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Progress in Computational Fluid Dynamics, Vol. 13, No. 5, 2013

Effects of stator clocking on unsteady rotor loading in a low-speed centrifugal compressor Hua Jiang School of Energy and Power Engineering, Xi’an Jiaotong University, Xian Ning Road No. 28, Xi’an, 710049, China and School of Energy, Xi’an University of Science and Technology, Xi’an, 710054, China E-mail: [email protected]

Wu Qi Gong* and Guang Xi School of Energy and Power Engineering, Xi’an Jiaotong University, Xian Ning Road No. 28, Xi’an, 710049, China E-mail: [email protected] E-mail: [email protected] *Corresponding author Abstract: This paper reports a numerical study on the effects of clocking between the inlet guide vane (IGV) and the stator (vaned diffuser) in a 1-1/2 stage low-speed centrifugal compressor. The unsteady flow field of the centrifugal compressor for different clocking configurations is investigated using a 3-dimantional unsteady viscous solver. The effects of stator clocking on the time-averaged pressure distribution and the unsteady pressure fluctuation on the rotor blade surface are discussed at a design operating point. The blade pressure forces are calculated and analysed, and the peak-to-peak amplitudes of the unsteady blade pressure forces are studied for different clocking configurations. It is found that the clocking effects on the time-averaged pressure distribution and the time-averaged pressure force of the rotor blades are very small at the operating point. However, the unsteady pressure fluctuation and the unsteady pressure force on the rotor blades are significantly influenced by the clocking of the stator blades. The maximal fluctuation amplitude of the unsteady loading can be reduced about 20% on the main blade and about 5% on the splitter blade with the IGV-stator clocking in the simulation. Keywords: centrifugal compressor; rotor-stator-interaction; clocking effect; pressure fluctuation; unsteady fluid; unsteady interaction; pressure distribution; compressor; turbomachine; numerical simulation. Reference to this paper should be made as follows: Jiang, H., Gong, W.Q. and Xi, G. (2013) ‘Effects of stator clocking on unsteady rotor loading in a low-speed centrifugal compressor’, Progress in Computational Fluid Dynamics, Vol. 13, No. 5, pp.312–321. Biographical notes: Hua Jiang is a Doctor at School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, China. She is also an Associate Professor at School of Energy, Xi’an University of Science and Technology, Xi’an, China. Wu Qi Gong is an Associate Professor and a Doctor at School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, China. Guang Xi is a Professor and a Doctor at School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, China.

Copyright © 2013 Inderscience Enterprises Ltd.

Effects of stator clocking on unsteady rotor loading in a low-speed centrifugal compressor

1

Introduction

There are inherently unsteady flow fields in turbomachines because of the relative motion between rotor and stator airfoils. This relative motion leads to viscous-inviscid interaction between blade rows. The continuing requirement to improve the durability and performance of turbomachinery has motivated designers to look into the mechanisms of unsteady interaction. Clocking is known as a possible method to reduce the unsteady blade loading and increase the performance of turbomachines, which uses the beneficial aspects of blade row interactions by altering the relative positions of the stator blade rows (and/or rotor blade rows). Experimental and numerical studies have been conducted on the effect of clocking on the performance of turbines and axial compressors. Huber et al. (1996), Griffin and Sharma (1996), Arnone et al. (2002), Dorney and Sharma (1996) and Dorney et al. (1998) suggested that clocking could improve the performance of a turbine by properly placing the downstream airfoils in the wakes of the upstream airfoils. Other studies (Yang et al., 2005; Chen et al., 2006; Bohn et al., 2005a, 2005b; Huang et al., 2004) yielded diverse results, indicating the need of further investigation. The unsteady blade pressure distribution is also significantly affected by clocking of the blade rows. Hsu and Wo (1998) performed an investigation of the effect of clocking on the unsteady blade loading in a low-speed axial compressor rig and found a 60% reduction in the stator unsteady force when clocking the downstream rotor for a rotor/stator/rotor configuration. The numerical studies of Jia et al. (2008, 2010) on the unsteady pressure fluctuation and blade pressure force of the rotor blades in a low-speed axial compressor, and the experimental studies of Mailach and Vogeler (2004) and Müller et al. (2010) in axial compressors showed that clocking had a very slight effect on the time-averaged pressure distribution and the time-averaged pressure force of the rotor blades, whereas the unsteady pressure fluctuation and the unsteady pressure force on the rotor blades were significantly influenced by the clocking of the stator blades. The unsteady blade loading could be Table 1

