Numerical investigation on the influence of nonuniform tip clearance on rotor tip-clearance flow field structure

Abstract

A numerical study was conducted on an axial compressor, the NASA Stage 35, with three nonuniform tip clearances to understand the effect of a different shape tip clearances rotor on the compressor's performance. The results demonstrated that by modifying the traditional parallel tip clearance to sine-type tip clearance (STC), hump-type tip clearance (HTC), and concave-type tip clearance (CTC), the compressor’s peak efficiency showed remarkable improvement and the SMI was significantly improved. In comparison to the design rotor, the SMI of STC, HTC, and CTC increased by 3.102 %, 2.672 %, and 0.645 %, respectively. The leakage distribution at the tip-clearance region from LE to TE exhibited an inverse pattern to that of the tip curve. The leakage’s magnitude could not reflect the TLF’s intensity, and the leakage in the middle of the blade tip had a role in the size of the TLV. Leakage at the tip’s TE influenced the corner separation’s scale in the casing. The STC and HTC schemes’ total pressure ratio improved, the low-velocity zone’s area, high-entropy area, and high absolute vorticity area at the LE of rotor’s PS decreased, the detached shock moved backward, the leading-edge spillage flow decreased, the shock action position of the rotor suction surface moved forward, the BLS increased, the TPL in the stator channel decreased, and the CTC exhibited the opposite trend. A nonuniform tip clearance was achieved by reducing the leakage and TLV intensity, thereby reducing the size of the TLV, induced vortex, or CSV in rotor passage and increasing the compressor’s SMI.

1. Introduction

The demand for modern compressor designs is shifting towards higher pressure ratios, improved efficiency, reduced stages, and lighter weight. Meeting these requirements necessitates the optimization of the airflow within the compressor. Among the different factors, the airflow at the tip gap significantly influences the compressor’s performance and is a critical focus in the optimization of compressor research. There are TLV, secondary flow, and leakage flow characteristics at the gap, all of which contribute to the complicated flow characteristics [[1], [2], [3]]. The TLF near the gap significantly influences the compressor efficiency and stability margin. It is estimated that flow at the tip gap contributes to up to 30 % of flow loss [4]. Consequently, this study is to enhance the compressor’s performance and stability margin by designing nonuniform tip-clearance shapes.

With technological advancements, the application of flow-control technology in turbomachinery has seen significant developments. This technology can be broadly categorized into active and passive control methods. Active control techniques include boundary-layer suction, end-wall air injection, and casing suction [[5], [6], [7]]. Passive flow-control methods include casing treatments, slotted blades, and curved/swept blades [[8], [9], [10], [11]]. Notably, the aforementioned flow-control method in the tip-clearance area requires substantial modifications to the compressor structure and internal aerodynamic layout, resulting in higher costs. To mitigate these expenses, an alternative approach is to adjust the rotor tip-clearance value. This adjustment helps suppress the intensity of the TLF, reduce the losses in the gap area, and ultimately improve the its performance [12,13]. Fu [14] proposed that decreasing the rotor tip gap could mitigate the occurrence of vortices in the blade gap region. This would result in a decrease in aerodynamic fluctuations of the rotor, subsequently enhancing its stability. Yamada [15] indicated that the development of the rotor stall is closely correlated with the blade gap size, with diverse values producing varied effects on the progress of the rotor rotational stall. According to Gao [16], lowering the gap’s value causes the TLF’s intensity in gap to drop, which in turn lowers loss and boosts compressor’s performance ultimately. Dong [17] discovered that altering the rotor blade tip’s value can influence the unsteady loading of the blade. It was discovered that a larger blade gap results in less aerodynamic dampening on the blade. Sakulkaew [18] considered multiple factors such as tip-leakage loss and fluid viscosity loss to determine the optimal clearance value for a compressor rotor at different clearance spans. Zhang [19] used unsteady methods to study the flow characteristics of transonic compressor. The abrupt breakup of the vortex causes a rapid rotational stall to begin in the tip region, which obstructs the channel. During compressor operation, uncertainty factors such as machining error and vibration during rotation change the blade tip gap to varying degrees. The tip value and shape inevitably change. However, the above studies all focused on uniform gap; there is limited research on the structure of nonuniform blade tip clearances. Therefore, studying the impact of nonuniform tip structures on the flow characteristics of rotor gap is significant.

