Authors Instructions

ISUAAAT13-S10-6

 UNSTEADY FLOW FIELD AND STRUCTURAL RESPONSE IN A TURBINE STAGE OF A ROCKET ENGINE Yuki TOKUYAMA Iwate University Morioka, Japan Email:[email protected]

Ken-ichi FUNAZAKI Iwate University Morioka, Japan

Hiromasa KATO Iwate University Morioka, Japan

Junya TAKIDA Japan Aerospace Exploration Agency Kakuda, Japan

Mitsuru SHIMAGAKI Japan Aerospace Exploration Agency Kakuda, Japan

Masaharu UCHIUMI Japan Aerospace Exploration Agency Kakuda, Japan

ABSTRACT This study treats the unsteady flow field in a partial admission supersonic turbine stage of a new rocket engine by Computational Fluid Dynamics (CFD) and response to unsteady aerodynamic force on rotor disk by Finite Element Method (FEM). In CFD analysis, quasi 3D full angular analysis is conducted in order to investigate unsteady flow field and low frequency unsteady aerodynamic forces on rotor. The results of CFD analysis indicate strong unsteady flow field having roots in partial admission turbine structure. The flow at blockage inlet is strongly accelerated and rotor load increases rapidly. Furthermore, high speed jet fluctuating in low frequency is observed at between blockage and rotor. FFT analysis results of unsteady rotor load indicate that frequency of maximum amplitude component, both tangential and axial direction, is close to blockage passing frequency. In FEM analysis, rotor blades are modeled as mass points on rotor disk hub. Rotor disk eigenfrequencies and mode shapes are calculated by modal analysis. In order to investigate response to unsteady aerodynamic force, transient structural analysis is also conducted using some unsteady aerodynamic load components calculated by CFD analysis, its frequency close to low eigenfrequencies, as input load. Rotor disk displacement is dominated by centrifugal force, and at the same time, unsteady aerodynamic load causes low nodal diameter mode shape. INTRODUCTION Rockets play increasingly important roles in our society, e.g., supply missions to the International Space Station (ISS) and artificial satellite launches. Increasing needs for future human transportation to the ISS after the retirement of the Space Shuttle also

requires further improvement in the reliability of launcher rocket. In general, rocket engine must create high thrust with one or two turbine stages because of its weight constraint. So turbines used in rocket engine system are often designed as supersonic turbine with partial admission. Nozzle (stator) – rotor interaction generates complex unsteady flow fields and partial admission configuration creates low frequency rotor force. [1, 2, 3] Moreover, unsteady shock waves interact with each other as well as with wakes and this shock system makes flow field more complex in supersonic turbine. These fluid dynamic phenomena are behind the unsteady aerodynamic forces experienced by rotor blades. [4, 5] These unsteady aerodynamic forces cause cyclic or noncyclic stress and resonance if frequencies of such forces coincide with turbine’s structural eigenfrequencies. Turbine used in rocket engine is often manufactured as brisk in order to increase its strength. This type of turbine has less structural damping due to the lack of frictions between turbine blades and disk. So predicting the influence of unsteady aerodynamic forces on turbine response at the design phase is very important to improve rocket reliability. Groth et al. [6] compared flutter limit computed by CFD with experimental results. Marcu et al. [7] conducted stress analysis and studied High Cycle Fatigue (HCF) using CFD. This study focuses on unsteady flow field in supersonic turbine stage with partial admission, especially the flow field around blockage, and response to unsteady aerodynamic force on rotor. Quasi 3D unsteady CFD analysis is conducted to investigate unsteady flow field. Furthermore, modal and transient structural analysis of rotor disk by FEM are also carried out using unsteady rotor blade load obtained by CFD analysis in order to study rotor disk mode shape and response.

Copyright © 2012 by ISUAAAT Scientific Committee with JSASS Publication

TARGET TURBINE Turbine considered in this study is a scaled model of the first stage of the FTP turbine for M-1 engine and the mean diameter is 152 mm. This turbine is a supersonic turbine with partial admission and consists of 44 nozzle blades and 94 rotor blades. 24 nozzle passages are actually considered, thus the

partiality is 54.5 percent. There are three admission regions with 8 passages each. Three blockage regions are such that two spanning 7 passages each and one spanning 6 passages. Figure 1 shows the blade profiles (both 1st and 2nd stages).

