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Effect of multiple DBD plasma actuators on the tip leakage flow structure and loss of a turbine cascade
International Journal of Heat and Fluid Flow 77 (2019) 377–387
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International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Effect of multiple DBD plasma actuators on the tip leakage flow structure and loss of a turbine cascade
T
Wang Zhao, Yu Jianyang , Chen Fu, Song Yanping ⁎
Harbin Institute of Technology, Harbin 150001, China
ARTICLE INFO
ABSTRACT
Keywords: Flow structure Tip leakage control DBD Plasma actuator Surrogate model
In this paper, the effects of multiple dielectric barrier discharge (DBD) plasma actuators on the leakage flow structures and loss conditions have been numerically studied in an axial turbine cascade. Kriging surrogate model is adopted to obtain the optimal cases. The physical mechanism of flow structures inside the gap that control leakage flow is presented, which is obtained by analyzing the flow topology, the evolution of the flow structures and its influence on the secondary velocity and loss conditions in the passage as well. The results show that the induced vortex caused by DBD actuators can change the leakage flow direction inside the tip gap and make the separation bubble break earlier, leading to a new type of the flow pattern. When the actuators are applied, the speed of leakage flow is significantly reduced and the angle between leakage flow and main flow has an obviously diminution, causing the reduction of mixing losses in the passage compared with the Baseline case. Furthermore, the comparison of secondary velocity shows that the tip leakage vortex (TLV) approaches the suction surface, resulting in reduced affected area and weakened loss strength. Plasma actuators can diminish the loss coefficient in both TLV and passage vortex near the casing (PVC) zones. The actuators arranged near the trailing edge mainly affect the strength of TLV, while the actuators in the leading edge area contribute to the loss reduction in the zone of PVC.
1. Introduction
loss. Kumada et al. (1994) found that TLF flows out of the gap in the back of the blade and a leakage vortex is recognized at the suction side near the trailing edge. Booth et al. (1982) found that the loss caused by TLF decreases as decreasing the gap height. But the minimum height of 1 to 2 percent of blade span is still required to tolerate the practical deformation of the rotor blade (Zhang et al., 2017). Thus, to further reduce the effect of the TLF, the research community has proposed numbers of shape designs such as squealer tips (Ma et al., 2017; Lee et al., 2011), winglets (O'Dowd et al., 2012; Lee et al., 2012), honeycomb tips (Fu et al., 2018), and other active blowing or suction (Volino, 2017), over the past few decades. These methods are all found to be particularly successful in reducing the tip leakage flow beneficially. There are three types of squealer tips, which are tips with a suction side squealer, a pressure side squealer and a double side squealer. Taking an example of double side squealer, its mechanism in improving aerodynamic performance is that this squealer tip can separate the flow at the top of the two squealers, meaning it could block the leakage flow twice and enhance the mixing of the leakage flow inside the cavity (Zou et al., 2017). The aerodynamic advantage of the pressure-side and the suction-side partial squealer tips has been confirmed (Lee S.E. and
In order to ensure the frictionless movement of the rotor blade relative to the casing and provide the space margin required for centrifugal stretching and thermal expansion, there is inevitably a tip gap in turbine cascade. However, under the pressure difference between both sides of the blade tip, the leakage fluid accelerates into the gap and flows from the pressure surface (PS) to the suction surface (SS), eventually rolling up the tip leakage vortex (TLV). When the TLV inevitably mixes with main flow and interacts with the passage vortex near the case (PVC), the leakage loss occurs, which accounts for about 1/3 of the total blade passage loss in unshrouded turbine (Denton, 1993). This large loss indicates the importance of controlling tip leakage flow (TLF) for increasing the whole turbine efficiency. The researchers conducted a detailed study of the flow structure of the TLF. Bindon (1989) found that all losses associated with the TLF are related to the separation bubble formed near the PS. Due to the pressure gradient, a large amount of low energy fluid is generated to form the separation bubble. But when the pressure gradient disappears, the separation bubble is broken. The TLF accelerates into the gap and is blended with the part of the low energy fluid, causing additional energy
⁎
Corresponding author. E-mail address: [email protected] (J. Yu).
https://doi.org/10.1016/j.ijheatfluidflow.2019.05.002 Received 5 December 2018; Received in revised form 2 March 2019; Accepted 2 May 2019 Available online 15 May 2019 0142-727X/ © 2019 Elsevier Inc. All rights reserved.
International Journal of Heat and Fluid Flow 77 (2019) 377–387
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Nomenclature a b C Cax Cp d DBD Di E E0 Eb ec fx, fy H k1 k2 m Ns p p* PVC PVH S SST TLF
TLMFR TLV U V y+
= 3, the height of the plasma [mm] = 10, the width of the plasma [mm] blade chord [m] blade axial chord [m] static pressure coefficient distance between the two samples dielectric barrier discharge dielectric barrier discharge plasma actuators Electric field [N/C] Electric field maximum [N/C] = 30, breakdown electric field strength [kv/cm] Elementary charge 1.602×10−19 [C] Body force vector [N/m3] in x direction and y direction, respectively blade span [m] Parameter defined in the plasma model Parameter defined in the plasma model mass flow rate number of actuators working static pressure [Pa] total pressure [Pa] passage vortex near the casing passage vortex near the hub entropy [J/mol] shear stress transport tip leakage flow
tip leakage mass flow rate tip leakage vortex AC voltage peak [V] velocity [m/s] dimensionless mesh height
Greek symbols α αk βk θ ρc τ ϑ ϖ ¯ ¯
= 1, factor to account for the collision efficiency inlet flow angle [°] outlet flow angle [°] the angle between main flow and leakage flow [°] charge density [C/m3] tip gap height [m] AC voltage frequency [Hz] The total pressure loss coefficient The pitch-wise averaged total pressure loss coefficient The mass flow averaged total pressure loss coefficient
Subscripts 0 1 a b CFD Kri L
Lee S.W., 2016; Lee S.W. and Lee S.E., 2014). Its effectiveness in reducing the tip leakage mass flow rate (TLMFR) was early confirmed by Heyes et al. (1992). Numerical studies conducted by Yang and Feng (2007) showed that TLMFR decreased as increasing the squealer depth until the depth reached 3 percent of the blade span. Nho et al. (2012) found that the double squealer tip and the grooved along pressure side tip performed best among the several tested blade tips and all tips showed smaller total pressure loss coefficients than that of the plane tip. Maesschalck et al. (2016) confirmed that the optimized blade tip design could further reduce TLMFR and improve the aerodynamic performance. Meanwhile, other researches emphasized that the sealing effect of the squealer tip originates from the internal flow structure within the gap. Krishnababu et al. (2009) found that the obstruction effect of the squealer tip became weaker as the contraction coefficient increased, which was in accordance with the prediction of the two-dimensional idealized model. Although the squealer tips are aerodynamically useful to alter the tip leakage flow, they do cause some negative effects on the heat transfer. So the DBD plasma actuator is currently of considerable interest as one of the active control techniques which provides a new approach to the tip leakage flow control. When sufficient voltages are applied to the actuator, plasma would be generated on the surface of the actuator along the exposed electrode. Compared with other active control methods, the DBD plasma actuator has its own distinct advantages such as it can be easily placed at most receptive locations (Cattafesta and Sheplak, 2011; Roth and Dai, 2006), and its low energy consumption (Traficante et al., 2016). This plasma-based technique has already been proved to be effective in enhancing lift (Post and Corke, 2004), exciting the boundary layer instabilities (Visbal, 2010), controlling dynamic stall (Post and Corke, 2006) on the airfoils and so on (Rizzetta and Visbal, 2010). In the present study, new designs with multiple DBD actuators are installed to evaluate the control effect at the suction side of the turbine blade tip. All actuators are continuous arrangement at the suction side in design and each actuator is independent of others. Kriging surrogate
cascade inlet cascade outlet y-coordinate of point A x-coordinate of point B obtained by CFD obtained by Kriging model leakage flow
model (Krige, 1994) is engaged by constructing an approximation model. And then several cases with good controlling effect are achieved by the approximation model predicting. Based on the previous research work, the current study aims to analyze the flow structure and loss condition with and without DBD actuators. The flow topology of blade tip are presented. Besides that, the flow structures and streamlines in the tip gap are considered. Finally, loss condition and the interaction between TLV and PVC are investigated preliminarily. 2. Calculation model and conditions In the present work, the high-pressure turbine cascade blade was obtained by stretching the tip profile of the LISA turbine rotor blade (Behr et al., 2007). Table 1 lists the geometric parameters of the cascade, which remain the same with the experiment by Fu et al. (2018). In detail, the gap height (τ) in all cases remains 1% of the blade span. The numerical simulation in this study uses the software CFX to solve the steady Navier-Stokes equations. The computational model as shown in Fig. 1. The inlet domain is extended to 1.5Cax upstream of the leading edge, while the outlet domain is extended to 2.0Cax downstream of the trailing edge. Total temperature, velocity and inflow Table 1 Geometric parameters.