313

remarkably reduced by the clocking of the stator blade row. The maximal fluctuation amplitude of the unsteady loading was reduced more than 50% in both the simulations and the experiments. A lot of other investigations on the effect of clocking on the blade loading have also been conducted (Benini and Toffolo, 2002; Bohn et al., 2005a, 2005b; Lee and Feng, 2004; Li and He, 2003; Reinmoller and Niehuis, 2002). It is obvious that most of the previous studies have been focused on the clocking phenomenon in axial compressors and turbines, while the effect of stator clocking in centrifugal compressors has been largely neglected. Jiang et al. (2008) performed an experimental study and Xi et al. (2008) carried out a numerical investigation on the effect of stator clocking on the performance of centrifugal compressors. To the authors’ knowledge, there has not been any attempt to study the effect of clocking on the dynamic loading imposed on the blade in a centrifugal compressor stage. The present study aims at investigating the IGV-stator clocking at a design operating point in a centrifugal compressor by a numerical method, focusing on the clocking effect on the unsteady blade pressure fluctuation and pressure force of the rotor blades. The IGV circumferential positions are varied in equidistant steps while the stator positions and the operating condition are fixed. The unsteady flow field is analysed at every different relative circumferential position between the IGV and stator rows.

2

Numerical method and data postprocessing

The compressor stage consists of a circumferentially rotatable IGV row, an unshrouded centrifugal impeller, and a stator (vaned diffuser) row. Table 1 shows the main geometric data and aerodynamic information of the test stage. The impeller is a typical modern three-dimensional unshrouded centrifugal impeller with 20 backward-swept blades, of which 10 are short (splitter) blades. The flow from the IGV enters the impeller axially. The IGV circumferential positions are changed continuously.

Geometric data and aerodynamic information Geometric data IGV

Impeller (rotor)

Aerodynamic information

Stator

Inlet diameter (mm)

222

210

940

Outlet diameter (mm)

422

796

1,250

Operating conditions Rotational speed

Inlet span (mm)

103.9

43.8

Flow rate coefficient

Outlet span (mm)

43.8

43.8

Inlet air reference conditions

Inlet blade angle (°)

28

17

Temperature

Outlet blade angle (°)

58

29

Air density

20

15

Number of blades

15

n = 3,100 rpm Φ = 0.185 T = 295 K ρ = 1.13 kg/m3

314

H. Jiang et al.

The unsteady flow field of the stage with stator clocking is analysed. Calculations presented here are implemented using the time-accurate, viscous flow solver FINE/Turbo. Figure 1 shows the computational mesh in the blade passage with a total number of 1,702,660 mesh cells. The governing equations are discretised in space using a finite volume scheme. The three-dimensional unsteady compressible Reynolds-averaged Navier-Stokes equations are solved using a time-marching four stage Runge-Kutta scheme. The dual time stepping method proposed by Jameson et al. (1981) is employed to calculate the time-dependent flow field. The Spalart and Allmaras one equation turbulence model (2001) is adopted to close the Reynolds averaged Navier-Stokes equations. Local time stepping, implicit residual smoothing and multigrids are used to accelerate the computational convergence. Figure 1