Currently, nonuniform blade gap structures can be broadly classified into three types. The first involves keeping the geometric structure of the casing unchanged and changing the shape of the rotor gap [20]; the second involves simultaneous changes in the structure of the rotor gap and casing [21]; and the third involves keeping the structure of the rotor gap unchanged and changing the structure of the shroud [22]. The first type of nonuniform blade tip clearance can be further divided into axial linear nonuniform and axial nonlinear nonuniform clearances. Because the rotor blades from the root to the tip are contracted, the trapezoidal shape and nonuniform blade tip clearance can reduce the amount of material used for blade manufacturing and reduce the weight of the compressor. Thompson [23] researched the impacts of dilation gap on the compressor’ performance. It demonstrated that at the rotor’s design rotation velocity, the stepped gap led to a significant enhancement in mass flow, with an increase of up to 2.0 %. Additionally, it also resulted in a notable improvement of 1.5 % in PEI. Zhang [24] found that the expansion-type tip clearance had the most significant performance improvement compared to the design and contraction-type tip gap for an axial linear nonuniform tip clearance. Li [25] optimized the tip clearance of a sloping blade by combining a proxy model with a multiobjective genetic algorithm. Research has demonstrated that a continuous axial linear nonuniform gap can reduce the TLF rate, inhibit flow separation. The aforementioned analysis indicates that in recent years, a great deal of research has focused on the first kind of linear nonuniform blade tip clearance. Consequently, in order to give more comprehensive information, extensive research on the axial nonlinear nonuniform blade tip clearance is required.

To explore the impact of the axial nonlinear nonuniform blade gap on the flow characteristics and TLF rate at rotor blade gap, this study uses a Stage 35 as the investigate object. First, we modify the uniform rotor blade tip clearance to obtain three axial nonlinear nonuniform clearance schemes. Then, through numerical simulation analysis, the flow characteristics of the nonuniform gap is analyzed, providing a basic research and design reference for the optimization of transonic compressor rotor tip-clearance modification in the future.

2. Calculation setting and investigate compressor structure

2.1. Investigate compressor structure

This study conducted a numerical investigation of the nonuniform gap of rotor using the Stage 35 as the investigate subject. With 36 rotor and 46 stator vanes, the compressor has a rotation velocity of 17,188 n/min. Flow rate is 20,188 g/s. A structural picture of the calculation channel and grid was built using the Autogrid5; this is displayed in Fig. 1. Every block utilized an O4H structure, whereas the blade gap employed a butterfly structure. A periodic complete matching connection was employed, and the blocks were perfectly matched. The first layer wall grid's width was established at 1 × 10⁻⁶ mm.

Fig. 1.

Computational grid.

2.2. Design scheme and three-dimensional structure of nonuniform blade tip clearance

According to Ref. [26], varying degrees of change were made in the tip-clearance value; it was found that the NASA Stage 35 compressor performed best at 0.204 mm. This study considers the axial nonlinear nonuniform blade tip clearance as the research objective. To control the circumferential leakage area of the tip gap to be the same as that of uniform type and to study its influence on blade tip leakage, three types of nonuniform clearances were designed. Fig. 2 displays a schematic of the nonuniform tip gap design scheme. In addition to the prototype rotor parallel tip clearance (PTC, clearance value 0.408 mm), three types of nonuniform clearances were also designed. These were the Sine-type tip clearance (STC, maximum clearance value 0.612 mm, minimum clearance value 0.204 mm), concave tip clearance (CTC, maximum clearance value 0.612 mm, minimum clearance value 0.204 mm), and hump tip clearance (HTC, maximum clearance value 0.612 mm, minimum clearance value 0.204 mm). The three-dimensional structure is displayed in Figs. 3(a)–(d).

Fig. 2.

Non-uniform blade tip clearance design scheme.

Fig. 3.

Grid structure of blade tip schemes: (a) PTC, (b) STC, (c) HTC, (d) CTC.

Fig. 3 depicts the three-dimensional grid structure of the different blade tip-clearance schemes, ensuring that the grid topology remained consistent for each scheme. Each tip-clearance scheme consisted of 17 radial and 73 axial nodes. Owing to millimeter-level changes in the tip’s value, the grid structure at the gap is not modified to maintain consistency among the nodes in each scheme.