378
Parameter
Value
Blade span H (mm) Chord length (mm) Blade axial chord Cax (mm) Blade pitch t (mm) Stager angle (°) Inlet flow angle αk (°) Outlet flow angle βk (°) Gap height τ Velocity of the inlet V0 (m/s) Chord/Pitch (°)
160 130 100 100 40.2 −40.7 67.4 1%H 10.0 1.30
International Journal of Heat and Fluid Flow 77 (2019) 377–387
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Fig. 1. Numerical calculation model.
angle are used as the inlet boundary conditions, and static pressure as the outlet boundary condition. The inlet velocity is 10 m/s. All the wall faces are set as adiabatic with no slip. Furthermore, total 15 DBD plasma actuators Di(i = 1,2,3…15) are continuous arrangement along the suction side from the 1% to 90%Cax in the present work. Each actuator is assumed to be the same electrodes
and performance. Every single actuator has two levels, which are 0 and 1 referring to the closed and open statuses of DBD plasma actuator respectively. There are a total of 215(32,768) combinations of difference actuators. To decrease the number of solutions, Kriging surrogate model is adopted, which would be introduced in Section 3.1. The grid system is present in Fig. 2a), an H-type grid topology is
Fig. 2. Grid and its independency validation:. 379
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used in the passage, and an O-type grid topology is used near the blade wall and inside the tip gap. Dimensionless mesh height y+ on the blade surface and other walls is close to 1 to capture high numerical accuracy. Grid independency analysis is performed to check the impact of mesh density on numerical results and to determine the appropriate mesh number. In this paper, grid independency validation is conducted by checking the effect of the span-wise node number above or below the blade tip, as marked in Fig. 2b). In the present study, the total pressure loss coefficient (ϖ) is defined as,
=
p0* p0*
p* (1)
p1
where p* and p0* are the total pressure of the local flow and the inflow, respectively. p1 is the static pressure of the outflow. The static pressure coefficient (Cp) are defined as,
Cp =
p p0*
p1 p1
(2)
Fig. 4. Velocity distribution after and before identification.
where p is the local static pressure. 2.1. The plasma model In this study, the flow field is described by the full Navier–Stokes equations, augmented by source terms standing for the local forcing induced the DBD actuators. And the local forcing is obtained by a new model illustrated in our previous work (Yu et al., 2019), which is improved from the model proposed by Shyy et al. (2002). The arrangement scheme of plasma actuator refers to the published paper (Baughn et al., 2006) in Fig. 3. In Shyy's model, the electric field strength expressions of the model are given as follows,
E = E0
k1 x
(3)
k2 y
(4)
E0 = U / d
where E is the electric field strength around actuator, E0 stands for the value at the coordinate origin which is set to be the maximum. k1 and k2 are defined in Eq. (5).
k1 = (E0
Eb)/ b k2 = (E0
Fig. 5. The electric field profile.
(5)
Eb)/a
improve the precision of the plasma model. Fig. 4 shows that the velocity distributions after and before identification. The differences between Shyy's model and the new model are that the boundary changes from a straight line to a broken line shown in Fig. 5 and a new expression of E0, which can be described by Eq. (8)–(10), (Table 2),
Eb is the electric field strength on the boundary line A-B; b and a correspond to the x-coordinate of point B and y-coordinate of point A, respectively. The value of the body force outside the triangular region is zero.
Ex = Ek2/ (k12 + k 22) Ey = Ek1/ (k12 + k 22)
(6)
E0 = 290U
The local forcing could be calculated in Eq. (7).
fx =
Ex c ec t fy =
E y c ec t
E = E0
(7)
k1(i) =
where ρc is the charge density and it is the same with the Shyy's computation. ec stands for the elementary charge. In the new model, a parameters identification method is adopted to
1.16 × 106
k1(i) x
(E0 yi x i
k 2(i) y
Eb)(x i x i 1) , i = 0, 1, 2, 3, 4 yi 1 x i 1
(8) (9) (10)
In the following work, the frequency (f) and applied voltage (U)
Fig. 3. Plasma actuators arrangement scheme and electric field configuration. 380
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The Kriging model regards the f(x) as the regression model.
Table 2 Endpoints of the segment lines. x0
y0
x1
y1
x2
y2
x3
y3
x4
y4
0
3
3.13056
2.91659
4.71031
1.34048
6.32863
0.68337
10
0
(12)
Cov [z (x (i ) ), z (x (j) )] = R (x (i ) , x (j ) ) 2
where, the z(x) is a Gaussian with zero mean and σ variance, and its covariance is, (13)
R (x i , x j ) = exp [ d (x (i) , x (j) )] used are 10 kHz and 40 kV, respectively.
d (xk(i ) , xk(j) ) =
2.2. The turbulence model
m k
xk(i)
xk(j)
2
(14)
k=1
R and d are the relation function and the distance of two samples, respectively. θk is the parametric components which is used fit the model. Based on the model, we can get the predicted value y^ (x 0 ) ,
Two-equation turbulence models are applied widely due to its good compromise between the numerical consumption and accuracy in CFD study. In our previous work (Wang et al., 2018), the numerical results with three turbulence models: k-ε, k-ω, and shear stress transport k-ω (SST) turbulence models have been compared. The experimental data obtained by Fu et al. (2018) is used to validate the aerodynamic parameters to compare calculation accuracy. The span-wise distributions of pitch-wise averaged total pressure coefficient ( ¯ ) shown in Fig. 6(a) reveals the superiority of k-ω turbulence model in loss estimation, especially in the zone of TLV. And k-ω turbulence model also plays an excellent performance in the axial distribution of Cp at 97%H on the blade surface presented in Fig. 6(b). Therefore, k-ω turbulence model is adopted in the current study.
y^ (x 0) = ^ + r T (x 0 ) R 1 (Y
I ^)
(15)
T
In the function, r (x) represents the correlations between the forecasted points and the sample points to
^ = (I T R 1Y )T (I T R 1I ) r T (x )
(16)
1
= [R (x , x1), R (x , x2),
(17)
, R (x , xm)]
3.2. The result of surrogate model
3. Surrogate model method and result
In the present work, the TLMFR of the Baseline case, which is without any plasma actuator, is 0.00423 kg/(s). All of the other TLMFRs (mKri ) could be obtained immediately when the surrogate model is completed. The distribution of mKri is shown in Fig. 7 varying with the number of actuators (Ns), which were active in each configuration. The optimal line, which marked blue in Fig. 7, connects all optimal cases in each Ns. In the following discussion, the mkri and the TLMFRs obtained from CFD results (mCFD ) are all dimensionless with the inlet mass flow rate m0.
3.1. Surrogate model method In this study, the Kriging surrogate model would be applied to decrease the number of solutions. The Kriging surrogate model is firstly proposed by Krige (1994) and widely used in geology. Then it was gradually applied to the optimization design. Applied in the present work, an appropriate number of samples were selected firstly by orthogonal experimental design, which can guarantee all the actuators in consideration efficiently. In order to ensure the accuracy and efficiency of the calculation, an orthogonal design table L60 is chosen. And then the 60 samples should be performed by numerical simulations to obtain their TLMFR accordingly. Finally, an approximation model which describes the relationship between the TLMFR and the configuration of the actuators is constructed using the surrogate model method. In this model, the known sample point is defined as X = [x (1) , x (2), , x (n) ], while the resulting is Y = [y (1) , y (2) , , y (n) ]T . The resulting refers to the TLMRF in the present work. The relation of the inputs and the resulting is given as follows,
m¯ Kri m¯ L = Kri mo
(18)
m¯ CFD m¯ L = CFD mo
(19)
(11)
¯m = (m¯ LKri L
y = f (x ) + z (x )
To further examine the accuracy of the surrogate model, the cases around the optimal line are re-calculation by CFD. Then the calculation CFD results and relative errors between m LKri and m¯ L are achieved and shown in Fig. 8 and Table 3. The relative errors are defined as follows,
m¯ LCFD)/ m¯ LCFD × 100
(20)
Table 3 Comparison of TLMFR between Kriging and CFD (✓: actuator turns on). D1 Baseline Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10 Case 11 Case 12 Case 13 Case 14 Case 15
✓
D2
✓ ✓
D3
✓ ✓ ✓
D4
✓ ✓ ✓ ✓
D5
✓ ✓ ✓ ✓ ✓
D6
✓ ✓ ✓ ✓ ✓ ✓
D7
D8
✓ ✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
D9
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
D10
D11
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
381
D12
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
D13
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
D14
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
D15
Kri m¯ L
CFD m¯ L
¯m
L
¯ CFD
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
/ 0.0270 0.0246 0.0232 0.0220 0.0210 0.0201 0.0191 0.0182 0.0174 0.0166 0.0161 0.0157 0.0153 0.0152 /
0.0285 0.0262 0.0247 0.0232 0.0219 0.0211 0.0201 0.0190 0.0185 0.0176 0.0166 0.0156 0.0154 0.0155 0.0156 0.0160
/ 3.08% 0.54% 0.29% 0.31% 0.64% 0 0.71% 1.45% 1.15% 0 3.46% 1.75% 1.30% 2.16% /
0.1298 0.1136 0.1165 0.1355 0.1164 0.1162 0.1195 0.1164 0.1133 0.1089 0.1126 0.1102 0.1094 0.1102 0.1029 0.1011
International Journal of Heat and Fluid Flow 77 (2019) 377–387
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Fig. 6. Aerodynamic validation of three turbulence models.
