# Effects of the Shock Wave Structure on the Tip Clearance Leakage Flow in Transonic Compressor Rotors

Authors: Xinlong Ye; Zhenggui Zhou
Journal: International Journal of Aerospace Engineering
 (2023)

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## Abstract

The interaction between shock waves and the tip clearance flow in a transonic compressor rotor has important effects on the tip clearance flow and the rotor aerodynamic performance. In this paper, two transonic rotors with a high pressure ratio are selected to study the tip flow, one of which has one normal shock wave at the blade tip and the other has two shock waves (an oblique shock wave and a normal shock wave) at the tip. The two rotors have the same meridional flow channel, design point flow rate, pressure ratio, and rotation speed to focus on the influence caused by the effect of the shock wave structure. The numerical results for the flow fields show the following conclusions. The strength of the two shock waves at the blade tip is weaker than that of one normal shock, and the former two shock waves are less stable than the latter. Therefore, with increasing tip clearance, the efficiency, pressure ratio, and stall margin of the rotor with the two shock waves decrease more rapidly. The static pressure difference between the pressure and suction sides of the tip clearance is the only driving factor of the tip clearance leakage flow, and the leakage flow depends on the local pressure difference and the secondary leakage caused by adjacent blades. The movement speed of the annular wall is less than that of the leakage flow, which has a minor blocking effect on the tip clearance leakage flow. The change in tip clearance has little effect on the chordal distribution of the static pressure difference and leakage flow rate per unit area, so the total leakage flow varies linearly with the tip clearance size.


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## Body

## 1. Introduction

Tip clearance flow and its mixing with main flow and spanwise transport flow, including the interaction with shock waves in supersonic or transonic rotors, form a complex three-dimensional flow [1]. Studies show that the flow loss in the tip region accounts for approximately one-third of the total loss of the compressor, so the rotor tip region is a high loss region, usually also the stall initiation region [2–7]. The tip region of the rotor is the only region with high flow loss regardless of whether the pressure ratio is high or low [8, 9]. The higher the rotor pressure ratio and the larger the tip clearance, the higher the flow loss in this region. In addition, tip leakage flow is also an important self-noise source [7, 10, 11].The effect of tip clearance on the performance of the compressor rotor of stage 9 of a 13-stage large IGT (industrial gas turbine) was studied using numerical simulations by Sakulkaew et al. [12]. The result shows three trends of efficiency variation with tip clearance. For a small tip clearance (<0.8% span), the trend of efficiency variation is nonmonotonic with an optimum tip clearance for maximum efficiency. The optimum gap exists because the trend of tip leakage mixing loss and viscous shear loss is opposite with increasing tip clearance. For a middle tip clearance (0.8%–3.4% span), the efficiency is monotone linear decreasing with increasing tip clearance. For a large tip clearance (>3.4% span), the change of tip clearance has no effect on efficiency due to the aft-loaded at the blade tip. The studies in references [13–15] show a similar trend.The change in tip clearance affects not only the rotor performance but also the downstream stator performance and the stability of the stage [15–18]. Howard and Puterbaugh [19] combined computational fluid dynamics(CFD) unsteady simulation calculations and experiments to study the effect of the tip clearance variation on the efficiency of the transonic small aspect ratio (average rotor aspect ratio=0.57) axial compressor stage. The results show that the flow rate, pressure ratio, and efficiency are reduced when the tip clearance is increased and that spanwise transport and/or mixing occur within the downstream stator, which causes an increase in the loss of the stator. Liu et al. [20] studied about the effect of the rotor tip clearance variation on performance of downstream stator of a highly loaded compressor stage by using oil flow visualizations, stereoscopic particle image velocimetry (SPIV), and five-hole probe measurements. The result shows that a critical rotor tip blockage usually occurs at a certain rotor tip clearance configuration, resulting in more fluid moving to the low blade region, which causes an increase in loading at the hub and a decrease in the tip corner separation of the rotor, and the opposite is true for the stator.More recent research was performed in the field of tip clearance leakage flow field structure and flow mechanism, which were the basis of understanding the effect of the tip clearance on compressor performance [21, 22]. The tip clearance flow of Rotor37 rotor was simulated and compared with the experiment by Chima [23]. The results show that there are three regions of different loads: a small, highly loaded region near the leading edge produced a strong clearance vortex; a large, moderately loaded region between the leading edge and the shock produced a wall jet; and a lightly loaded region downstream of the shock passed the clearance flow with little effect. Hoeger et al. [24] studied the inlet stage of a high-pressure compressor and found that there is obvious interaction between the tip leakage vortex and the shock wave in a transonic axial compressor, which greatly reduces the compressor efficiency and stability margin. Zhang et al. [25] measured the flow structure in tip region in a large-scale low-speed axial compressor and found a breakdown of the tip leakage vortex at the near stall condition for both of two tip clearances. And Wang et al. [26] found the breakdown of tip leakage vortex occurs in an intermittent manner. The tip leakage vortex (TLV) in a mixed flow pump as turbine at pump mode is decomposed and reconstructed by dynamic mode decomposition (DMD) by Liu and Tan and Han and Tan [27, 28]. The results show that the primary tip leakage vortex (PTLV) can be mainly classified as two parts, oscillating PTLV-A and shedding PTLV-B. PTLV-A starts near the blade leading edge and propagates downstream the passage with obvious oscillation, while PTLV-B stretches from the leading edge firstly and then sheds downstream. And the tip clearance has a slightly effect on the separation point of the PTLV but no effect on the separation angle.In this paper, the flow field of two transonic high pressure ratio compressor rotors with different shock wave structures at the blade tip is numerically calculated, the effect of different shock wave structures on the tip clearance flow is studied, and finally, the driving factors of the tip clearance flow are presented.