2.3. Numerical method validation

For the steady numerical simulation, NUMECA, a commercial program, was utilized. Using central-difference spatial discretization, the finite volume approach solves the three-dimensional Reynolds-averaged Navier-Stokes equations. Temporal discretization was done using a 4-stage Runge–Kutta method with local time-stepping and a CFL number of three. In this work, the Spalart–Allmaras turbulence model was employed for analysis. For rotor-stator interface, conservative coupling by the pitchwise row condition was utilized. Multigrid, local time step, implicit residual smoothing, and other technologies were applied to speed up the calculation's convergence. Every solid wall was constructed with no-slip and adiabatic conditions. For rotor inlet, the total temperature and the total pressure was set. For stage exit, the average static pressure was adjusted.

To ensure that the computation accuracy was not influenced by the number of grids, we established different numbers of passage grids. As the number of grids grows, Fig. 4 illustrates the compressor performance at the PEP, close to the NSP, and close to the choke point. Referring to Fig. 4, we can see that the three parameters didn't change much when the quantity of compressor channel grids got closer to 2,250,000. It is therefore acceptable to say that the requirement of grid independence was met.

Fig. 4.

Comparing the compressor's performance metrics over various grid numbers.

The correctness of the numerical computation is further confirmed by comparing the simulation data of 2,250,000 grids under various rotation velocity with experimental results [27,28] in Fig. 5, additional confirmation of the numerical simulation's accuracy. A comparison of the efficiency of the simulation and experimental data is shown in the upper part of Fig. 5. A comparison of the pressure ratios is shown in the lower half of Fig. 5. When comparing experimental results, the largest deviation in total pressure ratio were found to be 1.28 %, 1.63 %, and 1.49 %; The largest deviation in the efficiency were 0.0261, 0.0244, and 0.0233, respectively. All computational data came within an allowable margin of error using the technique with a 2,250,000 of grid number.

Fig. 5.

Comparing computational and experimental data at various design rotation speeds.

Fig. 6 shows a comparison of the experimental data at the design point with the simulation results. The comparison of the relative Ma at the rotor’s outlet along the span is shown in Fig. 6(a); The absolute Ma distribution comparison at the stage outlet along the span is shown in Fig. 6(b). The outcomes of the simulation matched the findings of the experiment. The absolute Ma of the stator outlet and the relative Ma of the rotor’s outlet close to the tip were both less than the experimental data. The major likely reason is the tendency of the turbulence model to overestimate the losses caused by TLF near the tip. As a result, there is a strong agreement between the computed data and the experiment's spanwise values.

Fig. 6.

Comparison of numerical data at the design point with experimental data: (a) Relative Ma number in rotor blade outlet, (b) Absolute Ma number in stator blade outlet.

Fig. 7 compares the experimental and CFD data for the ratios of total temperature and pressure at the rotor outflow at the design point. The figure displays similar distributions of both the computed and experimental data along the blade spanwise. The CFD data near the tip indicate higher temperature ratios and lower pressure ratios. The turbulence models' propensity to overestimate the losses resulting from TLF at the tip is one of the main causes of errors in the physical mechanism. The study's criterion for identifying compressor stall was the numerical divergence point. This study employed the static pressure dichotomy approach to calculate the NSP of the compressor near the near-stall flow in numerical calculations in order to precisely pick the crucial NSP within the compressor's operational range. The specific method is described in Reference [29]; the amplitude of the backpressure change near the stall point was 25 Pa.

Fig. 7.

Comparison of numerical data at the design point with experimental data.

The aforementioned analysis shows that, at 100 % n_cor, the isentropic efficiency numerical simulation data are higher than the experimental ones. The following are the reasons for the errors: (1) Time's impact on the flow characteristics is disregarded by the steady-state computation technique utilized in this investigation; (2) the CFD adopts a single-channel periodic method, and the data exchange at the rotor-stator interface is nonuniformly distributed, whereas in the simulation, the data exchange is uniform [30]; and (3) the disagreement between the steady-state simulation with ideal gas and adiabatic slip free-wall boundary conditions and the unstable flow within the compressor.

