Effects of upstream Rankine vortex on tip leakage vortex breakdown in a subsonic turbine

JID:AESCTE AID:105776 /FLA

[m5G; v1.282; Prn:13/02/2020; 9:47] P.1 (1-14)

Aerospace Science and Technology ••• (••••) ••••••

1

Contents lists available at ScienceDirect

67 68

2 3

Aerospace Science and Technology

4

69 70 71

5

72

6

www.elsevier.com/locate/aescte

7

73

8

74

9

75

10

76

11 12 13

Effects of upstream Rankine vortex on tip leakage vortex breakdown in a subsonic turbine

16 17

81

College of Engineering, Peking University, Beijing, 100871, China

83

82 84 85

19

a r t i c l e

i n f o

a b s t r a c t

86 87

21 22 23 24 25 26

79

Kai Zhou, Chao Zhou ∗

18 20

78 80

14 15

77

Article history: Received 15 May 2019 Received in revised form 9 February 2020 Accepted 9 February 2020 Available online xxxx Communicated by Qiulin Qu

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

In this paper, the tip leakage vortex was studied in a subsonic linear turbine cascade that was modeled from a typical high-pressure turbine stage. Due to a large streamwise adverse pressure gradient induced by the strong tip leakage vortex, vortex breakdown was observed and further examined. Vortex breakdown was accompanied by a sharp increase in swirl number and high mixing loss. In the vortex-vortex interaction simulations, a Rankine vortex flow was simulated numerically according to the complex turbine rotor inlet conditions. Ten quasi-steady, equally spaced phases between the incoming vortex and the blade in one pitch period were investigated for flow patterns and loss details. At a certain phase, the tip leakage vortex breakdown was significantly suppressed due to the vortex interactions. The upstream positive vortex was gradually entrained into the leakage flow. By improving the turbulent mixing between the leakage vortex and the incoming vortex, the streamwise momentum within the leakage core was enhanced. The vortex interactions weakened the adverse pressure gradient by eliminating the streamwise vorticity of the leakage vortex. By examining the vorticity transport equation, it’s found that the viscous diffusion of vorticity dominated the leakage vorticity damping. The dilation effect also contributed to vorticity transportation as it operated in the subsonic, hightemperature, high-pressure turbine environment. It was surprising that the baroclinicity effect caused by the transportation of low temperature fluid was found to take part in the vorticity evolution in the current subsonic cascade. The associated loss was significantly reduced when vortex breakdown was suppressed. Different incoming vortex strengths were also tested. As the positive vorticity magnitude increased, the incoming flow tended to amplify the vortex interaction effect. When the vorticity magnitude was beyond a certain threshold, the vortex breakdown was no more observed. However, when the negative vortex was imposed upstream of the blade, the incoming vortex flow was transported radially downward and hardly interacted with the tip leakage flow. The leakage vortex breakdown was almost the same as that in the uniform inlet case. The vortex transportation was clearly captured with the help of passive scalar released from the incoming vortex core. By simulating a triple-vortex interaction, we developed a better understanding of vortex kinematic evolution. © 2020 Elsevier Masson SAS. All rights reserved.

88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112

47

113

48

114

49

115

50 51

1. Introduction

52 53 54 55 56

To pursue higher efficient turbomachinery, advanced turbine designs are critically important in the industry. In the highpressure turbines of an aero-engine, a gap is left between the

57 58 59 60 61 62 63 64 65 66

*

Corresponding author at: Turbomachinery Laboratory, College of Engineering; BICESAT, Peking University, 100871 Beijing, China. E-mail addresses: [email protected] (K. Zhou), [email protected] (C. Zhou). https://doi.org/10.1016/j.ast.2020.105776 1270-9638/© 2020 Elsevier Masson SAS. All rights reserved.

rotor tip and casing to avoid scraping. For the unshrouded rotor tip in high-pressure turbines, the pressure gradient drives the flow across the gap from the pressure side to the suction side forming tip leakage flow, which is one of the most detrimental features of turbine blades. The turbulent mixing accounts for as much as one third of the total loss (Schaub et al. [1]). Besides, tip leakage flow causes heat loadings on the tip and suction side surface (Bunker et al. and Ameri [2,3]). After exiting the tip, the flow forms a streamwise vortex and induces itself by a low-pressure region at the core of the vortex. Towards the trailing edge, the flow at the vortex core suffers from an adverse pressure gradient and may be driven backward. This phenomenon is called vortex breakdown (Sell et al. [4]). Vortex breakdown leads to passage blockage and large mixing

116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE

AID:105776 /FLA

1 2 3 4 5 6 7

APG C Cx Cp C P0

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

67

Nomenclature

8 9

[m5G; v1.282; Prn:13/02/2020; 9:47] P.2 (1-14)

K. Zhou, C. Zhou / Aerospace Science and Technology ••• (••••) ••••••

2

HP NSV SP R Re S Y, Z P PSV PV p r TKE TLV u

Adverse pressure gradient . . . . . . . . . . . . . . . . . . . . . . . . pa/m Chord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m Axial chord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m Static pressure coefficient, C p = ( P 0,in − p )/( P 0,in − p ex ) . . . . . . . . . . . . . . . . . . . . . . . − Stagnation pressure coefficient, C P 0 = ( P 0,in − P 0 )/( P 0,in − p ex ) . . . . . . . . . . . . . . . . . . . . . − High pressure Negative swirling vortex Surface perpendicular to the suction surface Span in turbine stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − Swirl number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − Y, Z direction Total pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pa Positive swirling vortex Passage vortex Static pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pa Radius of the incoming vortices . . . . . . . . . . . . . . . . . . . . . m Turbulent kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . m2 /s2 Tip leakage vortex Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m/s

68

Subscript 0 C core d e ex in isen mar s v x, y

θ

Superscript

∗

Dimensionless parameter

α ω φ

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

72 73 74 75 76 77 78 79 80 81 82 83 84 85 86

89 90 91 92 93 94 95

29 31

71

88

Incidence angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ◦ Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/s Phase between the incoming vortex and the tip leakage vortex Circulation around the incoming vortex . . . . . . . . . m2 /s

28 30

70

87

Greek



27

Total pressure Chord Core of the vortex Datum Mathematical constant Exit Inlet Isentropic velocity Marginal of the integrated area Streamwise direction Incoming vortices Axial, tangential direction Tangential direction around the incoming vortices

69

losses. In compressors, vortex breakdown may be responsible for stalling, as argued by Schlechtriem and Lotzerich [5]. It is largely seen as a precursor to a stall. It has several undesirable effects, including pressure fluctuations and noise generation (Lee et al. [6], Tan et al. [7], You et al. [8], Wu et al. [9]). Furukama et al. [10,11] and Yamada et al. [12] found that there are different types of tip leakage vortex breakdown: under near-stall operating conditions, spiral type breakdown occurs; at near-peak efficiency, small bubble-type breakdown occurs periodically near the leading edge. However, the vortex breakdown is so weak and small that it is not observed in the time-averaged flow field at near-peak efficiency, which highlights the importance of the unsteady simulations. In turbines, tip leakage vortex has been widely studied to determine its effects on aerodynamic performance. Many methods towards reducing the tip leakage loss are broadly investigated, such as squealer tip geometries (Zhou and Hodson [13], Zou et al. [14]) and winglet tip geometries (Zhou and Zhong [15], Silva et al. [16]), tip coolant (Gao et al. [17]), casing contouring (Gao et al. [18], Kavurmacioglu et al. [19], Jiang et al. [20]) and many others. Huang et al. [21] explored the vortex characteristics and wake characteristics of tip leakage vortex in a simplified model, which gives an excellent prediction of how the vortex breakdown performs in different blade loadings. However, most of the current studies are considered in the relatively simple inlet condition, while it is still unknown how vortex breakdown performs in unsteady conditions, especially under the influence of complex vortex flow in turbine stages. The flow in a high-pressure turbine due to bladerow interaction is inherently unsteady, including the potential field, the shock wave, the upstream wake effect and the endwall secondary vortex. In a subsonic turbine, the potential pressure field decays very fast as the flow enters into the rotor passage (Dring et al. [22]). Shock waves, however, are only evident in transonic and supersonic turbines (Paniagua et al. [23], Hancock and Clark [24]). In the current subsonic turbine working environment, the shock effect can be neglected. The wake effect has been studied with a focus on boundary layer evolution, such as by Hodson and Howell