## 2. Two Transonic Rotors with Different Shock Wave Structures

In this paper, two transonic compressor rotors are studied: NASA Rotor37 and RotorSD designed by our own research group. The meridional channels of the two rotors are the same, and the design mass flow, total pressure ratio, and rotating speed are also the same, as shown in Table1. The tip profiles of the two rotors are shown in Figure 1. It shows that compared with Rotor37, the chord length of the RotorSD tip profile is longer and the curvature of the rear part is smaller. The relative Mach number contours at the 90% span of the two rotors at the design point are shown in Figure 2. From Figure 2, it can be seen that there is a single normal shock wave at the blade tip of Rotor37 and that there are one oblique and one normal shock wave at the tip of RotorSD. The profile at the blade tip of Rotor37 is foreloaded, and the load in front of the profile of RotorSD is low, as seen in Figure 3.Table 1
Main parameters of the two rotors.
ParametersDesign valuesRotor37RotorSDNumber of blades36Rotational speed (rpm)17188Tip speed (m/s)454Tip clearance (mm)0.356Mass flow rate (kg/s)20.19Total pressure ratio2.10Isentropic efficiency0.8680.907Surge margin (%)14.015.4Figure 1
Comparison of the geometry of the two rotors at 90% span.Figure 2
Contours of the relative Mach numbers at 90% span.
(a)
Rotor37(b)
RotorSDFigure 3
Static pressure distribution of two rotors at 90% span.


## 3. Computational Setup

The commercial NUMECA software is used to calculate the flow fields (see Figure4), and the flow fields are simulated using three-dimensional steady Reynolds-averaged Navier-Stokes (RANS) equations with the Spalart-Allmaras (S-A) turbulence model. The spatial second-order upwind difference scheme is used to discretize the governing equations. Boundary conditions are specified to match the conditions of the experiment in the literature. The inlet boundary condition is set to 288.15 K for the total temperature and 101325 Pa for the total pressure and axial for the velocity direction. The outlet boundary condition is set to the radial equilibrium equation, and the static pressure at root is given. No-slip boundary condition is used on the hub, shroud, and blade surface.Figure 4
Computational domain.
(a)
Rotor37(b)
RotorSDThe AutoGrid5 software is used to generate a grid, and the distance from the first grid to the wall is given as2×10−6 m(y+≈1) according to the Blasius equation. The O4H topology is used on the S1 surface, which is composed of an O-grid on the blade surface and four H-grids. The computational domain and three-dimensional topological structure of the grids of the two rotors are identical, as shown in Figures 5 and 6. To examine the dependence of the calculation results on the number of grid cells, five grids are generated, as shown in Table 2, in which Ni and Nk are the number of streamwise direction and azimuthal nodes, respectively (see Figure 3), and Nj is the number of spanwise nodes. Figure 7 shows that when the number of grid cells is more than 610,000, the calculation results almost do not change with the number of grid cells [29]. Therefore, a grid with 610,000 grid cells is used in this paper.Figure 5
Schematic diagram of the grid.Figure 6
O4H topology.Table 2
Number of grid points.
Grid pointsNiNjNk290,0004157109390,0004965109500,0005773109610,0006581109750,0006985109Figure 7
Performance of NASA Rotor37.
(a)
Total pressure ratio versus mass flow rate(b)
Isentropic efficiency versus mass flow rate