Nozzle 1st stage rotor

2nd stage stator 2nd stage rotor

Figure 1

Cross section of M-1 turbine scale model and turbine blade profiles

CFD PROCEDURE Computational domains Quasi 3D computational domains are shown in Figure 2. The span height is 4% of the actual height. Blockage region

Figure 2

and flow angle are imposed on the inlet boundary and static pressure is imposed on the outlet boundary. To conduct quasi 3D simulation, the hub and shroud walls are treated as slip wall with adiabatic conditions and no-slip condition was imposed on the other walls. Other boundary conditions, such as rotor rotational speed and total pressure etc., correspond to the experimental conditions in Ref. [8] and summarized in Table 1.

Computational domains

Computational mesh Structured meshes for CFD analysis were generated using Pointwise Gridgen. An O-block is placed around the blade and H-blocks are used for the passage. The nozzle domain has 80,000 nodes and the rotor domain has 64,000 nodes per passage respectively. As a result, the total number of nodes becomes 9.3 million. The generated mesh is shown in Figure 3. Computational conditions URANS simulation was performed by ANSYS CFX 13. Temporal discretization is the second-order backward Euler method and the convective terms are discretized by a high resolution TVD scheme. The shear stress transport (SST) model was used as turbulence model. Total pressure, total temperature

Figure 3

Computational mesh (CFD)

Table 1

Boundary condition values

Working fluid Rotational speed

N2 ideal gas 10,080 [rpm] 2.35 [MPa]

Inlet conditions Outlet condition

Total pressure Total temperature Inflow angle Static pressure

273 [K] Normal to boundary 0.645 [MPa]

FEM PROCEDURE Rotor disk model and computational mesh The FEM model of the rotor disk for structural analysis is shown in Figure 4. The rotor blades are modeled as 94 mass points on the hub surface. The fluid dynamic force on the rotor blades are obtained from the CFD analysis and applied at these mass points. The mesh consists of 20,000 hexahedral cells and 20-node hexagonal element was used. Mass points

≪CFD anaysis≫ Unsteady aerodynamic load

Load components Amp

FFT

Freq

Load Time

≪Modal analysis≫ Eigenfrequencies Eigenfreq Mode Choice near

freq component

Define load with interblade phase angle

𝐹𝑛 = 𝐴 ∙ sin {2𝜋 (𝜔 ∙ 𝑡 + Figure 4 Rotor disk model and computational mesh (FEM) Material properties and load definition Material properties are set as titanium alloy. All 6 degrees of freedom, rotational and translation, of the nodes located on the center hole are constrained. Two kinds of load are defined in transient structural analysis as follows. Ⅰ：Angular velocity Ⅱ：Unsteady aerodynamic load components Ⅰ）Angular velocity Angular velocity value in FEM analysis is applied as 5236 rad/s, i.e. 50,000 rpm, different from the value in CFD analysis (10,080 rpm). The reason of this is that the value in CFD analysis is not actual operating speed (50,000 rpm) but chosen for aerodynamic experiment. Ⅱ）Unsteady aerodynamic load components Figure 5 illustrates a flow chart of unsteady aerodynamic load components definition. These load components are calculated by FFT analysis of tangential and axial load on one representative rotor blade obtained from CFD analysis. After conducting of modal analysis, some load components, close to low eigenfrequencies, are chosen and applied on each mass point with interblade phase angle as shown in equation (1). These load components are result of quasi 3D analysis (one four hundredth), so their amplitudes are simply scaled up by multiplying 400 in order to define full span height load.