[25] and Coull and Hodson [26]. Ciorciari et al. [27] experimentally studied the effects of unsteady wakes on the secondary flows in the linear T106 turbine cascade, and they found that the wake induced by the moving bars periodically reduces the strength of the passage vortex, the trailing edge wake and corner vortices. However, in other experiments by Volino et al. [28], they used PIV experimentally to explore the wake influence on the tip leakage flow. The results showed that, although the instantaneous flow field varies, only slight differences can be observed due to the time-average field, which implies that the wake and leakage vortex interaction is relatively small. An intuitive understanding is that the wake consists of spanwise vorticity, which is perpendicular to the streamwise vorticity in the tip leakage flow. However, this does not address the interaction between the upstream secondary flow generated near the stator casing and the downstream tip leakage flow, which are both of similar order of magnitude with the streamwise components. When the upstream casing secondary vortex convects downstream to the following blade passage, the vortex is chopped by the blade row in a similar way as the wake transportation (Stieger and Hodson [29], Zhou and Zhou [30]). Vortex distortion, stretching and shearing are encountered because of the unsteady blade motion, leaving the vortex tube concentrated on the suction side. This may affect the downstream tip leakage flow. Zeschky and Gallus [31] found that, at the exit of the rotor, the flow near the tip region suffers the highest fluctuations in velocity and flow angle, but the flow mechanism within the blade passage is not clear. Although no direct visualization was performed, some indirect measurements showed significant pressure perturbation on the suction surface close to the tip (Dunn et al. [32]). In the research of Liu et al. [33], where the main focus was on the unsteady effects on the tip heat transfer on the turbine stage, they found significant pressure variation at 95% radial span of the blade, which is possibly caused by the unsteady development of tip leakage vortex. The flow physics of the vortex-vortex interaction in the tip region of the rotor tip are still not clear, and the loss mechanism is not well understood. Ma and Devenport ([34–36]) conducted a series

96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE AID:105776 /FLA

[m5G; v1.282; Prn:13/02/2020; 9:47] P.3 (1-14)

K. Zhou, C. Zhou / Aerospace Science and Technology ••• (••••) ••••••

1 3

Axial chord, C x Chord, C Pitch Casing moving velocity Flow coefficient, u x /u y Span Tip gap Incidence angle, α Exit isentropic mach number ReC ,isen

4 5 6 7 8 9 10 11

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

of cascade experiments. They used delta-shaped vortex generators to interrupt with the tip leakage vortex. Although the inflow vortices have a circulation about two orders smaller than that of the tip leakage vortex, this flow has a dramatic influence on the phaseaveraged structure of the tip-leakage vortex. The tip leakage vortex reaches maxima when the inflow vortex is near the middle of its pitchwise excursions, which implies that vortex-vortex interaction is significant. In a previous work by Zhou and Zhou [37], they mentioned that unsteadiness near the casing is beneficial, compared with the unsteady loss in the hub region. The tip leakage vortex breakdown is significantly suppressed due to the passing of upstream stator passage vortex (Zhou and Zhou [38]). However, the detailed flow mechanism is not clearly discussed. The current work is considered as a fundamental study based on the findings of unsteady vortex interaction in the real HP turbine (Zhou and Zhou [38]). To understand the interaction between the upstream secondary vortex and the tip leakage vortex, we simulated a subsonic turbine cascade numerically by using a structured 3-D Navier-Stokes solver, using the rotor inlet conditions modeled from a subsonic high-pressure turbine stage. A Rankine vortex was imposed on the incoming flow to study the vortexvortex interaction. The effects of different incoming vortex characteristics on the flow patterns were examined comprehensively. The vorticity transport evolution of the tip leakage vortex was found to be affected the decomposed terms, including the viscous diffusion effect, the dilation effect and the baroclinicity effect. The vortex transportation was clearly visualized with the help of a passive scalar. Based on these findings, it helps to estimate the tip leakage loss accurately and future work on three-dimensional turbine design can make full use of the aerodynamic and thermal benefits of vortex unsteadiness.

45 46

2. CFD settings and grid independence study

47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

68

mm mm mm m/s / mm mm degree / /

34.4 45.3 43.1 620 0.34 40.0 0.6 33.3 0.80 1.18 × 106

The current linear cascade geometry was modeled based on the rotor tip section of a typical subsonic high-pressure turbine. Detailed parameters of the turbine stage could be found in a previous publication (Zhou and Zhou [38]). The turbine stage consisted of 28 stators and 56 rotors. The objective of the current research was to explore the effects of incoming vortex on the suppression of the tip leakage vortex breakdown. A linear cascade with a flat tip gap was used, as shown in Fig. 1. The tip clearance gap was 0.6 mm, which was a typical value for a real turbine stage, i.e., 1% of the rotor span. To simulate a flow coefficient of 0.34 in the rotating machine, the casing wall was set to move relatively. In the uniform inlet simulation, a total pressure and a static back pressure is imposed on the inlet and exit boundary condition. The exit isentropic Mach number was 0.80. The Reynolds number based on the chord and exit isentropic velocity was 1.18 × 106 . The blade span was 40 mm, which was 66.7% of the rotor span of a real turbine stage. The bottom surface of the computational domain was set to be symmetric to reduce the computational consumptions. The span was large enough that the effect of the secondary

69 70 71 72 73 74 75 76 77 78

Fig. 1. Turbine rotor and cascade geometry.

13 14

67

Table 1 Cascade geometrical parameters.

2

12

3

flow near the casing on the bottom surface could be ignored. As the turbine cascade was simulated in engine-representative conditions, the high-temperature-dependence parameters were used by performing linear interpolation, including specific heat capacity, thermal conductivity and viscosity, as shown in Fig. 2. The inlet total temperature was set to 1700 K, and the turbulence intensity was 5%. Ideal gas property is applied to the working air. An extra domain was added upstream to naturally develop the inlet boundary layer on the casing, and the upstream domain was set to move relatively so that different phases of vortex-blade interaction were achieved. More geometrical parameters are listed in Table 1. The computational mesh, as shown in Fig. 3, was generated by ICEM CFD with structured hexahedron grids. The overall dimensions were 2.7C x × 1.3C x × 1.2C x . The distance between the inlet and the blade leading edge is 0.7C x . For the convenience of definition of vortex inlet condition, we used velocity inlet condition. Detailed velocity profile can be found in later section ‘5. Modeling of vortex interaction’. The boundary layer grows naturally from the domain inlet. Non-reflecting pressure conditions were applied at the exit boundary. In order to minimize the computational consumption, the ‘hub’ surface is set to be symmetric. The mesh around the blade was modified with an O-grid topology to obtain a finer mesh. The maximum y + near the blade is below 2 and the area-weighted average y + is 1.4. The growth ratio of the mesh was controlled to be less than 1.2. The commercial solver FLUENT was used to solve the discrete Reynolds-averaged Navier–Stokes (RANS) equations. For the current high-speed, compressible flow, the governing equations for the mass, the momentum and the energy were solved. Secondorder spatial discretization was used with an implicit scheme. To study the grid dependence, we estimated the discretization error by changing the overall mesh number from cases V1 (2.43 million), V2 (3.08 million), V3 (4.22 million) and V4 (5.58 million). Grid independence was conducted for the cases with uniform inlet conditions. The total-to-total pressure ratio was monitored for the convergence level. According to the recommended method [40], by Richardson’s extrapolation, the extrapolated value for the total pressure ratio was 0.973269. The relative error was shown in Table 2. The overall error converged in a satisfactory level with the increase of the node number. With 4.22 million nodes (case V3), the relative error was 0.09%. Due to considerations of computational cost and numerical accuracy, case V3 was used for further analysis, and the discretization error was controlled to be below 0.1%.