4. Result and discussion 4.1. The flow topology on the blade tip To illustrate the mechanisms of the different combinations of plasma actuators in the tip leakage flow control, limited streamlines on the tip of blade are drawn in Fig. 9 and the topological analysis method was initially adopted. Flow structures of the reattachment line, separation line, the saddle and nodal point are spotted in Fig. 9. In the Baseline case, the reattachment line exists from the leading edge to the trailing edge continuously. In general, a separation would generates when the flow entries the tip gap from the pressure-side. And the existence of a reattachment line presents that the flow attaches to the tip surface. When the plasma actuator of D8 turns on in Case 1, a small perturbation happens at zone①. In Case 3, the reattachment line becomes shorter than Baseline case. And described in Table 3, the plasma actuators of D8, D14 and D15 are open in Case 3. As mentioned above, the D8 would cause the obstacle at zone①. Similarly, the action of D14 and D15 would induce two structures which exist in ②−1 and ②−2, respectively. At zone ②−1, a small vortex is discovered. At zone ②−2, the flow direction is changed compared with the Baseline case. Shown in Case 5, the flow structures are the same with the Case 3 but more obvious. In Case 8, the small vortices which found at zone ① and zone ②−1 disappear and they are replaced by a separation line observed at zone④. Normally, the existence of the separation line indicates the happening of a big scale separation. This flow structure would bring more perturbation to the TLF. As more actuators turn on, the separation line is extended to the leading edge. However, it would reach its limits by Case 12. Shown in Fig. 9, the separation line stops extending and the flow structures are almost the same between the Case 12 and Case 14. What's more, the Case 12 plays the most performance in decreasing the TLMFR as shown in Table 3. Different flow topologies indicate that different flow structures exist in the tip gap. To further study the effect of multiple DBD plasma actuators on TLF, several representative cases, which are Baseline case, case 5, 8, and 12, are compared and discussed with detail in the following discussion.
Fig. 7. Obtained by the surrogate model.
Fig. 8. Distribution of mLKri , mLCFD and ¯mL.
4.2. Flow structures inside the tip gap
The all relative errors are below 3.5% except Baseline case and Case 15 which are included in the samples. Therefore, the Kriging model used in this work is reasonable and reliable.
Three-dimensional streamlines passing the tip gap are shown in Fig. 10. In the Baseline case, separation bubble formed by a large 382
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Fig. 9. Limiting streamlines on the blade tip.
on three axial sections. In Baseline case, the development of separation bubble is easy to observe. In 50%Cax section, the separation bubble takes up half height of the gap and TLF flows above it. But in 70%Cax section, the separation bubble has a bigger scale and takes up nearly the whole space in height. On the one hand, separation bubble obstructs TLF passing through the tip gap. On the other hand, when the separation bubble broken, the fluid with low level energy mix with TLF, causing additional energy loss. In 90%Cax sections, the separation bubble disappears, but the fluid with low level energy still exists. When the actuators turns on gradually, the separation bubble has a slight transformation in Case 5. In Case 8, the flow structure has almost no change in 50%Cax section. But in 70%Cax section, the separation bubble moves up and an induced vortex appears near the SS, which rotates in the same direction with the separation bubble. Meanwhile, a small vortex appears between the separation bubble and induced vortex. The three vortexes lead to a new type of the flow pattern appeared in the evolution. In Case 12, the induced vortex appears earlier than other cases, meaning it can block the leakage flow twice and enhance the mixing of the leakage flow inside the gap earlier. In addition, the velocity distribution becomes more uniform and the velocity magnitude decreases significantly because of the effects of actuators.
Fig. 10. Three-dimensional streamlines passing the tip gap.
amount of low energy fluid exists from the leading edge to trailing edge near the PS. TLF flows out of the gap above the separation bubble and eventually involved by TLV. In case 5, the yellow streamline, which is separate into two parts, shows the effect of single actuator D8. The separation bubble breaks earlier and this part of fluid with low energy level flows out of the gap in the middle of blade, as shown with blue streamline. Another part of the fluid shown in purple changes its flow direction under the influence of the induced force by actuators, and flows out of the gap near the trailing edge eventually. In case 8 and 12, the induced vortex is easily spotted near the SS. In case 8, separation bubble breaks earlier than that in the Baseline case, but under the influence of actuators D8∼D15, the broken fluid is involved by the induced vortex and eventually flows out of the gap near the trailing edge. In case 12, actuators drive more fluid near the leading edge change direction so that the strength of the induced vortex is increased. Fig. 11 shows two-dimensional streamlines and velocity distribution
4.3. Effects of different flow structures 4.3.1. Velocity distribution contours The primary influence brought about by changes of flow structure is the reduction of speed. Fig. 12 delineates the velocity contours near the blade tip and positions of actuators which are in active of the four selected cases. In the Baseline case, a part of fluid accelerates into the tip gap from the PS under the pressure gradient. When the actuators are applied, it can be observed that the speed is significantly reduced at the corresponding position. 4.3.2. Comparison of mixing losses In this study, the TLF is considered as jet, which flows out of the tip gap and mixes with mainstream. As shown in Fig. 13, the angle between leakage flow and main flow is defined as θ, which varies with axial chord. Obviously, if θ is smaller, leakage flow and main flow are easier 383
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Fig. 11. Two-dimensional streamlines and velocity distribution at 50%, 70% and 90% axial chords.
to mix so that the tip leakage losses can be decreased. Therefore, angle θ can be used as a criterion for measuring the tip leakage losses. In this study, an assumption is made that the direction of the main flow is parallel to the suction side. Therefore, the angle θ between main flow and leakage flow is equal to the angle between leakage flow and the suction side. θ is calculated by
= arctan
parallel because the angle θ is very small. In the downstream of the 20% streamwise position, the angle θ begins to increase rapidly and reaches nearly 65° in 70% streamwise position, meaning that a large amount of mixing loss will be generated due to speed discrepancy especially in the normal direction. But in case 12, the angle θ has a significant reduction in the downstream of the 20% streamwise position. What's more, there is a diminution of up to 31.77° in 90% streamwise position and then the diminution disappears rapidly, which is consistent with the arrangement of the plasma shown in Fig. 1. In order to quantify the effect of plasma actuators on mixing losses, a mixing loss model presented by Young and Wilcock (2002) is adopted in this study. According to the mixing loss model, the amount of mixing loss between leakage flow and main flow can be estimated by
N Vleakage T Vleakage
(21)
In the following discussion, the tangential direction is defined as the direction parallel to the suction side, and the normal direction is defined as the direction perpendicular to the tangential and the radial direction. The distribution of the angle θ between main flow and leakage flow along the streamwise direction is shown in Fig. 14. In the Baseline case, θ decreases first and reaches its minimum in 20% streamwise position. At this position, the mainstream and the leakage flow are almost
T S=
mleakage mpassage
N [(Vleakage
N Vpassage )2
T (Vleakage
T Vpassage )2]
(22)
which demonstrates that, the mixing loss is dependent on the leakage flow rate and the velocity differences in the normal and the tangential
Fig. 12. Velocity magnitude contours near the blade tip. 384
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case and case 12. The distribution of normal and tangential momentum difference per unit area along the streamwise direction and their massflow averaged values are shown in Fig. 15. The momentum difference is calculated by
(m f V ) =
T Vleakage Vdiff dA e
dAe
(23)
Here, the normal velocity of the main flow is zero. As shown in Fig. 15, plasma actuators have significant influence on both normal and tangential momentum differences. Compared to the Baseline case, the averaged value of tangential momentum reduces by 64.7% and the averaged value of normal momentum reduces by 41.1% in case 12. It means that the plasma actuators arranged in case 12 can significantly decrease the momentum differences between main flow and leakage flow especially in normal direction. 4.3.3. Total pressure loss and secondary velocity in the passage Fig. 16 delineates the contours of ϖ and secondary flow streamlines on three axial sections. In the current study, the main flow direction is defined as the pitch-wise averaged flow direction at 50%H, and then the secondary flow velocity is obtained by projecting the local velocity onto the plane perpendicular to the main flow direction. There is no significant effects on the flow field below 50%H between these four cases. In the 50%Cax section of Baseline case, the TLV, the PVC, and the passage vortex near the hub (PVH) have already formed. The TLV is in a small scale while the PVC occupies most of the space along the pitchwise direction. In the 70%Cax and 90%Cax sections, the TLV expands by entraining other fluid in the passage so that the PVC is extruded and separated into a vortex pair. The distance between the vortex cores of the PVC1 and the PVC2 can be used as a criterion of the strength of the TLV. As the plasma actuators turn on gradually, a significant change is that the TLV core moves towards the SS gradually. This can be induced by the change of TLF in flow direction and the decrease of velocity magnitude. As shown in Fig. 14, the angle θ between main flow and leakage flow has a big diminution in case 12 compared with the Baseline case. Furthermore, the area of high loss coefficient associated with the TLV is reduced a lot especially in the 90%Cax section shown in Fig. 16(d). As the TLF flows out of the gap with a lower velocity and less angle θ, the TLV is closer to the SS and less fluid in the passage is rolled up into the TLV, presenting that the region of high loss reduces. Since the TLV is weaken, the extrusion action that the PVC bearing is reduced. The PVC expands and the core moves towards the casing, which results
Fig. 13. Schematic diagram of the angle between leakage flow and mainstream.