Figure 5

𝑛−1 )} 94

(1)

Flow chart of unsteady aerodynamic load components definition

CFD RESULTS AND DISCUSSION Pressure histories Figure 6(a) shows static pressure histories on representative four points, nozzle outlet, rotor inlet, blockage inlet and blockage center shown in the upper side of that figure. Only rotor inlet point is on rotational domain and the others are on stationary domain, so rotor inlet point moves with rotor blades during unsteady calculation. Vertical axis indicates nondimensionalized static pressure by inlet total pressure and horizontal axis shows normalized time by rotor rotation period. Characteristic pressure variations on each point are summarized as follows. 【Nozzle outlet】 ・Rotor blade passing frequency (BPF) variation. 【Blockage inlet】 ・Rotor BPF variation. 【Blockage center】 ・Rotor BPF and low frequency variations. 【Rotor inlet】 ・Nozzle BPF variation and large variation caused by blockage. On nozzle outlet and blockage inlet points, the pressure variation is governed by rotor leading edge shock wave. Static pressure contours around admission region at one-moment is presented in Figure 6(b). Rotor leading edge shock waves are observed as close-set constant-pressure lines. Because nozzle outlet and blockage inlet points are in stationary domain, shock waves pass through these points and rotor BPF pressure variation is caused.

Flow fields around blockage are discussed in a later

section.

Nozzle outlet point (stationary) Blockage inlet point (stationary) Blockage center point (stationary)

Rotor inlet point (moving)

Static pressure ( 𝑷𝒔 /𝑷𝒕 𝒊𝒏 )

𝑷𝒔 /𝑷𝒕 𝒊𝒏

0.35

0.60

0.30

0.37

0.25

0.15

0.20 0.15 0.10 0.05 0

0.2

0.4

0.6

0.8

1

Rotation time ( 𝒕/𝒕𝒓𝒐𝒕 ) (a)

Figure 6

Pressure histories (a) and contours around admission region (b)

Flow field around blockage region Blockage inlet. Flow field around blockage inlet region is very complex in partial admission turbine. We can see that rotor inlet point experiences large pressure drop at blockage inlet from Figure 6(a). In general, partial admission turbine, nozzle jet at the end of admission region is sucked into blockage region and flow is accelerated at blockage inlet. [9] As a result of this, pressure decrease is caused. Furthermore, in this study case, nozzle jet is supersonic flow, hence supersonic flow characteristics also relate to this acceleration. Figure 7 depicts an instantaneous absolute mach number (Mabs) contours around blockage inlet with streamlines (solid lines) and Mrel = 1.0 lines, i.e. sonic lines in relative frame, (broken lines). Nozzle jet turns corner of blockage and expansion wave occurs, thus the flow is accelerated. At downstream of expansion wave, flow slows down because of rotor leading edge shock wave, but channel formed by blockage wall and rotor leading edge acts as divergent nozzle and causes re-acceleration. Accelerated by divergent channel

(b)

Accelerated by expansion wave

Existing of rotor leading edge shock wave created in this high velocity region makes flow field more complex and causes separations. Boundary layer thickness on blockage wall is increased by impingement of shock waves. Finally, the flow separates from blockage wall at the end of high velocity region. This rotor leading edge shock wave also causes very large separation on neighboring rotor suction surface in comparison with shock wave created when rotor blade is in admission region. Figure 8 and 9 shows rotor reading edge shock waves when rotor blade is in admission region and in blockage inlet respectively. As shown in figure 8, when rotor blade is in admission region, rotor leading edge shock wave is almost normal shock wave and flow does not much change its direction at upstream and downstream of shock wave. On the other hand in Figure 9, at t/tr_pitch = 0 indicates that intensity of shock wave is increased by flow acceleration and separation region becomes slightly large. As a result of this, the shock wave near rotor suction surface becomes oblique shock wave. At t/tr_pitch = 0.25, this oblique shock wave region accounts for about half of rotor passage and flow changes its direction toward pressure side at downstream of shock wave, so the separation region is expanded. And then at t/tr_pitch = 0.50, shock wave is almost completely oblique shock wave and separation region becomes more large.

Normal shock wave

Mabs 0.0

1.0

2.0

Figure 7 An instantaneous absolute mach number contours around blockage inlet, streamlines (solid lines) and Mrel = 1.0 lines (broken lines)

Mabs 0.0

1.0

2.0

Figure 8 Rotor reading edge shock wave when rotor blade is in admission region

＊ Separation

＊

0.0

𝑡/𝑡𝑟_𝑝𝑖𝑡𝑐ℎ = 0

Mabs

1.0

2.0

＊

＊

Oblique shock wave Flow direction changes

𝑡/𝑡𝑟_𝑝𝑖𝑡𝑐ℎ = 0.25

Figure 9

𝑡/𝑡𝑟_𝑝𝑖𝑡𝑐ℎ = 0.5

Rotor reading edge shock wave when rotor blade is in blockage inlet

Rotation time ( 𝒕/𝒕𝒓𝒐𝒕 )