Fig. 8 displays the tangential velocity distribution in the section near the 33 %, 55 %, and 72 % chord lengths of the rotor under the peak efficiency conditions. The experimental data were measured by the laser Doppler velocimeter discussed in literature [[31], [32], [33]]. The comparison revealed that the shock wave positions obtained from the experiments and numerical calculations were consistent and the main TLF, and the shock location before and after the shock were accurately predicted. Based on Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, it can be concluded that this study's numerical approach can consistently capture the flow-field characteristics in the channel and estimate overall aerodynamic performance with high accuracy. As a result, the outcomes are trustworthy.

Fig. 8.

Comparison of numerical and experiment parameters nephogram on rotor chord slice: (a) 33 %chord, (b) 55 %chord, (c) 72 %chord.

3. Discussion and analysis for CFD results

3.1. Comparison of compressor’s performance

The characteristic curves of compressor with various tip gap shape schemes are displayed in Fig. 9. After modification, the nonuniform tip method improved peak efficiency, expanded the stability margin, raised the mass flow at the choking condition, and markedly changed the overall pressure ratio. It also greatly improved the flow structure in the tip area. In Fig. 9(a), the pressure ratio exhibits an ascending trend in both the STC and HTC schemes, with the increase being more notable under near-stall conditions. In Fig. 9(b), the peak efficiency achieved by nonuniform tip-clearance scheme improved to 84.842 %, 84.661 %, and 84.655 %, respectively. However, the change near the stall point was not particularly significant.

Fig. 9.

Characteristic curves of different tip shape: (a) Stage pressure ratio, (b) Stage efficiency.

Fig. 10 compares the compressor SMI and PEI. They are defined as follows (Eq. (1) and Eq. (2):

$$\text{SMI}=\left[{\left(\frac{{\mathrm{\pi }}_{\text{CS}}}{{\mathrm{\pi }}_{\mathrm{B}}}\right)}_{\text{NS}}\times {\left(\frac{{\mathrm{m}}_{\mathrm{B}}}{{\mathrm{m}}_{\text{CS}}}\right)}_{\text{NS}}-1\right]\times 100\%$$ (1)

$$\text{PEI}=\left[{\left(\frac{{\mathrm{\eta }}_{\text{CS}}}{{\mathrm{\eta }}_{\mathrm{B}}}\right)}_{\text{PEP}}-1\right]\times 100\%$$ (2)

where CS is the nonuniform scheme, B is the design scheme, and NS is the minimum stall point. Based on the data, compared with the parallel tip gap PTC, the SMI of the STC, HTC, and CTC increased by 3.102 %, 2.672 %, and 0.645 %, respectively; the PEI increase by 0.278 %, 0.064 %, and 0.057 %, respectively. From Fig. 9, Fig. 10, the compressor performance was shown to be greatly enhanced by the nonuniform blade tip clearance. The following section analyzes the TLF rate and flow-field structure for nonuniform clearances.

Fig. 10.

Comparison of SMI and PEI for different schemes with baseline schemes.

3.2. Comparison and analysis of rotor tip clearance leakage

To further explore the effect of nonuniform gap schemes on the flow structure in clearance area, the clearance leakage of different clearance schemes was studied; the gap leakage is defined as (Eq. (3)):

$$M={\int }_S\left[\rho \left(t\right)·V_P\left(t\right)\right]ds$$ (3)

Fig. 11 displays the chordwise distribution of the leakage based on the altered coordinate system. S represents the radial area of the tip clearance, ρ represents density, and Vp represents the leakage flow’s velocity. V_y and V_z in the figure represent the tangential and axial velocities, respectively, and V_t represents the chordwise velocity. Vp represents the chordwise velocity component that is perpendicular to the blade. As depicted in Fig. 11, for every chord length, there was a positive association between the leakage and tip-clearance values at the modification position. Furthermore, this trend was generally contrary to the tip clearance’s shape. Within the 0 %–5 % chord length at the blade tip LE, both the STC and HTC experienced an increase in leakage, whereas the CTC exhibited a decrease. Within the middle 50 % of the gap, the leakage was minimized owing to the smallest clearance value for the HTC. However, as the STC and CTC blade tip-clearance values increased, the leakage also increased. At the TE of the blade tip, an increase in the STC and HTC clearance values corresponded to an increase in the TLF rate, whereas the CTC scheme values decreased, thereby causing the TLF rate to drop.

