the flow direction in the mixing layer is obtained for different contraction angles. As a result, the pattern of energy variation within the mixing layer can be clearly gained. From that, the pattern of energy variation within the mixing layer can be clearly acquired.
From
Figure 14, compared to
φ = 0°, the pressurization enhancement within the mixing layer is greater when
φ is larger than 0°.
P0δ/P0s decreases when
φ = 4° and 6°, which is different from the later stages of the mixing layer development for
φ = 4° and 6°. The main reason is that the pressurization enhancement from the primary flow is already smaller than the pressure loss caused by the wave structure within the mixing layer. Again, it shows that the energy transfer from the primary flow to the mixing layer is also gradually weakening. In the growth phase of
P0δ/P0s, larger
φ leads to larger values of
P0δ/P0s. On the contrary, in the decreasing phase of
P0δ/P0s, a larger
φ results in a smaller value of
P0δ/P0s. The reason for that is that the contraction of the mixing chamber promotes the mixing and pressurization of the primary and secondary flows. With further development of flow mixing, a larger mixing chamber contraction angle results in more pressure loss. Thus, the pressurization effect in the later stages of mixing layer development is weakened when
φ is larger.
3.3. Performance Variation: Entrainment Ratio and Total Pressure Loss
The development process of the boundary and thickness of the supersonic mixing layer and the pressurization pattern within the mixing layer are described in
Section 3.1 and
Section 3.2, respectively. Directly responsible for the change in the patterns of supersonic mixing layer development is the variation of
φ. Developments in the supersonic mixing layer intrinsically affect the variation of the secondary stream mass flow rate (
ms). Therefore, with the perspective of the supersonic mixing layer, the variation pattern of
ms is explored in this section.
The mixing process of the primary and secondary flows at different contraction angles is depicted in
Figure 15. When the contraction angle of the mixing chamber is larger, i.e.,
φn >
φo, the secondary flow boundary will develop to the wall more quickly. Namely, point n is upstream of point o. Additionally, the non-mixed length
l decreases, as shown in
Figure 8a. Due to the convective and viscous shear effects of the primary and secondary flows, mass diffusion and transfer take place mainly within the mixing layer. For the high-energy primary flow region, the secondary flow can be neglected for the mass transfer into it. Therefore, the secondary flow passes through the nozzle exit section (m-section) and still develops in a contracting flow channel, i.e., the region of mpqn or mpro. The secondary flow increases in velocity and decreases in static pressure along its path as it develops in the contracted flow channel. The flow mixing area of the secondary flow is smaller when
φ is larger, i.e.,
Asn <
Aso. At this time, the secondary flow channel has a greater shrinkage ratio for a value of
Asm/Asn larger than
Asm/Aso, resulting in a higher static pressure (
Figure 8b) and lower velocity of the secondary flow in the m cross section. Additionally, when
φ becomes larger, the minimum circulation area (
At1) also decreases, resulting in a decline in
ms.
The secondary stream mass flow rate is significantly inhibited by a 35.02% reduction when
φ increases from 2° to 6°. However, the primary stream mass flow rate (
mp) is not affected during this process. Consequently, a large mixing chamber contraction angle results in a lower entrainment ratio (ER). The above results can be observed in
Figure 16a. The total pressure loss at different locations is depicted in
Figure 16b for various
φ. As
φ increases from 2° to 6°, the mixing chamber length subsequently decreases, and the total pressure loss at the secondary throat inlet gradually falls. At the same time, the total pressure loss at the outlet is also reduced. At the entrance of the secondary throat, the total pressure loss is the largest at
φ = 2°, which is 2.89 times of the smallest (
φ = 6°). The total pressure loss at the outlet position of the secondary throat is the largest at the mixing chamber contraction angle
φ = 2°, with 1.23 times of the smallest (
φ = 6°). The presence of contraction produces new oblique shock waves (
Figure 13), resulting in additional total pressure loss. Thus, the total pressure loss of the ejector will be greater when the mixing chamber has a contraction angle.