Figure 6 shows the design method of the bow blade and the computational meshes of the two types of bow tandem blades. As shown in
Figure 6a, the bow blade adopted in this paper consists of two second-order Bezier curves of the end wall and a straight line in the middle span of the blade. The control parameters of the bow blade include: the angle of the two end wall curves
, the angle of the middle straight segment
, the spanwise ratio of the two end wall curves
, and the radial height of the second control point of the end wall two curves
. The design method of bow blade has many control parameters, and can flexibly apply different bow blade formations to the blade hub section, the blade middle section, and the blade tip section.
Considering that this paper mainly studies the influence of different forms of bow blades on the flow field structure of supersonic tandem blades, the design method of bow blades is simplified, and the angle of the middle straight section is taken as 0°. The spanwise ratio and the angle of the two curves of the end wall are equal, and the position of the second control point of the two curves of the end wall is the midpoint of the curve. The design parameters of the simplified bow blade are composed of two control parameters: the bow angle of the end wall curve and the spanwise ratio of the end wall curve. For the convenience of research, when taking the spanwise ratio of the two curves of the end wall, the bow height is 0.4. In this paper, three kinds of positive bow blades (bow to the suction surface of the blade) and three kinds of negative bow blades (bow to the pressure surface of the blade) are studied, respectively, to find out the influence on the flow field structure of supersonic tandem blades with different bow angles. The bow angle corresponding to three positive bow blades and three negative bow blades is 10°, 20°, and 30°, respectively. For the convenience of description, PB10, PB20, PB30 and NB10, NB20, NB30 are used to represent the three positive bow blades and the three negative bow blades, respectively.
4.1. Influence of a Negative Bow Blade on Supersonic Tandem Rotors
Figure 7 shows the comparison of isentropic efficiency and total pressure ratio characteristics of NB10, NB20, and NB30, and
Table 3 shows the comparison of design point performance and the stall margin of NB10, NB20, and NB30. It can be seen that, compared with the original tandem rotor (ORG), the negative bow blade increases the pressure ratio and efficiency at the design point, and the mass flow rate at the design point decreases slightly.
The efficiency of the design point of the tandem rotor with NB10 is increased by 0.37%, the stall margin of the rotor is increased, and the stall margin reaches 20.71%. With the increase of the negative bow angle, the stall margin of the tandem rotor gradually decreases, and the stall margin of the tandem rotor with NB30 decreases to 13.52%. From the comparison between the isentropic efficiency and the total pressure ratio characteristics, it can be seen that negative bow blades improve the efficiency and pressure ratio of the tandem rotor under all operating conditions, and improve the flow field structure of the tandem rotor. With the increase of the bow angle, the efficiency and total pressure ratio of the tandem rotor under the same mass flow conditions are improved more significantly.
Figure 8 shows the comparison of the spanwise distribution of the total pressure ratio and the efficiency of NB10, NB20, and NB30. Compared with ORG, negative bow blades significantly increase the efficiency and total pressure ratio of the rotor from a 20% span to a 85% span. With the increase of the bow angle, the increase in efficiency and total pressure ratio from a 20% to a 85% span is more significant. In addition, negative bow rotors deteriorate the flow field structure in the blade tip region, and at the same time, negative bow rotors also increase the total pressure loss of the hub end wall. With the increase of the bow angle, the total pressure loss in the blade tip region and the hub region gradually increases, and the decrease degree in the total pressure ratio also increases gradually.
The negative bow blades increase the end wall loss of the case and hub, but improve the performance of most spans of the rotor blade. Therefore, the overall performance of the tandem rotor is improved, the average total pressure loss at the rotor outlet is reduced, and the average efficiency of the rotor is increased. This is consistent with the affect mechanism of negative bow blades on the flow field of the transonic single rotor. Previous studies have shown that negative bow blades generate pressure from the blade towards the upper and lower end wall by exerting radial force on the fluid in the blade channel. It is conducive to the migration of low-energy fluid near the middle of the blade to the end wall, which improves the flow field in the middle region of the blade span, and increases the blockage in the end wall. Therefore, the flow in middle span is improved, the efficiency in the middle span is increased, and the efficiency in the hub and case is decreased. However, the total pressure loss in the middle span of the original supersonic tandem rotor is larger and the efficiency is lower, so the three kinds of negative bow blades (NB10, NB20, and NB30) can obviously improve the flow field at the middle span of the blade.