## 4. Effect of the Tip Clearance on the Aerodynamic Performance of the Two Rotors

The blade tip clearances of Rotor37 and RotorSD were set to 0.25 mm, 0.50 mm, 0.75 mm, and 1.00 mm, and the effects of the blade tip clearances on the aerodynamic performance of the two rotors and flow fields in the tip regions were analyzed by numerical calculation.Figure8 shows that the total pressure ratio, isentropic efficiency, and mass flow rate of the two rotors decrease with increasing tip clearance in the whole flow rate range. The effect of the tip clearance on the values of the aerodynamic performance parameters is shown in Figure 9. Figure 9(a) shows the relationship between the tip clearance and the relative variation of the peak efficiency ΔEFF (based on the efficiency at the tip clearance of 0.25 mm, Figures 8(c) and 8(d) and 9(b) are the same); Figure 9(b) shows the relationship between the tip clearance and relative variation of the peak total pressure ratio ΔTPR; Figure 9(c) shows the relationship between the tip clearance and the relative variation of the nondimensional flow rate of the tip clearance leakage ΔTCL (ratio of the flow rate of the clearance leakage to the main flow rate); Figure 9(d) shows the relationship between the tip clearance and the relative variation of the stall margin. It can be seen from Figure 8 that with an increase in the tip clearance and a decrease in the isentropic efficiency, the total pressure ratio and the stall margin of RotorSD are more severe than those of Rotor37, and the leakage flow rate increases more. Therefore, with an increase in the tip clearance, the aerodynamic performance of RotorSD deteriorates more than that of Rotor37, although at the design point, the performance of the former is better than that of the latter. Figures 10 and 11 show that the efficiency and the total pressure ratio near the blade tip region of RotorSD decrease more than those of Rotor37 as the tip clearance increases.Figure 8
Comparison of the characteristics of the two rotors with different clearances.Figure 9
Comparison of the effect of the tip clearance on the performance of the two rotors.
(a)
Isentropic efficiency(b)
Total pressure ratio(c)
Tip clearance leakage(d)
Stall marginFigure 10
Isentropic efficiency distribution along the span.
(a)
Rotor37(b)
RotorSDFigure 11
Total pressure ratio distribution along the span.
(a)
Rotor37(b)
RotorSD