79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123

3. Validations of numerical tools

124 125

Cascade study is widely used in the turbomachinery community (Gao et al. [42], Kavurmacioglu et al. [43]). It’s based on the physical comparability with similar Reynolds number, exit Mach number (compressibility) and the flow physics (tip leakage vortex driven by the pressure difference). The validations of the current numerical simulations were conducted in two typical working conditions. The first one was a low speed cascade situation, where

126 127 128 129 130 131 132

JID:AESCTE

AID:105776 /FLA

[m5G; v1.282; Prn:13/02/2020; 9:47] P.4 (1-14)

K. Zhou, C. Zhou / Aerospace Science and Technology ••• (••••) ••••••

4

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79 80

14

Fig. 2. High-temperature-dependence parameters.

15

81

16

82

17

83

18

84

19

85

20

86

21

87

22

88

23

89

24

90

25

91

26 27

92

Fig. 3. Mesh near the tip clearance and the trailing edge.

93 94

28 29 30 31 32 33 34

vortex-vortex interaction was simulated by elaborately imposing a swirl generator. This was to highlight the flow physics of vortex evolution. The other was a transonic cascade condition, which was to strengthen the validation of the turbulence models in predicting the compressible flow with similar Reynolds number and exit Mach number.

(Zhou and Zhou [44]). The turbulence model was able to capture the basic vortex characteristics and provided satisfactory accuracy.

3.1. Validations in a low-speed cascade

3.2. Validations in a transonic cascade

Fig. 4. Diagram of blade profile and an upstream swirl generator.

37

98

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

60

63 64 65 66

103 104

By installing a swirl generator upstream of the blade cascade, as shown in Fig. 4 (Zhou and Zhou [44]), the vortex incoming flow was expected to interact with the tip leakage flow. This was examined by the same authors in a low-speed, large scale linear cascade in the Turbomachinery Laboratory at the Peking University. The operating Reynolds number based on the blade chord is 4.1 × 105 . Quasi-steady interaction of vortex was investigated at ten equallyspaced phases. A Rankine-like vortex was simulated numerically to guarantee similar inlet conditions. The numerical results agreed well with the experimental measurement of the total pressure coefficient near the casing, where the tip leakage flow dominated the local loss, as shown in Fig. 5. The numerical and experimental results showed that the imposed incoming positive vortex was responsible for the aerodynamic benefit when the swirl generator was imposed at phase 9/10. The mixing of the incoming vortex and the tip leakage vortex increased the streamwise momentum in the tip leakage vortex core, thus decreasing the mixing loss. The improved loss agreed well with the predictions of a simple one-dimensional mixing model as explained in the previous work

For the current turbine working in a subsonic, compressible environment, we validated the ability of the CFD solver to predict flow under transonic conditions based on published experimental results by Zhang et al. [45,46]. The operating Reynolds number based on the blade chord is 1.27 × 106 . The exit isentropic Mach number is 1.0. Fig. 6 showed that the isentropic Mach number was predicted accurately around the blade mid-span and near the tip. Fig. 7 showed the measured and predicted heat transfer coefficient on the tip surface. The effects of shock wave reflections within the tip gap on the heat transfer of the blade tip were clearly observed in the numerical simulations. Fig. 8 presented the stagnation pressure coefficient on the cutting plane of one axial chord downstream of the trailing edge at g / S = 1.5%. In general, the turbulence model captured most of the flow characters and achieved a satisfactory agreement. The results provided by the CFD tools agreed well with the experimental data under transonic conditions. Thus, shear stress transport turbulence models can be used for compressible flow.

105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125

Table 2 Discretization error.

61 62

100 102

58 59

99 101

38 39

96 97

35 36

95

V1 V2 V3 V4 Extrapolated value

126

Overall nodes (million)

Total pressure ratio

Relative error

127

2.43 3.08 4.22 5.58 /

0.971792 0.972143 0.972398 0.972546 0.973269

0.15% 0.12% 0.09% 0.07% /

128 129 130 131 132

JID:AESCTE AID:105776 /FLA

[m5G; v1.282; Prn:13/02/2020; 9:47] P.5 (1-14)

K. Zhou, C. Zhou / Aerospace Science and Technology ••• (••••) ••••••

5

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77 78

12 13

Fig. 5. Mass-weighted-averaged C P 0 along the spanwise direction.

79

14

80

15

81

16

82

17

83

18

84

19

85

20

86

21

87

22

88

23

89

24

90

25

91 92

26 27

Fig. 8. Stagnation pressure coefficient on plane of one axial chord downstream of the blade (a) upper, experiment (from Virdi et al. [46]) and (b) down, CFD.

28

93 94

29

95

30

96 97

31 32 33

Fig. 6. Isentropic Mach number distribution along the blade middle span and near the tip region (experimental data from Zhang et al. [45]). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

98 99

34

100

35

101

36

102

37

103

38

104 105

39

Fig. 9. C p distribution at 0.5 Span and 0.93 span around the turbine blade surface.

40 42 43 44 45 46 47 48 49 50 51 52 53

Fig. 7. Heat transfer coefficient (W/m2 K) on the blade tip by (a) experiment (from Zhang et al. [45]) and (b) CFD.

54 55

4. Flow patterns in the case of a uniform inlet

56 57 58 59 60 61 62 63 64 65 66

106 107

41

The datum case with a uniform inlet condition is examined first. The static pressure coefficient C p distribution around the blade surface at 0.5 span and 0.93 span is shown in Fig. 9. The cut plane of 0.93 span cuts across the tip leakage core, as shown in Fig. 9, where the region surrounded by the dark iso-surface represents the streamwise reversed flow (iso-surface of streamwise velocity zero, also named as vortex breakdown). When the flow passes along the suction surface in the middle span, it reaches maximum velocity at approximately 0.2C x and then begins to decelerate. It experiences a large adverse pressure gradient from

0.2C x towards the trailing edge. However, in the 0.93 span, the flow decelerates less because of its induced low-pressure region due to the tip leakage vortex. The detailed dynamic process is described later associated with the vortex breakdown. Several cut planes are defined in Fig. 10. For example, SP70 represents the cut plane perpendicular to the suction side surface at the axial intersection located at 0.7C x . The plane is similarly defined for SP80 and SP95, and SP110 is parallel to SP95. As the tip leakage vortex travels along the suction surface, the cut planes of SP70, SP80, SP95 and SP110 are roughly perpendicular to the streamwise direction of the tip leakage vortex filament. In the following analysis, the direction perpendicular to those surfaces is treated as the streamwise direction. As Hall [46] expressed for inviscid flow in Equation (1), the streamwise pressure gradient within the vortex core is the result of the imposed pressure gradient (the first term on the right side) plus the induced effect of the swirling flow (the second term on the right side). If the vortex core experiences a large adverse pressure gradient on the main flow and/or if the vortex circulation is strong enough, the flow within the core may be driven backward. This is referred to as vortex breakdown. Usually, vortex breakdown is accompanied by reversed flow. Adding to the complexity, there are seven types of vortex breakdown, including double helix type, spiral type, and bubble type (Lucca-Negro and O’Doherty [48]).

108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE

AID:105776 /FLA

[m5G; v1.282; Prn:13/02/2020; 9:47] P.6 (1-14)

K. Zhou, C. Zhou / Aerospace Science and Technology ••• (••••) ••••••

6

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

16

82

17

83 84

18

Fig. 12. Iso-surface of reversed flow region.

19

85 86

20

parameters can be referred to the previous work by the same authors (Zhou and Zhou [44]). A monitor cutting plane was located perpendicular to the inlet direction, 0.1C x to the leading edge in Fig. 13a. The incoming streamwise vorticity distribution (nondimensionalized by the axial chord and exit isentropic velocity as defined in Equation (2)) is shown in Fig. 13c. At 0.9 span, evident positive streamwise vorticity is observed in Fig. 13c, which is considered as the upstream stator secondary passage vortex. In order to simplify the simulation of the incoming conditions, a Rankine vortex is used with comparable vorticity magnitude and vortex size, based on the inlet flow conditions in the turbine stage, as shown in Fig. 13b. A streamwise Rankine vortex is superimposed on a uniform inlet flow as defined in Equation (2) Negative vorticity within the trailing edge vortex is not simulated. The core center of the incoming vortex is located at the 0.9 span of the cascade, a similar spanwise location in the turbine stage. In this way, only the vortex-vortex interaction is considered and all other influences are excluded, such as wake and potential pressure waves.