Fig. 11.

The distribution of leakage rate at different chord lengths.

3.3. Analysis of the flow characteristic in tip gap

Fig. 12 displays the relative Ma and limiting streamline on the rotor SS. Based on the figure, it is evident that when the PTC scheme (Fig. 12(a)) is changed to the STC (Fig. 12(b)) and HTC (Fig. 12(c)) schemes, the relative Ma near the rotor’ SS trailing edge decreases, the shock-boundary layer interaction location moves upstream, and the BLS at the TE increases. When PTC scheme is changed to the CTC (Fig. 12(d)) scheme, the relative Ma near the rotor SS’s trailing edge increases, the shock/boundary layer interaction location moves toward to TE, and the BLS at the TE decreases.

Fig. 12.

Limiting streamlines and relative Mach number on the SS with different tip shape: (a) PTC, (b) STC, (c) HTC, (d) CTC.

Fig. 13 displays the relative velocity streamlines and entropy distribution in the tip gap near the NSP at the tip’s LE. The red line represents the direction of the TLF and the arrow indicates the TLV. The high-entropy region created by the blending of the TLV and spanwise underflow at the tip’s TE is shown by red box. Compared with the PTC scheme (Fig. 13(a)), the overflow phenomenon at the LE of the STC (Fig. 13(b)) and HTC (Fig. 13(c)) schemes vanished completely. Additionally, the high-entropy zone near the rotor's LE decreased as a result of the interaction between the TLV and disconnected shock. The BLS at the TE of the blade’s SS and entropy value increased. In the CTC scheme (Fig. 13(d)), the rotor LE pressure side's high-entropy region's area grew, the leading-edge spillage flow increased, and the BLS on the TE suction surface decreased. Owing to the reduced leakage rate at the TE of the CTC scheme and reduced TLF mixing with the spanwise underflow in the casing corner area, the entropy value decreased. A comparative analysis revealed that an increase in the TLF rate weakened the LEe spillage flow phenomenon, whereas an increase in the TE leakage rate increased the casing CSV. Furthermore, the leakage rate within the middle tip gap had a notable influence on the TLV’s size, breaking at the rotor PS’s leading edge.

Fig. 13.

The streamlines and entropy distribution in the tip gap: (a) PTC, (b) STC, (c) HTC, (d) CTC.

Fig. 14 shows the nephogram of the absolute vorticity in the gap [33], which expresses the vorticity intensity. The formula used is as follows (Eq. (4)):

$${\xi }_n=\frac{\left|\overrightarrow{\xi }\right|}{2\omega }$$ (4)

Fig. 14.

The absolute vorticity distribution in the tip region: (a) PTC, (b) STC, (c) HTC, (d) CTC.

The black line in the enlarged area represents the TLV’s edge; the white arrow indicates the vortex core. As indicated in Fig. 14, the vorticity at the tip clearance was significantly higher than that at other positions. After modification, compared with the PTC scheme (Fig. 14(a)), the vorticity values at the tip region of the STC (Fig. 14(b)) and HTC (Fig. 14(c)) decreased, the axial expansion range decreased, the vortex core moved backward, and TLV’s edge moved toward downstream. Therefore, the leading-edge spillage flow phenomenon disappeared. However, an opposing trend was observed for the CTC (Fig. 14(d)), which is consistent with the description in Fig. 13.

The upstream deflection of the TLF was the main cause of rotor blade LE spillage flow. The larger the TLA between the rotor blade TLF and axial direction, the greater degree of leakage-flow upstream deflection and the more significant the rotor LE spillage flow. To express the degree of leakage-flow upstream deflection intuitively, the TLA between the TLF and axial direction is defined [34]. The formula used is as follows (Eq. (5)):

$${\beta }_a=\text{arc}\cos \left(\frac{W_a}{\sqrt{W^2-W_r^2}}\right)$$ (5)

Fig. 15 displays the curve of the TLA between the TLF and axial direction along the chordwise direction. It was discovered that throughout the whole chord-length region, the TLA of the STC and HTC schemes reduced. Therefore, the suppression effect on the mainstream of the TLF in the tip gap at the rotor’s LE at 0–5 % chord length was weakened and the overflow phenomenon at the rotor leading edge was improved. At 5–100 % chord length, the suppression effect on the upstream airflow was weakened, the compressor’s loss was reduced, and the stability margin was increased. The CTC scheme exhibited the opposite trend.