## 5. Effect of the Tip Clearance on the Flow Fields near the Blade Tip at the Peak Efficiency Point

For the two rotors with tip clearances of 0.25 mm, 0.5 mm, 0.75 mm, and 1.0 mm, the relative Mach number contours of theS1 surface at 90%, 94%, and 98% spans and at the middle of the tip clearance are shown in Figure 12. From the relative Mach number contours of the S1 surface at the 90% span for Rotor37 at the tip clearance of 0.25 mm (seen in Figure 12(a)), the Mach number before the normal shock is 1.49; for RotorSD in the same tip clearance, the Mach number before the oblique shock is 1.47 and the Mach number before the ending normal shock is 1.3. Therefore, the intensities of the two shock waves at the 90% span of RotorSD are weaker than that of the normal shock wave at the 90% span of Rotor37. Additionally, from Figure 12(a), with the increase in the tip clearance, the shock wave structures of the two rotors are basically unchanged at the 90% span. Comparing Figures 12(a)–12(c), for the same rotor, the closer to the tip, the greater the change of shock wave intensity or structure with the increase of the clearance, and the change for Rotor37 is much less than that for RotorSD, which shows that the stronger the shock wave is, the less easily it is disturbed. At the peak efficiency point, the contours of entropy at 95% span and tip vortex structures in the tip fields of the two rotors at two tip clearances are shown in Figures 12 and 13, respectively. The Q in Figure 14 represents the second invariants of the velocity gradient tensor which defines as
(1)Q=‐12UijUji=14ΩiΩi−2SijSij,where Uij=∂ui/∂xj is the velocity gradient tensor, Ωi=εijk∂uj/∂xk is the velocity field, and Sij=1/2∂ui/∂xj+∂uj/∂xi is the rate-of-strain tensor. It can be seen from Figure 13 that with an increase of the clearance, the tip leakage mass flow rate increases, resulting in the increase in tip leakage vortex intensity especially the first tip leakage vortex intensity (shown in Figure 14), which cause more low-momentum fluid accumulates in the pressure sides (shown in Figure 13). From Figure 8(c), the tip leakage mass flow rates of the two rotors at the minimal tip clearance are nearly the same; but with the increase of the clearance, the increase of the tip leakage mass flow rate of RotorSD is greater than that of Rotor37. Therefore, the efficiency drop near the tip region of the RotorSD is greater than that of Rotor37(seen in Figure 11). However, the more shock waves there are, the weaker the intensity of each shock wave, and the more easily they are disturbed. Therefore, the greater the number of shock waves is, the more sensitive the flow field near the blade tip is to the tip clearance, and the faster the performance decays as the tip clearance increases. Figures 15 and 16 show the entropy contours and streamlines near the blade tip of Rotor37 and RotorSD with the four tip clearances, respectively. The comparison between Figures 15 and 16 shows that with an increase in the tip clearance, the entropy increase of RotorSD is significantly greater than that of Rotor37.Figure 12
Contours of the relative Mach number.
(a)
90% span(b)
98% span(c)
Middle of the tip clearanceFigure 13
Contours of entropy (J/(kg·K)) in 95%span.Figure 14
Tip vortex structure (Q=9×107).Figure 15
Contours of TCL streamlines and entropy near the blade tip of Rotor37.Figure 16
Contours of flow streamlines and entropy near the blade tip of RotorSD.To further verify that the decrease in aerodynamic performance of the two rotors with the increase of the tip clearance is mainly due to the change of shock waves, the rotation speed of the two rotors is reduced to 70% of the design speed (12032 rpm) to eliminate the channel shock waves of the two rotors, and the effect of the increase of the blade tip clearance on the performance of the two rotors is investigated. The overall performance comparison of the two rotors at 70% of the design speed is shown in Table3. The efficiency variation ΔEFF of the two rotors with the tip clearance is very small at low speeds without shock waves near the blade tip region. From Figures 17 and 18, the distributions of the efficiency and total pressure ratio along the span also vary very little with increasing tip clearance. From Figure 19, there is no shock wave in the passages at the 90% span for both two rotors. These results support the above argument from the negative side: as the tip clearance increases, the decrease in the aerodynamic performance of the rotors is mainly due to changes in the shock waves.Table 3
Performance of the two rotors at 70% design speed.
Bladeδ (mm)ṁ (kg/s)πk∗ηk∗ΔEFFṁleakage (%)Rotor370.2515.8891.3770.9173/0.510.5015.9981.3650.9171-0.02%0.990.7515.9941.3630.9157-0.14%1.501.0015.9041.3590.9142-0.29%2.03RotorSD0.2514.6511.3180.8884/0.490.5014.6051.3170.8883-0.01%1.030.7514.5931.3140.8881-0.02%1.631.0014.5671.3110.8877-0.06%2.27Figure 17
Distribution of the isentropic efficiency along the span at 70% of the design speed.
(a)
Rotor37(b)
RotorSDFigure 18
Distribution of the total pressure ratio along the span at 70% of the design speed.
(a)
Rotor37(b)
RotorSDFigure 19
Contours of the relative Mach number at 70% of the design speed.


## 6. Driving Factors of the Tip Clearance Leakage Flow

The tip clearance leakage flow is the main factor causing the increase in flow loss in the tip area. The above results show that the tip clearance leakage flow rate varies linearly with the tip clearance size at the design rotating speed and 70% of the design rotating speed (seen in Table3). Therefore, the leakage flow rate per unit flow area at different clearance sizes is unchanged. In the following, the driving factors of leakage flow per unit flow area are demonstrated.As shown in Figure20, A represents a microelement of the tip clearance area at a certain chord position, which is composed of n mesh cells, and ρi, Wni, and ΔAi represent the flow density at the ith cell, the relative velocity perpendicular to the cell, and the area of the cell, respectively. ṁ¯leakage represents the average value of the leakage flow rate per unit area at the position, and ΔP¯ represents the average value of the pressure difference between the pressure and suction sides of the clearance, which are defined as
(2)ṁ¯leakage=∑i=1nρiWniΔAi∑i=1nΔAi,ΔP¯=P¯PS−P¯SS.Figure 20
Schematic of parameter definitions at the tip clearance.P¯PS and P¯SS are the average pressures on the pressure surface and the suction sides in the tip clearance at the same chord position, respectively, and the average pressure is defined as
(3)P¯=∑i=1nρiWniPiΔAi∑i=1nρiWniΔAi.For the two rotors with a clearance of 0.25 mm, the distributions of the static pressures on the pressure and suction surfaces at the 95% span andP¯PS and P¯SS (the average pressure on the pressure surface and suction sides, respectively, at the tip clearance) along the chord are shown in Figure 21. Figure 21 shows that the static pressures on both sides of the tip clearance are nearly equal to the static pressures of the corresponding sides at the 95% span. Figures 22 and 23 show distributions of the average pressure difference between the pressure and suction sides of the clearances ΔP¯ and the average leakage flow rate per unit area ṁ¯leakage along the chord for the two rotors with a clearance of 0.25 mm. Comparing the two figures, the variation trend of ΔP¯ is the same as that of ṁ¯leakage, indicating that the pressure difference is one of the main driving forces of the tip leakage flow. Figures 24 and 25 show that the distributions of ΔP¯ and ṁ¯leakage along the chord are affected quite slightly by the tip clearance, so the total leakage flow rate changes linearly with the size of the tip clearance.Figure 21
Distributions of the static pressures on the pressure and suction surfaces at the 95% span and along the chord.Figure 22
Pressure differences between the pressure and suction sides of the clearances along the chords at a tip clearance of 0.25 mm.Figure 23
Distributions of the average leakage flow rate at a tip clearance of 0.25 mm.Figure 24
Pressure differences between the pressure and suction sides of the clearances along the chords at the four tip clearances.
(a)
Rotor37(b)
RotorSDFigure 25
Distributions of the average leakage flow rate at the four tip clearances.
(a)
Rotor37(b)
RotorSDFigures26 and 27 show the contours of the pressure difference ΔP¯ and relative velocity perpendicular to the pressure surface at the blade tip Wn, where δ¯ is the dimensionless tip clearance, 0 for the blade surface and 1 for the annular wall. From Figures 26(a) and 27(a), when the tip clearance is small, the isopleths of ΔP¯ are approximately perpendicular to the annular wall and the blade tip end surface, indicating that the flow in the clearance is approximately parallel to the annular wall and blade tip end surface, and the pressure gradient normal to the wall and the surface is approximately zero. As the tip clearance increases, the orthogonality between the contours of ΔP¯ near the blade tip end surface becomes weaker, but the orthogonality between the contours and the annular wall is still stronger. This result implies that as the tip clearance increases, the three-dimensionality of the tip clearance flow near the blade tip end becomes stronger, and the tip clearance flow near the annular wall is still approximately parallel to the wall. From Figures 26(b) and 27(b), as the tip clearance increases, the contours of the normal velocity Wn change little, which again shows that the leakage flow is approximately proportional to the tip clearance size. Comparing Figures 26(a) and 26(b) and 27(a) and 27(b), it can be seen that the region with a larger pressure difference ΔP¯ has a higher normal velocity Wn (larger leakage flow rate) and a region with a smaller ΔP¯ has a smaller Wn. However, there is inconsistency between the contours of ΔP¯ and Wn, especially Wn is quite large in the region near the blade trailing edges where ΔP¯ is close to 0. Therefore, there are other drivers for the leakage flow.Figure 26
Distribution of flow parameters at the tip clearance of Rotor37. (a) Distributions of the pressure difference. (b) Distributions of relative velocity perpendicular to the pressure surface at the blade tip.Figure 27
Distribution of the flow parameters at the tip clearance of RotorSD. (a) Distributions of the pressure difference. (b) Distributions of relative velocity perpendicular to the pressure surface at the blade tip.The relative movement direction of the annular wall is the same as the direction of the leakage flow. Due to the viscous effect of the boundary layer attached to the wall, the leakage flow in the blade tip clearance is affected. Figure28 shows the distribution of the normal velocity Wn along the tip height at 10%, 50%, and 90% chords. It can be seen from the figure that the tip clearance has little effect on the distribution of Wn. This result is consistent with the above results that the leakage flow rate is proportional to the blade tip clearance size. It can also be seen from Figure 28 that the velocity component of the annular wall motion perpendicular to the chord is smaller than the normal velocity Wn slightly away from the wall. Therefore, the annular wall motion has a blocking effect on the tip leakage flow; however, as seen from the figure, the affected region is small. Figure 29 shows that the increase in entropy in the tip clearance is very small, which implies that the flow in the tip clearance is low loss. From the above analysis, it can be concluded that the relative movement of the annular wall can reduce the leakage flow, but the effect is small.Figure 28
Distribution of the normal velocity along the gap height at 10%, 50%, and 90% chord lengths.
(a)
Rotor37(b)
RotorSDFigure 29
Contour of entropy at the tip clearances at 10%, 50%, and 90% chords.
(a)
Rotor37(b)
RotorSDThe leakage flow streamlines at 97% clearance height under the minimum and maximum tip clearances (0.25 mm and 1.00 mm) are shown in Figure30. There is an obvious double-leakage tip clearance flow (that is, the leakage flow generated by adjacent blades flows into the local tip clearance [21, 27, 30]). At the same time, Figures 15 and 16 show that there is a secondary leakage phenomenon in the leakage flow. This means that the tip clearance leakage flow is driven not only by the local tip pressure difference but also by the adjacent blade tip pressure difference, which leads to inconsistency between the contours of the pressure difference ΔP¯ and the normal velocity Wn in Figures 26 and 27.Figure 30
Leakage flow streamlines at 97% clearance height.
(a)
Rotor37(b)
RotorSD