Figure 9 shows the comparison of spanwise distribution of the inlet flow angle and the axial velocity density (AVD) of NB10, NB20, and NB30. Compared with ORG, the inlet flow angle of the tandem rotor with NB10 is reduced about 1° across the full span, and the flow moves along the direction of the negative incidence angle, resulting in a decrease of blade aerodynamic load across the full span. However, the total pressure at the rotor outlet increases because the efficiency of the middle region of the tandem rotor is improved. The tandem rotor with 20°and 30° negative bow increases the inlet flow angle in the hub region, resulting in an increment of blade aerodynamic load in this region. In addition, the inlet flow angle in the middle span is reduced by the tandem rotor with NB20, whereas the inlet flow angle in the middle span is basically unchanged by the tandem rotor with NB30. It can be seen from the AVD comparison diagram that the AVD in the hub and case is reduced by the three negative bow tandem rotors, whereas the AVD in middle span is increased. This is because the three negative bow tandem rotors migrate the low-energy fluid in middle span to the hub and case region, increase the amount of low-energy fluid in the hub and case region, and improve the flow field in the middle span.
Figure 10 shows the contrast diagram of surface static pressure distribution at different spans of NB10, NB20, and NB30. Compared with the ORG rotor, in the hub span region, the negative bow blades reduce the aerodynamic load on the leading edge and middle of the front blade, and also reduce the aerodynamic load on the middle of the rear blade. Therefore, the negative bow blades reduce the aerodynamic load on the hub region of tandem rotors, which is also illustrated in
Figure 8b. In the middle span region, the negative bow blades increase the load on the front of the front blade and decrease the load on the front of the rear blade. The negative bow blades do not change the position of shock wave, but reduce the adverse pressure gradient behind the shock wave and the separation of boundary layer behind the shock wave. In the tip span region, the negative bow blades reduce the aerodynamic load on the front and middle of the front blade, but have little influence on the aerodynamic load on the rear blade. Similar to the hub span region, the negative bow blades make the position of the shock wave moving forward to the front of the front blade, increase the Mach number of the wave and the inverse pressure gradient after the shock wave, and result in an increment of shock wave loss and the viscous loss of the boundary layer.
Figure 11 shows the comparison of static pressure and the limit flow line diagram on the suction surface of NB10, NB20, and NB30. There are obvious separation lines at the hub and middle of the suction surface of ORG, and the separation line extends from 10% to 60% of the blade spread. The separation line is generated by boundary layer separation caused by shock waves. The negative bow blades increase the boundary layer separation range in hub region of front blade, and the separation line caused by shock waves extends to the middle span. The range and intensity of the boundary layer separation in the hub region of the rear blade are also obviously increased. The negative bow blades reduce the radial migration range of low-energy fluid in the middle region of the rear blade. However, the radial migration intensity of low-energy fluid in the blade tip region was increased. With the increase of the bow angle, there is no obvious radial migration of low-energy fluid in the middle region, but a small range of corner separation region with high intensity appeared near the trailing edge of the blade tip, which increases the total pressure loss in the blade tip region.
Figure 12 shows the comparison of flow line and static pressure on the end-wall surface of the hub of NB10, NB20, and NB30. The flow field in the hub region is deteriorated due to the negative bow blades, and the deterioration degree increases with the increase of the bow angle. Firstly, the separation range and intensity of the boundary layer of the front blade and rear blade are increased due to the intersection of the suction surface branch of the horseshoe vortex at the leading edge of the blade and suction surface. When the negative bow angle is 30°, an obvious vortex zone is formed at the trailing edge of the rear blade. Secondly, the branch of the horseshoe vortex on the front blade pressure surface is mixed with the low-energy fluid separated from the corner region of the rear blade. According to past research, when the horseshoe vortex of the blade pressure surface develops downstream, it intertwines with the end wall boundary layer to form passage vortex. Therefore, the negative bow blades increase the mixing degree of the blade passage vortex and low energy fluid in the corner region, and further increase the deterioration degree of the flow field in the hub region.