The computational mesh

Inlet boundary conditions consist of specifying stagnation pressure, stagnation temperature, and the inlet flow angles. The pitchwise-averaged static pressure is assigned at the exit and adjusted to achieve the desired mass flow rate. These boundary conditions values are based on measurements from the experimental rig for the selected compressor stage. The direct periodic conditions are applied on the surfaces of revolution that bound the rotor and stator passages. Walls are specified as no slip surfaces. At the rotor/stator interfaces, mixing plane treatment is used for the steady simulations, while a direct interpolation on sliding meshes is employed for the unsteady calculations. The steady state calculation is used as initial solution for unsteady time accurate calculations. In one simulation period, 100 physical time steps are used. In one simulation period, a rotor passes three stator blades. In the simulation, the ratio of IGV, rotor and stator is 3:4:3. The stator clocking effect is investigated by varying the IGV circumferential positions in equidistant steps while the stator positions and the operating condition are fixed. The unsteady flow field is simulated at every different relative circumferential position between the IGV and stator rows. A schematic view of the IGV positions is given in Figure 2(a). Calculations are carried out at seven IGV circumferential positions over one IGV pitch under the designed operating condition. The unsteady blade pressure force coefficient for different circumferential positions is calculated from the unsteady profile pressure distribution along the whole blade. The unsteady blade pressure force F(t) and the force coefficient CF(t) are computed by the following formulas: G F(t) = ( P(t) ⋅ dS ) (1)

∫

s

CF(t) =

Figure 2

F(t) ρ / 2 ⋅ w ∞2 ⋅ A

(2)

IGV position and force component definition, (a) schematic view of IGV position (b) force component definition in the blade coordinate system

stator

z

y

rotor

IGV

x

clocking position 1

0.5

(a)

0

(b)

315

Effects of stator clocking on unsteady rotor loading in a low-speed centrifugal compressor Figure 3

Comparison of efficiency and pressure ratio between computational and experimental data 1.0

1.3

0.9

1.2

0.8

1.1

0.6

1.0

0.5

0.9

computation experiment

0.4 0.3

0.8

0.2 1.0

1.5

2.0

2.5

3

Qv/m /s

3.0

3.5

4.0

RMS=

1 M

∑ ( p (r) − p )

2

j

(3)

j= 0

The parameters pj(t) and p are the instantaneous pressure and the time-averaged pressure at a given chord point, respectively. Figure 3 gives the numerical performance maps and the experimental ones. It can be observed that the performance graph in simulation is very similar to that in experiment. The results show that the efficiency of the compressor changes with the flow rate. The stage efficiency gets to be the maximum near the design point; and the pressure ratio decreases slowly with the increase of flow rate.

3

0.7 1.0

Figure 4

Results and discussion

3.1 IGV wake In order to accurately identify the circumferential position of the IGV wake in stator passages, Figure 4 gives the time-averaged entropy value at different IGV positions in the middle position between two neighbouring leading edges of the stator blades at mid-span. When the wake of the IGV just passes the middle position between two downstream stator blades, the time-averaged entropy value, which is induced by the wake of the IGV, should reach its maximum.

1.5

2.0

2.5

3

Qv/m /s

3.0

3.5

4.0

Time-averaged entropy value at different IGV positions in the middle position between two neighbouring leading edges of the stator blades 0.005

_

The components of the force F, denoted as Fx, Fy and Fz, are along the blade chord direction, perpendicular to Fx on the plane that consists of the blade chord line and the camber line of the blade, and perpendicular to the plane that consists of Fx and Fy, respectively. Figure 2(b) shows the force component definition in the blade coordinate system. It should be noticed that the friction force on the blade surface is not considered. The root-mean-square (RMS) value reveals the fluctuation of pressure, and it is calculated by M −1

computation experiment

S-Sref / kJ*(kg*K)-1

η

ε

0.7

0.004 0.003 0.002 0.001 0.000 0.00 0.17 0.33 0.50 0.67 0.83 1.00

IGV clocking position

It can be seen that the time-averaged entropy value is maximal at IGV position 0.67, i.e., the wake of IGV passes the middle position of the stator passages at the IGV position 0.67. The wake of IGV impinges the leading edge of the downstream stator blades at IGV position 0.17, and it is close to the pressure side (PS) of the downstream stator blades at IGV position 0.0. The wake position of IGV relative to the leading edge of the downstream stators affects the time-averaged flow field of the centrifugal compressor, and it thus has an essential effect on the performance of the compressor. The stator clocking effect on the performance of a centrifugal compressor can be referenced in the experimental investigation of Jiang et al. (2008), in which it was found that the efficiency changes with the change of IGV circumferential position and the pressure ratio also varies correspondingly, but the change is very slight and close to measurement errors. In this simulation, the changes in efficiency and pressure ratio at different IGV circumferential positions are also very small and close to the computational errors (no figures), which suggests that the effect of stator clocking on the performance of a centrifugal compressor is mild at the design point. These results are similar to the results reported on an axial compressor by Jia et al. (2008).

316

H. Jiang et al.

3.2 Time-averaged pressure and RMS

the blade contour. Table 2 shows the blade pressure force parameters on the rotor blades. The force component coefficients on the main blade are almost equivalent in the three directions because the rotor blades are three-dimensional and screwy; whereas, on the splitter blade, the coefficient in y direction is the dominating force component and is about two times of that in x or z direction. The force coefficient CF on the main blade is nearly equal to that on the splitter blade; however, the component coefficients in the three directions vary between the main blade and the splitter blade. These results show that the time-averaged blade pressure force on the main blade equals that on the splitter blade in magnitude but not in direction. The reason may be that the splitter blade is 2/3 the length of the main blade and the blade pressure alters only slightly in the first 30% chord on the main blade. Figure 6 shows the RMS value of rotor blades at midspan at the clocking position 0.0. It can be seen that RMS values vary from the leading edge to the trailing edge. The implication is that the pressure fluctuations around the mean pressure distribution are different at different chord points.

The time-averaged pressure on the rotor blades is analysed at the design point in the simulation. The time-averaged pressure distribution is presented in Figure 5 and the time-averaged blade pressure force parameters are given in Table 2. The results show similar trends as reported in Jia et al. (2008, 2010), Mailach and Vogeler (2004) and Müller et al. (2010). The effect of clocking on the time-averaged pressure distribution is very small and the change of the time-averaged pressure force at different IGV positions cannot be detected. In Figure 5 and Table 2, only IGV position 0.0 is selected to show the pressure distribution and pressure force parameters. Figure 5 shows the time-averaged pressure distribution at the mid-spans of the main blade and the splitter blade at IGV position 0.0. The chord length x (x-axis) is related to the total chord length l of the main blade at the mid-span. The time-averaged pressure on the main blade is almost equal to that on the splitter blade at the same chordwise point. On the main blade, the pressure increases from the leading edge towards the trailing edge, and the amount of change is about three times of the incoming dynamic head. The pressure alters first slightly from the leading edge until the 30% point of the chord, and then increases quickly after that. On the splitter blade, the increment of the timeaveraged pressure from the leading edge towards the trailing edge is about 2.5 times of the incoming dynamic head. The time-averaged blade pressure force is calculated by integrating the profile pressure distribution with respect to Figure 5

Table 2

Time-averaged blade pressure force parameters on rotor blades CFz

CF

Main blade

0.3

–0.25

–0.30

0.49

Splitter blade

0.17

–0.37

–0.19

0.45

Time-averaged pressure distribution of rotor blades at mid-span at IGV 0.0, (a) on the main blade (b) on the splitter blade 3.0

3.0 main PS main SS

1.0 0.0

-

-1.0 -2.0 -3.0 0.0

Splitter PS Splitter SS

2.0

(P-P0 )/Pdyn / -

(P-P0 )/Pdyn / -

2.0

1.0 0.0 -1.0 -2.0

0.2

0.4

0.6

x/l / -

0.8

-3.0 0.0

1.0

0.2

0.4

(a)

0.6

x/l / -

0.8

1.0

(b)

RMS value of rotor blades at mid-span at IGV 0.0, (a) on the main blade (b) on the splitter blade 150

150

main PS main SS

100

RMS / Pa

RMS / Pa

Figure 6

CFy

CFx

50

0

splitter PS splitter SS

100

50

0 0.00

0.25

0.50

x/l / (a)

0.75

1.00

0.00

0.25

0.50

x/l / (b)

0.75

1.00

317

Effects of stator clocking on unsteady rotor loading in a low-speed centrifugal compressor

Unsteady profile pressure distributions dependent on IGV positions at leading edge, the 30% chord point and trailing edge on PS and SS of the main blade at mid-span

1.5 B

1.0

A

0.5

IGV

2.0

IGV clocking position

25.4 20.2 15.1 9.9 4.7 -0.4 -5.6 -10.8 -15.9 -21.1 -26.2

A

0.0 0.0

0.5

1.0

t/tstator /-

PS, 30%

6.0 4.8 3.6 2.4 1.2 0.0 -1.2 -2.4 -3.6 -4.8 -6.0

A

0.5

0.5

1.0

t/tstator /-

IGV

PS, out

1.5

6.0 4.8 3.6 2.4 1.2 0.0 -1.2 -2.4 -3.6 -4.8 -6.0

A

1.5 B

1.0 A

0.5 0.0 0.0

IGV

0.5

1.0

t/tstator /-

1.5

2.0

It can be seen from Figure 6(a) that on the main blade the amplitudes of the pressure fluctuations on both the PS and the suction side (SS) increase acutely from the leading edge towards the trailing edge. The extrema of the fluctuations on both PS and SS are found at the leading edge (maximum) and trailing edge (minimum). Each of the two maxima is about four times the corresponding minimum, and the difference between the extrema on PS or SS is more than 100 Pa. Between the 30% and the 55% chord points, the fluctuation amplitudes on SS are larger than those on PS, whereas at other chord points, the amplitudes on PS are larger. Figure 6(b) shows that the amplitudes of the pressure fluctuations alter gently on the splitter blade from the leading edge towards the trailing edge, and the amplitudes on PS are invariably larger than those on SS. The maximum of the pressure fluctuation on either PS or SS occurs at the trailing edge, and the minimum appears at the 60% chord point on PS and the 50% chord point on SS. The difference between the maximum and the minimum on both PS and SS is less than 50 Pa.

P-Pm / Pa

A

6.0 4.8 3.6 2.4 1.2 0.0 -1.2 -2.4 -3.6 -4.8 -6.0

B

1.0 A

0.5

IGV

0.5

SS, 30%

1.0

1.5

t/tstator /-

2.0

P-Pm / Pa

S

6.0 4.8 3.6 2.4 1.2 0.0 -1.2 -2.4 -3.6 -4.8 -6.0

A

1.5 B

1.0 A

0.5 0.0 0.0

2.0

S

1.5

2.0

P-Pm / Pa

S

SS, in

0.0 0.0

2.0

B

1.0

2.0

P-Pm / Pa

S

1.5

2.0

1.5

A

0.0 0.0

IGV clocking position

P-Pm / Pa

S

IGV clocking position

PS, in

IGV clocking position

IGV clocking position

2.0

2.0

IGV clocking position

Figure 7

IGV

0.5

SS, out

1.0

t/tstator /-

1.5

2.0

P-Pm / Pa

S

6.0 4.8 3.6 2.4 1.2 0.0 -1.2 -2.4 -3.6 -4.8 -6.0

A

1.5 B

1.0

A

0.5 0.0 0.0

IGV

0.5

1.0

t/tstator /-

1.5

2.0

The RMS value reveals information about the pressure fluctuation which is brought by the viscous wake and the potential flow interaction from upstream and downstream blade rows. The above results represent that the distribution characteristics of the unsteady pressure are different between the main blade and the splitter blade.

3.3 Unsteady pressure distribution The simulation results of unsteady pressure distribution at the mid-span of the rotor blades are presented in Figures 7 and 8. The unsteady profile pressure distribution on the rotor blades is influenced by the superimposed effect of the incoming wake of the upstream IGV and the potential effect of the downstream stator blade row. Theoretically, when the potential effect reaches the observed point on the rotor blades, the static pressure should be larger than that at other instantaneous times; when the wake from the IGV arrives at the observed point, both the velocity and static pressure should be smaller than those at other time points.

Unsteady profile pressure distributions dependent on IGV positions at the 30% and the 70% chord points on PS and SS of the splitter blade at mid-span

IGV clocking position

2.0

PS, 30%

6.0 4.8 3.6 2.4 1.2 0.0 -1.2 -2.4 -3.6 -4.8 -6.0

A

1.5

B

1.0 A

0.5 0.0 0.0

IGV

2.0

IGV clocking position

P-Pm / Pa

S

0.5

PS, 70%

1.0

1.5

t/tstator /-

2.0

A

6.0 4.8 3.6 2.4 1.2 0.0 -1.2 -2.4 -3.6 -4.8 -6.0

B

1.0 A

0.5 0.0 0.0

IGV

0.5

1.0

t/tstator /-

1.5

2.0

Figure 7 shows the unsteady profile pressure distributions dependent on the IGV positions at the leading edge, the 30% chord point and the trailing edge at mid-span of the main blade, respectively. In Figure 7, the x-axis gives the time relative to the stator blade passing period tstator, and the y-axis is the IGV position. The pressure difference p – pm is shown as greyscale, the parameter p is the instantaneous pressure at a given chord point, and the parameter pm is the averaged pressure at different IGV positions at a given chord point. It can be seen that the unsteady profile pressure distributions change regularly along with the IGV positions at these considered points. The change of the static pressure as the IGV is clocked is the largest at the leading edge on PS of the main blade, and it decreases from the leading edge to the trailing edge. These results indicate that different clocking positions of IGV can change the unsteady profile pressure distributions on the main blade. The influence weakens when the distance from the rotor inlet increases, but the wake of IGV can still reach the rotor outlet. In the simulation, the effect of stator clocking is studied by varying the IGV circumferential position in equidistant steps while the downstream stator blade row remains at a fixed position. Thus changes of unsteady pressure distributions on the rotor are caused by the stepwise shifted wake of IGV for the clocking configuration. At an observed point for different clocking positions, the effect of the IGV wake appears at different instantaneous times, while the potential effect of the fixed stator row appears at the same instantaneous times of each cycle.

SS, 30%

P-Pm / Pa

S

6.0 4.8 3.6 2.4 1.2 0.0 -1.2 -2.4 -3.6 -4.8 -6.0

A

1.5

B

1.0 A

0.5 0.0 0.0

P-Pm / Pa

S

1.5

2.0

IGV clocking position

Figure 8

H. Jiang et al.

2.0

IGV clocking position

318

IGV

0.5

SS, 70%

1.0

1.5

t/tstator /-

2.0

P-Pm / Pa

S

6.0 4.8 3.6 2.4 1.2 0.0 -1.2 -2.4 -3.6 -4.8 -6.0

A

1.5

B

1.0 A

0.5 0.0 0.0

IGV

0.5

1.0

t/tstator /-

1.5

2.0

The effect of a smaller static pressure due to the IGV wake is highlighted by a white dashed line in Figure 7. The potential effect of the downstream stator row is marked as ‘S’ with an arrow. The determination of the time points of the potential effect will be discussed in Section 6. On the main blade, if the influences of the IGV wake and the potential effect of the downstream stator row appear simultaneously at certain positions including leading edge, the 30% chord point and trailing edge, the minimum pressure caused by IGV wake coincides with the maximum pressure caused by the downstream stator row, and the amplitude of unsteady pressure decreases. This point is marked as A in Figure 7. At the point A, the IGV wake and the potential effect of the stator row superimpose each other at the IGV positions 0.5, 1.5, etc. At these positions, the unsteady pressure fluctuations on the main blade are obviously lower than those at other clocking positions. Contrarily, if the maximum pressure caused by the clocked IGV and that caused by the downstream stator row superimpose at the considered positions, the pressure amplitude increases. This point is marked as B in Figure 7. At the point B, the maximum pressure caused by the clocked IGV and that by the downstream stator row superimpose each other at the IGV positions 0.0, 1.0, etc. At these positions, the unsteady pressure fluctuations on the main blade are apparently larger. Figure 8 presents the unsteady profile pressure distributions dependent on the IGV positions at the 30% and the 70% chord points at mid-span of the splitter blade. The unsteady profile pressure distributions on the splitter blade change regularly with the change of IGV positions. Similar

319

Effects of stator clocking on unsteady rotor loading in a low-speed centrifugal compressor

to those in Figure 7, the points A and B in Figure 8 denote the same phenomena on the splitter blade. It is observed that the IGV positions of the points A and B on the splitter blade are different from those on the main blade. For the splitter blade, the point A, at which the IGV wake and the potential effect of the stator row superimpose each other, is at the IGV positions 0.67, 1.67, etc., where the unsteady pressure fluctuations are remarkable lower than those at other IGV positions. The point B, at which the maximum pressure caused by the clocked IGV and the maximum pressure by the downstream stator row superimpose each other, is at the IGV positions 0.17, 1.17, etc., where the amplitudes of the unsteady pressure fluctuations are evidently larger. The pressure distributions at other chord points are similar to that at these pointes given. It was no longer listed because of limitations of space.

different downstream stator positions, the time point of the potential effect of the downstream stator blade row can be determined. The effect of a larger static pressure due to the potential effect of the downstream stator is highlighted by a black dashed line. At the stator position 0.0, the downstream stator position is the same as that in Section 5. Thus, at the stator position 0.0, the time point of the downstream stator potential effect for an observed point is also the same as that in Section 5. This time, marked as ‘S’ with an arrow on the x-axis, is the time point of the potential effect at the 30% chord point on SS of the main blade at mid-span in Figure 7. The other time points of the potential effect for other observed points in Figures 7 and 8 can be determined in the same way (no figures). Figure 9

In Figures 7 and 8, the potential effect of the downstream stator row appears at different time for different observation points. For the same observed point at different IGV positions, the potential effect of the fixed stator row appears at invariable instantaneous times. In order to determine the time points of the effect of the downstream stator, the unsteady flow field is simulated by varying the circumferential position of the downstream stator row in equidistant steps while keeping the IGV position unchanged; the other conditions remain the same as in the previous simulations. Thus the changes of the unsteady pressure distributions on the rotor are caused by the stepwise shift of the potential of the clocking stator. Contrarily, at an observed point for different downstream stator positions, the potential effect of the stator row appears at different time points, while the effect of the fixed IGV wake appears at constant time points. Figure 9 shows the unsteady profile pressure distributions depended on the downstream stator positions at the 30% chord point on SS of the main blade at mid-span. The x-axis gives the time relative to the stator blade passing period tstator, and the y-axis is the downstream stator position. In the space-time contour of the unsteady pressure fluctuation at one considered chord point for

main SS, 30%

2.0

P/ Pa 95318.5 95304.8 95291.1 95277.4 95263.7 95250.0 95236.3 95222.6 95208.9 95195.2

1.5 1.0 0.5 0.0 0.0

0.5

1.0

1.5

t/tstator /- S

2.0

3.5 Unsteady pressure forces The unsteady blade pressure force is calculated by integrating the unsteady profile pressure on the whole blade. The blade pressure force on the rotor blade is not only dependent on the amplitude of the unsteady pressure, but also on the phase shift between the unsteady pressure on PS and SS. Figure 10 shows the unsteady pressure force coefficients on the main blade and the splitter blade at the IGV position 0.0. It can be seen that the pressure force changes periodically in every stator passing period tstator because of the identical blade numbers of the IGV and the stator blades. The fluctuation amplitude on the main blade is close to that on the splitter blade.

Unsteady pressure force coefficient on rotor blades at IGV 0.0

main

0.505

CF / -

0.450

0.495

0.490 0.0

splitter

0.455

0.500

CF / -

Figure 10

stator clocking position

3.4 Determination of time points for the potential effect of the downstream stator row

Unsteady profile pressure distributions dependent on downstream stator positions at the 30% chord point on SS of the main blade at mid-span

0.5

1.0

t/tstator / -

1.5

2.0

0.445

0.440 0.0

0.5

1.0

t/tstator / -

1.5

2.0

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H. Jiang et al.

Figure 11 Peak-to-peak amplitude of the unsteady pressure force coefficient on rotor blades for different IGV positions

main

splitter 0.012

peak-to-peak / -

peak-to-peak / -

0.011 0.010 0.009 0.008

0.010 0.009

0

0.2

0.4

0.6

0.8

IGV clocking position

1

Figure 11 shows the fluctuation amplitude (peak-to-peak), which is the difference between the maximum and the minimum of the unsteady pressure force coefficient CF(t) on the main blade and the splitter blade for different IGV positions. On the main blade, the maximal and minimal fluctuation amplitudes of the unsteady rotor loading appear at the IGV positions 0.0 and 0.5, respectively; whereas on the splitter blade, the maximal and minimal fluctuation amplitudes appear at the IGV positions 0.17 and 0.67, respectively. That is, the clocking positions with the maximal and minimal fluctuation amplitudes on the splitter blade shift about 1/6 stator pitch relative to those on the main blade. The minimal fluctuation amplitude on the main blade is ±1.6% of the time-averaged value at the clocking position IGV 0.5, while the maximal is ±2% of the mean value at IGV 0.0. On the splitter blade, the minimal and maximal fluctuation amplitudes are ±2.1% at the clocking position IGV 0.67 and ±2.2% of the mean value at IGV 0.17, respectively. From these results it is calculated that the maximal fluctuation amplitude of the unsteady loading is reduced about 20% on the main blade and about 5% on the splitter blade with the IGV clocking in the simulation. Compared with the results in literatures (Jia et al., 2008, 2010; Mailach and Vogeler, 2004; Müller et al., 2010), the influence of the stator clocking on the unsteady loading on the rotor in a centrifugal compressor is smaller than that in a axial compressor.

4

0.011

Summary

The clocking effect between IGV and downstream stator row in a centrifugal compressor is numerically investigated. The effects of clocking on the time-averaged pressure distribution and the unsteady pressure fluctuation on the rotor blades are discussed at a design operating point. The blade pressure forces are calculated and analysed and the peak-to-peak fluctuation amplitudes of the unsteady blade pressure forces are studied for different clocking positions. It is found that the clocking effects on the time-averaged pressure distribution and the time-averaged pressure force of the rotor blades are very small at the operating point. However, the unsteady pressure fluctuation and the

0

0.2

0.4

0.6

0.8

IGV clocking position

1

unsteady pressure force on the rotor blades can be significantly influenced by the clocking of the stator blades. The maximal fluctuation amplitude of the unsteady loading can be reduced about 20% on the main blade and about 5% on the splitter blade with the IGV clocking in the simulation. This investigation has revealed the effect of clocking on unsteady rotor loading in a centrifugal compressor. It can help us better understand the mechanisms of unsteady interaction. Further experimental study is needed on the effects associated with airfoil clocking in centrifugal compressors.

Acknowledgements The authors wish to thank the financial support of the National Science Foundation of China (No. 50676072, No. 50725621).

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Appendix Nomenclature A

Area of blade surface (m2)

c

Absolute velocity (m/s)

CF

Pressure force coefficient

l

Chord length of the main blade (m)

P

Pressure (Pa)

RMS

Root-mean-square value of pressure (Pa)

S

Entropy (kg/(kg*K))

t

Time (s)

trotor

Rotor blade passing period (s)

u2

Peripheral velocity at the impeller outlet (m/s)

w

Relative velocity (m/s)

ρ

Density (kg/m3)

Superscripts ~

The ensemble-averaged quantities

―

Time average

Subscripts 0

Reference pressure

∞

Average of the velocity vectors of the incoming and outgoing flow

dyn

incoming dynamic head

m

average of different IGV positions

Abbreviations IGV

inlet guide vane

LE

leading edge

MS

mid-span

PS

pressure side

SS

suction side

TE

trailing edge