## 7. Conclusion

The existence of tip leakage flow in a compressor rotor has a great impact on the aerodynamic performance of the compressor. This paper focuses on the tip clearance flows of two transonic rotors with the same design index but different shock wave structures. The following conclusions are drawn through the analysis of their aerodynamic performances and flow field characteristics via a numerical calculation method:(1)
For supersonic and transonic rotors, achieving the same total pressure ratio at the design point, the more shock waves in the flow field near the blade tip, the weaker the intensity of each shock wave, and the more easily they are disturbed. The efficiency of the rotor with a normal shock wave is lower than that of the rotor with two or more shock waves because the shock loss of the former is larger than that of the latter. However, the greater the number of shock waves, the more sensitive the flow field near the blade tip is to the tip clearance, and the faster the performance decays as the clearance increases(2)
The pressure difference between the pressure and suction sides of the tip clearance is the only driving factor for the tip clearance leakage flow. The leakage flow depends not only on the local pressure difference but also on the secondary tip leakage generated by adjacent blades. Although the movement direction of the annular wall is the same as the direction of the leakage flow, the movement speed of the annular wall is lower than the leakage flow speed, which has a minor blocking effect on the tip clearance leakage flow(3)
With an increase in the tip clearance, between the pressure and suction sides of the tip clearance, the static pressure difference and the leakage flow rate per unit area along the chord direction change little. Therefore, the total leakage flow rate changes linearly with the size of the tip clearance



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*Source: 1477251-2023-02-20.xml*