Blockage behind. Pressure history on blockage center point in Figure 6(a) indicates low frequency variation, about eight variations in one rotor rotation period, in addition to rotor BPF variation. Figure 10 shows pressure history contours on blockage behind line illustrated in the right side of that figure as solid red line. Three broken lines are trajectory of different three rotor blades, blade1, blade7 and blade13 respectively. Unsteady rotor blade loads on these three blades during passing blockage behind are illustrated in Figure 12. Scales on horizontal axis in Figure 10 and 12 are drawn every nozzle pitch. 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 te

We can see from Figure 10 that pressure increase by rotor leading edge shock wave is observed around 2nd scale mark from blockage inlet. This shock wave obviously moves toward blockage inlet and outlet at low frequency, about eight variations in one rotor rotation period. At between 3rd scale mark from blockage inlet and blockage outlet, low pressure regions appear when shock wave moves toward blockage outlet and disappear when shock wave moves toward blockage inlet. However, we can also see almost no variational regions.

𝑷𝒔 /𝑷𝒕 𝒊𝒏 0.043

0.43

te

out

Trajectories Blade1 Blade7 Blade13

Plot line out

in

in Figure 10

Pressure history contours on blockage behind line

Figures 11 (a) and (b) show absolute Mach number contours in the region behind the blockage when low pressure regions disappear (a : t/trot = 0 to 0.0186) and appear (b : t/trot = 0.0612 to 0.0798), these time frames are equal to rotor two pitch rotation. Solid lines are Mabs = 1.0 lines, i.e. sonic lines in absolute frame. There are supersonic jets extending circumferential direction as far as near blockage outlet in both (a) and (b), and jet in (b) is slightly thicker than that of in (a). These jets sneak

into the blockage region and repeat breakage and reconnection, periodically at rotor BPF, around the separation region on the blockage wall. A piece of high velocity area (indicated by the red arrow) appears when the jet connects with high velocity region around the blockage inlet and the speed of this area in (b) is higher than in (a). Furthermore in (b), we can also see intermittent high velocity regions inside the jet. These regions are observed as low pressure regions in Figure 10.

0.0

1.0

2.0

Mabs ＊

＊

𝑡/𝑡𝑟_𝑝𝑖𝑡𝑐ℎ = 0

𝑡/𝑡𝑟𝑜𝑡 = 0

𝑡/𝑡𝑟𝑜𝑡 = 0.0612

＊

＊

𝑡/𝑡𝑟_𝑝𝑖𝑡𝑐ℎ = 0.25

＊

＊

𝑡/𝑡𝑟_𝑝𝑖𝑡𝑐ℎ = 0.5

＊

＊

𝑡/𝑡𝑟_𝑝𝑖𝑡𝑐ℎ = 0.75

＊

＊

𝑡/𝑡𝑟_𝑝𝑖𝑡𝑐ℎ = 0.1

＊

＊

𝑡/𝑡𝑟_𝑝𝑖𝑡𝑐ℎ = 1.25

＊

＊

𝑡/𝑡𝑟_𝑝𝑖𝑡𝑐ℎ = 1.5

＊

＊

𝑡/𝑡𝑟_𝑝𝑖𝑡𝑐ℎ = 1.75

Figure 11

𝑡/𝑡𝑟𝑜𝑡 = 0.0186

𝑡/𝑡𝑟𝑜𝑡 = 0.0798

(a)

(b)

Absolute mach number contours around blockage behind and M abs = 1.0 lines (solid lines) (a) : t/trot = 0 to 0.0186, (b) : t/trot = 0.0612 to 0.0798

field. On the other hand, blade7 experiences pressure field whose phase is different 180 degrees. The influences of this difference on rotor unsteady load are illustrated in Figure 12. Unsteady rotor blade loads on blade1 and blade13 agree well and that of blade7 differs in both tangential and axial directions. Thus, pressure field differences caused by the movement of rotor leading edge shock wave influence rotor blade load.

0.1

0.05

0.08

Axial load ( 𝑳𝒂𝒙𝒊 [N] )

Tangential load ( 𝑳𝒕𝒂𝒏 [N] )

The detailed mechanisms of the shock wave movement and the low frequency pressure variations are at present not understood. But these pressure variations at low frequency mean that each rotor blade experiences different pressure field, i.e. the timings at which rotor blades experience low pressure regions are different as they travel the blockage region. As shown in Figure 10 (trajectories), blade1 and blade13 experience almost the same pressure

Blade1 Blade7 Blade13

0.06 0.04 0.02 0.0 -0.02 -0.04 65 te

56.875 out 48.75 40.625 32.5 24.375 16.25 8.125

Figure 12

0 in

Rotor load ( 𝑳 − 𝑳𝒂𝒗𝒆 [N] )

0.08 0.08

Tan Axi

0.06 0.06 0.04 0.04 0.02 0.02 0.00 0.0 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06 0.2 72

0.03 0.02 0.01 0.0 -0.01 -0.02

65 te

56.875 out 48.75 40.625 32.5 24.375 16.25 8.125

0 in

Unsteady rotor blade loads during passing blockage behind left : tangential direction, right : axial direction

Unsteady rotor blade load Figure 13 shows tangential and axial unsteady rotor blade loads on blade1 over rotor one revolution. Vertical axis indicates difference of unsteady load and its time average. Horizontal axis indicates normalized time by rotor rotation period. Here, we must take care that rotor moves from left to right in this figure differently from Figure10 and 12. Rapid load increases, in the direction of rotation and upstream respectively, are caused at near blockage inlet. Immediately after entrance of rotor into blockage region, rotor passage gets empty. However, neighboring passage is still filled up with high pressure gas, so tangential load increases in the direction of rotation occur. Meanwhile, as previously mentioned, deep static pressure drops are caused by flow acceleration around blockage inlet and this makes axial load increases in the direction of upstream.

0

0.04

0.4 0.6 0.8 144 216 288 Rotation time ( 𝒕/𝒕𝒓𝒐𝒕 )

Figure 13 Unsteady rotor blade loads over rotor one rotation

1 360

In addition to these load variations, we can see that appreciable tangential load fluctuations at blockage behind, although axial load fluctuations are very small. These load fluctuations are maybe caused by inflow of high momentum fluid into rotor passages. However, the details are at present not understood, so we will conduct more detail study in the future. Figure 14 shows FFT analysis results of unsteady rotor blade load on blade1. Vertical axis indicates amplitude and horizontal axis indicates nondimensionalized frequency by frequency of rotor rotation. Distinct components in tangential load are about 3, 4, 10, and 15.5 multiples of rotation frequency. On the other hand in axial load, distinct components exist in wide frequency range. Only from this result, it is difficult to judge each load component comes from the influence of blockages or not. So, additional simulations, such as full admission and different blockage count, are needed and will be conducted. However, considered turbine stage in this study case has three blockages, so load component at near 3 multiples of rotation frequency can be considered as the influence of blockages. This load component can be clearly observed in Figure 14 and its amplitude is the maximum in both tangential and axial direction. These low frequency load components can cause low nodal diameter mode and fatal vibrations, thus rotor disk response to these load components are investigated and discussed in later section.

0.030 0.030

0.006 0.006

0.025 0.025

0.005 0.005

Amplitude

0.007 0.007

Amplitude

0.035 0.035

0.020 0.020

0.004 0.004

0.015 0.015

0.003 0.003

0.010 0.010

0.002 0.002

0.000 0.000 1.41 2.81 4.22 5.63 7.03 8.44 9.84 11.25 12.66 14.06 15.47 16.88 18.28 19.69 21.09 22.50 23.91 25.31 26.72 28.13 29.53 30.94 32.34 33.75 35.16 36.56 37.97 39.38 40.78 42.19 43.59 45.00

0.001 0.001

0.000 0.000 1.41 2.81 4.22 5.63 7.03 8.44 9.84 11.25 12.66 14.06 15.47 16.88 18.28 19.69 21.09 22.50 23.91 25.31 26.72 28.13 29.53 30.94 32.34 33.75 35.16 36.56 37.97 39.38 40.78 42.19 43.59 45.00

0.005 0.005

Frequency / Frequency of rotation

Frequency / Frequency of rotation

Figure 14 Unsteady rotor blade load in frequency domain left : tangential direction, right : axial direction FEM RESULTS AND DISCUSSION Modal analysis Figure 15 depicts rotor disk eigenfrequencies and modal shapes calculated by modal analysis conducted before unsteady structural analysis. Important modes in rocket engine turbine are generally of low nodal diameters such as 1, 2 and 3 nodal diameter modes. Their frequencies are below 4000 Hz in this study case. Based on this result, three unsteady rotor blade load components on blade1 (Force1, Force2 and Force3) are chosen for unsteady structural analysis

and applied on all the mass points with inter-blade phase angles. Force1 and Force2 are the maximum amplitude component in tangential and axial load respectively. Their frequency is about 2341 Hz. Force3 is axial load component at about 4 multiples of rotation frequency and its frequency is about 3515 Hz. This frequency is close to the eigenfrequency of the 3 nodal diameter mode (3340 Hz).

2ND

3ND

Thrust

Thrust

1ND

Eigenfrequency [Hz]

10000 8000 6000 4000 2000 0 1 Figure 15

2

3

4

5

6

7

8

9 10 11 12

Eighenfrequencies and mode shapes of rotor disk

Response to unsteady load Axial displacement at one node placed on hub edge under Force1 and Force2 load conditions are plotted in Figure 16 and 17 respectively. Negative value means displacement toward the downstream direction. In both load cases, this node fluctuates around -85.8 μm and this displacement is the influence of centrifugal force. The amplitude does not amplify in both load cases and the amplitude value is

about 0.8 μm under Force1 condition and 40 μm under Force2 condition. Thus, although Force1 load amplitude is about 4.8 times greater than Force2, vibration amplitude is much greater under Force2 condition. This means that tangential unsteady force does not much influence axial vibration amplitude in comparison with axial unsteady force.

Time [s]

Time [s] 0

0.1

0.2

0.3

0.4

0.2

0.5

0.2025

0.205

0.2075

0.21

-8.53E-05 -85.3

Axial directional displacement [μm]

Axial directional displacement [μm]

0.00E+00 0.0

-8.55E-05 -85.5

-2.00E-05 -20

-8.57E-05 -85.7

-40 -4.00E-05 -60 -6.00E-05

-85.9 -8.59E-05 -8.61E-05 -86.1

-80 -8.00E-05

-86.3 -8.63E-05

-100 -1.00E-04

Figure 16

Axial displacement under Force1 load condition (right : expanded) Time [s]

0.1

0.2

0.3

0.4

Time [s]

0.5

0.2025

0.205

0.2075

0.21

-60 -6.00E-05

Axial directional displacement [μm]

Axial directional displacement [μm]

0.0 0.00E+00

0.2

-2.00E-05 -20

-70 -7.00E-05

-4.00E-05 -40

-80 -8.00E-05

-6.00E-05 -60

-90 -9.00E-05

-8.00E-05 -80

-1.00E-04 -100

-1.00E-04 -100 -1.20E-04 -120

-1.10E-04 -110

Axial displacement under Force2 load condition (right : expanded)

These vibrations in frequency domain are shown in Figure19 and Figure18 illustrates an instantaneous displacement contours under Force2 condition. Tow clear peaks at the same frequency can be observed from Figure 19 in both load cases. The maximum peak is 1 nodal diameter mode and this can be observed in Figure18. The second is forced vibration because its frequency completely agrees with unsteady load frequency. These results indicate that excited mode shapes by unsteady rotor blade load component at blockage passing frequency are unrelated to load direction in this study case rotor disk. And response to tangential load is very small, so this load component does not have need of consideration for low nodal diameter vibrations. Of course this does not mean that this load component can be completely ignorable in turbopump system because it may cause significant shaft vibration. On the other hand, axial load component should be reduced its amplitude, i.e. reduction of pressure drop at blockage inlet, because it causes 1 nodal diameter vibration with comparatively large amplitude. The rotor disk response to Force3 is plotted in Figure 20 and 21. The amplitude is about 12 μm and does not amplify as well as Force1 and Force2 load cases. 2nd peak in Figure 21 indicates forced response. Although the frequency of Force3 is close to 3 nodal diameter mode eigenfrequency, that mode does not appear and only 1 nodal diameter mode is excited. The amplitude of 1 nodal diameter mode component is about 1.5 times larger than forced response component under Force2 load condition. On

[μm] 0.0

20

Figure 18 An instantaneous displacement contours under Force2 condition 2.50E-07

1ND

2.00E-07

Amplitude

Figure 17

Forced response

1.50E-07 1.00E-07 5.00E-08 0.00E+00 0

1000 2000 3000 4000 5000 6000 7000 8000

Frequency [Hz] 1.25E-05

Amplitude

0

1ND

1.00E-05

Forced response

7.50E-06 5.00E-06 2.50E-06 0.00E+00 0

1000 2000 3000 4000 5000 6000 7000 8000

Frequency [Hz]

Figure 19 Axial displacement in frequency domain, upside : Force1, downside : Force2

the other hand, in Force3 load condition, 1 nodal diameter mode’s amplitude is 4 times larger than forced response amplitude. Thus, it can be considered that 1 nodal diameter mode is more sensitive to

Force3 and its lode should be reduced as well as Force2. In order to this, it is necessary that more investigations of relationship between Force3 and unsteady flow field. Time [s]

Time [s] 0.2

0.3

0.4

0.5

0.20 -7.8E-05 -78

-100 -1.0E-04

Axial directional displacement [μm]

0.1

Axial directional displacement [μm]

0.0 0.0E+00

0.0

-1.2E-04 -120

-9.4E-05 -94

-20 -2.0E-05

0.21

0.21

0.21

-8.2E-05 -82

-40 -4.0E-05

-8.6E-05 -86

-60 -6.0E-05 -80 -8.0E-05

-9.0E-05 -90

Figure 20

5.00E-06

Amplitude

0.20

Axial displacement under Force3 load condition (right : expanded)

1ND

4.00E-06 3.00E-06 2.00E-06

Forced response

1.00E-06

0.00E+00 0

1000 2000 3000 4000 5000 6000 7000 8000

and may be ignorable. However, amplitude caused by axial load was comparatively large, so axial road amplitude should be reduced. Rotor blade load component close to 3 nodal diameter mode eigenfrequency does not excite its mode shape. But this load component excites 1 nodal diameter mode and more sensitive than blockage passing frequency load component.

Frequency [Hz]

Figure 21 Axial displacement in frequency domain under Force3 condition

CONCLUSIONS Quasi 3D unsteady CFD analysis of the flow through a supersonic partial admission turbine and investigation of rotor disk response to unsteady blade load by FEM were carried out. Characteristics caused by partial admission structure were observed from CFD analysis. At blockage inlet region, the flow was strongly accelerated and strong rotor leading edge shock wave was created. This shock wave impinged on blockage wall and neighboring rotor suction surface and caused large separation. Supersonic jet extended circumferential direction at blockage behind and fluctuated its shape and velocity at low frequency. Furthermore, because of this low frequency fluctuation, each rotor blade experienced different load history during passing through blockage behind. When rotor enters blockage passage, large and dramatic rotor blade load variations were caused both tangential and axial direction and the maximum unsteady rotor blade load component was blockage passing frequency. In FEM analysis, 1 nodal diameter mode shape was excited by unsteady rotor blade load component at blockage passing frequency. The vibration amplitude caused by tangential load was very small

NOMENCLATURE F = force A = amplitude of unsteady load component ω = frequency of unsteady load component t = time trot = period of rotor one rotation tr_pitch = period of rotor one pitch rotation n = rotor blade number (1, 2, …, 94) P = pressure M = mach number L = rotor blade load

SUBSCRIPTS s = static t = total in = nozzle inlet abs = absolute value rel = relative value tan = tangential direction axi = axial direction ave = time average

REFERENCES [1] Miller, R. J., Moss, R. W., Ainsworth, R. W., and Harvey, N. W., 2003, ”Wake, Shock, and Potential Field Interactions in a 1.5 Stage Turbine – PartⅠ: Vane-Rotor and Rotor-Vane Interaction,” ASME J. Turbomachinery, 125, pp. 33-39.

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