21 22 23

Fig. 10. Sketch of cut planes.

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

Fig. 11. Streamwise velocity and limiting streamlines on the suction surface.

39 40 41 42 43 44 45 46 47 48

When the Reynolds number is high enough, taking the current turbine cascade for example, the only characteristic geometric forms are the bubble and the spiral types. Due to the limitations of RANS methods, no detailed unsteady flow characteristics can be distinguished. However, the reversed flow patterns are still creditable.

 dp   ds 

r =0

 dp   = ds 

r =∞

∞ + 2ρ 0

 ∂ Vr dr r3 ∂ r V s

(1)

49 50 51 52 53 54 55 56 57 58 59 60

In Fig. 11, the streamwise velocity normalized by the exit isentropic velocity is shown on SP70, SP80, SP95 and SP110. The velocity pointing streamwise backward is cut off. The first occurrence of the reversed flow is on SP80. When the flow travels downstream to SP95, the reversed region expands quickly. After the trailing edge, the reversed region becomes small. Fig. 12 compares the iso-surface region of reversed flow in a full fledged turbine rotor and in this current cascade model. As shown in Fig. 12, both cases present similar vortex breakdown pattern and length scale. In this way, this cascade model is presentative for the tip leakage vortex characteristics in the subsonic turbine rotor.

61 62

5. Modeling for vortex interactions

63 64 65 66

To examine the vortex-vortex interaction, we simulate a typical subsonic turbine stage with bladerow interaction in advance. The URANS equations are solved with the S S T k − ω model. More stage

ωs∗ =

ωs C x u isen

87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106

(2)

The velocity profile of the vortex model is defined using a polar coordinate system, which is shown in Equation (3), where  is the circulation of the incoming vortex and r v is the radius of the vortex. The circulation and radius of the incoming swirl flow is carefully chosen so that the vorticity magnitude and size is similar to the inlet condition in the turbine stage. The maximum nondimensional streamwise vorticity at the core of the swirl flow is ωd = 5.5. In order to simulate the streamwise velocity deficit in the passage vortex core, which is approximately 80% of the main flow velocity, the streamwise velocity profile is defined in Equation (4). To trace the path of the swirling vortex in the blade passage, the passive scalar released by the User Defined Function (UDF) in FLUENT is used within the incoming vortex region. This requires solving an additional conservation equation for the passive scalar . As indicated in Equation (3), the region within the swirling vortex corresponds to = 1, and the flow outside the vortex has = 0. Thus, the transportation of the swirl flow within the blade passage can be identified. Similar method with passive scalar was also used by Pullan and Denton [50] to visualize the vortex-blade interaction near the hub region. The assumption of the incoming flow condition as a Rankine vortex is also widely adopted in open publications by research groups from Whittle Lab at Cambridge University and MIT ([21,49,50]).

107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE AID:105776 /FLA

[m5G; v1.282; Prn:13/02/2020; 9:47] P.7 (1-14)

K. Zhou, C. Zhou / Aerospace Science and Technology ••• (••••) ••••••

7

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

16

82

17

83

18

84

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

⎧ ·r ⎪ ⎪ ; and = 1; ⎨ uθ = 2 2π r v

86 87

for r < r v

⎪  · rv ⎪ ⎩ uθ = ; and = 0; for r ≥ r v 2π r  ⎧ ⎨ u = 9 u + 1 u × sin π × r − π ; s s0 s0 10 10 rv 2 ⎩ u s = u s0 ;

46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

89 90 91

for r < r v

92

(4)

93

for r ≥ r v

Note that in the current study, ‘quasi-steady’ vortex-vortex interaction is studied, which means that the incoming vortex is fixed in the upstream domain and no relative motion is considered between the upstream domain and the blade domain. However, by changing the blade domain phase in pitchwise direction, different interaction phase is achieved. In total, there are ten equallyspaced phases within one blade pitch considered. Although the quasi-steady vortex interaction is different from the unsteady vortex transportation in the downstream passage with the absence of the vortex bending and stretching (Zhou and Zhou [38,37]), the basic flow patterns are similar and worthy of fundamental investigation. 6. Effects of vortex interaction

44 45

88

(3)

42 43

85

Fig. 13. Non-dimensional vorticity at the cut plane of 0.1C x upstream of the leading edge in streamwise direction.

With the incoming vortex conditions imposed, the effects of different phases of vortex-vortex interaction can be assessed. In total, ten-equally-spaced-phases of vortex interaction are investigated in this research. Fig. 14 presents several typical phases between inlet Rankine vortex and blade. φ = 0/10 is defined at the stagnation point of the leading edge and φ = 9/10 is defined right upstream of the tip leakage flow. Fig. 15 shows the phase where the incoming swirling flow interacts with the tip leakage vortex. The scalar density distribution is presented in Fig. 15a on the cut plane of 0.1C x upstream of the leading edge, SP70, SP80, SP95 and SP105. Of the ten phases considered, this phase has the most significant effect on the tip leakage vortex, which is defined as φ = 9/10. The streamwise vorticity, as defined in Equation (2), is shown in Fig. 15b. The tip leakage vortex implied by the positive vorticity and the passage vortex implied by the negative vorticity is distinguished clearly. The four pathlines released from the incoming swirling region show that when the swirling flow convects downstream in the blade passage, the flow swirls around itself and is partly transported into the tip leakage vortex. Note that the incoming vortex flow has positive vorticity magnitude, which is approximately 20% of the vorticity magnitude of the tip leakage vortex on SP70. Part of the swirling

94 95 96 97 98 99 100 101 102 103

Fig. 14. Sketch of several typical phases between inlet Rankine vortex and blade.

104 105

flow is gradually entrained into the leakage core, and the other part coincides with the passage vortex, which has the opposite vorticity magnitude. Two typical phases with contours of streamwise vorticity to highlight the vortex-vortex interaction are shown in Fig. 16: (a) φ = 4/10, meaning the phase where the tip leakage vortex is least affected by the incoming swirling flow; and (b) φ = 9/10, meaning the phase where the tip leakage vortex is most affected by the incoming flow. The region with reversed streamwise flow is circled in black. The flow patterns at φ = 4/10 are similar to the datum case with uniform inlet condition. A slight delay of breakdown is observed. The tip leakage vortex breaks down clearly on SP95 and SP105. However, at φ = 9/10, the reversed region is not observed, which means that the vortex breakdown is totally suppressed. The swirl flow initiates its breakdown through a very complicated mechanism, as explained in previous papers (Lucca-Negro and O’Doherty [48], Darmofal et al. [49]). However, from the perspective of the Reynolds averaged flow field, swirl number is considered as a reference value to judge whether breakdown occurs. The swirl number is defined as follows:



ωmar

S= 

ωs ds

ωmar u s dl

106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128

(5)

where the integrated area is circled by the iso-value of ωmar . ωmar = ωcore /e and ωcore is the streamwise vorticity perpendicu-

129 130 131 132

JID:AESCTE

8

AID:105776 /FLA

[m5G; v1.282; Prn:13/02/2020; 9:47] P.8 (1-14)

K. Zhou, C. Zhou / Aerospace Science and Technology ••• (••••) ••••••

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81 82

16 17

Fig. 15. Phase φ = 9/10 on cut planes of 0.1C x upstream, SP70, SP80, SP95 and SP105; and pathlines released from the swirling region.

83

18

84

19

85

20

86

21

87

22

88

23

89

24

90

25

91

26

92

27

93

28

94

29

95

30

96

31

97 98

32 33

Fig. 16. Non-dimensional streamwise vorticity distribution on SP70, SP80, SP95 and SP110 for two phases.

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

99 100

34

lar to the surface located at the core of the tip leakage vortex and e is the mathematical constant. In the unconfined flow, the swirl number represents the stability state of the vortex to determine whether it breaks√down or not. As argued by Darmofal et al. [49], the threshold of 2√shows the criterion of breakdown. When the swirl number is over 2, it’s taken as vortex breakdown, accompanied by reversed flow. Though the flow in the turbine cascade is technically confined, a similar trend is observed (Huang et al. [21]). Fig. 17 shows the swirl number along the axial distance for three cases: the datum case with a uniform inlet, phase φ = 4/10 and phase φ = 9/10. From √ 0.5C x to 0.75C x , the swirl number increases slowly around 2. In the case with a uniform inlet, right across 0.8C x , the swirl number increases dramatically. The vortex breakdown is identified with a sharp increase in the swirl number. Associated with the start of the reversed flow, 0.8C x is roughly taken as the breakdown starting point. When the vortex transports downstream to SP95, large expansion of reversed area can be distinguished because of flow diffusion after the throat. After the trailing edge, as the large turbulent mixing happens due to the highly 3-D flow, the reversed flow gradually recovers its streamwise momentum and the corresponding reversed region is smaller. The limiting streamlines on the suction surface are also presented in Fig. 11. The limiting streamlines close to the tip region are induced by the tip leakage vortex and tend to reverse when the tip leakage vortex transports towards the trailing edge, which also implies the occurrence of vortex breakdown. Cases of the uniform inlet situation and phase φ = 4/10 experience the sharp increase of the swirl number after 0.8C x and vortex breakdown is observed. At phase φ = 9/10, the swirl number is al-

101 102 103 104 105 106 107 108 109 110 111 112 113 114 115

Fig. 17. Swirl number along the axial direction.

116 117

ways below the threshold and vortex breakdown is significantly suppressed. The stagnation pressure coefficient is shown in Fig. 18 for phase φ = 4/10 and φ = 9/10. When the tip leakage vortex is unaffected by the incoming vortex at φ = 4/10, the induced pressure at the core of the tip leakage vortex is low. High stagnation pressure coefficient is observed within the vortex core. However, when the tip vortex breakdown is suppressed in Fig. 18b, the tip leakage vortex is not as strong as that at φ = 4/10. The vorticity magnitude within the vortex core at φ = 9/10 is reduced. As a result, the high loss within the vortex core is also eliminated. This indicates that when the vortex breakdown is suppressed, the loss is also reduced. The area-weighted-average viscous dissipation is evaluated on the cut plane of 0.93 span. Compared to the value in the uniform datum case, the case at φ = 4/10 increases the dissipation by 2.6%. This

118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE AID:105776 /FLA

[m5G; v1.282; Prn:13/02/2020; 9:47] P.9 (1-14)

K. Zhou, C. Zhou / Aerospace Science and Technology ••• (••••) ••••••

9

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78 79

13

Fig. 19. C p distribution on 0.93 span of the blade for two phases.

14 15

82

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

80 81

Fig. 18. C p0 distribution on SP70, SP80, SP95 and SP105 for two phases.

effect is mainly contributed to the viscous shear stress of incoming vortex in the blade passage. At φ = 9/10, the viscous dissipation is reduced by 4.9%, compared to the datum case. By introducing the incoming vortex, although the viscous dissipation induced by the vortex itself is increased, there is an overall reduction of mixing loss due to the suppression of vortex breakdown. The overall mixed-out entropy is reduced by 2.3% at φ = 9/10, compared with the entropy at φ = 4/10. It’s interesting to point out that the vorticity magnitude of the incoming vortex is only 20% of the vorticity magnitude of the tip leakage vortex on cutting plane SP70, which implies that we can control the aerodynamic loss with relatively small incoming vortex magnitude. The static pressure coefficient around the blade surface on 0.93 span is shown in Fig. 19. Around the pressure side surface, the static pressure coefficient distributes similarly between φ = 4/10 and φ = 9/10. However, on the suction side surface towards the trailing edge, slight difference is observed. The imposed adverse pressure gradient becomes larger at φ = 9/10. As indicated by Equation (1), the adverse pressure gradient along the vortex core is the result of the imposed pressure gradient and the vortex induced pressure gradient. In this case, the static pressure around the blade surface is roughly taken as the imposed pressure gradient. It can be concluded that the vortex induced pressure gradient is much larger than the imposed pressure gradient because at phase φ = 4/10, the tip leakage vortex suffers more severe adverse pressure gradient. The streamwise pressure gradient non-dimensionalized by Equation (6) is shown in Fig. 20. The regions circled in red correspond to high positive streamwise vorticity (greater than 10). At φ = 4/10, when the upstream incoming vortex is away from the tip leakage vortex, the peak adverse pressure gradient (APG) is induced by the streamwise vorticity, followed by the reversed flow (black iso-surface region). At φ = 9/10 when the incoming vortex is partly entrained into the leakage flow, the region of high streamwise vorticity (tip leakage vortex) expands. The peak positive streamwise vorticity is reduced and pushed further downstream. The peak APG is also reduced significantly and pushed downstream close to the trailing edge. As a result, the leakage vortex endures the moderate adverse pressure gradient to resist to reverse. No vortex breakdown is observed. This is mainly contributed by the vorticity elimination within the tip leakage vortex.



∂p ∂s

∗

=

∂p ∂ x ux



+

u 2x

∂p ∂y uy

+ u 2y

/

P 0inlet − p ex Cx

(6)

The non-dimensional turbulent kinetic energy (TKE) is shown on the same cutting planes in Fig. 21. When the incoming vortex is away from the tip leakage vortex at φ = 4/10, the peak TKE region ‘A’ coincides with the high streamwise vorticity region, where the

83 84 85 86 87 88 89 90 91 92 93 94 95

Fig. 20. Non-dimensional streamwise pressure gradient, iso-line of high streamwise vorticity (red lines), reversed flow (black surface), on cut plane of 0.93 span.

96 97 98

tip leakage vortex first develops at 0.5C x . The reversed flow acts like a ‘separation bubble’, followed by two high TKE regions ‘B’. The flow convects around the reversed flow region and converges together, and high TKE is unavoidable because of strong shear stress effect in 3D way. At φ = 9/10, the imposed incoming vortex, however, enhances the local TKE in ‘A’, making the leakage flow mix with the swirl flow. As a result, the streamwise momentum is replenished into the tip leakage vortex core flow. At the same time, the leakage vortex circulation and the adverse pressure gradient are significantly reduced. The vortex breakdown is thus avoided. Due to the suppression of the reversed bubble, the high TKE region in ‘B’ is also removed, which greatly eliminates the mixing loss. In other words, by introducing a pre-mixing at the leakage vortex near 0.5C x , we successfully improve the downstream flow field. Note that in the simulation of the incoming vortex, no extra turbulence intensity is defined within the core. That means that the improvement of TKE in ‘A’ is the result of shear stress, not the transportation of TKE from the incoming vortex, because TKE of incoming vortex is two orders of magnitude lower than that of the leakage vortex. As discussed above, the vortex-induced pressure term dominates the local adverse pressure gradient. In order to explain the difference in streamwise vorticity distribution between two typical phases, the vorticity evolution is described by the vorticity transport equation, expressed based on the mean velocity field given in Equation (7) (Wu et al. [51]). Note that turbulent vorticity may also be involved in the turbine environment. To simplify the calculations without losing generality, we use the mean flow field, which includes the basic flow information. The rate of change in vorticity has several contributing terms: 1. the stretching and tilting of the vorticity by the mean velocity gradient (also called the convective term, the first term on the right side); 2. the dilation effect due to compressibility (the second term); 3. the unconser-

99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE

AID:105776 /FLA

[m5G; v1.282; Prn:13/02/2020; 9:47] P.10 (1-14)

K. Zhou, C. Zhou / Aerospace Science and Technology ••• (••••) ••••••

10

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79 80

14

Fig. 23. Convective term, on cut plane of 0.93 span.

15 16

81 82

Fig. 21. Non-dimensional turbulent kinetic energy, on cut plane of 0.93 span.

17

83

18

84

19

85

20

86

21

87

22

88

23

89

24

90

25

91

26

92

27

93

28

94 95

29

Fig. 24. Dilation term, on cut plane of 0.93 span.

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

vative force part (the third part), especially for Coriolis force in rotationary frame. Here, it is treated as zero in the stationary linear cascade; 4. the baroclinicity effect due to coexistence of the effects of density and pressure (the fourth term); and 5. the viscous diffusion due to the viscous fluid (the last term). Except the third term, four other terms will be discussed in detail. The positive magnitude in these terms means the vorticity enhancement, and vice versa. All the values are non-dimensionalized by the exit velocity and axial chord length. The magnitude are evaluated in streamwise direction on the cut plane of 0.93 span, which means the spanwise vorticity transportation is neglected. Only the 2-D effect is considered. 

48

dω

49

dt

50











= (ω •∇) u − ω (∇• u )+∇× f −∇

 1

ρ

2

×∇ p + νe f f ∇ ω (7)

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

96 97

Fig. 22. Non-dimensional streamwise vorticity, on cut plane of 0.93 span.

The streamwise vorticity on cut plane of 0.93 span is shown in Fig. 22. The region with positive vorticity region greater than 10 is circled in black. As discussed above, at phase φ = 9/10, the positive streamwise vorticity is reduced in magnitude and enhanced in size. There’s a small area ‘A’ of high vorticity corresponding to the incoming vortex. As the flow accelerates through the passage, the incoming vortex is also stretched. As the vortex tube is stretched, streamwise vorticity increases as marked by ‘A’. When the mean flow field decelerates after the peak loading, the vortex tube is eliminated. The first convective term in Equation (7) is shown in Fig. 23. Comparing the two phases, it is found that the vortex interaction has little influence on vortex dynamic evolution, which indicates that the mean flow field is little affected. It only affects some small region outside of the tip leakage vortex core as marked by ‘D’.

The dilation term is compared in Fig. 24. At phase φ = 9/10, a significant negative region marked as ‘B’ is observed, which means the dilation effect eliminates the streamwise vorticity as the vortex core expands. The vortex interaction reduces the leakage vorticity by the dilation effect. While at phase φ = 4/10, the dilation effect is neglected as the vortex core is strong enough to remain in size. The baroclinic term is compared in Fig. 25. Note that the legend range of the baroclinic value is smaller than the streamwise vorticity. It’s taken for granted that the only way to address the baroclinicity effect is caused by an oblique shock wave, which is observed in transonic and supersonic turbines. In the current subsonic cascade environment, the baroclinicity effect is usually neglected. At phase φ = 4/10, the numerical evaluation shows that the baroclinic effect is small. A surprising finding is that the baroclinicity effect cannot be neglected in the current subsonic turbine due to vortex-vortex interaction. At phase φ = 9/10, as marked by ‘C’, the incoming vortex significantly changes the local density gradient and pressure gradient, inducing the negative term. The dimensionless static temperature is shown in Fig. 26. The gray iso-surface of passive scalar 0.2 indicates the transportation of the incoming vortex. Within the leakage core, the temperature is very high, as circled by a red dashed line at phase φ = 4/10 in Fig. 26a. When the main flow accelerates through the passage, the static temperature drops around the peak loading. At phase φ = 9/10, when the incoming vortex passes through the passage, the low temperature fluid is transported by the incoming vortex and mixes with the tip leakage vortex as marked by ‘C’ in Fig. 26b, inducing the density gradient nonparallel to the pressure gradient. The baroclinicity effect tends to reduce the streamwise vorticity. To the best of our knowledge, this is the first report of a baroclinicity effect caused by temperature non-uniformity in the turbine without an oblique shock wave. This implies that in a low-speed, large-scale cascade, the baroclinicity effect cannot be modeled as the temperature field varies slightly. Another potential benefit of vortex transportation

98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE AID:105776 /FLA

[m5G; v1.282; Prn:13/02/2020; 9:47] P.11 (1-14)

K. Zhou, C. Zhou / Aerospace Science and Technology ••• (••••) ••••••

11

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14 15

Fig. 25. Baroclinicity term, on cut plane of 0.93 span.

Fig. 27. Viscous diffusion term, on cut plane of 0.93 span.

80 81

16

82

17

83

18

84

19

85

20

86

21

87

22

88

23

89

24

90

25

91

26

92

27

93 94

28 29 30

Fig. 26. Non-dimensional static temperature, iso-surface of scalar 0.2 (gray), on cut plane of 0.93 span.

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

of low-temperature fluid is that the blade surface suffers less from high-temperature exposure at this particular phase. In the region circled by red dashed lines in Fig. 26b, due to the low temperature fluid entrained into the tip leakage vortex core and near the suction side surface, the blade surface temperature is significantly reduced. Potential thermal management may benefit from the vortex interaction. Overall thermal performance due to unsteady interaction should be included in future research. The viscous diffusion term is shown in Fig. 27. Of these four terms, the viscous diffusion dominates the local vorticity damping. Comparing the two phases, at phase φ = 9/10, we find that the diffusion term is much larger than that at phase φ = 4/10. The main reason is that the vortex interaction increases the local TKE, and the viscous diffusion tends to eliminate the leakage vortex vorticity. This is consistent with the TKE distribution in Fig. 21. To conclude, the four terms are investigated to contribute to the leakage vorticity damping. The viscous diffusion effect dominates the evolution of leakage vorticity. The convective term due to mean velocity field contributes little to the vorticity change. The dilation and baroclinicity effects are small, but still cannot be neglected.

52 53

7. Effects of different vortex characters

54 55 56 57 58 59 60 61 62 63 64 65 66

In this section, different incoming vortex characters are examined at phase φ = 9/10. If the incoming vorticity magnitude ωd = 5.5 is taken as a reference, four extra cases with different vorticity magnitude can be defined: ωd /4 = 1.4, ωd /2 = 2.8, 2ωd = 11.0 and −ωd = −5.5. Each incoming vortex is kept the same vortex diameter and is imposed at the same spanwise location. The swirl number along the axial direction is shown in Fig. 28 for five cases. With the increase of positive incoming vorticity from 1.4 to 11.0, the swirl number decreases gradually, which implies that the incoming vortex flow tends to suppress the tip leakage vortex breakdown. When the incoming swirl flow is strong enough, e.g., ωd ≥ 5.5, tip leakage vortex breakdown can be avoided. However, if

95 96

Fig. 28. Swirl number along the axial direction for different inlet vortex characteristics, at φ = 9/10.

97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114

Fig. 29. Scalar iso-surface of value 0.3 for two cases, both at φ = 9/10.

115 116 117

the negative incoming vortex is imposed upstream of the blade, tip leakage vortex breakdown is encountered, which means that the opposite vortex direction to the tip leakage vortex does not reduce the tip leakage loss. The iso-surface value 0.3 of scalar density released from the inlet swirling region is shown in Fig. 29. The cases with opposite incoming vortex directions are compared with an incoming vorticity magnitude of 5.5. When positive swirling flow convects downstream, part of the flow is entrained into the tip leakage vortex, while the negative swirling flow is transported radially inward, far away from the tip leakage vortex, with little flow entrained into the tip leakage vortex. This results in less vortex interaction between the swirling flow and the tip leakage vortex. At the same time, the swirling flow coincides with the passage vortex, both having negative vorticity. The vortex dynamic evolution is explained with a simulation of a triple-vortex interaction model.

118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE

AID:105776 /FLA

12

[m5G; v1.282; Prn:13/02/2020; 9:47] P.12 (1-14)

K. Zhou, C. Zhou / Aerospace Science and Technology ••• (••••) ••••••

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

16

82

17

Fig. 30. A simple simulation of triple-vortex interaction model.

18 19 20

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

86 87

Similar flow patterns corresponding to the vortex-vortex interaction are validated in a previous experimental work by the same authors [44]. In a low-speed, large-scale linear cascade in the Turbomachinery Laboratory at Peking University, an incoming swirling flow is simulated by swirl generators used upstream of the cascade. The incoming positive swirling flow is observed to be gradually entrained into the tip leakage core, reducing the leakage loss significantly, while the negative swirling flow has little influence on the leakage loss. In order to explain the flow mechanisms, a triple-vortex interaction model is examined. A simple numerical simulation is conducted by modelling three vortices at the inlet of a tube in Fig. 30. According to the vortex characters in the cascade, the non-dimensional vorticity within the tip leakage vortex (TLV) is set to be +1, the passage vortex (PV) is −1/2 and the positive swirling vortex (PSV) is +1/4, which is typical value for the subsonic cascade simulation. PSV or NSV (negative swirling vortex) is imposed by a passive scalar, which is circled in black. As the vortices move downstream, the positive inlet Rankine vortex flow (PSV) is gradually entrained into tip leakage vortex (TLV). If the negative inlet Rankine vortex (NSV) is imposed, it finally merges into the passage vortex (PV). The numerical results agree well with the findings in the subsonic cascade. The flow mechanism is explained with the help of the diagram in Fig. 31. When PSV and TLV have the same vortex direction, the shear stress effect on the interface between TLV and PSV induces flow instability, which enhances TKE. High TKE makes PSV and TLV mix together. Because the vorticity magnitude of TLV is much higher than that of PSV (about four times greater), PSV is gradually entrained by TLV. The mixing process enhances the local TKE and eliminates the streamwise vorticity, as shown in Fig. 31 (a) Type A. When negative swirling vortex is imposed upstream, NSV and PV have the same vortex direction, and the shear stress effect makes the two vortices mix together, as shown in Fig. 31(b) Type B. As a result, TKE within TLV remains almost unchanged. This explains the vortex dynamic process in Fig. 29.

57 58

9. Conclusions

59 60 61 62 63 64 65 66

84 85

8. A triple-vortex interaction model

21 22

83

In the current research, the numerical methods are used to identify the vortex breakdown characteristics of the tip leakage vortex in a linear subsonic turbine cascade with the influence of the three-dimensional incoming vortex flow. The CFD tools are validated with a satisfactory level. A Rankine vortex, modeled from the complex inlet condition in a subsonic turbine stage, is imposed on the inlet at different interaction phases by a quasi-steady

88 89 90 91 92 93 94 95 96 97 98 99 100 101 102

Fig. 31. Two types of a triple-vortex interaction kinetic model (TLV: Tip Leakage Vortex; PV: Passage Vortex; PSV: Positive Swirling Vortex; PS: Pressure Side; SS: Suction Side).

103 104 105 106

method. The flow mechanisms due to the interaction between the incoming vortex and the tip leakage vortex are comprehensively investigated. The main findings are listed below: 1. In the basic case with a uniform inlet condition, the tip leakage vortex is highly unstable. The flow can’t endure the severe adverse pressure gradient and begins to reverse. Vortex breakdown occurs at 0.8C x , accompanied by a dramatic increase in the swirl number. The associated loss is large because of the shear stress effect downstream of the reversed bubble. The adverse pressure gradient within the vortex core along the streamwise direction is mainly contributed by the induction of vortex circulation, not the exposed pressure gradient. 2. The positive incoming vortex has the same vortex direction as the tip leakage vortex. When these two vortices interact with each other at φ = 9/10, the maximum streamwise vorticity within the tip leakage vortex is reduced. As a result, the vortex-induced adverse pressure gradient within the tip leakage vortex core is relatively moderate. The leakage vortex is stable enough to endure the adverse pressure gradient. Vortex breakdown is not observed. The pre-mixing at the very beginning of the leakage vortex at 0.5C x is enhanced, promoting turbulent mixing between the leakage core and the incoming vortex. Because of the suppression of the separation bubble, the high TKE region after the bubble is also removed. By examining the vorticity transport equation, it’s found that the viscous diffusion effect dominates the vorticity damping. The dilation effect should not be neglected as the flow is com-

107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE AID:105776 /FLA

[m5G; v1.282; Prn:13/02/2020; 9:47] P.13 (1-14)

K. Zhou, C. Zhou / Aerospace Science and Technology ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

pressible. It’s surprising to find that the baroclinicity effect also contributes to the decay of vorticity because the low temperature fluid is transported into the leakage core. This also helps to prevent thermal fatigue of blade. At φ = 4/10, there is no evident vortex interaction observed. The flow pattern is quite similar to the basic case with uniform inlet conditions. The mixed-out entropy at φ = 9/10 decreases by 2.3%, compared to the entropy at φ = 4/10. 3. Tip leakage vortex breakdown tends to be suppressed as the incoming positive vorticity increases from 1.4 to 11.0. However, the direction of the incoming vortex has a significant effect on the tip leakage vortex breakdown. When the negative vortex is imposed upstream, the swirling flow is transported radially inward and little flow is entrained into the tip leakage flow. A triple-vortex interaction model is proposed to explain the vortex kinetic process and is validated by numerical examinations. The suppression of vortex breakdown is preferred due to aerodynamic performance consideration, and it is also advantageous for flow uniformity when it convects downstream. Moreover, turbine blade thermal loading may also be improved by the vortex interaction. Although only a quasi-steady vortex-vortex interaction is considered in the current work, unsteady vortex interaction is expected to be similarly important as discussed in the previous work (Zhou and Zhou [38]). In the future of advanced turbomachinery design, three-dimensional turbine design may be considered for a better aero-thermal performance, and complicated vortex interaction should be involved.

27 28

Declaration of competing interest

29 30

There is no conflict of interest.

31 32

Acknowledgements

33 34 35 36 37 38

The authors would like to acknowledge the support of the National Natural Science Foundation of China (NSFC), Grant No. 91752106 and No. 51576003. The authors would also thank Prof. Liping Xu for suggestions during the discussions. The first author also thanks Dr. Lichao Jia for his valuable suggestions.

39 40

Uncited references

41 42

[39] [41] [47]

43 44

References

45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

[1] U.W. Schaub, E. Vlasic, S.H. Moustapha, Effect of tip clearance on the performance of a highly loaded turbine stage, Technology Requirements for Small Gas Turbine, AGARD CP-537, Paper 29, 1993. [2] R.S. Bunker, J.C. Bailey, A.A. Ameri, Heat transfer and flow on the first-stage blade tip of a power generator gas turbine: part 1-experimental results, J. Turbomach. 122 (2) (2000) 263–271. [3] A.A. Ameri, R.S. Bunker, Heat transfer and flow on the first-stage blade tip of a power generator gas turbine: part 2 – simulation results, J. Turbomach. 122 (2) (2000) 272–277. [4] M. Sell, M. Treiber, C. Casciaro, G. Gyarmathy, Tip-clearance-affected flow fields in a turbine blade row, Proc. Inst. Mech. Eng. A 23 (4) (1999) 309–318. [5] S. Schlechtriem, M. Lötzerich, Breakdown of tip leakage vortices in compressors at flow conditions close to stall, ASME Paper 1997, 97-GT-41 1997. [6] J.H. Lee, J.M. Han, H.G. Park, J.S. Seo, Application of signal processing techniques to the detection of tip vortex cavitation noise in marine propeller, J. Hydrodyn. 25 (3) (2013) 440–449. [7] C.S. Tan, I. Day, S. Morris, A. Wadia, Spike-type compressor stall inception, detection, and control, Annu. Rev. Fluid Mech. 42 (1) (2010) 275–300. [8] D. You, M. Wang, P. Moin, R. Mittal, Effects of tip-gap size on the tip-leakage flow in a turbomachinery cascade, Phys. Fluids 18 (2006) 105102-1-14. [9] Y. Wu, G. An, B. Wang, Numerical investigation into the underlying mechanism connecting the vortex breakdown to the flow unsteadiness in a transonic compressor rotor, Aerosp. Sci. Technol. 86 (2019) 106–118. [10] M. Furukawa, M. Inoue, K. Saiki, K. Yamada, The role of tip leakage vortex breakdown in compressor rotor aerodynamics, ASME Paper 1998, 98-GT-239, 1998.

13

[11] M. Furukawa, K. Saiki, K. Yamada, M. Inoue, Unsteady flow behavior due to breakdown of tip leakage vortex in an axial compressor rotor at near-stall condition, ASME Paper 2000, 2000-GT-0666, 2000. [12] K. Yamada, M. Furukawa, T. Nakano, M. Inoue, K. Funazaki, Unsteady threedimensional flow phenomena due to breakdown of tip leakage vortex in a transonic axial compressor rotor, ASME Paper 2004, GT2004-53745, 2004. [13] C. Zhou, H. Hodson, Squealer geometry effects on aerothermal performance of tip-leakage flow of cavity tips, J. Propuls. Power 28 (3) (2012) 556–567. [14] Z.P. Zou, F. Shao, Y.R. Li, W.H. Zhang, A. Berglund, Dominant flow structure in the squealer tip gap and its impact on turbine aerodynamic performance, Energy 138 (2017) 167–184. [15] C. Zhou, F.P. Zhong, A novel suction side winglet design method for high pressure turbine rotor tips, J. Turbomach. 139 (2017) 111001-1-11. [16] L. Silva, J. Tomita, C. Bringhenti, Numerical investigation of a HPT with different rotor tip configurations in terms of pressure ratio and efficiency, Aerosp. Sci. Technol. 63 (2016) 33–40. [17] J. Gao, Q. Zheng, P. Dong, X. Zha, Control of tip leakage vortex breakdown by tip injection in unshrouded turbines, J. Propuls. Power 30 (6) (2014) 1510–1519. [18] J. Gao, Q. Zheng, T. Xu, P. Dong, Inlet conditions effect on tip leakage vortex breakdown in unshrouded axial turbines, Energy 91 (2015) 255–263. [19] L. Kavurmacioglu, C. Senel, H. Maral, C. Camci, Casing grooves to improve aerodynamic performance of a HP turbine blade, Aerosp. Sci. Technol. 76 (2018) 194–203. [20] S. Jiang, F. Chen, J. Yu, S. Chen, Y. Song, Treatment and optimization of casing and blade tip for aerodynamic control of tip leakage flow in a turbine cascade, Aerosp. Sci. Technol. 86 (2019) 704–713. [21] A.C. Huang, E.M. Greitzer, C.S. Tan, E.F. Clemens, S.G. Gegg, E.R. Turner, Blade loading effects on axial turbine tip leakage vortex dynamic and loss, J. Turbomach. 135 (5) (2013) 051012-1-11. [22] R.P. Dring, H.D. Joslyn, L.W. Hardin, J.H. Wagner, Turbine rotor-stator interaction, J. Eng. Power 104 (4) (1982) 729–742. [23] G. Paniagua, T. Yasa, A. de la Loma, L. Castillon, T. Coton, Unsteady strong shock interactions in a transonic turbine experimental and numerical analysis, J. Propuls. Power 24 (4) (2008) 722–731, https://doi.org/10.2514/1.34774. [24] B. Hanlock, J. Clark, Reducing shock interactions in transonic turbine via threedimensional aerodynamic shaping, J. Propuls. Power 30 (5) (2014) 1248–1256. [25] H. Hodson, R. Howell, Bladerow interactions, transition, and high-lift aerofoils in low pressure turbines, Annu. Rev. Fluid Mech. 37 (2015) 71–98. [26] J.D. Coull, H.P. Hodson, Unsteady boundary-layer transition in low-pressure turbines, J. Fluid Mech. 681 (2011) 370–410. [27] R. Ciorciari, I. Kirik, R. Niehuis, Effects of unsteady wakes on the secondary flows in the linear T106 turbine cascade, J. Turbomach. 136 (9) (2014) 0910101-11. [28] R.J. Volino, C.D. Galvin, C.J. Brownell, Effects of unsteady wakes on flow through high pressure turbine passage with and without tip gaps, ASME Paper 2014, GT2014-27006, 2014. [29] R.D. Stieger, H.P. Hodson, The unsteady development of a turbulent wake through a downstream low-pressure turbine blade passage, J. Turbomach. 127 (2) (2005) 388–394. [30] K. Zhou, C. Zhou, Transport mechanism of hot streaks and wakes in a turbine cascade, J. Propuls. Power 32 (5) (2016) 1045–1054. [31] J. Zeschky, H.E. Gallus, Effects of stator wakes and spanwise nonuniform inlet conditions on the rotor flow of an axial turbine stage, in: International Gas Turbine and Aeroengine Congress and Exposition, 1991. [32] M.G. Dunn, W.J. Rae, J.L. Holt, Measurement and analyses of heat flux data in a turbine stage: part II—discussion of results and comparison with predictions, J. Eng. Gas Turbines Power 106 (1) (1984) 234–240. [33] Z. Liu, Z. Liu, Z. Feng, Unsteady analysis on the effects of tip clearance height on hot streak migration across rotor blade tip clearance, J. Eng. Gas Turbines Power 126 (8) (2014) 082605. [34] R. Ma, W. Devenport, Unsteady periodic behavior of a disturbed tip-leakage flow, AIAA J. 44 (5) (2006) 1073–1086. [35] R. Ma, W. Devenport, Tip gap effects on the unsteady behavior of a tip-leakage vortex, AIAA J. 45 (7) (2007) 1713–1724. [36] R. Ma, W. Devenport, Unsteady aperiodic behavior of a periodically disturbed tip leakage vortex, AIAA J. 46 (5) (2008) 1025–1038. [37] K. Zhou, C. Zhou, Unsteady aerodynamics of a flat tip and a winglet tip in a high-pressure turbine, ASME Paper 2016, GT2016-56845, 2016. [38] K. Zhou, C. Zhou, Unsteady effects of vortex interaction on tip leakage vortex breakdown and its loss mechanism, Aerosp. Sci. Technol. 82–83 (2018) 363–371. [39] F. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, AIAA J. 32 (8) (1994) 1598–1605. [40] F. Menter, M. Kuntz, R. Langtry, Ten years of experience with the SST turbulence model, Turbul. Heat Mass Transf. 4 (2003) 625. [41] P.J. Roache, Quantification of uncertainty in computational fluid dynamics, Annu. Rev. Fluid Mech. 29 (1997) 123–160. [42] J. Gao, M. Wei, W.L. Fu, Q. Zheng, G.Q. Yue, Experimental and numerical investigations of trailing edge injection in a transonic turbine cascade, Aerosp. Sci. Technol. 92 (2019) 258–268.

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE

14

1 2 3 4 5 6 7 8 9

AID:105776 /FLA

[m5G; v1.282; Prn:13/02/2020; 9:47] P.14 (1-14)

K. Zhou, C. Zhou / Aerospace Science and Technology ••• (••••) ••••••

[43] L. Kavurmacioglu, C. Senel, H. Maral, C. Camci, Casing grooves to improve aerodynamic performance of a HP turbine blade, Aerosp. Sci. Technol. 76 (2018) 194–203. [44] K. Zhou, C. Zhou, Aerodynamic interaction between an incoming vortex and tip leakage flow in a turbine cascade, J. Turbomach. 140 (11) (2018) 111004-1-12. [45] Q. Zhang, D.O. O’Dowd, A.PS. Wheeler, Overtip shock wave structure and its impact on turbine blade tip heat transfer, J. Turbomach. 133 (4) (2011) 0410011-8. [46] A.S. Virdi, Q. Zhang, L. He, H.D. Li, R. Hunsley, Aerothermal performance of shroudless turbine blade tips with relative casing movement effects, J. Propuls. Power 31 (2) (2015) 527–536.

[47] M.G. Hall, A new approach to vortex breakdown, Proc. Heat Transfer Fluid Mech. Inst. (1967) 319. [48] O. Lucca-Negro, T. O’Doherty, Vortex breakdown: a review, Prog. Energy Combust. 27 (4) (2001) 431–483. [49] D. Darmofal, R. Khan, E. Greitzer, C. Tan, Vortex core behaviour in confined and unconfined geometries: a quasi-one-dimensional model, J. Fluid Mech. 449 (2001) 61–84. [50] G. Pullan, J.D. Denton, Numerical simulations of vortex-turbine blade interaction, in: Fifth European Turbomachinery Conference, 2003, Prague, Czech Republic, Mar. 7–11, 2003, pp. 1049–1059. [51] J. Wu, H. Ma, M. Zhou, Vorticity and Vortex Dynamics, Springer, 2006.

67 68 69 70 71 72 73 74 75

10

76

11

77

12

78

13

79

14

80

15

81

16

82

17

83

18

84

19

85

20

86

21

87

22

88

23

89

24

90

25

91

26

92

27

93

28

94

29

95

30

96

31

97

32

98

33

99

34

100

35

101

36

102

37

103

38

104

39

105

40

106

41

107

42

108

43

109

44

110

45

111

46

112

47

113

48

114

49

115

50

116

51

117

52

118

53

119

54

120

55

121

56

122

57

123

58

124

59

125

60

126

61

127

62

128

63

129

64

130

65

131

66

132