Figure 13 shows the contrast diagram of the static pressure coefficient near the casing wall of NB10, NB20, and NB30. According to existing studies, the tip leakage vortex trajectory corresponds to the static pressure chute of the casing wall. Therefore, the static pressure isoline chute line near the casing end wall is used in this paper to approximate the trajectory of the tip leakage vortex, as shown by the red dotted line. It can be seen that the negative bow blade does not significantly change the initial position of the tip leakage flow in the front blade, but increases the intensity of the tip leakage flow. The circumferential deflection and axial length of the tip leakage flow of the front blade increase significantly, whereas the negative bow blade makes the tip leakage flow of the rear blade move to the trailing edge, reducing the intensity and influence range of the tip leakage flow of the rear blade. However, compared with the tip leakage flow of front blade, the intensity and range of the tip leakage flow of the rear blade are significantly smaller. Therefore, the negative bow blade increases the intensity of the tip leakage flow and deteriorates the flow field in the tip region.
Figure 14 shows the comparison of the entropy diagram of the casing wall of NB10, NB20, and NB30. It can be seen that the negative bow blade worsens the flow field in the tip region and increases the total pressure loss in the tip region. As can be seen from the above analysis, this is because the negative bow blade pushes the low-energy fluid into in the tip region, which intensifies the blockage of the tip region passage and makes the shock wave in the tip region move towards the leading edge of blade. Therefore, the intensity of shock wave, the boundary layer separation, and the intensity of the tip leakage flow is increased.
4.2. Influence of a Positive Bow Blade on Supersonic Tandem Rotors
Figure 15 shows the comparison of isentropic efficiency and total pressure ratio characteristics of tandem rotors with different positive bow angles, including PB10, PB20, and PB30.
Table 4 shows the comparison of design point performance and the stall margin of PB10, PB20, and PB30. It can be seen that, compared with the original rotor, the positive bow blade decreases the pressure ratio and efficiency at the design point and also reduces the stall margin of the tandem rotor. However, the positive bow blade increases the flow capacity of the tandem rotor. Compared with the ORG, the design point efficiency of the PB10 is reduced by 0.3%, the stall margin of the PB10 is reduced by 1%, and the design point flow rate of the PB10 is increased by 0.38%. With the increase of the angle, the effect of the positive bow on the column rotor is more remarkable.
Figure 16 shows the comparison of efficiency and total pressure ratio spanwise distribution of PB10, PB20, and PB30. Compared with the ORG, positive bow blade reduces the efficiency from 10% to 50% of the span of the tandem rotors, but the efficiency of the hub region and the casing region is increased. In addition, the positive bow blade increases the total pressure ratio in the hub region and the casing region, but decreases the total pressure ratio in middle span of the tandem rotor. It is inconsistent with the variation of efficiency along the spanwise distribution. From
Table 4, we can see that the positive bow blades decrease the maximum efficiency and worsen the flow field of the tandem rotor, which reduces the back pressure corresponding to the maximum efficiency point of the rotor. Therefore, the distribution of the total pressure ratio decreases significantly along the spanwise direction. In general, although the positive bow blade increases the efficiency of the end-wall region of the hub and casing, the overall performance of the tandem rotor is reduced because the flow field of the rotor with 10–50% blade span is deteriorated at the same time. This is consistent with the action mechanism of the positive bow blade on the flow field of the transonic single rotor.
Past studies have shown that the positive bow blade exerts radial force on the fluid in the blade passage to generate pressure pointing from the end wall to the middle span, which is conducive to the migration of low-energy fluid in the hub and the casing region to the middle span and improves the flow field in the hub and the casing region. But at the same time, the efficiency of the positive bow blades is reduced. The total pressure loss in the middle span of the ORG is larger. Therefore, the positive bow blade further worsened the flow in the middle span. At the same time, the influence of the positive bow blade on the flow field of the supersonic tandem rotor has some different characteristics. The three positive bow blades do not deteriorate the performance at the high blade span region, among which the positive bow blade with an angle of 10° and 20° can improve the performance of the tandem rotor from a 50% to a 80% span.
Figure 17 shows the comparison of spanwise distribution of the inlet flow angle and the axial velocity density (AVD) of PB10, PB20, and PB30. Compared with ORG, the positive bow blade reduces the inlet flow angle of the whole span of the tandem rotor, which makes the flow move along the direction of the negative incidence angle. In addition, the AVD in the hub and the casing region is increased and the low-energy fluid in the hub and the casing region is reduced by the three positive bow blades. At the same time, the positive bow blade also reduces the AVD in most spans of the tandem blades, and reduces the diffuser capacity of the corresponding span.
Figure 18 shows the Comparison of surface static pressure distribution at different spans of PB10, PB20, and PB30. Compared with the ORG, the positive bow blade increases the aerodynamic load in the hub region and the tip region, and reduces the aerodynamic load in the middle span region. This is because the positive bow blade promotes the migration of the low-energy fluid in the hub region and the tip region to the middle span, reduces the low-energy fluid in the blade hub region and the tip region, and increases the diffusing capacity of the blade hub region and the tip region. In addition, in the blade hub region, the positive bow blade does not change the position of shock waves significantly, and the adverse pressure gradient after shock waves is basically unchanged. In the middle span, the positive bow blade obviously increases the adverse pressure gradient behind the shock wave of the front blade and the rear blade, and increases the separation loss of the boundary layer behind the shock wave. At the blade tip region, the positive bow blade migrates the position of the shock wave to the trailing edge, and meanwhile reduces the adverse pressure gradient behind the shock wave of the front blades. Therefore, the shock loss and separation loss of the boundary layer behind the shock wave is decreased.
Figure 19 shows the comparison of static pressure and the limit flow line diagram on the suction surface of PB10, PB20, and PB30. The positive bow blade increases the radial migration degree of low-energy fluid from the hub region to the middle span, and improves the flow field in the hub region. The positive bow blade has less influence on the corner separation range and strength of the hub region; this is why the corner separation range and strength of the rear blade are small. At the same time, the positive bow blade improves the flow field in the tip region of the rear blade and reduces the intensity of the radial migration of low-energy fluid in the middle span region. However, the positive bow blade worsens the flow field and increases the separation intensity of the boundary layer in the middle span.
Figure 20 shows the comparison of flow line and the static pressure diagram on the hub end wall of PB10, PB20, and PB30. It can be seen that the positive bow blade improves the flow field in the hub region. With the increase of positive bow angle, the saddle point of the horseshoe vortex at the leading edge (the red circle point in
Figure 20) gradually moves away from the leading edge of the blade to the middle blade passage. The intersection of the suction surface branch of the hoof vortex and the suction surface of the blade moves towards the leading edge of the blade. The intersection of the pressure surface branch of the horseshoe vortex and the suction surface of the blade also gradually moves towards the leading edge of the blade. This is because the positive bow blade improves the flow field in the hub region, the diffusing capacity of the hub region is increased, and the lateral and flow pressure gradients are significantly increased, which results in the saddle point of the horseshoe vortex separation at the leading edge of the blade to move to the middle of the blade channel. At the same time, under the action of the large adverse pressure gradient, the pressure surface branch of the horseshoe vortex and the blade suction surface intersect earlier. It can be seen from the limiting streamline diagram of the suction surface that the lateral flow of the end wall is enhanced, and the flow field in the hub region is improved, but the separation range of the low-span boundary layer of the front rotor is slightly increased.
Figure 21 shows the comparison of the static pressure coefficient diagram near the casing end wall of PB10, PB20, and PB30. This paper uses the static pressure contour line inclined groove connection near the casing end wall to approximate the blade. The trajectory of the tip leakage vortex is shown as the red dotted line in the figure. In this paper, the static pressure isoline inclined groove connection near the end wall of the casing is used to approximate the trajectory of the tip leakage vortex, as shown by the red dotted line in the figure. It can be seen that the positive bow blade does not significantly change the starting position of the tip leakage flow of the front and rear blades, but slightly increases the circumferential deflection of the tip leakage flow trajectory and increases the strength of the tip leakage flow.
Figure 22 shows the comparison of the entropy diagram of the casing wall of PB10, PB20, and PB30. The positive bow blade reduces the total pressure loss in the tip region. It can be seen from the previous analysis that this is because the positive bow blade promotes the movement of the low-energy fluid in the blade tip region to the low blade span, which increases the flow capacity of the blade tip channel and changes the shock wave structure in the blade tip region. The intensity of the shock wave and the range of the low-energy high-entropy fluid after the shock wave is also reduced.