Fig. 15.

Distribution of TLA.

Fig. 16 displays the nephogram of the relative Mach numbers at 99 % rotor height, with the black line representing the shock wave line. It can be observed from the figure that there are two low-velocity zones in the different tip shape schemes, namely, the black-dotted line and red-circled area. The black-dotted low-speed Zone 1 was formed by the vortex breakdown of the TLF and double leakage flow after mixing and passing through the detached shock. The low-velocity Zone 2 in the red circle was caused by the mixing and accumulation of the spanwise underflow and trailing-edge leakage flow at the casing. It was found that the detached shock positions of the STC (Fig. 16(b)) and HTC (Fig. 16(c)) schemes moved backward, the low-velocity Zone 1’s area decreased, the shock position at rotor suction surface moved forward, and the BLS increased; thus, the low-velocity Zone 2’s area increased. More importantly, after modification, the BLS on the stator blade’s SS decreased, as indicated by red-dashed circle in the figure. The CTC scheme (Fig. 16(d)) exhibited the opposite trend, and the BLS intensity of the stator SS remained unchanged. The reason is the upstream rotor TLV and BLS (wake) collide with the stator’s LE, causing flow separation to appear at the corner region of the casing and expand towards the TE. When PTC scheme (Fig. 16(a)) was changed to the STC and HTC schemes, the low-momentum zone area caused by the blade TLV decreased significantly and the BLS increased (as indicated in Fig. 16(b) and (c)). However, the decrease in the low-momentum zone far exceeded the increase in the BLS. Therefore, the influence of the LE of the stator weakened, and BLS decreased. The low-momentum zone area caused by the TLV rupture in the CTC scheme increased; however, BLS weakened, and the two canceled out. Therefore, the BLS of the stator SS in the CTC scheme remained unchanged.

Fig. 16.

The nephogram of relative Ma at the 99 % span: (a) PTC, (b) STC, (c) HTC, (d) CTC.

Fig. 17 displays the distribution of the absolute vorticity in the rotor channel and leakage flow streamline within a 20 % chord-length range of the blade LE. As indicated in Fig. 17, the TLV is the likely reason for formation of the red high-vorticity zone at the rotor SS’s LE. The interaction between detached shock and the TLF and secondary leakage flow results in Zone 1, resulting in the rupture of the TLV. Zone 2, a high-vorticity zone in the TE of the rotor suction surface, is formed by the mixing of the spanwise underflow and TLF in the casing corner region. After changing the PTC (Fig. 17(a)) to a nonuniform tip-clearance STC (Fig. 17(b)) and HTC (Fig. 17(c)), the size of the high vorticity zone and Zone 1 area decreased, indicating a decreasing in the occurrence of TLV breakdown and channel blockage and an improvement in the rotor’s flow path. Compared with the PTC scheme, HTC scheme demonstrated an increased area in Zone 2, followed by the STC scheme. Because the location of the shock on the rotor’s SS moved forward after modification, thereby increasing the BLS on the SS. Changes in the CTC scheme (Fig. 17(d)) exhibited the opposite trend.

Fig. 17.

The distribution of absolute vorticity and the leakage flow streamline in the rotor channel: (a) PTC, (b) STC, (c) HTC, (d) CTC.

Fig. 18 displays the distribution of the Cp at 99 % blade height of the rotor, which reflects the changes in loading and diffusion capacity. Based on the picture, the nonuniform tip clearance significantly affects the distribution of the Cp on the rotor’s PS and SS. Both the STC and HTC schemes demonstrated an increasing trend in the Cp of the rotor both side’s surface, and the enveloping area of the static pressure coefficient at the blade tip LE increased. This implies that the leakage flow was driven by a rising pressure differential, the rotor blade loading increased, and the diffusion capacity increased. Therefore, the TLF intensity increased, whereas the CTC scheme exhibited the opposite trend. The variation in shock location on the rotor SS is indicated in the enlarged area of the figure. The shock action location of the STC and HTC schemes moved forward, and the interaction between the blade tip airflow and channel airflow led to this phenomenon, whereas the CTC scheme demonstrated the opposite trend.