Fig. 14. The distribution of θ along the streamwise direction.
direction between leakage flow and main flow. The influences of the leakage flow rate and the velocity difference are comprehensively represented by momentum difference. Therefore, normal momentum difference and tangential momentum difference are chosen as another two criterions to explain the mixing loss discrepancy of the Baseline
Fig. 15. Distribution of momentum difference between leakage flow and mainstream. 385
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Fig. 16. Contours of ϖ and secondary velocity streamlines in the passage.
in that partial fluid of the TLV with low energy level escapes and enters into the PVC affected zone shown in Fig. 16(d). Fig. 17 delineates the three-dimensional streamlines flowing out of the gap and the contour of ϖ in the 120%Cax section plane. It can be observed that there is a clear dividing line in the streamlines. The fluid flowing out from the rear part of the blade converges into TLV and is shown in red streamlines, while the fluid flowing out from the front part converges into PVC and is shown in blue. In Baseline case, the leakage flow in the downstream of 66%Cax is rolled up into the TLV. As the plasma actuators turn on in case 12, the direction of leakage flow changes and the dividing line moves downstream to 74%Cax. This indicates that the plasma actuators in case 12 not only diminish the m LCFD as mentioned in Table 3, but also change the distribution of leakage flow, causing the reduction of TLV. Fig. 18 plots the pitch-wise averaged total pressure loss coefficient ( ¯ ) in the 120%Cax section plane. The curve can be divided into three regions, which indicate the locations of the TLV and the PVC and the interaction zone between these two adjacent vortices respectively. In current study, TLV and PVC have been labeled in Fig. 18. It can be observed that there is less loss production in zone of TLV in cases 5, 8, and 12 than in Baseline case, while more loss is contained in zone of PVC. But these changes are not continuously increasing or decreasing. In the zone of PVC, the maximum of ¯ gradually increases at the beginning and eventually fall above the initial value in Baseline case presenting that strength of PVC increases first and then decreases, as the plasma actuators turn on gradually. And in the zone of TLV, the maximum of ¯ declines sharply at first, while there is almost no change between case 8 and case 12. This is because that the opened plasma actuators D12∼D15 in case 5 are arranged near the trailing edge, which coincides with location of composition fluid of TLV shown in
Fig. 17. D12∼D15 accurately block the leakage flow near the trailing edge so that causing a sharply decrease of ¯ in the zone of TLV. Comparing with case 8, the newly opened actuators D4∼D7 in case 12 have almost no effect on TLV, but a decline of the maximum of ¯ occurs in the zone of PVC. The reason is that actuators D4∼D7 are located at the upstream of 74%Cax and block the fluid which composes the PVC. So the TLV is hardly affected, while the strength of PVC is reduced. In addition, the position of the maximum of ¯ , which indicates the PVC core, moves to the top of the blade continuously. This can be induced by the reduction of TLV as mentioned above.
Fig. 17. Contours of ϖ and tip leakage flow streamlines. 386
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doi.org/10.1115/1.2447707. Bindon, J.P., 1989. The measurement and formation of tip clearance loss. J. Turbomach. 111 (3), 257–263. https://doi.org/10.1115/1.3262264. Booth, T.C., Dodge, P.R., Hepworth, H.K., 1982. Rotor-tip leakage: part I—basic methodology. J. Eng. Power 104, 154–161. https://doi.org/10.1115/1.3227244. Cattafesta III, L.N., Sheplak, M., 2011. Actuators for active flow control. Ann. Rev. Fluid Mech. 43, 247–272. https://doi.org/10.1146/annurev-fluid-122109-160634. Denton, J.D., 1993. Loss mechanisms in turbomachines. J. Turbomach. 115 (4), 621–656. https://doi.org/10.1115/1.2929299. Fu, Y.F., Chen, F., Liu, H.P., Song, Y.P., 2018. Experimental and numerical study of honeycomb tip on suppressing tip leakage flow in turbine cascade. J. Turbomach. 140 (6), 061006. https://doi.org/10.1115/1.4039049. Heyes, F.J.G., Hodson, H.P., Dailey, G.M., 1992. The effect of blade tip geometry on the tip leakage flow in axial turbine cascades. J. Turbomach. 114, 643–651. https://doi. org/10.1115/1.2929188. Krige, D.G., 1994. A statistical approach to some basic mine valuation problems on the witwatersrand. J. South. Afr. Inst. Min. Metall. 94 (3), 95–111. Krishnababu, S.K., Newton, P.J., Dawes, W.N., Lock, G.D., Hodson, H.P., Hannis, J., Whitney, C., 2009. Aerothermal investigations of tip leakage flow in axial flow turbines-part I: effect of tip geometry and tip clearance gap. J. Turbomach. 131 (1), 11006. https://doi.org/10.1115/1.2950068. Kumada, M., Iwata, S., Obata, M., Watanabe, O., 1994. Tip clearance effect on heat transfer and leakage flows on the shroud-wall surface in an axial flow turbine. J. Turbomach. 116 (1), 39–45. https://doi.org/10.1115/1.2928276. Lee, S.E., Lee, S.W., 2016. Over-tip leakage flow and loss in a turbine cascade equipped with suction-side partial squealers. Int. J. Heat Fluid Flow 61, 575–584. https://doi. org/10.1016/j.ijheatfluidflow.2016.07.001. Lee, S.E., Sang, W.L., Kwak, H.S., 2011. 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Heterogeneous optimization strategies for carved and squealer-like turbine blade tips. J. Turbomach. 138, 121011. https://doi.org/10.1115/1.4033975. Nho, Y.C., Park, J.S., Lee, Y.J., Kwak, J.S., 2012. Effects of turbine blade tip shape on total pressure loss and secondary flow of a linear turbine cascade. Int. J. Heat Fluid Flow 33 (1), 92–100. https://doi.org/10.1016/j.ijheatfluidflow.2011.12.002. O'Dowd, D.O., Zhang, Q., He, L., Cheong, B.C.Y., Tibbott, I., 2012. Aerothermal performance of a cooled winglet at engine representative Mach and Reynolds numbers. J. Turbomach. 135 (1), 011041. https://doi.org/10.1115/1.4006537. Post, M.L., Corke, T.C., 2004. Separation control on high angle of attack airfoil using plasma actuators. AIAA J 42 (11), 2177–2184. https://doi.org/10.2514/1.2929. Post, M.L., Corke, T.C., 2006. Separation control using plasma actuators: dynamic stall vortex control on oscillating airfoil. AIAA J 44 (12), 3125–3135. https://doi.org/10. 2514/1.22716. Rizzetta, D.P., Visbal, M.R., 2010. Large-eddy simulation of plasma-based turbulent boundary-layer separation control. AIAA J 48 (12), 2793–2810. https://doi.org/10. 2514/1.J050014. Roth, J.R., Dai, X., 2006. Optimization of the aerodynamic plasma actuator as an electrohydrodynamic (EHD) electrical device. 44th AIAA Aerosp. Sci. Meet. Exhibi. https://doi.org/10.2514/6.2006-1203. Shyy, W., Jayaraman, B., Andersson, A., 2002. Modeling of glow discharge-induced fluid dynamics. J. Appl. Phys. 92 (11), 6434–6443. https://doi.org/10.1063/1.1515103. Traficante, S., De Giorgi, M.G., Ficarella, A., 2016. Flow separation control on a compressor-stator cascade using plasma actuators and synthetic and continuous jets. J. Aerosp. Eng. 29 (3), 04015056. https://doi.org/10.1061/(ASCE)AS.1943-5525. 0000539. Visbal, M.R., 2010. Strategies for control of transitional and turbulent flows using plasmabased actuators. Int. J. Comput. Fluids Dyn. 24 (7), 237–258. https://doi.org/10. 1080/10618562.2010.533123. Volino, R.J., 2017. Control of tip leakage in a high-pressure turbine cascade using tip blowing. J. Turbomach. 139 (6), 061008. https://doi.org/10.1115/1.4035509. Wang, Y.B., Song, Y.P., Yu, J.Y., Chen, F., 2018. Effect of cooling injection on the leakage flow of a turbine cascade with honeycomb tip. Appl. Therm. Eng. 133, 690–703. https://doi.org/10.1016/j.applthermaleng.2018.01.090. Yang, D.L., Feng, Z.P., 2007. Tip leakage flow and heat transfer predictions for turbine blades. ASME Paper No. GT-2007-27728. https://doi.org/10.1115/GT2007-27728. Young, J.B., Wilcock, R.C., 2002. Modeling the air-cooled gas turbine: part2-coolant flows and losses. J. Turbomach. 124 (2), 214–221. https://doi.org/10.1115/1.1415038. Yu, J.Y., Yu, J.N., Chen, F., Wang, C., 2019. Numerical study of tip leakage flow control in turbine cascades using the DBD plasma model improved by the parameter identification method. Aerosp. Sci. Technol. 84, 856–864. https://doi.org/10.1016/j.ast. 2018.11.020. Zhang, M., Liu, Y., Zhang, T.L., Zhang, M.C., He, Y., 2017. Aerodynamic optimization of a winglet-shroud tip geometry for a linear turbine cascade. J. Turbomach. 139 (10), 101011. https://doi.org/10.1115/1.4036647. Zou, Z.P., Shao, F., Li, Y.R., Zhang, W.H., Berglund, A.B., 2017. Dominant flow structure in the squealer tip gap and its impact on turbine aerodynamic performance. Energy 138, 167–184. https://doi.org/10.1016/j.energy.2017.07.047.
Fig. 18. Span-wise distributions of ¯ (120%Cax).
5. Conclusion In the current study, the numerical investigations of high pressure turbine cascade with multiple DBD actuators applied in the blade tip are carried out. Several cases selected by Kriging model are compared to analyze the flow structures and loss conditions with and without DBD actuators. The changes of flow topology and flow structures inside the tip gap are summarized. The mixing loss conditions are investigated by considering the leakage flow as a jet. Furthermore, the secondary velocity distributions and total pressure loss conditions are analyzed. Conclusions are made as follows, 1) The induced vortex induced by DBD actuators can change the leakage flow direction and flow structures inside the tip gap. Due to the presence of induced vortices, the separation bubble becomes labile and breaks earlier, leading to a new type of the flow pattern. 2) When the actuators are applied, the speed of leakage flow is significantly reduced and the angle between leakage flow and main flow has an obviously diminution, causing the reduction of mixing losses in the passage. 3) As the TLF flows out of the gap with a lower velocity and less angle θ, the TLV core is closer to the SS and less fluid in the passage is rolled up into the TLV, presenting that the region of high loss reduces. 4) The actuators arranged near the trailing edge can diminish the strength of TLV, while the actuators arranged in the front of blade leads to the reduction of ¯ in the zone of PVC. Declarations of interest None. Acknowledgments The authors would like to thank the China Postdoctoral Science Foundation funded project (Grant No.2018M631928) for its financial support. References Baughn, J.W., Porter, C.O., Peterson, B.L., Mclaughlin, T.E., Enloe, C.L., Font, G.I., Baird, C., 2006. Momentum transfer for an aerodynamic plasma actuator with an imposed boundary layer. 44th AIAA Aerosp. Sci. Meet. Exhibi. https://doi.org/10.2514/6. 2006-168. Behr, T., Kalfas, A.I., Abhari, R.S., 2007. Unsteady flow physics and performance of a oneand-1/2-stage unshrouded high work turbine. J. Turbomach. 129, 348–359. https://
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International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Effect of multiple DBD plasma actuators on the tip leakage flow structure and loss of a turbine cascade
T
Wang Zhao, Yu Jianyang , Chen Fu, Song Yanping ⁎
Harbin Institute of Technology, Harbin 150001, China
ARTICLE INFO
ABSTRACT
Keywords: Flow structure Tip leakage control DBD Plasma actuator Surrogate model
In this paper, the effects of multiple dielectric barrier discharge (DBD) plasma actuators on the leakage flow structures and loss conditions have been numerically studied in an axial turbine cascade. Kriging surrogate model is adopted to obtain the optimal cases. The physical mechanism of flow structures inside the gap that control leakage flow is presented, which is obtained by analyzing the flow topology, the evolution of the flow structures and its influence on the secondary velocity and loss conditions in the passage as well. The results show that the induced vortex caused by DBD actuators can change the leakage flow direction inside the tip gap and make the separation bubble break earlier, leading to a new type of the flow pattern. When the actuators are applied, the speed of leakage flow is significantly reduced and the angle between leakage flow and main flow has an obviously diminution, causing the reduction of mixing losses in the passage compared with the Baseline case. Furthermore, the comparison of secondary velocity shows that the tip leakage vortex (TLV) approaches the suction surface, resulting in reduced affected area and weakened loss strength. Plasma actuators can diminish the loss coefficient in both TLV and passage vortex near the casing (PVC) zones. The actuators arranged near the trailing edge mainly affect the strength of TLV, while the actuators in the leading edge area contribute to the loss reduction in the zone of PVC.
1. Introduction
loss. Kumada et al. (1994) found that TLF flows out of the gap in the back of the blade and a leakage vortex is recognized at the suction side near the trailing edge. Booth et al. (1982) found that the loss caused by TLF decreases as decreasing the gap height. But the minimum height of 1 to 2 percent of blade span is still required to tolerate the practical deformation of the rotor blade (Zhang et al., 2017). Thus, to further reduce the effect of the TLF, the research community has proposed numbers of shape designs such as squealer tips (Ma et al., 2017; Lee et al., 2011), winglets (O'Dowd et al., 2012; Lee et al., 2012), honeycomb tips (Fu et al., 2018), and other active blowing or suction (Volino, 2017), over the past few decades. These methods are all found to be particularly successful in reducing the tip leakage flow beneficially. There are three types of squealer tips, which are tips with a suction side squealer, a pressure side squealer and a double side squealer. Taking an example of double side squealer, its mechanism in improving aerodynamic performance is that this squealer tip can separate the flow at the top of the two squealers, meaning it could block the leakage flow twice and enhance the mixing of the leakage flow inside the cavity (Zou et al., 2017). The aerodynamic advantage of the pressure-side and the suction-side partial squealer tips has been confirmed (Lee S.E. and
In order to ensure the frictionless movement of the rotor blade relative to the casing and provide the space margin required for centrifugal stretching and thermal expansion, there is inevitably a tip gap in turbine cascade. However, under the pressure difference between both sides of the blade tip, the leakage fluid accelerates into the gap and flows from the pressure surface (PS) to the suction surface (SS), eventually rolling up the tip leakage vortex (TLV). When the TLV inevitably mixes with main flow and interacts with the passage vortex near the case (PVC), the leakage loss occurs, which accounts for about 1/3 of the total blade passage loss in unshrouded turbine (Denton, 1993). This large loss indicates the importance of controlling tip leakage flow (TLF) for increasing the whole turbine efficiency. The researchers conducted a detailed study of the flow structure of the TLF. Bindon (1989) found that all losses associated with the TLF are related to the separation bubble formed near the PS. Due to the pressure gradient, a large amount of low energy fluid is generated to form the separation bubble. But when the pressure gradient disappears, the separation bubble is broken. The TLF accelerates into the gap and is blended with the part of the low energy fluid, causing additional energy
⁎
Corresponding author. E-mail address: [email protected] (J. Yu).
https://doi.org/10.1016/j.ijheatfluidflow.2019.05.002 Received 5 December 2018; Received in revised form 2 March 2019; Accepted 2 May 2019 Available online 15 May 2019 0142-727X/ © 2019 Elsevier Inc. All rights reserved.
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Nomenclature a b C Cax Cp d DBD Di E E0 Eb ec fx, fy H k1 k2 m Ns p p* PVC PVH S SST TLF
TLMFR TLV U V y+
= 3, the height of the plasma [mm] = 10, the width of the plasma [mm] blade chord [m] blade axial chord [m] static pressure coefficient distance between the two samples dielectric barrier discharge dielectric barrier discharge plasma actuators Electric field [N/C] Electric field maximum [N/C] = 30, breakdown electric field strength [kv/cm] Elementary charge 1.602×10−19 [C] Body force vector [N/m3] in x direction and y direction, respectively blade span [m] Parameter defined in the plasma model Parameter defined in the plasma model mass flow rate number of actuators working static pressure [Pa] total pressure [Pa] passage vortex near the casing passage vortex near the hub entropy [J/mol] shear stress transport tip leakage flow
tip leakage mass flow rate tip leakage vortex AC voltage peak [V] velocity [m/s] dimensionless mesh height
Greek symbols α αk βk θ ρc τ ϑ ϖ ¯ ¯
= 1, factor to account for the collision efficiency inlet flow angle [°] outlet flow angle [°] the angle between main flow and leakage flow [°] charge density [C/m3] tip gap height [m] AC voltage frequency [Hz] The total pressure loss coefficient The pitch-wise averaged total pressure loss coefficient The mass flow averaged total pressure loss coefficient
Subscripts 0 1 a b CFD Kri L
Lee S.W., 2016; Lee S.W. and Lee S.E., 2014). Its effectiveness in reducing the tip leakage mass flow rate (TLMFR) was early confirmed by Heyes et al. (1992). Numerical studies conducted by Yang and Feng (2007) showed that TLMFR decreased as increasing the squealer depth until the depth reached 3 percent of the blade span. Nho et al. (2012) found that the double squealer tip and the grooved along pressure side tip performed best among the several tested blade tips and all tips showed smaller total pressure loss coefficients than that of the plane tip. Maesschalck et al. (2016) confirmed that the optimized blade tip design could further reduce TLMFR and improve the aerodynamic performance. Meanwhile, other researches emphasized that the sealing effect of the squealer tip originates from the internal flow structure within the gap. Krishnababu et al. (2009) found that the obstruction effect of the squealer tip became weaker as the contraction coefficient increased, which was in accordance with the prediction of the two-dimensional idealized model. Although the squealer tips are aerodynamically useful to alter the tip leakage flow, they do cause some negative effects on the heat transfer. So the DBD plasma actuator is currently of considerable interest as one of the active control techniques which provides a new approach to the tip leakage flow control. When sufficient voltages are applied to the actuator, plasma would be generated on the surface of the actuator along the exposed electrode. Compared with other active control methods, the DBD plasma actuator has its own distinct advantages such as it can be easily placed at most receptive locations (Cattafesta and Sheplak, 2011; Roth and Dai, 2006), and its low energy consumption (Traficante et al., 2016). This plasma-based technique has already been proved to be effective in enhancing lift (Post and Corke, 2004), exciting the boundary layer instabilities (Visbal, 2010), controlling dynamic stall (Post and Corke, 2006) on the airfoils and so on (Rizzetta and Visbal, 2010). In the present study, new designs with multiple DBD actuators are installed to evaluate the control effect at the suction side of the turbine blade tip. All actuators are continuous arrangement at the suction side in design and each actuator is independent of others. Kriging surrogate
cascade inlet cascade outlet y-coordinate of point A x-coordinate of point B obtained by CFD obtained by Kriging model leakage flow
model (Krige, 1994) is engaged by constructing an approximation model. And then several cases with good controlling effect are achieved by the approximation model predicting. Based on the previous research work, the current study aims to analyze the flow structure and loss condition with and without DBD actuators. The flow topology of blade tip are presented. Besides that, the flow structures and streamlines in the tip gap are considered. Finally, loss condition and the interaction between TLV and PVC are investigated preliminarily. 2. Calculation model and conditions In the present work, the high-pressure turbine cascade blade was obtained by stretching the tip profile of the LISA turbine rotor blade (Behr et al., 2007). Table 1 lists the geometric parameters of the cascade, which remain the same with the experiment by Fu et al. (2018). In detail, the gap height (τ) in all cases remains 1% of the blade span. The numerical simulation in this study uses the software CFX to solve the steady Navier-Stokes equations. The computational model as shown in Fig. 1. The inlet domain is extended to 1.5Cax upstream of the leading edge, while the outlet domain is extended to 2.0Cax downstream of the trailing edge. Total temperature, velocity and inflow Table 1 Geometric parameters.
378
Parameter
Value
Blade span H (mm) Chord length (mm) Blade axial chord Cax (mm) Blade pitch t (mm) Stager angle (°) Inlet flow angle αk (°) Outlet flow angle βk (°) Gap height τ Velocity of the inlet V0 (m/s) Chord/Pitch (°)
160 130 100 100 40.2 −40.7 67.4 1%H 10.0 1.30
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Fig. 1. Numerical calculation model.
angle are used as the inlet boundary conditions, and static pressure as the outlet boundary condition. The inlet velocity is 10 m/s. All the wall faces are set as adiabatic with no slip. Furthermore, total 15 DBD plasma actuators Di(i = 1,2,3…15) are continuous arrangement along the suction side from the 1% to 90%Cax in the present work. Each actuator is assumed to be the same electrodes
and performance. Every single actuator has two levels, which are 0 and 1 referring to the closed and open statuses of DBD plasma actuator respectively. There are a total of 215(32,768) combinations of difference actuators. To decrease the number of solutions, Kriging surrogate model is adopted, which would be introduced in Section 3.1. The grid system is present in Fig. 2a), an H-type grid topology is
Fig. 2. Grid and its independency validation:. 379
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used in the passage, and an O-type grid topology is used near the blade wall and inside the tip gap. Dimensionless mesh height y+ on the blade surface and other walls is close to 1 to capture high numerical accuracy. Grid independency analysis is performed to check the impact of mesh density on numerical results and to determine the appropriate mesh number. In this paper, grid independency validation is conducted by checking the effect of the span-wise node number above or below the blade tip, as marked in Fig. 2b). In the present study, the total pressure loss coefficient (ϖ) is defined as,
=
p0* p0*
p* (1)
p1
where p* and p0* are the total pressure of the local flow and the inflow, respectively. p1 is the static pressure of the outflow. The static pressure coefficient (Cp) are defined as,
Cp =
p p0*
p1 p1
(2)
Fig. 4. Velocity distribution after and before identification.
where p is the local static pressure. 2.1. The plasma model In this study, the flow field is described by the full Navier–Stokes equations, augmented by source terms standing for the local forcing induced the DBD actuators. And the local forcing is obtained by a new model illustrated in our previous work (Yu et al., 2019), which is improved from the model proposed by Shyy et al. (2002). The arrangement scheme of plasma actuator refers to the published paper (Baughn et al., 2006) in Fig. 3. In Shyy's model, the electric field strength expressions of the model are given as follows,
E = E0
k1 x
(3)
k2 y
(4)
E0 = U / d
where E is the electric field strength around actuator, E0 stands for the value at the coordinate origin which is set to be the maximum. k1 and k2 are defined in Eq. (5).
k1 = (E0
Eb)/ b k2 = (E0
Fig. 5. The electric field profile.
(5)
Eb)/a
improve the precision of the plasma model. Fig. 4 shows that the velocity distributions after and before identification. The differences between Shyy's model and the new model are that the boundary changes from a straight line to a broken line shown in Fig. 5 and a new expression of E0, which can be described by Eq. (8)–(10), (Table 2),
Eb is the electric field strength on the boundary line A-B; b and a correspond to the x-coordinate of point B and y-coordinate of point A, respectively. The value of the body force outside the triangular region is zero.
Ex = Ek2/ (k12 + k 22) Ey = Ek1/ (k12 + k 22)
(6)
E0 = 290U
The local forcing could be calculated in Eq. (7).
fx =
Ex c ec t fy =
E y c ec t
E = E0
(7)
k1(i) =
where ρc is the charge density and it is the same with the Shyy's computation. ec stands for the elementary charge. In the new model, a parameters identification method is adopted to
1.16 × 106
k1(i) x
(E0 yi x i
k 2(i) y
Eb)(x i x i 1) , i = 0, 1, 2, 3, 4 yi 1 x i 1
(8) (9) (10)
In the following work, the frequency (f) and applied voltage (U)
Fig. 3. Plasma actuators arrangement scheme and electric field configuration. 380
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The Kriging model regards the f(x) as the regression model.
Table 2 Endpoints of the segment lines. x0
y0
x1
y1
x2
y2
x3
y3
x4
y4
0
3
3.13056
2.91659
4.71031
1.34048
6.32863
0.68337
10
0
(12)
Cov [z (x (i ) ), z (x (j) )] = R (x (i ) , x (j ) ) 2
where, the z(x) is a Gaussian with zero mean and σ variance, and its covariance is, (13)
R (x i , x j ) = exp [ d (x (i) , x (j) )] used are 10 kHz and 40 kV, respectively.
d (xk(i ) , xk(j) ) =
2.2. The turbulence model
m k
xk(i)
xk(j)
2
(14)
k=1
R and d are the relation function and the distance of two samples, respectively. θk is the parametric components which is used fit the model. Based on the model, we can get the predicted value y^ (x 0 ) ,
Two-equation turbulence models are applied widely due to its good compromise between the numerical consumption and accuracy in CFD study. In our previous work (Wang et al., 2018), the numerical results with three turbulence models: k-ε, k-ω, and shear stress transport k-ω (SST) turbulence models have been compared. The experimental data obtained by Fu et al. (2018) is used to validate the aerodynamic parameters to compare calculation accuracy. The span-wise distributions of pitch-wise averaged total pressure coefficient ( ¯ ) shown in Fig. 6(a) reveals the superiority of k-ω turbulence model in loss estimation, especially in the zone of TLV. And k-ω turbulence model also plays an excellent performance in the axial distribution of Cp at 97%H on the blade surface presented in Fig. 6(b). Therefore, k-ω turbulence model is adopted in the current study.
y^ (x 0) = ^ + r T (x 0 ) R 1 (Y
I ^)
(15)
T
In the function, r (x) represents the correlations between the forecasted points and the sample points to
^ = (I T R 1Y )T (I T R 1I ) r T (x )
(16)
1
= [R (x , x1), R (x , x2),
(17)
, R (x , xm)]
3.2. The result of surrogate model
3. Surrogate model method and result
In the present work, the TLMFR of the Baseline case, which is without any plasma actuator, is 0.00423 kg/(s). All of the other TLMFRs (mKri ) could be obtained immediately when the surrogate model is completed. The distribution of mKri is shown in Fig. 7 varying with the number of actuators (Ns), which were active in each configuration. The optimal line, which marked blue in Fig. 7, connects all optimal cases in each Ns. In the following discussion, the mkri and the TLMFRs obtained from CFD results (mCFD ) are all dimensionless with the inlet mass flow rate m0.
3.1. Surrogate model method In this study, the Kriging surrogate model would be applied to decrease the number of solutions. The Kriging surrogate model is firstly proposed by Krige (1994) and widely used in geology. Then it was gradually applied to the optimization design. Applied in the present work, an appropriate number of samples were selected firstly by orthogonal experimental design, which can guarantee all the actuators in consideration efficiently. In order to ensure the accuracy and efficiency of the calculation, an orthogonal design table L60 is chosen. And then the 60 samples should be performed by numerical simulations to obtain their TLMFR accordingly. Finally, an approximation model which describes the relationship between the TLMFR and the configuration of the actuators is constructed using the surrogate model method. In this model, the known sample point is defined as X = [x (1) , x (2), , x (n) ], while the resulting is Y = [y (1) , y (2) , , y (n) ]T . The resulting refers to the TLMRF in the present work. The relation of the inputs and the resulting is given as follows,
m¯ Kri m¯ L = Kri mo
(18)
m¯ CFD m¯ L = CFD mo
(19)
(11)
¯m = (m¯ LKri L
y = f (x ) + z (x )
To further examine the accuracy of the surrogate model, the cases around the optimal line are re-calculation by CFD. Then the calculation CFD results and relative errors between m LKri and m¯ L are achieved and shown in Fig. 8 and Table 3. The relative errors are defined as follows,
m¯ LCFD)/ m¯ LCFD × 100
(20)
Table 3 Comparison of TLMFR between Kriging and CFD (✓: actuator turns on). D1 Baseline Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10 Case 11 Case 12 Case 13 Case 14 Case 15
✓
D2
✓ ✓
D3
✓ ✓ ✓
D4
✓ ✓ ✓ ✓
D5
✓ ✓ ✓ ✓ ✓
D6
✓ ✓ ✓ ✓ ✓ ✓
D7
D8
✓ ✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
D9
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
D10
D11
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
381
D12
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
D13
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
D14
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
D15
Kri m¯ L
CFD m¯ L
¯m
L
¯ CFD
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
/ 0.0270 0.0246 0.0232 0.0220 0.0210 0.0201 0.0191 0.0182 0.0174 0.0166 0.0161 0.0157 0.0153 0.0152 /
0.0285 0.0262 0.0247 0.0232 0.0219 0.0211 0.0201 0.0190 0.0185 0.0176 0.0166 0.0156 0.0154 0.0155 0.0156 0.0160
/ 3.08% 0.54% 0.29% 0.31% 0.64% 0 0.71% 1.45% 1.15% 0 3.46% 1.75% 1.30% 2.16% /
0.1298 0.1136 0.1165 0.1355 0.1164 0.1162 0.1195 0.1164 0.1133 0.1089 0.1126 0.1102 0.1094 0.1102 0.1029 0.1011
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Fig. 6. Aerodynamic validation of three turbulence models.
4. Result and discussion 4.1. The flow topology on the blade tip To illustrate the mechanisms of the different combinations of plasma actuators in the tip leakage flow control, limited streamlines on the tip of blade are drawn in Fig. 9 and the topological analysis method was initially adopted. Flow structures of the reattachment line, separation line, the saddle and nodal point are spotted in Fig. 9. In the Baseline case, the reattachment line exists from the leading edge to the trailing edge continuously. In general, a separation would generates when the flow entries the tip gap from the pressure-side. And the existence of a reattachment line presents that the flow attaches to the tip surface. When the plasma actuator of D8 turns on in Case 1, a small perturbation happens at zone①. In Case 3, the reattachment line becomes shorter than Baseline case. And described in Table 3, the plasma actuators of D8, D14 and D15 are open in Case 3. As mentioned above, the D8 would cause the obstacle at zone①. Similarly, the action of D14 and D15 would induce two structures which exist in ②−1 and ②−2, respectively. At zone ②−1, a small vortex is discovered. At zone ②−2, the flow direction is changed compared with the Baseline case. Shown in Case 5, the flow structures are the same with the Case 3 but more obvious. In Case 8, the small vortices which found at zone ① and zone ②−1 disappear and they are replaced by a separation line observed at zone④. Normally, the existence of the separation line indicates the happening of a big scale separation. This flow structure would bring more perturbation to the TLF. As more actuators turn on, the separation line is extended to the leading edge. However, it would reach its limits by Case 12. Shown in Fig. 9, the separation line stops extending and the flow structures are almost the same between the Case 12 and Case 14. What's more, the Case 12 plays the most performance in decreasing the TLMFR as shown in Table 3. Different flow topologies indicate that different flow structures exist in the tip gap. To further study the effect of multiple DBD plasma actuators on TLF, several representative cases, which are Baseline case, case 5, 8, and 12, are compared and discussed with detail in the following discussion.
Fig. 7. Obtained by the surrogate model.
Fig. 8. Distribution of mLKri , mLCFD and ¯mL.
4.2. Flow structures inside the tip gap
The all relative errors are below 3.5% except Baseline case and Case 15 which are included in the samples. Therefore, the Kriging model used in this work is reasonable and reliable.
Three-dimensional streamlines passing the tip gap are shown in Fig. 10. In the Baseline case, separation bubble formed by a large 382
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Fig. 9. Limiting streamlines on the blade tip.
on three axial sections. In Baseline case, the development of separation bubble is easy to observe. In 50%Cax section, the separation bubble takes up half height of the gap and TLF flows above it. But in 70%Cax section, the separation bubble has a bigger scale and takes up nearly the whole space in height. On the one hand, separation bubble obstructs TLF passing through the tip gap. On the other hand, when the separation bubble broken, the fluid with low level energy mix with TLF, causing additional energy loss. In 90%Cax sections, the separation bubble disappears, but the fluid with low level energy still exists. When the actuators turns on gradually, the separation bubble has a slight transformation in Case 5. In Case 8, the flow structure has almost no change in 50%Cax section. But in 70%Cax section, the separation bubble moves up and an induced vortex appears near the SS, which rotates in the same direction with the separation bubble. Meanwhile, a small vortex appears between the separation bubble and induced vortex. The three vortexes lead to a new type of the flow pattern appeared in the evolution. In Case 12, the induced vortex appears earlier than other cases, meaning it can block the leakage flow twice and enhance the mixing of the leakage flow inside the gap earlier. In addition, the velocity distribution becomes more uniform and the velocity magnitude decreases significantly because of the effects of actuators.
Fig. 10. Three-dimensional streamlines passing the tip gap.
amount of low energy fluid exists from the leading edge to trailing edge near the PS. TLF flows out of the gap above the separation bubble and eventually involved by TLV. In case 5, the yellow streamline, which is separate into two parts, shows the effect of single actuator D8. The separation bubble breaks earlier and this part of fluid with low energy level flows out of the gap in the middle of blade, as shown with blue streamline. Another part of the fluid shown in purple changes its flow direction under the influence of the induced force by actuators, and flows out of the gap near the trailing edge eventually. In case 8 and 12, the induced vortex is easily spotted near the SS. In case 8, separation bubble breaks earlier than that in the Baseline case, but under the influence of actuators D8∼D15, the broken fluid is involved by the induced vortex and eventually flows out of the gap near the trailing edge. In case 12, actuators drive more fluid near the leading edge change direction so that the strength of the induced vortex is increased. Fig. 11 shows two-dimensional streamlines and velocity distribution
4.3. Effects of different flow structures 4.3.1. Velocity distribution contours The primary influence brought about by changes of flow structure is the reduction of speed. Fig. 12 delineates the velocity contours near the blade tip and positions of actuators which are in active of the four selected cases. In the Baseline case, a part of fluid accelerates into the tip gap from the PS under the pressure gradient. When the actuators are applied, it can be observed that the speed is significantly reduced at the corresponding position. 4.3.2. Comparison of mixing losses In this study, the TLF is considered as jet, which flows out of the tip gap and mixes with mainstream. As shown in Fig. 13, the angle between leakage flow and main flow is defined as θ, which varies with axial chord. Obviously, if θ is smaller, leakage flow and main flow are easier 383
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Fig. 11. Two-dimensional streamlines and velocity distribution at 50%, 70% and 90% axial chords.
to mix so that the tip leakage losses can be decreased. Therefore, angle θ can be used as a criterion for measuring the tip leakage losses. In this study, an assumption is made that the direction of the main flow is parallel to the suction side. Therefore, the angle θ between main flow and leakage flow is equal to the angle between leakage flow and the suction side. θ is calculated by
= arctan
parallel because the angle θ is very small. In the downstream of the 20% streamwise position, the angle θ begins to increase rapidly and reaches nearly 65° in 70% streamwise position, meaning that a large amount of mixing loss will be generated due to speed discrepancy especially in the normal direction. But in case 12, the angle θ has a significant reduction in the downstream of the 20% streamwise position. What's more, there is a diminution of up to 31.77° in 90% streamwise position and then the diminution disappears rapidly, which is consistent with the arrangement of the plasma shown in Fig. 1. In order to quantify the effect of plasma actuators on mixing losses, a mixing loss model presented by Young and Wilcock (2002) is adopted in this study. According to the mixing loss model, the amount of mixing loss between leakage flow and main flow can be estimated by
N Vleakage T Vleakage
(21)
In the following discussion, the tangential direction is defined as the direction parallel to the suction side, and the normal direction is defined as the direction perpendicular to the tangential and the radial direction. The distribution of the angle θ between main flow and leakage flow along the streamwise direction is shown in Fig. 14. In the Baseline case, θ decreases first and reaches its minimum in 20% streamwise position. At this position, the mainstream and the leakage flow are almost
T S=
mleakage mpassage
N [(Vleakage
N Vpassage )2
T (Vleakage
T Vpassage )2]
(22)
which demonstrates that, the mixing loss is dependent on the leakage flow rate and the velocity differences in the normal and the tangential
Fig. 12. Velocity magnitude contours near the blade tip. 384
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case and case 12. The distribution of normal and tangential momentum difference per unit area along the streamwise direction and their massflow averaged values are shown in Fig. 15. The momentum difference is calculated by
(m f V ) =
T Vleakage Vdiff dA e
dAe
(23)
Here, the normal velocity of the main flow is zero. As shown in Fig. 15, plasma actuators have significant influence on both normal and tangential momentum differences. Compared to the Baseline case, the averaged value of tangential momentum reduces by 64.7% and the averaged value of normal momentum reduces by 41.1% in case 12. It means that the plasma actuators arranged in case 12 can significantly decrease the momentum differences between main flow and leakage flow especially in normal direction. 4.3.3. Total pressure loss and secondary velocity in the passage Fig. 16 delineates the contours of ϖ and secondary flow streamlines on three axial sections. In the current study, the main flow direction is defined as the pitch-wise averaged flow direction at 50%H, and then the secondary flow velocity is obtained by projecting the local velocity onto the plane perpendicular to the main flow direction. There is no significant effects on the flow field below 50%H between these four cases. In the 50%Cax section of Baseline case, the TLV, the PVC, and the passage vortex near the hub (PVH) have already formed. The TLV is in a small scale while the PVC occupies most of the space along the pitchwise direction. In the 70%Cax and 90%Cax sections, the TLV expands by entraining other fluid in the passage so that the PVC is extruded and separated into a vortex pair. The distance between the vortex cores of the PVC1 and the PVC2 can be used as a criterion of the strength of the TLV. As the plasma actuators turn on gradually, a significant change is that the TLV core moves towards the SS gradually. This can be induced by the change of TLF in flow direction and the decrease of velocity magnitude. As shown in Fig. 14, the angle θ between main flow and leakage flow has a big diminution in case 12 compared with the Baseline case. Furthermore, the area of high loss coefficient associated with the TLV is reduced a lot especially in the 90%Cax section shown in Fig. 16(d). As the TLF flows out of the gap with a lower velocity and less angle θ, the TLV is closer to the SS and less fluid in the passage is rolled up into the TLV, presenting that the region of high loss reduces. Since the TLV is weaken, the extrusion action that the PVC bearing is reduced. The PVC expands and the core moves towards the casing, which results
Fig. 13. Schematic diagram of the angle between leakage flow and mainstream.
Fig. 14. The distribution of θ along the streamwise direction.
direction between leakage flow and main flow. The influences of the leakage flow rate and the velocity difference are comprehensively represented by momentum difference. Therefore, normal momentum difference and tangential momentum difference are chosen as another two criterions to explain the mixing loss discrepancy of the Baseline
Fig. 15. Distribution of momentum difference between leakage flow and mainstream. 385
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Fig. 16. Contours of ϖ and secondary velocity streamlines in the passage.
in that partial fluid of the TLV with low energy level escapes and enters into the PVC affected zone shown in Fig. 16(d). Fig. 17 delineates the three-dimensional streamlines flowing out of the gap and the contour of ϖ in the 120%Cax section plane. It can be observed that there is a clear dividing line in the streamlines. The fluid flowing out from the rear part of the blade converges into TLV and is shown in red streamlines, while the fluid flowing out from the front part converges into PVC and is shown in blue. In Baseline case, the leakage flow in the downstream of 66%Cax is rolled up into the TLV. As the plasma actuators turn on in case 12, the direction of leakage flow changes and the dividing line moves downstream to 74%Cax. This indicates that the plasma actuators in case 12 not only diminish the m LCFD as mentioned in Table 3, but also change the distribution of leakage flow, causing the reduction of TLV. Fig. 18 plots the pitch-wise averaged total pressure loss coefficient ( ¯ ) in the 120%Cax section plane. The curve can be divided into three regions, which indicate the locations of the TLV and the PVC and the interaction zone between these two adjacent vortices respectively. In current study, TLV and PVC have been labeled in Fig. 18. It can be observed that there is less loss production in zone of TLV in cases 5, 8, and 12 than in Baseline case, while more loss is contained in zone of PVC. But these changes are not continuously increasing or decreasing. In the zone of PVC, the maximum of ¯ gradually increases at the beginning and eventually fall above the initial value in Baseline case presenting that strength of PVC increases first and then decreases, as the plasma actuators turn on gradually. And in the zone of TLV, the maximum of ¯ declines sharply at first, while there is almost no change between case 8 and case 12. This is because that the opened plasma actuators D12∼D15 in case 5 are arranged near the trailing edge, which coincides with location of composition fluid of TLV shown in
Fig. 17. D12∼D15 accurately block the leakage flow near the trailing edge so that causing a sharply decrease of ¯ in the zone of TLV. Comparing with case 8, the newly opened actuators D4∼D7 in case 12 have almost no effect on TLV, but a decline of the maximum of ¯ occurs in the zone of PVC. The reason is that actuators D4∼D7 are located at the upstream of 74%Cax and block the fluid which composes the PVC. So the TLV is hardly affected, while the strength of PVC is reduced. In addition, the position of the maximum of ¯ , which indicates the PVC core, moves to the top of the blade continuously. This can be induced by the reduction of TLV as mentioned above.
Fig. 17. Contours of ϖ and tip leakage flow streamlines. 386
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Fig. 18. Span-wise distributions of ¯ (120%Cax).
5. Conclusion In the current study, the numerical investigations of high pressure turbine cascade with multiple DBD actuators applied in the blade tip are carried out. Several cases selected by Kriging model are compared to analyze the flow structures and loss conditions with and without DBD actuators. The changes of flow topology and flow structures inside the tip gap are summarized. The mixing loss conditions are investigated by considering the leakage flow as a jet. Furthermore, the secondary velocity distributions and total pressure loss conditions are analyzed. Conclusions are made as follows, 1) The induced vortex induced by DBD actuators can change the leakage flow direction and flow structures inside the tip gap. Due to the presence of induced vortices, the separation bubble becomes labile and breaks earlier, leading to a new type of the flow pattern. 2) When the actuators are applied, the speed of leakage flow is significantly reduced and the angle between leakage flow and main flow has an obviously diminution, causing the reduction of mixing losses in the passage. 3) As the TLF flows out of the gap with a lower velocity and less angle θ, the TLV core is closer to the SS and less fluid in the passage is rolled up into the TLV, presenting that the region of high loss reduces. 4) The actuators arranged near the trailing edge can diminish the strength of TLV, while the actuators arranged in the front of blade leads to the reduction of ¯ in the zone of PVC. Declarations of interest None. Acknowledgments The authors would like to thank the China Postdoctoral Science Foundation funded project (Grant No.2018M631928) for its financial support. References Baughn, J.W., Porter, C.O., Peterson, B.L., Mclaughlin, T.E., Enloe, C.L., Font, G.I., Baird, C., 2006. Momentum transfer for an aerodynamic plasma actuator with an imposed boundary layer. 44th AIAA Aerosp. Sci. Meet. Exhibi. https://doi.org/10.2514/6. 2006-168. Behr, T., Kalfas, A.I., Abhari, R.S., 2007. Unsteady flow physics and performance of a oneand-1/2-stage unshrouded high work turbine. J. Turbomach. 129, 348–359. https://
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