Fig. 18.

The Cp at the 99 % blade height of the rotor.

Fig. 19 displays the vortex structure model of the baseline rotor channel. The distribution and formation reasons of the vortices in the channel are detailed in Fig. 19; these are also applicable to nonuniform tip-clearance rotors. The TLV, induced vortex, CSV, and LE spillage flow are indicated in the figure. The TLVE, induced vortex, and shock interfered with each other to form the main vortex. After the main vortex rupture, a large range of low-momentum airflow vortex clusters (blue area) formed near the PS of the alongside rotor blades. Besides, owing to the impact of the centrifugal force of the rotor, the radial vortex moved towards the casing wall, which caused a large area of spanwise underflow. After being blocked by the casing wall, it mixed with the separation vortex, forming a low-momentum zone (blue-dotted line) near the rotor SS’s TE, as displayed in Fig. 19.

Fig. 19.

Vortex structure model of rotor channel under near-stall conditions.

The distribution of the channel vortices of the four schemes is displayed in Fig. 20; the vortex structures are displayed in color with dimensionless helicity. The Q criterion [35] is defined to capture the vortex edge; the Q criterion and dimensionless helicity [31] are defined as follows (Eq. (6) and Eq. (7)):

$$Q=-\frac{1}{2}\left(e_{ij}{e}_{ji}+{\mathrm{\Omega }}_{ij}{\mathrm{\Omega }}_{ji}\right)=\frac{1}{2}\left({\Vert \mathrm{\Omega }\Vert }^2-{\Vert E\Vert }^2\right)$$ (6)

$$H_n=\frac{\overrightarrow{\xi }\cdot \overrightarrow{w}}{\left|\overrightarrow{\xi }\right|\cdot \left|\overrightarrow{w}\right|}$$ (7)

where e_ij and Ω_ij are strain rate tensors and vorticity tensors, and the region with Q > 0 is defined as a vortex structure. Compared with Fig. 20(a), it indicates that the spillage flow phenomenon of the nonuniform clearance STC (Fig. 20(b)) and HTC (Fig. 20(c)) schemes was weakened, the size of the TLV and induced vortex decreased, and the shock moved backward. However, the size of the radial vortex and casing CSV on the rotor SS increased. The stability margins of the STC and HTC schemes significantly improved, whereas that of the CTC scheme (Fig. 20(d)) demonstrated the opposite trend.

Fig. 20.

The distribution of channel vortices of the four schemes: (a) PTC, (b) STC, (c) HTC, (d) CTC.

Fig. 21 displays the radial distribution of airflow angle at the rotor’s outlet. The formula of airflow angle β is as follows (Eq. (8)):

$$\beta =-\mathrm{a}\cos \left(\frac{c_{2a}}{c_2}\right)$$ (8)

where C₂ is the absolute velocity and C_2a is the axial velocity. Based on the figure, the airflow angle of the nonuniform tip-clearance STC and HTC schemes increased in 0–85 % blade height, whereas the airflow angle of the CTC scheme changed in the opposite trend. The airflow angles of the STC and HTC schemes decreased within 85–100 % blade span, whereas the airflow angle of the CTC scheme increased. The increase in the scale of the TLV in the CTC scheme significantly influenced the airflow in the middle and bottom blade spans of the rotor. Therefore, the airflow angle in the range of the middle and bottom blades decreased, and the loading on the middle and bottom blade spans of the rotor decreased. Owing of the increase of rotor trailing edge leakage-flow rate in the STC and HTC schemes and the accumulation and mixing in the casing corner area, the low-momentum region near the rotor tip SS increased and the BLS near the tip SS increased, causing the airflow angle to drop between 85 % and 100 % of the blade span; the CTC scheme presented an opposite trend.

Fig. 21.

The radial distribution of the airflow angle.

Fig. 22 displays the distribution of the ω for the three-dimensional slice in the stator channel, as well as the distribution of streamlines at each slice. The ω [36] is shown as follows (Eq. (9)):

$$\omega =\frac{P_{t2}-P_{t3}}{P_{t2}-P_2}$$ (9)

where, P₂ is the static pressure at the rotor’s outlet, P_t2 is the total pressure at the rotor’s outlet, and P_t3 is local total pressure. The red area is the core loss. Based on the figure, the change in the ω on the stator SS under the nonuniform tip gap scheme was consistent with the change in the BLS of the stator SS in Fig. 15. Comparing the high-loss areas of Fig. 22(a), it was found that the angle between the chordwise and vortex cores near the stator’s SS of the STC (Fig. 22(b)) and HTC (Fig. 22(c)) schemes decreased, and the area of the loss region decreased, whereas the CTC scheme (Fig. 22(d)) remained unchanged.

Fig. 22.

Total pressure loss coefficient distribution on stator channel: (a) PTC, (b) STC, (c) HTC, (d) CTC.

4. Conclusions

Numerical simulations were conducted on the NASA Stage 35 compressor with axial nonlinear nonuniform blade tip clearance to analyze the leakage at the gap and changes in the flow structure in the gap area. The conclusions are as follows.

• 1.The stability margin of the rotor with the STC scheme was higher than those of the other tip-clearance schemes. This was because the loading was higher near the blade’s LE, and the leakage-flow rate in this area was generally lower under the STC scheme. Compared with the HTC and CTC schemes, the STC scheme demonstrated less circumferential leakage area in the high loading area, and the leakage-flow rate was low within the 10–50 % chord length range. The size of the tip LE's low-momentum zone was the smallest, and the leakage flow in the middle of the tip gap was less mixed with the TLF, leading to the highest stability margin in the STC scheme.
• 2.The shock/boundary layer interaction location on the rotor SS of all schemes varied significantly at 99 % blade span. The shock/boundary layer interaction of the CTC scheme moved downstream, whereas under the STC and HTC schemes, the shock/boundary layer interaction location moved upstream. The interaction between the TLF and main channel flows leads to this phenomenon. The channel area also changed owing to changes in the geometric shape of the tip.
• 3.Axial nonlinear nonuniform tip clearance was achieved by reducing the leakage and TLV intensity at the gap, thereby reducing the size of the TLV, induced vortex, or CSV in the rotor passage and increasing the compressor’s SMI. Compared to PTC scheme, under nonuniform tip-clearance STC and HTC schemes, the BLS on the rotor trailing-edge SS increased. Compressor stall was caused by the combined effect of the TLV and BLS on the rotor SS. Under the nonuniform tip-clearance CTC scheme, the BLS on the rotor SS improved significantly. The reason for the compressor stall was the same as that in the PTC scheme, which was caused by the TLV.

Data availability statement

The data are available from the corresponding author on reasonable request. Email: lizhipeng@stu.sau.edu.cn.

CRediT authorship contribution statement

Guochen Zhang: Writing – review & editing, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. Zhipeng Li: Writing – review & editing, Writing – original draft, Validation, Software, Resources, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Zhiyuan Cao: Resources, Funding acquisition. Zhihui Xu: Validation, Supervision. Weihang Liu: Software, Data curation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Notation

Abbreviations

LE

Leading edge

PTC

Parallel tip clearance

STC

Sine-type tip clearance

HTC

Hump tip clearance

CTC

Concave tip clearance

NSP

Near stall point

PEP

Peak efficiency point

SMI

Stability margin improvement

PEI

Peak efficiency improvement

PS

Pressure surface

SS

Suction surface

TLA/V/F

Tip leakage angle/vortex/flow

CSV

Corner separation vortex

Cal

Calculation

CS

Changed scheme

P

Static pressure

P_t

Total pressure

SBLI

Shock/boundary layer interaction

BLS

Boundary layer separation

Symbols

n_cor

Corrected rotation speed

ρ

Destiny

m

Mass flow

π

Total pressure ratio

V

Absolute velocity

$H_n$

Normalized helicity

$\overrightarrow{\xi }$

Vorticity vector

${\xi }_{\mathrm{n}}$

Absolute vorticity

$\overrightarrow{w}$

Relative velocity vector

Cp

Static pressure coefficient

t

Chordwise

P

Perpendicular direction to t

a

Axial direction

r

Radial direction

ω

Rotation angular velocity

e_ij

Strain rate tensors

Ω_ij

Vorticity tensors

Q

Q criterion

TPL

Total pressure loss

Appendix A. Supplementary data

The following is the Supplementary data to this article: