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Numerical investigation of tip clearance effects on propulsion performance and pressure fluctuation of a pump-jet propulsor
Ocean Engineering 192 (2019) 106500
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Numerical investigation of tip clearance effects on propulsion performance and pressure fluctuation of a pump-jet propulsor Haiting Yu a, b, Zhenguo Zhang a, b, *, Hongxing Hua a, b, ** a b
Institute of Vibration, Shock & Noise, State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai, 200240, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai, 200240, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Pump-jet propulsor Tip clearance Propulsion performance Pressure fluctuation
Rotor-stator interaction and tip clearance between rotor blade tip and shroud of a pump-jet propulsor can induce strong pressure fluctuations and vibrations, which may cause damage of the propulsion system at some severe working conditions. This paper investigates numerically the effects of tip clearance on propulsion performance and pressure fluctuations in a pump-jet propulsor. The shear-stress transport (SST) k–omega turbulence model together with the sliding mesh technique are adopted to simulate the three-dimensional unsteady flow. The accuracy of the computation results is verified through available experimental data of propulsion performance with a tip clearance of 1 mm. The comparison of hydrodynamic performance for pump-jet propulsor with tip clearance of 0, 1 mm, 2 mm and 4 mm shows that the thrust and propulsion efficiency can be reduced signifi cantly due to the increase of tip clearance sizes. Further results indicate that the decrease of thrust and torque of the pump-jet propulsor with tip clearance is mainly caused by the change of time-averaged pressure distributions on the rotor blade in the range of 0.8R–1.0R (where R is the rotor diameter). In addition, a method based on Statistics is applied to reveal the characteristics of pressure fluctuations at all grid nodes on the whole blade surface, and the result shows that the amplitudes of pressure fluctuations increase significantly on the blade surface when the tip clearance size is enlarged, especially on the blade tip, blade root and the leading edge. Moreover, it is found that the tip clearance effects do not change the main frequency of thrust and torque fluctuations, while it has a significant impact on the amplitudes of fluctuations, especially in the higher frequency band.
1. Introduction Over the past few decades, pump-jet propulsor (PJP) has been widely used in underwater vehicles, such as torpedoes and submarines, owing to its high propulsion efficiency and low noise property at medium and high speed. Pump-jet propulsors are typically composed of a stationary vane system (stator), a rotating vane system (rotor), and an axisym metric shroud completely surrounding the rotor and stator with an airfoil section, which prevents the inner noise emitting into the outside flow field (Bruce et al., 1974; Furuya and Chiang, 1986). However, the application of shroud around the rotor usually cause a complex problem of tip-clearance flow, which has a considerable effect on propulsion performance, pressure fluctuation, vibration, and radiated noise of pump-jet propulsor (Liu et al., 2018). The tip-clearance flow problems, such as evolution of tip-leakage
vortex (TLV) and flow mechanism in gap region, have motivated a lot of experimental and numerical studies (Liu et al., 2018; Zhang et al., 2018). Limited by the complexity of flow characteristics and mechanical structures, many experimental studies on TLV and on the associated cavitation problem focus mainly on the fixed hydrofoil with one end fixed and with a gap near the other end (Decaix et al., 2015; Dreyer et al., 2014; Gopalan et al., 2002; Guo et al., 2016; Higashi et al., 2002; Murayama et al., 2005; Watanabe et al., 2009). Based on these experi mental studies, the flow mechanism in gap region, including the flow pattern, the temporal and spatial evolution of various vortices, and the cavitation characteristics, have been well identified. Further studies on the influence of gap flow point out that TLV and its associated cavitation are very sensitive to the tip-clearance size, tip geometry, surface roughness and its operating environment (Dreyer et al., 2014; Gopalan et al., 2002; Katz and Galdo, 1989; Lei et al., 2017; Liu and Tan, 2018;
* Corresponding author. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai, 200240, China. ** Corresponding author. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai, 200240, China. E-mail addresses: [email protected] (Z. Zhang), [email protected] (H. Hua). https://doi.org/10.1016/j.oceaneng.2019.106500 Received 29 March 2019; Received in revised form 25 August 2019; Accepted 28 September 2019 Available online 30 October 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
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Ocean Engineering 192 (2019) 106500
Thapa et al., 2017; Zhang et al., 2015), and part of these research results are summarized in Liu et al. (2018) and Zhang et al. (2018). By using numerical simulation methods, the influence of tip clearance on pressure fluctuations of pumps and turbines has been extensively investigated. Zhang et al. (2017) investigated the characteristics of pressure fluctua tion in a low specific speed mixed-flow pump with different tip clearance sizes by using the Shear Stress Transport (SST) k-omega turbulence model, and concluded that the pressure fluctuation in the whole passage of the pump will increase significantly if the gap increases beyond a certain range. Feng et al. (2016) proposed a new method based on pressure statistics to evaluate pressure fluctuations on all grid nodes inside an axial flow pump. The result indicated that the gap flow in troduces some lower multiples of the rotation frequency to the pressure spectrum and magnifies the pressure fluctuation in the whole impeller region greatly. Further studies based on numerical simulation show that the influence of tip clearance on pressure fluctuations is ubiquitous in pumps and turbines, and this influence varies greatly with the changing of operation conditions (such as non-cavitation condition or cavitation condition), the tip clearance sizes, and especially the geometrical con struction (Liu et al., 2017; Tan et al., 2015; Xu et al., 2017). However, different from pumps and turbines, the length of axialsymmetry shroud of pump-jet propulsor is usually finite, and thus the flow outside the shroud may interact with the internal flow, which leads to more complex flow field and pressure fluctuations. Several studies have been conducted on the performance and flow characteristics of pump-jet propulsors. Furuya and Chiang (1986) combined the stream line curvature method (SCM) with blade to blade flow theory and then proposed a three-dimensional design method for pump-jet propulsor system. By using momentum defect principle for propulsion perfor mance in a wind tunnel facility, the performance and the complex inner flow field of a pump-jet propulsor for high Reynolds number are experimentally studied (Suryanarayana et al., 2010). Ahn and Kwon (2015) numerically studied the propulsion performance and flow fields of a pump-jet propulsor with ring rotor at the blade tip by using an incompressible RANS flow solver. Their results suggested that the addition of tip ring at the blade tip may reduce tip vortex strength without losing any propulsion efficiency even though it could cause a slight drop of thrust. Tip clearance effect on the propulsion performance of pump-jet propulsor have been investigated by using numerical simulation method based on the Reynolds Averaged Navier Stokes (RANS) model, and the simulation results shown that the efficiency of pump-jet propulsor drop sharply with the increase of tip clearance size (Lu et al., 2016). In engineering practice, the vibration of hull caused by the fluctuations of thrust and torque of the propeller usually contributes most to the radiated noise for underwater vehicles compared with other factors. Therefore, reducing the fluctuations of rotor thrust and torque is of great significance to improve the acoustic radiation performance and the operating stability for underwater vehicles. However, above mentioned researches mainly focused on propulsion performance and flow characteristics of pump-jet propulsors, relatively scarce work has been devoted to the effects of tip clearance on the unsteady pressures and forces which have great effects on the operation stability and un derwater radiation noise for pump-jet propulsors. Aiming at overcoming the limitations highlighted above, this paper contributes a numerical investigation on the influence of the tip clear ance on propulsion performance and pressure fluctuations in pump-jet propulsor by using the shear-stress transport (SST) k–omega turbu lence model. Significant contribution is achieved by verifying the computation result through the experimental data of the pump-jet pro pulsor with a tip clearance of 1 mm. Moreover, the unsteady flow fields, hydrodynamic performance, time-averaged pressure distributions, pressure fluctuations and also the fluctuations of thrust and torque in pump-jet propulsor with tip clearance of 0, 1 mm, 2 mm and 4 mm are numerically studied and compared.
2. Research object 2.1. Pump-jet configuration A physical scale model of pump-jet propulsor according to the experiment conditions of the cavitation tunnel is designed in the present work to explore the influence rule of the tip clearance. The basic configuration of the pump-jet propulsor consists of three parts: an inlet guide vane system (Stator), an impeller (Rotor), and an axisymmetric shroud (see in Fig. 1). The length of the shroud is L ¼ 230 mm, with an inlet diameter of D1 ¼ 316 mm and an outlet diameter of D2 ¼ 284 mm. The stators have a right-turn angle of 7.5� , with a chord length of 60 mm. The axial distance between the leading edge of shroud and the leading edge of stator blades is 20 mm. The detailed parameters of the sections of shroud and stator blade are listed in Table 1, where yu and yl are the dimensionless coordinate values of the upper and lower surface for the airfoil in Cartesian coordinate system, respectively. The diameter of the rotor hub is Dh ¼ 74 mm and the outer diameter of the rotor blades is DR ¼ 294 mm, with a tip clearance of 1 mm to the inner wall of shroud. The rotor has five three-dimensional blades, equipped with seven guide vanes upstream the rotor. The axial distance between the rotor and stator is 1.5 times the chord-length of guide vanes. Table 2 gives the detailed parameters of the rotor blade, where LR is the chordwise distance from the leading edge of blade sections to the blade reference line. Usually, the rake and skew of the rotor blade are composed of two parts: the initial rake and skew of the blade reference line, and the rake and skew in the chordwise direction relative to the blade reference line which is given by LR. In the present research object, the initial rake and skew of the blade reference line (which is perpen dicular to the rotor axis) of the rotor blade are all zero at all spanwise sections. Under the design operating condition, the inlet flow velocity is 3 m/s, and the rotor blades rotate at 600 rpm with a speed of 9.72 m/s at the blade tip. The local Reynolds number based on the chord length of blade tip and the inlet flow velocity at the blade tip is 5.8 � 105. To evaluate the influence of tip clearance on propulsion performance and pressure fluctuations in the pump-jet propulsor, tip clearances of 2 mm and 4 mm, along with no tip clearance, are also employed. All the pump-jet propulsors have the same geometric parameters with the pump-jet propulsor shown in Fig. 1. For pump-jet propulsor without tip clearance, its shroud diameter is slightly reduced to eliminate the tip clearance between rotor blade tip and the inner wall of the shroud, and for pump-jet propulsor with tip clearance of 2 mm or 4 mm, the shroud diameter is slightly increased. In this way, the parameters of rotor blades are kept unchanged, and hence the possible influence induced by the variation of rotor parameters can be avoided. 2.2. Meshing The shear-stress transport (SST) k-omega turbulence model (Menter, 1994) combines the advantages of stability of the original k-omega turbulence model and independent of the external boundary k-ε turbu lence model and leads to major improvements in the prediction of adverse pressure gradient flows. According to the existing study by Ji et al. (2010) and Huang et al. (2010), the SST k-omega turbulence model with the sliding mesh technique are employed to simulate the unsteady flows through the pump-jet propulsor with and without tip clearance with the commercial code ANSYS Fluent in the present computations. Considering the rotation of rotor in unsteady flow computation, the whole computation domain is separated into two parts as show in Fig. 2 (a), namely the rotating domain and the stationary domain. The rotating domain is a cylinder around the propeller, and the two bottom surfaces of the cylinder, which act as the interface of rotating domain and sta tionary domain, are located upstream and downstream of the rotor respectively. The stationary domain is a large cylinder that surrounds the geometric model of pump-jet propulsor with a diameter much larger than the shroud. 2
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(a) The stators in shroud
(c) Computational model
(b) Rotor blades
Fig. 1. Geometric model of pump-jet propulsor. Table 1 Parameters of shroud and stator. Shroud
x yu yl x yu yl
0 0 0 0.480 0.091 0.039
0.005 0.009 0.009 0.540 0.083 0.039
0.010 0.017 0.013 0.600 0.074 0.039
0.030 0.030 0.022 0.660 0.065 0.043
0.060 0.048 0.026 0.720 0.052 0.043
0.090 0.061 0.030 0.780 0.035 0.048
0.120 0.074 0.035 0.840 0.013 0.052
0.180 0.087 0.039 0.900 0.013 0.061
0.240 0.096 0.039 0.940 0.035 0.065
0.300 0.100 0.039 0.960 0.043 0.070
0.360 0.100 0.039 0.980 0.057 0.074
0.420 0.096 0.039 1.000 0.065 0.074
Stator
x yu yl x yu yl
0 0 0 0.560 0.028 0.087
0.005 0.008 0.005 0.640 0.012 0.095
0.010 0.012 0.008 0.720 0.012 0.105
0.020 0.018 0.012 0.800 0.040 0.115
0.035 0.023 0.015 0.880 0.077 0.128
0.050 0.028 0.018 0.920 0.098 0.135
0.080 0.037 0.023 0.940 0.108 0.137
0.160 0.048 0.035 0.960 0.118 0.140
0.240 0.055 0.047 0.970 0.125 0.142
0.320 0.053 0.057 0.980 0.130 0.142
0.400 0.050 0.067 0.990 0.135 0.143
0.480 0.042 0.077 1.000 0.140 0.143
areas at the duct wall near the blade tip, the maximum values of yþ is found slightly greater than 1 (about 1.6), but it has little effect on the convergence of the calculation. 20 layers of prismatic cells are placed around the rotor blade surface and the thickness of the first layer is 0.002 mm where the value of yþ is about 1. To evaluate the influence of the grids density on computation results, three sets of meshes (see in Table 3) are selected to conduct the nu merical simulation and the variation of unsteady pressures at some arbitrarily selected monitoring points (see in Fig. 5) on rotor blade surface are compared as shown in Fig. 4 (T is the rotor rotation period). The comparisons for other monitoring points are similar. It is suggested that the grids density of Case 2 has shown good convergence. Even the number of grids increased by 50 percent as in Case 3, no more significant changes of the pressure fluctuation curves can be found at both posi tions. Considering the balance of the accuracy and the computational efficiency, the grids of Case2 are employed in the subsequent computations.
Table 2 Detailed dimensions of the rotor blade. r/R
Pitch (mm)
Chord Length (mm)
Thickness (mm)
LR (mm)
0.25 0.35 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.975 1
403.25 401.90 402.93 405.92 406.87 401.31 383.81 347.34 316.93 298.15 276.09
64.17 68.49 70.37 73.80 76.43 77.74 76.78 71.85 66.38 62.44 56.67
12.78 11.26 10.46 8.79 7.22 5.89 4.91 4.3 4.13 4.08 4.04
29.02 32.968 34.087 34.547 32.11 26.265 16.629 2.604 6.705 12.51 20.721
In general, the convergence of computations and the accuracy of computation results are strongly dependent on the quality of the computational meshes. Both the hexa-structured and hybridunstructured meshes can obtain similar levels of accuracy during nu merical simulation of flow field but the hybrid-unstructured meshes are less suitable for detailed investigations of the flow field (Morgut and Nobile, 2012). Because of the complexity of twisted geometry and the efforts of grids generation, the hexa-structured meshes are combined with the hybrid-unstructured meshes to solve the Navier-Stokes equa tions (Ji et al., 2010; Huang et al., 2010). As show in Figs. 2(b) and Fig.3, the stationary domain is discretized by hexa-structured meshes and the rotating domain is discretized by hybrid-unstructured meshes. Consid ering that the flow fields vary violently near the blade tip, a local mesh refinement scheme has been applied to the mesh generation at blade tip. Because the numerical results of wall pressure fluctuations are very sensitive to the size and quality of near wall meshes, boundary layer elements are applied in the solid surface to get the flow and pressure fields near the wall accurately. The first layer of boundary layers in the duct and stator blade wall is 0.01 mm, with a maximum yþ of approx imately 1 in the stator blades and most areas of the duct wall. In some
2.3. Boundary conditions As stated in Menter (1994) and Huang et al. (2010), the reliability and the convergence speed of the CFD simulation depends tremendously on the definition of boundary conditions, and in this work the parallel algorithm based on a domain decomposition method is adopted to maintain a balance between efficiency and accuracy. The inlet of the computation domain is set as velocity inlet boundary condition where the velocity components are set as axial velocity with no radial and circumferential components, and the initial turbulence intensity is assumed to be 2.83% with the external diameter of the shroud as the characteristic length. In order to reduce the influence of far-field boundary of computation domain, the same velocity boundary condition with the inlet boundary is also applied to the far-field boundary (see Fig. 2(a)). For the outlet boundary, the static pressure outlet boundary condition with a reference pressure of 0 Pa is defined. No slip boundary condition is applied to solid surfaces such as shroud 3
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Ocean Engineering 192 (2019) 106500
10D
INLET
OUTLET
Stationary Domain
6D
6D
Interface
Shroud
Propeller
Stator
(a) Computational domain
(b) Grids of stator blade
Fig. 2. Computational domain of pump-jet propulsor and computational grids of stator blade.
(a) Grids of rotor blade for PJP without tip clearance
(b) Grids of rotor blade for PJP with tip clearance
Fig. 3. Computational grids of rotor blades.
wall and rotor blades, in this way all components of velocity and the normal gradient of static pressure would vanish on solid surfaces. In addition, the shroud inner wall around the rotor blades in the rotating domain is given a reverse rotational speed (which is equal to the rota tional speed of the rotational domain) relative to the adjacent cell zone to keep the shroud wall absolutely stationary in the absolute coordinate system. The overlapped surfaces (the interface of the rotating domain and the stationary domain) between the stationary and the rotating domain are defined as sliding interfaces in order to exchange the flow field information between the stationary and the rotating domains simultaneously. The convergence criterion is set as follows: the continuity and
Table 3 Three sets of meshes for numerical calculation. Grid numbers
Case 1 Case 2 Case 3
Maximum grid size in Rotating domain
Stationary domain
Rotating domain
5.64 million
8.95 million
12 mm
10.85 million
20.45 million
8 mm
15.26 million
30.65 million
5 mm
4
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Ocean Engineering 192 (2019) 106500
1.8
104 Case1 Case2 Case3
PS-4 (r/R=0.9)
1.6 1.5 1.4 1.3 1.2 1.0 0.0
0.4
0.8
t/T
1.2
1.6
-2.0
SS-4 (r/R=0.9)
-4.0
SS-2 (r/R=0.9)
0.0
2.0
-1.0
Case1 Case2 Case3
PS-4 (r/R=0.6)
0.4
0.8
t/T
1.2
1.6
PS-2 (r/R=0.6)
2.0
103 Case1 Case2 Case3
-2.0 Pressure /Pa
Pressure /Pa
0.0
-8.0
103
9.3 9.0 8.7 8.4 8.1 7.8 7.5 7.2 6.9 6.6 0.0
Case1 Case2 Case3
-6.0
PS-2 (r/R=0.9)
1.1
103
2.0 Pressure /Pa
Pressure /Pa
1.7
4.0
-3.0
SS-4 (r/R=0.6)
-4.0 -5.0 -6.0
SS-2 (r/R=0.6)
-7.0 0.4
0.8
t/T
1.2
1.6
2.0
0.0
0.4
0.8
t/T
1.2
1.6
2.0
Fig. 4. Comparison of unsteady pressure variations under different grid density.
Fig. 5. Position of monitoring points at six selected spanwise sections.
velocity residual error are less than 10 4, the energy residual error is less than 10 7, and the relative error of mass flow rate at the inlet and outlet of pump-jet propulsor is less than 0.1%. In the present work, the convergence usually occurs after 3000 steps in steady calculation, and for unsteady calculation, due to the initial condition is the result of steady calculation, the convergence usually occurs after one rotor rev olution. After the process converges, the continuity residual error is less than 10 4, the velocity residual error is close to 10 6, the energy re sidual error is less than 10 7, and the relative error of mass flow rate is about 0.1%.
addition, the monitoring points are equidistant in the chordwise direc tion in every spanwise sections. The points LE and TE represent the monitoring points at leading edge and trailing edge, respectively. 3. Experimental verification of the numerical method To assure the reliability and efficiency of the developed numerical model, experimental data available for the pump-jet propulsor with a tip clearance of 1 mm (0.676%R, where R is the radius of the rotor) is used to validate the numerical results. Specially, the propulsion performance is tested in the cavitation tunnel at Shanghai Jiao Tong University (SJTU). The cavitation tunnel is composed of a closed loop with a rectangular test section of 1 m � 1 m � 6.1m as inner dimensions. A metal screen is installed in the contraction section upstream of the test section to avoid flow rotation and to reduce the turbulence intensity. In general, the velocity non-uniformity is less than 1%, and the turbulence intensity at the inlet of test section is less than 0.5%. The maximum free stream velocity at the inlet of test section, V∞, is 15 m/s. The propeller dynamometer is installed downstream the pump-jet propulsor model,
2.4. Monitoring points setting To obtain the characteristic of fluctuating pressures on the blade surface, six sets of monitoring points are set in the rotor blade. Fig. 5 shows the locations of monitoring points in the rotor blade surfaces. For each spanwise section, points from SS-1 to SS-5 are the points on suction surface listed from blade leading edge to trailing edge, and on the pressure surface, there are corresponding points PS-1 to PS-5. In 5
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Ocean Engineering 192 (2019) 106500
and its maximum revolution speed is 1200 revolutions per minute (rpm). Different from the conventional experiments of propellers, the pro pulsion performance experiment for pump-jet propulsor includes the tests of forces acting on both the rotor and the stator as well as the shroud. The rotor is fixed on the horizontal axis of propeller dyna mometer (see in Fig. 6), and the horizontal thrust and torque acted on the rotor blades can be directly measured in real time. In order to measure the resistance acted on stator and shroud, a three-component balance was installed in the equipment box above the experimental model by a high-strength stainless steel rod. By using a unified measuring control system, the required operating condition could be satisfied easily and the forces acted on the shroud (including stator) and rotor could be measured simultaneously by the three-component bal ance and the propeller dynamometer. For the convenience of expression, the non-dimensional thrusts and torque can be defined: KTR ¼ KT ¼
TR
ρn2 D4
; KTS ¼
TS
ρn2 D4
TR þ TS ¼ KTR þ KTS ρn2 D4
10KQ ¼
are nearly the same for pump-jet propulsor with and without tip clear ance. Unlike conventional highly skewed propellers (Tian et al., 2017) which can only maintain high efficiency in a small range of advance ratio near the design operating point, the pump-jet propulsor can maintain a very high propulsion efficiency at a large range of advance ratio for both cases with or without tip clearance. Compared with the case of Gap ¼ 0, the rotor thrust of the case of Gap ¼ 1 mm dropped by 6%–9%, and the rotor torque dropped by 3%–5%. Consequently, the propulsion efficiency dropped by about 5.8% at the design operating point. For the case of Gap ¼ 2 mm and Gap ¼ 4 mm, the decrease of rotor thrust and torque is more significant, and eventually lead to a more significant reduction of propulsion efficiency relative to case of Gap ¼ 1 mm. However, for the resistance of shroud and stator blades, the influence of tip clearance is relatively small. The existence of tip clearance leads to a significant decrease in rotor thrust and torque, but the propulsion efficiency almost remains unchanged at lower advance ratio and decreases significantly at high advance ratio. Furthermore, the four cases all reach the maximum efficiency at about J ¼ 1.0, which indicate that the existence of tip clearance and its sizes do not change the advance ratio corresponding to the maximum efficiency point of the propeller.
(1)
10QR
4.2. Time-averaged pressure distributions on the rotor blade
ρn2 D5
where J ¼ VnDA is the advance ratio, and VA is the inlet velocity. The comparison of numerical and experimental results for propul sion performance is shown in Fig. 7. It is observed that the maximum numerical error of rotor torque relative to the experimental result is 2.7%, and those for the thrust on rotor and the thrust on stator blades and shroud are respectively, 4.6% and 3.2%. Relatively small tolerable error illustrates the consistency between the simulation results and the experimental results, which demonstrates feasibility of the present nu merical calculation.
To investigate the influence of tip clearance on time-averaged pres sure distributions and pressure fluctuations on rotor blades, all unsteady computation results presented are obtained at the design operating point. The total time for the unsteady computation is set for the rotor to rotate 6 revolutions after the convergence of computation, and the total flow time is 0.6 s. The time step is Δt ¼ 2*10 5, which means that there are 5000 sampling points for every monitoring points for every revo lution of the rotor. The computation results of propulsion performance show that the existence of tip clearance mainly changes the thrust and torque of the rotor blades, and the resistance on shroud and stator blades is almost unchanged. Therefore, the influence of tip clearance on propulsion performance can be explained by the analysis of circumferentiallyaveraged (time-averaged) pressure distributions (eq. (4)) on rotor blade surface. Fig. 9 shows the comparison of dimensionless circumferentially-averaged pressure distributions (one revolution) on rotor blade surface at selected spanwise sections. The dimensionless pressure is defined as:
4. Results and discussions
Cp ¼
where TR is the thrust of rotor; TS is the thrust (resistance in high advance ratio) of shroud and stator; QR is the rotor toque; and ρ, n and D represent the fluid density, the revolution speed and the rotor diameter respectively. And then the efficiency of pump-jet propulsor can be defined:
η0 ¼
ðTR þ TS ÞVA J KT ¼ 2π KQ 2πnQ
(2)
4.1. Propulsion performance
P 1 2
P∞
ρV 2A
(3)
As can be seen from the figure, the circumferentially-averaged pressure distributions on the rotor blade are almost the same in the range of r � 0.8R for the four cases. However, in the range of r > 0.8R, the pressures on pressure surface for pump-jet propulsor with tip clearance are much lower than that on pressure surface for pump-jet propulsor without tip clearance, and the pressures near the blade
Based on the same numerical method, Fig. 8 shows the comparison of propulsion performance for pump-jet propulsor with tip clearance of 0, 1 mm, 2 mm and 4 mm. As can be seen, the variation trends of perfor mance curves, such as dimensionless forces and propulsion efficiency,
Fig. 6. Pump-jet propulsor model fixed on the Propeller Dynamometer in cavitation tunnel. 6
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1.2
Experiments
Computed
1.0 10KQ
KT,10KQ,
0
0.8 0.6 KTR
0.4
KT
0.2 0.0
KTS
-0.20.2
0.4
0.6
0.8
1.0
1.2
J=V/nD Fig. 7. Comparison between the numerical and experimental results.
Gap=0 Gap=1mm Gap=2mm Gap=4mm
1.2 1.0
1.0
10KQ
0.8
0
0.6
0.6
0.4
KT-Rotor
0
0.4
0.2 0.0 -0.2 0.2
10KQ
KT,10KQ,
KT,10KQ
0.8
Gap=0 Gap=1 mm Gap=2 mm Gap=4 mm
1.2
KT-Shroud & Stator
0.4
0.6
0.8 J=V/nD
1.0
KT
0.2 0.2
1.2
0.4
0.6
0.8
J=V/nD
1.0
1.2
Fig. 8. Comparison of propulsion performance for pump-jet propulsor with and without tip clearance.
trailing edge on suction surface for pump-jet propulsor with tip clear ance are much lower than that on suction surface for pump-jet propulsor without tip clearance, but overall, the larger the tip clearance size, the smaller the pressure difference between pressure surface and suction surface of the rotor blade. It shows that the decrease of thrust and torque of the pump-jet propulsor with tip clearance is mainly caused by the change of circumferentially-averaged pressure distributions on the rotor blade in the range of 0.8R–1.0R. Fig. 10 shows the instantaneous streamline distributions at the blade tip, and Fig. 11 shows the instantaneous pressure distributions near the blade tip. For the case of Gap ¼ 0, the main flow is free from the influ ence of gap flow, therefore the flow near the blade tip is relatively stable and the streamline distribution is uniform along the blade surface. For the case of Gap ¼ 1 mm,2 mm and 4 mm, because of the interaction of the gap inverse flow and the main flow, the flow near the blade tip is chaotic and partial streamlines shed from the pressure surface to the suction surface near the leading edge of the blade tip and mingle with each other near the blade trailing edge. It shows that the fluid near the
blade tip flows from the pressure surface to the suction surface under the pressure difference between pressure surface and suction surface, thus neutralizing the pressures on both sides of the rotor blade, which in turn leads to the significant decrease of pressure difference between pressure surface and suction surface (see in Fig. 11). From Fig. 9(a–c) and Fig.10, the larger the tip clearance size, the more significant decrease of pres sure difference between the two sides of the rotor blade (see in Fig. 9 (a–b)). In addition, for pump-jet propulsor with tip clearance, the negative pressure is mainly observed at the core area of tip-leakage vortex which has an angle of about 15� with the chord direction of blade tip, and the larger the tip clearance size, the lower the negative pressure at the core area of tip-leakage vortex. As can be seen from Fig. 9 (a), the influence of tip leakage vortices on the circumferentiallyaveraged pressure distributions (especially the circumferentiallyaveraged pressures on suction surface) near the trailing edge of the blade tip is very significant, mainly because the tip-leakage vortex is fully formed in the suction surface near the trailing edge of the blade tip.
7
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5
(a) r/R=1.0
5
Gap=0 Gap=1mm Gap=2mm Gap=4mm
3
(b) r/R=0.95
Gap=0 Gap=1mm Gap=2mm Gap=4mm
3 1
Cp
Cp
1 -1
-1
-3
-3
-5
0
5
0.2
0.4
0.6 x/C
(c) r/R=0.9
0.8
-5
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Gap=0 Gap=1mm Gap=2mm Gap=4mm
3
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(d) r/R=0.8
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3 1
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Cp
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-3
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(e) r/R=0.7
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(f) r/R=0.6
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Fig. 9. Comparison of circumferentially-averaged pressure distributions on rotor blade surface at selected spanwise sections, J ¼ 1.0.
(a)Gap=0
(b)Gap=1mm
(c)Gap=2mm
Fig. 10. Instantaneous streamline distributions at the blade tip, J ¼ 1.0. 8
(d)Gap=4mm
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Ocean Engineering 192 (2019) 106500
(a)Gap=0
(b)Gap=1mm
(c)Gap=2mm
(d)Gap=4mm
Fig. 11. Instantaneous pressure distributions on the sections perpendicular to the chord direction of blade tip, J ¼ 1.0.
4.3. Influence of tip clearance on pressure fluctuations
is the product of stator numbers NS and the rotor revolution frequency fn, are the main frequency of pressure fluctuations on rotor blade for both propulsors. Although tip clearance significantly increases the ampli tudes of pressure fluctuations on the rotor blade surface, it does not change the main frequency of these pressure fluctuations. The spectrum analysis only gives the local spectrum characteristics of pressure fluctuations at selected monitoring points. To reveal the characteristics of pressure fluctuations on the whole blade surface, the statistical method, which can intuitively show the fluctuation of instantaneous pressures at each grid node, is applied to the analysis in present work. In general, the unsteady pressure at each grid node con sists of two parts, namely the circumferentially-averaged part depending on the location of grid node (eq. (4)), and the fluctuating part changing periodically with the relative position of grid node during the rotation of rotor (eq. (5))(Feng et al., 2016).
Fig. 12 shows the frequency spectrogram of pressure fluctuations at representative monitoring points on blade surface from r/R ¼ 0.6 to r/ R ¼ 1.0 for the case of Gap ¼ 0 and Gap ¼ 1 mm. The monitoring points located on the position as given in Fig. 4. It is observed that the am plitudes of pressure fluctuations for pump-jet propulsor with tip clear ance of 1 mm increased significantly at most of the monitoring points compared with the pump-jet propulsor without tip clearance, especially at the monitoring points located on the leading edge. Furthermore, points LE at all selected spanwise sections of pump-jet propulsor, with or without tip clearance, have the biggest amplitudes of fluctuating pres sures, and it is mainly due to that the points located at the leading edge are affected most significantly by the interaction of rotor and stator. It suggests that although the pre-swirl stators are generally considered to make the incoming flow more uniform, they are also an inducing factors of non-uniform flow field for the inflow of rotor blades. The stator blade passing frequency fs (relative to the rotating coordinate system), which
pð! r Þ¼
9
Nt 1 X pð! r ; i ⋅ ΔtÞ Nt i¼1
(4)
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Ocean Engineering 192 (2019) 106500
(a) Monitoring points on pressure surface at r/R=1.0
(b) Monitoring points on suction surface at r/R=1.0
(c) Monitoring points on pressure surface at r/R=0.80
(d) Monitoring points on suction surface at r/R=0.80
(e) Monitoring points on pressure surface at r/R=0.60
(f) Monitoring points on suction surface at r/R=0.60
Fig. 12. Comparison of frequency spectra of pressure fluctuations on representative monitoring points, J ¼ 1.0.
b p ð! r ; tÞ ¼ pð! r ; tÞ
pð! rÞ
Hence, the standard deviation of fluctuating pressure can be defined as Eq. (6) to evaluate the amount of pressure variation for a serial of fluctuating pressures at each grid node.
(5)
where ! r ¼ ðx; r; θÞ is the coordinates of grid nodes in rotating coordi nate system, Nt is the number of time steps for one revolution, and Δt is the time step used for the computation. 10
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Ocean Engineering 192 (2019) 106500
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Nt u Nt u1X u1 X b p 2 ð! σ ð! r Þ¼t ðpð! r ; i⋅ΔtÞ pð! r ÞÞ2 ¼ t r ; i⋅ΔtÞ Nt i¼1 Nt i¼1
(6)
KTR ¼
Based on the method shown in eq. (6), Fig. 13 and Fig. 14 give the contour of standard deviation of pressure fluctuations on the whole rotor blade surface colored with the standard deviation of fluctuating pres sures. The figures plot a detailed information of pressure variations on the rotor blade. It is obvious that the standard deviation σ generally increases with the increase of tip clearance sizes on most areas of the blade surface, which illustrates that despite the influence of tip clear ance on static pressure distributions is mainly focused on the blade tip region, the influence of tip clearance on pressure fluctuations works throughout the rotor blade surface. In general, for the four cases with advance ratio of J ¼ 1.0, the standard deviation σ on blade tip and blade leading edge is much larger than that on other areas, and the standard deviation σ is larger on the suction surface than that on the pressure surface. Moreover, the standard deviation σ on the blade root increase sharply with the increase of tip clearance, mainly because the increasing of tip clearance sizes reduces the effective velocities at the rotor plane (According to the calculation result of time-averaged pressure distri butions on the rotor blade, when the tip clearance size increases, the blade loads will decrease, and in turn leads to the decrease of induced velocities at the rotor plane) and thus promotes the evolution of blade root vortices. The integral of unsteady pressures along the blade surface is the blade thrust and torque of the rotor. The unsteady thrust and torque usually consist of two parts, namely the time-averaged part (eq. (7)) depending on the operating condition, and the fluctuating part (eq. (8)) changing periodically with the rotation of rotor blade. The engineering practice shows that the time-averaged parts of thrust and torque are presented as the propulsion performance of a pump-jet propeller, and the fluctuation parts of thrust and torque are usually the main factor inducing the hull vibration for modern underwater vehicles.
Nt Nt 1 X 1 X Fði ⋅ ΔtÞ ; KQ ¼ Mði ⋅ ΔtÞ Nt i¼1 Nt i¼1
~ ¼ FðtÞ Fx
~ ¼ MðtÞ KTR ; Mx
KQ
(7) (8)
where, F and M are the dimensionless unsteady thrust and torque of the rotor blade, respectively. According to eqs. (7) and (8), the fluctuation of thrust and torque for a single blade of rotor and their frequency spectra are shown in Figs. 13 and 14. Both the thrust fluctuation amplitudes and torque fluctuation amplitudes increase significantly with the increase the tip clearance sizes, and the stator blade passing frequency fs is the main frequency. For the case of Gap ¼ 0, the fluctuation of thrust and torque is relatively regular, and the amplitude of the third-order frequency is significantly less than that of the first-order and second-order frequencies. However, for pump-jet propulsor with tip clearance, the amplitude of the thirdorder frequency increases significantly with the increase of tip clear ance sizes, and the amplitude of the third-order frequency is approxi mately equal to the amplitude of the second-order frequency for case of Gap ¼ 2 mm and Gap ¼ 4 mm. It shows that the tip clearance effects do not change the main frequency of thrust and torque fluctuations, but it has a significant impact on the amplitudes of fluctuations, especially in the higher frequency band (f � 3fs). 5. Conclusions This study presents numerical investigation of tip clearance effects on propulsion performance and pressure fluctuations of a pump-jet propulsor. The SST k–omega turbulence model based on RANS method and the sliding mesh technique are employed to compute the unsteady flow at the design operating point. Propulsion performance experiment for pump-jet propulsor with tip clearance of 1 mm is also
(a) Gap=0
(b) Gap=1mm
(c) Gap=2mm
(d) Gap=4mm
Fig. 13. Comparison of pressure fluctuation contours for rotor pressure surface, J ¼ 1.0. 11
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Ocean Engineering 192 (2019) 106500
(a) Gap=0
(b) Gap=1mm
(c) Gap=2mm
(b) Gap=4mm
Fig. 14. Comparison of pressure fluctuation contours for rotor suction surface, J ¼ 1.0.
carried out to validate the numerical computation method. The propulsion performance of pump-jet propulsor with tip clear ance of 1 mm, 2 mm and 4 mm are computed and compared with that without tip clearance. The results reveal that although the tip clearance size is very small with respect to the rotor diameter (only 0.676–2.7 percent of the rotor radius), the reduction of rotor thrust and propulsion efficiency caused by the tip clearance is very significant, and especially when the tip clearance size is enlarged, these effects will be more sig nificant. For the case of Gap ¼ 1 mm, due to the influence of tip clear ance, the rotor thrust and torque dropped by 6%–9% and 3%–5% respectively compared with the case of Gap ¼ 0, and consequently, the propulsion efficiency reduced by about 5.8% at the design operating point. The decrease of thrust and torque of the pump-jet propulsor with tip clearance is mainly caused by the change of time-averaged pressure distributions on the rotor blade in the range of 0.8R–1.0R, and the larger the tip clearance size, the more significant decrease of pressure differ ence between the two sides of the rotor blade. The influence of tip clearance on pressure fluctuations can be observed throughout the whole rotor blade surface. The amplitudes of pressure fluctuations increase significantly on the rotor blade surface when the tip clearance is enlarged, especially on the blade tip, blade root and the leading edge, but the main frequency of the pressure fluctua tions is not affected. In addition, for thrust and torque fluctuations, the tip clearance has magnified greatly the amplitudes of force fluctuations, especially in the higher frequency band. The amplitudes of thrust and torque fluctuations increase along with the increase of tip clearance sizes, and the amplitudes corresponding to the third-order, fourth-order, and even the fifth-order frequencies are obviously increasing more sig nificant. If the tip clearance effects can be eliminated or suppressed effectively, the pressure and force fluctuations will decrease significantly.
Acknowledgments The authors gratefully acknowledge the financial support provided by the National Natural Science Foundation of China (NSFC, Grant No. 51505281). References Ahn, S.J., Kwon, O.J., 2015. Numerical investigation of a pump-jet with ring rotor using an unstructured mesh technique. J. Mech. Sci. Technol. 29 (7), 2897–2904. Bruce, E.P., Gearhart, W., Ross, J., Treaster, A., 1974. The design of pumpjets for hydrodynamic propulsion. Fluid Mech. Acoust. Des. Thrbomachinery Part I 795–839. NASA SP-304. Decaix, J., Balarac, G., Dreyer, M., Farhat, M., Münch, C., 2015. RANS and LES computations of the tip-leakage vortex for different gap widths. J. Turbul. 16 (4), 309–341. Dreyer, M., Decaix, J., Münch-Allign� e, C., Farhat, M., 2014. Mind the gap: a new insight into the tip leakage vortex using stereo-PIV. Exp. Fluid 55 (11), 1–13, 1849. Feng, J., Luo, X., Guo, P., Wu, G., 2016. Influence of tip clearance on pressure fluctuations in an axial flow pump. J. Mech. Sci. Technol. 30 (4), 1603–1610. Furuya, O., Chiang, W.L., 1986. A New Pumpjet Design Theory. Report No. TC-3037. Tetra Tech, Inc., Pasadena, CA. Gopalan, S., Katz, J., Liu, H.L., 2002. Effect of gap size on tip leakage cavitation inception, associated noise and flow structure. J. Fluids Eng. 124 (4), 994–1004. Guo, Q., Zhou, L., Wang, Z., 2016. Numerical evaluation of the clearance geometries effect on the flow field and performance of a hydrofoil. Renew. Energy 99, 390–397. Higashi, S., Yoshida, Y., Tsujimoto, Y., 2002. Tip leakage vortex cavitation from the tip clearance of a single hydrofoil. JSME Int. J. - Ser. B Fluids Therm. Eng. 45 (3), 662–671. Huang, S., Mohamad, A., Nandakumar, K., Ruan, Z.Y., 2010. Numerical simulation of unsteady flow in a multistage centrifugal pump using sliding mesh technique. Prog. Comput. Fluid Dyn. Int. J. 10 (4), 239–245. Ji, B., Luo, X.-w., Wu, Y.-l., Liu, S.-h., Xu, H.-y., Oshima, A., 2010. Numerical investigation of unsteady cavitating turbulent flow around a full scale marine propeller. J. Hydrodyn. Ser. B 22 (5, Suppl. 1), 747–752. Katz, J., Galdo, J.B., 1989. Effect of roughness on rollup of tip vortices on a rectangular hydrofoil. J. Aircr. 26 (3), 247–253. Lei, T., Zhifeng, X., Yabin, L., Yue, H., Yun, X., 2017. Influence of T-shape tip clearance on performance of a mixed-flow pump. In: Proceedings of the Institution of Mechanical Engineers, Part A. Journal of Power and Energy, vol. 232, pp. 386–396, 4.
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Ocean Engineering 192 (2019) 106500 Tian, J., Zhang, Z., Ni, Z., Hua, H., 2017. Flow-induced vibration analysis of elastic propellers in a cyclic inflow: an experimental and numerical study. Appl. Ocean Res. 65, 47–59. Thapa, B.S., Dahlhaug, O.G., Thapa, B., 2017. Sediment erosion induced leakage flow from guide vane clearance gap in a low specific speed francis turbine. Renew. Energy 107, 253–261. Watanabe, S., Suefuji, T., Ikeda, A., Furukawa, A., Nishii, K., 2009. Experimental study of unsteady cavitating flow around flat plate hydrofoil with/without tip clearance. In: Proceedings of the ASME Fluids Engineering Division Summer Conference, Vail, Colorado, United States, 2, pp. 387–391, 5. Xu, Y., Tan, L., Liu, Y., Cao, S., 2017. Pressure fluctuation and flow pattern of a mixedflow pump with different blade tip clearances under cavitation condition. Adv. Mech. Eng. 9 (4), 1–12. Zhang, D., Shi, W., Pan, D., Dubuisson, M., 2015. Numerical and experimental investigation of tip leakage vortex cavitation patterns and mechanisms in an axial flow pump. J. Fluids Eng. 137 (12), 1–14, 121103. Zhang, W., Yu, Z., Zhu, B., 2017. Influence of tip clearance on pressure fluctuation in low specific speed mixed-flow pump passage. Energies 10 (2), 1–16, 148. Zhang, Y., Liu, K., Xian, H., Du, X., 2018. A review of methods for vortex identification in hydroturbines. Renew. Sustain. Energy Rev. 81, 1269–1285.
Liu, Y., Tan, L., 2018. Method of C groove on vortex suppression and energy performance improvement for a NACA0009 hydrofoil with tip clearance in tidal energy. Energy 155, 448–461. Liu, Y., Tan, L., Hao, Y., Xu, Y., 2017. Energy performance and flow patterns of a mixedflow pump with different tip clearance sizes. Energies 10 (2), 1–15, 191. Liu, Y., Tan, L., Wang, B., 2018. A review of tip clearance in propeller, pump and turbine. Energies 11 (9), 1–30, 2202. Lu, L., Pan, G., Wei, J., Pan, Y., 2016. Numerical simulation of tip clearance impact on a pumpjet propulsor. Int. J. Nav. Archit. Ocean Eng. 8 (3), 219–227. Menter, F.R., 1994. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J. 32 (8), 1598–1605. Morgut, M., Nobile, E., 2012. Influence of grid type and turbulence model on the numerical prediction of the flow around marine propellers working in uniform inflow. Ocean Eng. 42, 26–34. Murayama, M., Yoshida, Y., Tsujimoto, Y., 2005. Unsteady tip leakage vortex cavitation originating from the tip clearance of an oscillating hydrofoil. J. Fluids Eng. 128 (3), 421–429. Suryanarayana, C., Satyanarayana, B., Ramji, K., Saiju, A., 2010. Experimental evaluation of pumpjet propulsor for an axi-symmetric body in wind tunnel. Int. J. Nav. Archit. Ocean Eng. 2 (1), 24–33. Tan, L., Zhu, B., Wang, Y., CAO, S., Gui, S., 2015. Numerical study on characteristics of unsteady flow in a centrifugal pump volute at partial load condition. Eng. Comput. 32 (6), 1549–1566.
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Numerical investigation of tip clearance effects on propulsion performance and pressure fluctuation of a pump-jet propulsor Haiting Yu a, b, Zhenguo Zhang a, b, *, Hongxing Hua a, b, ** a b
Institute of Vibration, Shock & Noise, State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai, 200240, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai, 200240, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Pump-jet propulsor Tip clearance Propulsion performance Pressure fluctuation
Rotor-stator interaction and tip clearance between rotor blade tip and shroud of a pump-jet propulsor can induce strong pressure fluctuations and vibrations, which may cause damage of the propulsion system at some severe working conditions. This paper investigates numerically the effects of tip clearance on propulsion performance and pressure fluctuations in a pump-jet propulsor. The shear-stress transport (SST) k–omega turbulence model together with the sliding mesh technique are adopted to simulate the three-dimensional unsteady flow. The accuracy of the computation results is verified through available experimental data of propulsion performance with a tip clearance of 1 mm. The comparison of hydrodynamic performance for pump-jet propulsor with tip clearance of 0, 1 mm, 2 mm and 4 mm shows that the thrust and propulsion efficiency can be reduced signifi cantly due to the increase of tip clearance sizes. Further results indicate that the decrease of thrust and torque of the pump-jet propulsor with tip clearance is mainly caused by the change of time-averaged pressure distributions on the rotor blade in the range of 0.8R–1.0R (where R is the rotor diameter). In addition, a method based on Statistics is applied to reveal the characteristics of pressure fluctuations at all grid nodes on the whole blade surface, and the result shows that the amplitudes of pressure fluctuations increase significantly on the blade surface when the tip clearance size is enlarged, especially on the blade tip, blade root and the leading edge. Moreover, it is found that the tip clearance effects do not change the main frequency of thrust and torque fluctuations, while it has a significant impact on the amplitudes of fluctuations, especially in the higher frequency band.
1. Introduction Over the past few decades, pump-jet propulsor (PJP) has been widely used in underwater vehicles, such as torpedoes and submarines, owing to its high propulsion efficiency and low noise property at medium and high speed. Pump-jet propulsors are typically composed of a stationary vane system (stator), a rotating vane system (rotor), and an axisym metric shroud completely surrounding the rotor and stator with an airfoil section, which prevents the inner noise emitting into the outside flow field (Bruce et al., 1974; Furuya and Chiang, 1986). However, the application of shroud around the rotor usually cause a complex problem of tip-clearance flow, which has a considerable effect on propulsion performance, pressure fluctuation, vibration, and radiated noise of pump-jet propulsor (Liu et al., 2018). The tip-clearance flow problems, such as evolution of tip-leakage
vortex (TLV) and flow mechanism in gap region, have motivated a lot of experimental and numerical studies (Liu et al., 2018; Zhang et al., 2018). Limited by the complexity of flow characteristics and mechanical structures, many experimental studies on TLV and on the associated cavitation problem focus mainly on the fixed hydrofoil with one end fixed and with a gap near the other end (Decaix et al., 2015; Dreyer et al., 2014; Gopalan et al., 2002; Guo et al., 2016; Higashi et al., 2002; Murayama et al., 2005; Watanabe et al., 2009). Based on these experi mental studies, the flow mechanism in gap region, including the flow pattern, the temporal and spatial evolution of various vortices, and the cavitation characteristics, have been well identified. Further studies on the influence of gap flow point out that TLV and its associated cavitation are very sensitive to the tip-clearance size, tip geometry, surface roughness and its operating environment (Dreyer et al., 2014; Gopalan et al., 2002; Katz and Galdo, 1989; Lei et al., 2017; Liu and Tan, 2018;
* Corresponding author. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai, 200240, China. ** Corresponding author. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai, 200240, China. E-mail addresses: [email protected] (Z. Zhang), [email protected] (H. Hua). https://doi.org/10.1016/j.oceaneng.2019.106500 Received 29 March 2019; Received in revised form 25 August 2019; Accepted 28 September 2019 Available online 30 October 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
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Thapa et al., 2017; Zhang et al., 2015), and part of these research results are summarized in Liu et al. (2018) and Zhang et al. (2018). By using numerical simulation methods, the influence of tip clearance on pressure fluctuations of pumps and turbines has been extensively investigated. Zhang et al. (2017) investigated the characteristics of pressure fluctua tion in a low specific speed mixed-flow pump with different tip clearance sizes by using the Shear Stress Transport (SST) k-omega turbulence model, and concluded that the pressure fluctuation in the whole passage of the pump will increase significantly if the gap increases beyond a certain range. Feng et al. (2016) proposed a new method based on pressure statistics to evaluate pressure fluctuations on all grid nodes inside an axial flow pump. The result indicated that the gap flow in troduces some lower multiples of the rotation frequency to the pressure spectrum and magnifies the pressure fluctuation in the whole impeller region greatly. Further studies based on numerical simulation show that the influence of tip clearance on pressure fluctuations is ubiquitous in pumps and turbines, and this influence varies greatly with the changing of operation conditions (such as non-cavitation condition or cavitation condition), the tip clearance sizes, and especially the geometrical con struction (Liu et al., 2017; Tan et al., 2015; Xu et al., 2017). However, different from pumps and turbines, the length of axialsymmetry shroud of pump-jet propulsor is usually finite, and thus the flow outside the shroud may interact with the internal flow, which leads to more complex flow field and pressure fluctuations. Several studies have been conducted on the performance and flow characteristics of pump-jet propulsors. Furuya and Chiang (1986) combined the stream line curvature method (SCM) with blade to blade flow theory and then proposed a three-dimensional design method for pump-jet propulsor system. By using momentum defect principle for propulsion perfor mance in a wind tunnel facility, the performance and the complex inner flow field of a pump-jet propulsor for high Reynolds number are experimentally studied (Suryanarayana et al., 2010). Ahn and Kwon (2015) numerically studied the propulsion performance and flow fields of a pump-jet propulsor with ring rotor at the blade tip by using an incompressible RANS flow solver. Their results suggested that the addition of tip ring at the blade tip may reduce tip vortex strength without losing any propulsion efficiency even though it could cause a slight drop of thrust. Tip clearance effect on the propulsion performance of pump-jet propulsor have been investigated by using numerical simulation method based on the Reynolds Averaged Navier Stokes (RANS) model, and the simulation results shown that the efficiency of pump-jet propulsor drop sharply with the increase of tip clearance size (Lu et al., 2016). In engineering practice, the vibration of hull caused by the fluctuations of thrust and torque of the propeller usually contributes most to the radiated noise for underwater vehicles compared with other factors. Therefore, reducing the fluctuations of rotor thrust and torque is of great significance to improve the acoustic radiation performance and the operating stability for underwater vehicles. However, above mentioned researches mainly focused on propulsion performance and flow characteristics of pump-jet propulsors, relatively scarce work has been devoted to the effects of tip clearance on the unsteady pressures and forces which have great effects on the operation stability and un derwater radiation noise for pump-jet propulsors. Aiming at overcoming the limitations highlighted above, this paper contributes a numerical investigation on the influence of the tip clear ance on propulsion performance and pressure fluctuations in pump-jet propulsor by using the shear-stress transport (SST) k–omega turbu lence model. Significant contribution is achieved by verifying the computation result through the experimental data of the pump-jet pro pulsor with a tip clearance of 1 mm. Moreover, the unsteady flow fields, hydrodynamic performance, time-averaged pressure distributions, pressure fluctuations and also the fluctuations of thrust and torque in pump-jet propulsor with tip clearance of 0, 1 mm, 2 mm and 4 mm are numerically studied and compared.
2. Research object 2.1. Pump-jet configuration A physical scale model of pump-jet propulsor according to the experiment conditions of the cavitation tunnel is designed in the present work to explore the influence rule of the tip clearance. The basic configuration of the pump-jet propulsor consists of three parts: an inlet guide vane system (Stator), an impeller (Rotor), and an axisymmetric shroud (see in Fig. 1). The length of the shroud is L ¼ 230 mm, with an inlet diameter of D1 ¼ 316 mm and an outlet diameter of D2 ¼ 284 mm. The stators have a right-turn angle of 7.5� , with a chord length of 60 mm. The axial distance between the leading edge of shroud and the leading edge of stator blades is 20 mm. The detailed parameters of the sections of shroud and stator blade are listed in Table 1, where yu and yl are the dimensionless coordinate values of the upper and lower surface for the airfoil in Cartesian coordinate system, respectively. The diameter of the rotor hub is Dh ¼ 74 mm and the outer diameter of the rotor blades is DR ¼ 294 mm, with a tip clearance of 1 mm to the inner wall of shroud. The rotor has five three-dimensional blades, equipped with seven guide vanes upstream the rotor. The axial distance between the rotor and stator is 1.5 times the chord-length of guide vanes. Table 2 gives the detailed parameters of the rotor blade, where LR is the chordwise distance from the leading edge of blade sections to the blade reference line. Usually, the rake and skew of the rotor blade are composed of two parts: the initial rake and skew of the blade reference line, and the rake and skew in the chordwise direction relative to the blade reference line which is given by LR. In the present research object, the initial rake and skew of the blade reference line (which is perpen dicular to the rotor axis) of the rotor blade are all zero at all spanwise sections. Under the design operating condition, the inlet flow velocity is 3 m/s, and the rotor blades rotate at 600 rpm with a speed of 9.72 m/s at the blade tip. The local Reynolds number based on the chord length of blade tip and the inlet flow velocity at the blade tip is 5.8 � 105. To evaluate the influence of tip clearance on propulsion performance and pressure fluctuations in the pump-jet propulsor, tip clearances of 2 mm and 4 mm, along with no tip clearance, are also employed. All the pump-jet propulsors have the same geometric parameters with the pump-jet propulsor shown in Fig. 1. For pump-jet propulsor without tip clearance, its shroud diameter is slightly reduced to eliminate the tip clearance between rotor blade tip and the inner wall of the shroud, and for pump-jet propulsor with tip clearance of 2 mm or 4 mm, the shroud diameter is slightly increased. In this way, the parameters of rotor blades are kept unchanged, and hence the possible influence induced by the variation of rotor parameters can be avoided. 2.2. Meshing The shear-stress transport (SST) k-omega turbulence model (Menter, 1994) combines the advantages of stability of the original k-omega turbulence model and independent of the external boundary k-ε turbu lence model and leads to major improvements in the prediction of adverse pressure gradient flows. According to the existing study by Ji et al. (2010) and Huang et al. (2010), the SST k-omega turbulence model with the sliding mesh technique are employed to simulate the unsteady flows through the pump-jet propulsor with and without tip clearance with the commercial code ANSYS Fluent in the present computations. Considering the rotation of rotor in unsteady flow computation, the whole computation domain is separated into two parts as show in Fig. 2 (a), namely the rotating domain and the stationary domain. The rotating domain is a cylinder around the propeller, and the two bottom surfaces of the cylinder, which act as the interface of rotating domain and sta tionary domain, are located upstream and downstream of the rotor respectively. The stationary domain is a large cylinder that surrounds the geometric model of pump-jet propulsor with a diameter much larger than the shroud. 2
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(a) The stators in shroud
(c) Computational model
(b) Rotor blades
Fig. 1. Geometric model of pump-jet propulsor. Table 1 Parameters of shroud and stator. Shroud
x yu yl x yu yl
0 0 0 0.480 0.091 0.039
0.005 0.009 0.009 0.540 0.083 0.039
0.010 0.017 0.013 0.600 0.074 0.039
0.030 0.030 0.022 0.660 0.065 0.043
0.060 0.048 0.026 0.720 0.052 0.043
0.090 0.061 0.030 0.780 0.035 0.048
0.120 0.074 0.035 0.840 0.013 0.052
0.180 0.087 0.039 0.900 0.013 0.061
0.240 0.096 0.039 0.940 0.035 0.065
0.300 0.100 0.039 0.960 0.043 0.070
0.360 0.100 0.039 0.980 0.057 0.074
0.420 0.096 0.039 1.000 0.065 0.074
Stator
x yu yl x yu yl
0 0 0 0.560 0.028 0.087
0.005 0.008 0.005 0.640 0.012 0.095
0.010 0.012 0.008 0.720 0.012 0.105
0.020 0.018 0.012 0.800 0.040 0.115
0.035 0.023 0.015 0.880 0.077 0.128
0.050 0.028 0.018 0.920 0.098 0.135
0.080 0.037 0.023 0.940 0.108 0.137
0.160 0.048 0.035 0.960 0.118 0.140
0.240 0.055 0.047 0.970 0.125 0.142
0.320 0.053 0.057 0.980 0.130 0.142
0.400 0.050 0.067 0.990 0.135 0.143
0.480 0.042 0.077 1.000 0.140 0.143
areas at the duct wall near the blade tip, the maximum values of yþ is found slightly greater than 1 (about 1.6), but it has little effect on the convergence of the calculation. 20 layers of prismatic cells are placed around the rotor blade surface and the thickness of the first layer is 0.002 mm where the value of yþ is about 1. To evaluate the influence of the grids density on computation results, three sets of meshes (see in Table 3) are selected to conduct the nu merical simulation and the variation of unsteady pressures at some arbitrarily selected monitoring points (see in Fig. 5) on rotor blade surface are compared as shown in Fig. 4 (T is the rotor rotation period). The comparisons for other monitoring points are similar. It is suggested that the grids density of Case 2 has shown good convergence. Even the number of grids increased by 50 percent as in Case 3, no more significant changes of the pressure fluctuation curves can be found at both posi tions. Considering the balance of the accuracy and the computational efficiency, the grids of Case2 are employed in the subsequent computations.
Table 2 Detailed dimensions of the rotor blade. r/R
Pitch (mm)
Chord Length (mm)
Thickness (mm)
LR (mm)
0.25 0.35 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.975 1
403.25 401.90 402.93 405.92 406.87 401.31 383.81 347.34 316.93 298.15 276.09
64.17 68.49 70.37 73.80 76.43 77.74 76.78 71.85 66.38 62.44 56.67
12.78 11.26 10.46 8.79 7.22 5.89 4.91 4.3 4.13 4.08 4.04
29.02 32.968 34.087 34.547 32.11 26.265 16.629 2.604 6.705 12.51 20.721
In general, the convergence of computations and the accuracy of computation results are strongly dependent on the quality of the computational meshes. Both the hexa-structured and hybridunstructured meshes can obtain similar levels of accuracy during nu merical simulation of flow field but the hybrid-unstructured meshes are less suitable for detailed investigations of the flow field (Morgut and Nobile, 2012). Because of the complexity of twisted geometry and the efforts of grids generation, the hexa-structured meshes are combined with the hybrid-unstructured meshes to solve the Navier-Stokes equa tions (Ji et al., 2010; Huang et al., 2010). As show in Figs. 2(b) and Fig.3, the stationary domain is discretized by hexa-structured meshes and the rotating domain is discretized by hybrid-unstructured meshes. Consid ering that the flow fields vary violently near the blade tip, a local mesh refinement scheme has been applied to the mesh generation at blade tip. Because the numerical results of wall pressure fluctuations are very sensitive to the size and quality of near wall meshes, boundary layer elements are applied in the solid surface to get the flow and pressure fields near the wall accurately. The first layer of boundary layers in the duct and stator blade wall is 0.01 mm, with a maximum yþ of approx imately 1 in the stator blades and most areas of the duct wall. In some
2.3. Boundary conditions As stated in Menter (1994) and Huang et al. (2010), the reliability and the convergence speed of the CFD simulation depends tremendously on the definition of boundary conditions, and in this work the parallel algorithm based on a domain decomposition method is adopted to maintain a balance between efficiency and accuracy. The inlet of the computation domain is set as velocity inlet boundary condition where the velocity components are set as axial velocity with no radial and circumferential components, and the initial turbulence intensity is assumed to be 2.83% with the external diameter of the shroud as the characteristic length. In order to reduce the influence of far-field boundary of computation domain, the same velocity boundary condition with the inlet boundary is also applied to the far-field boundary (see Fig. 2(a)). For the outlet boundary, the static pressure outlet boundary condition with a reference pressure of 0 Pa is defined. No slip boundary condition is applied to solid surfaces such as shroud 3
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10D
INLET
OUTLET
Stationary Domain
6D
6D
Interface
Shroud
Propeller
Stator
(a) Computational domain
(b) Grids of stator blade
Fig. 2. Computational domain of pump-jet propulsor and computational grids of stator blade.
(a) Grids of rotor blade for PJP without tip clearance
(b) Grids of rotor blade for PJP with tip clearance
Fig. 3. Computational grids of rotor blades.
wall and rotor blades, in this way all components of velocity and the normal gradient of static pressure would vanish on solid surfaces. In addition, the shroud inner wall around the rotor blades in the rotating domain is given a reverse rotational speed (which is equal to the rota tional speed of the rotational domain) relative to the adjacent cell zone to keep the shroud wall absolutely stationary in the absolute coordinate system. The overlapped surfaces (the interface of the rotating domain and the stationary domain) between the stationary and the rotating domain are defined as sliding interfaces in order to exchange the flow field information between the stationary and the rotating domains simultaneously. The convergence criterion is set as follows: the continuity and
Table 3 Three sets of meshes for numerical calculation. Grid numbers
Case 1 Case 2 Case 3
Maximum grid size in Rotating domain
Stationary domain
Rotating domain
5.64 million
8.95 million
12 mm
10.85 million
20.45 million
8 mm
15.26 million
30.65 million
5 mm
4
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1.8
104 Case1 Case2 Case3
PS-4 (r/R=0.9)
1.6 1.5 1.4 1.3 1.2 1.0 0.0
0.4
0.8
t/T
1.2
1.6
-2.0
SS-4 (r/R=0.9)
-4.0
SS-2 (r/R=0.9)
0.0
2.0
-1.0
Case1 Case2 Case3
PS-4 (r/R=0.6)
0.4
0.8
t/T
1.2
1.6
PS-2 (r/R=0.6)
2.0
103 Case1 Case2 Case3
-2.0 Pressure /Pa
Pressure /Pa
0.0
-8.0
103
9.3 9.0 8.7 8.4 8.1 7.8 7.5 7.2 6.9 6.6 0.0
Case1 Case2 Case3
-6.0
PS-2 (r/R=0.9)
1.1
103
2.0 Pressure /Pa
Pressure /Pa
1.7
4.0
-3.0
SS-4 (r/R=0.6)
-4.0 -5.0 -6.0
SS-2 (r/R=0.6)
-7.0 0.4
0.8
t/T
1.2
1.6
2.0
0.0
0.4
0.8
t/T
1.2
1.6
2.0
Fig. 4. Comparison of unsteady pressure variations under different grid density.
Fig. 5. Position of monitoring points at six selected spanwise sections.
velocity residual error are less than 10 4, the energy residual error is less than 10 7, and the relative error of mass flow rate at the inlet and outlet of pump-jet propulsor is less than 0.1%. In the present work, the convergence usually occurs after 3000 steps in steady calculation, and for unsteady calculation, due to the initial condition is the result of steady calculation, the convergence usually occurs after one rotor rev olution. After the process converges, the continuity residual error is less than 10 4, the velocity residual error is close to 10 6, the energy re sidual error is less than 10 7, and the relative error of mass flow rate is about 0.1%.
addition, the monitoring points are equidistant in the chordwise direc tion in every spanwise sections. The points LE and TE represent the monitoring points at leading edge and trailing edge, respectively. 3. Experimental verification of the numerical method To assure the reliability and efficiency of the developed numerical model, experimental data available for the pump-jet propulsor with a tip clearance of 1 mm (0.676%R, where R is the radius of the rotor) is used to validate the numerical results. Specially, the propulsion performance is tested in the cavitation tunnel at Shanghai Jiao Tong University (SJTU). The cavitation tunnel is composed of a closed loop with a rectangular test section of 1 m � 1 m � 6.1m as inner dimensions. A metal screen is installed in the contraction section upstream of the test section to avoid flow rotation and to reduce the turbulence intensity. In general, the velocity non-uniformity is less than 1%, and the turbulence intensity at the inlet of test section is less than 0.5%. The maximum free stream velocity at the inlet of test section, V∞, is 15 m/s. The propeller dynamometer is installed downstream the pump-jet propulsor model,
2.4. Monitoring points setting To obtain the characteristic of fluctuating pressures on the blade surface, six sets of monitoring points are set in the rotor blade. Fig. 5 shows the locations of monitoring points in the rotor blade surfaces. For each spanwise section, points from SS-1 to SS-5 are the points on suction surface listed from blade leading edge to trailing edge, and on the pressure surface, there are corresponding points PS-1 to PS-5. In 5
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and its maximum revolution speed is 1200 revolutions per minute (rpm). Different from the conventional experiments of propellers, the pro pulsion performance experiment for pump-jet propulsor includes the tests of forces acting on both the rotor and the stator as well as the shroud. The rotor is fixed on the horizontal axis of propeller dyna mometer (see in Fig. 6), and the horizontal thrust and torque acted on the rotor blades can be directly measured in real time. In order to measure the resistance acted on stator and shroud, a three-component balance was installed in the equipment box above the experimental model by a high-strength stainless steel rod. By using a unified measuring control system, the required operating condition could be satisfied easily and the forces acted on the shroud (including stator) and rotor could be measured simultaneously by the three-component bal ance and the propeller dynamometer. For the convenience of expression, the non-dimensional thrusts and torque can be defined: KTR ¼ KT ¼
TR
ρn2 D4
; KTS ¼
TS
ρn2 D4
TR þ TS ¼ KTR þ KTS ρn2 D4
10KQ ¼
are nearly the same for pump-jet propulsor with and without tip clear ance. Unlike conventional highly skewed propellers (Tian et al., 2017) which can only maintain high efficiency in a small range of advance ratio near the design operating point, the pump-jet propulsor can maintain a very high propulsion efficiency at a large range of advance ratio for both cases with or without tip clearance. Compared with the case of Gap ¼ 0, the rotor thrust of the case of Gap ¼ 1 mm dropped by 6%–9%, and the rotor torque dropped by 3%–5%. Consequently, the propulsion efficiency dropped by about 5.8% at the design operating point. For the case of Gap ¼ 2 mm and Gap ¼ 4 mm, the decrease of rotor thrust and torque is more significant, and eventually lead to a more significant reduction of propulsion efficiency relative to case of Gap ¼ 1 mm. However, for the resistance of shroud and stator blades, the influence of tip clearance is relatively small. The existence of tip clearance leads to a significant decrease in rotor thrust and torque, but the propulsion efficiency almost remains unchanged at lower advance ratio and decreases significantly at high advance ratio. Furthermore, the four cases all reach the maximum efficiency at about J ¼ 1.0, which indicate that the existence of tip clearance and its sizes do not change the advance ratio corresponding to the maximum efficiency point of the propeller.
(1)
10QR
4.2. Time-averaged pressure distributions on the rotor blade
ρn2 D5
where J ¼ VnDA is the advance ratio, and VA is the inlet velocity. The comparison of numerical and experimental results for propul sion performance is shown in Fig. 7. It is observed that the maximum numerical error of rotor torque relative to the experimental result is 2.7%, and those for the thrust on rotor and the thrust on stator blades and shroud are respectively, 4.6% and 3.2%. Relatively small tolerable error illustrates the consistency between the simulation results and the experimental results, which demonstrates feasibility of the present nu merical calculation.
To investigate the influence of tip clearance on time-averaged pres sure distributions and pressure fluctuations on rotor blades, all unsteady computation results presented are obtained at the design operating point. The total time for the unsteady computation is set for the rotor to rotate 6 revolutions after the convergence of computation, and the total flow time is 0.6 s. The time step is Δt ¼ 2*10 5, which means that there are 5000 sampling points for every monitoring points for every revo lution of the rotor. The computation results of propulsion performance show that the existence of tip clearance mainly changes the thrust and torque of the rotor blades, and the resistance on shroud and stator blades is almost unchanged. Therefore, the influence of tip clearance on propulsion performance can be explained by the analysis of circumferentiallyaveraged (time-averaged) pressure distributions (eq. (4)) on rotor blade surface. Fig. 9 shows the comparison of dimensionless circumferentially-averaged pressure distributions (one revolution) on rotor blade surface at selected spanwise sections. The dimensionless pressure is defined as:
4. Results and discussions
Cp ¼
where TR is the thrust of rotor; TS is the thrust (resistance in high advance ratio) of shroud and stator; QR is the rotor toque; and ρ, n and D represent the fluid density, the revolution speed and the rotor diameter respectively. And then the efficiency of pump-jet propulsor can be defined:
η0 ¼
ðTR þ TS ÞVA J KT ¼ 2π KQ 2πnQ
(2)
4.1. Propulsion performance
P 1 2
P∞
ρV 2A
(3)
As can be seen from the figure, the circumferentially-averaged pressure distributions on the rotor blade are almost the same in the range of r � 0.8R for the four cases. However, in the range of r > 0.8R, the pressures on pressure surface for pump-jet propulsor with tip clearance are much lower than that on pressure surface for pump-jet propulsor without tip clearance, and the pressures near the blade
Based on the same numerical method, Fig. 8 shows the comparison of propulsion performance for pump-jet propulsor with tip clearance of 0, 1 mm, 2 mm and 4 mm. As can be seen, the variation trends of perfor mance curves, such as dimensionless forces and propulsion efficiency,
Fig. 6. Pump-jet propulsor model fixed on the Propeller Dynamometer in cavitation tunnel. 6
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1.2
Experiments
Computed
1.0 10KQ
KT,10KQ,
0
0.8 0.6 KTR
0.4
KT
0.2 0.0
KTS
-0.20.2
0.4
0.6
0.8
1.0
1.2
J=V/nD Fig. 7. Comparison between the numerical and experimental results.
Gap=0 Gap=1mm Gap=2mm Gap=4mm
1.2 1.0
1.0
10KQ
0.8
0
0.6
0.6
0.4
KT-Rotor
0
0.4
0.2 0.0 -0.2 0.2
10KQ
KT,10KQ,
KT,10KQ
0.8
Gap=0 Gap=1 mm Gap=2 mm Gap=4 mm
1.2
KT-Shroud & Stator
0.4
0.6
0.8 J=V/nD
1.0
KT
0.2 0.2
1.2
0.4
0.6
0.8
J=V/nD
1.0
1.2
Fig. 8. Comparison of propulsion performance for pump-jet propulsor with and without tip clearance.
trailing edge on suction surface for pump-jet propulsor with tip clear ance are much lower than that on suction surface for pump-jet propulsor without tip clearance, but overall, the larger the tip clearance size, the smaller the pressure difference between pressure surface and suction surface of the rotor blade. It shows that the decrease of thrust and torque of the pump-jet propulsor with tip clearance is mainly caused by the change of circumferentially-averaged pressure distributions on the rotor blade in the range of 0.8R–1.0R. Fig. 10 shows the instantaneous streamline distributions at the blade tip, and Fig. 11 shows the instantaneous pressure distributions near the blade tip. For the case of Gap ¼ 0, the main flow is free from the influ ence of gap flow, therefore the flow near the blade tip is relatively stable and the streamline distribution is uniform along the blade surface. For the case of Gap ¼ 1 mm,2 mm and 4 mm, because of the interaction of the gap inverse flow and the main flow, the flow near the blade tip is chaotic and partial streamlines shed from the pressure surface to the suction surface near the leading edge of the blade tip and mingle with each other near the blade trailing edge. It shows that the fluid near the
blade tip flows from the pressure surface to the suction surface under the pressure difference between pressure surface and suction surface, thus neutralizing the pressures on both sides of the rotor blade, which in turn leads to the significant decrease of pressure difference between pressure surface and suction surface (see in Fig. 11). From Fig. 9(a–c) and Fig.10, the larger the tip clearance size, the more significant decrease of pres sure difference between the two sides of the rotor blade (see in Fig. 9 (a–b)). In addition, for pump-jet propulsor with tip clearance, the negative pressure is mainly observed at the core area of tip-leakage vortex which has an angle of about 15� with the chord direction of blade tip, and the larger the tip clearance size, the lower the negative pressure at the core area of tip-leakage vortex. As can be seen from Fig. 9 (a), the influence of tip leakage vortices on the circumferentiallyaveraged pressure distributions (especially the circumferentiallyaveraged pressures on suction surface) near the trailing edge of the blade tip is very significant, mainly because the tip-leakage vortex is fully formed in the suction surface near the trailing edge of the blade tip.
7
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5
(a) r/R=1.0
5
Gap=0 Gap=1mm Gap=2mm Gap=4mm
3
(b) r/R=0.95
Gap=0 Gap=1mm Gap=2mm Gap=4mm
3 1
Cp
Cp
1 -1
-1
-3
-3
-5
0
5
0.2
0.4
0.6 x/C
(c) r/R=0.9
0.8
-5
1
5
Gap=0 Gap=1mm Gap=2mm Gap=4mm
3
0
0.2
0.4
0.6 x/C
(d) r/R=0.8
1
Gap=0 Gap=1mm Gap=2mm Gap=4mm
3 1
Cp
Cp
1
0.8
-1
-1
-3
-3
-5
0
5
0.2
0.4
0.6 x/C
(e) r/R=0.7
0.8
-5
1
5
Gap=0 Gap=1mm Gap=2mm Gap=4mm
3
0
0.2
0.4
0.6 x/C
(f) r/R=0.6
1
Gap=0 Gap=1mm Gap=2mm Gap=4mm
3
1
0.8
Cp
Cp
1
-1
-1
-3
-3
-5
0
0.2
0.4
0.6 x/C
0.8
-5
1
0
0.2
0.4
0.6 x/C
0.8
1
Fig. 9. Comparison of circumferentially-averaged pressure distributions on rotor blade surface at selected spanwise sections, J ¼ 1.0.
(a)Gap=0
(b)Gap=1mm
(c)Gap=2mm
Fig. 10. Instantaneous streamline distributions at the blade tip, J ¼ 1.0. 8
(d)Gap=4mm
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Ocean Engineering 192 (2019) 106500
(a)Gap=0
(b)Gap=1mm
(c)Gap=2mm
(d)Gap=4mm
Fig. 11. Instantaneous pressure distributions on the sections perpendicular to the chord direction of blade tip, J ¼ 1.0.
4.3. Influence of tip clearance on pressure fluctuations
is the product of stator numbers NS and the rotor revolution frequency fn, are the main frequency of pressure fluctuations on rotor blade for both propulsors. Although tip clearance significantly increases the ampli tudes of pressure fluctuations on the rotor blade surface, it does not change the main frequency of these pressure fluctuations. The spectrum analysis only gives the local spectrum characteristics of pressure fluctuations at selected monitoring points. To reveal the characteristics of pressure fluctuations on the whole blade surface, the statistical method, which can intuitively show the fluctuation of instantaneous pressures at each grid node, is applied to the analysis in present work. In general, the unsteady pressure at each grid node con sists of two parts, namely the circumferentially-averaged part depending on the location of grid node (eq. (4)), and the fluctuating part changing periodically with the relative position of grid node during the rotation of rotor (eq. (5))(Feng et al., 2016).
Fig. 12 shows the frequency spectrogram of pressure fluctuations at representative monitoring points on blade surface from r/R ¼ 0.6 to r/ R ¼ 1.0 for the case of Gap ¼ 0 and Gap ¼ 1 mm. The monitoring points located on the position as given in Fig. 4. It is observed that the am plitudes of pressure fluctuations for pump-jet propulsor with tip clear ance of 1 mm increased significantly at most of the monitoring points compared with the pump-jet propulsor without tip clearance, especially at the monitoring points located on the leading edge. Furthermore, points LE at all selected spanwise sections of pump-jet propulsor, with or without tip clearance, have the biggest amplitudes of fluctuating pres sures, and it is mainly due to that the points located at the leading edge are affected most significantly by the interaction of rotor and stator. It suggests that although the pre-swirl stators are generally considered to make the incoming flow more uniform, they are also an inducing factors of non-uniform flow field for the inflow of rotor blades. The stator blade passing frequency fs (relative to the rotating coordinate system), which
pð! r Þ¼
9
Nt 1 X pð! r ; i ⋅ ΔtÞ Nt i¼1
(4)
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Ocean Engineering 192 (2019) 106500
(a) Monitoring points on pressure surface at r/R=1.0
(b) Monitoring points on suction surface at r/R=1.0
(c) Monitoring points on pressure surface at r/R=0.80
(d) Monitoring points on suction surface at r/R=0.80
(e) Monitoring points on pressure surface at r/R=0.60
(f) Monitoring points on suction surface at r/R=0.60
Fig. 12. Comparison of frequency spectra of pressure fluctuations on representative monitoring points, J ¼ 1.0.
b p ð! r ; tÞ ¼ pð! r ; tÞ
pð! rÞ
Hence, the standard deviation of fluctuating pressure can be defined as Eq. (6) to evaluate the amount of pressure variation for a serial of fluctuating pressures at each grid node.
(5)
where ! r ¼ ðx; r; θÞ is the coordinates of grid nodes in rotating coordi nate system, Nt is the number of time steps for one revolution, and Δt is the time step used for the computation. 10
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vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Nt u Nt u1X u1 X b p 2 ð! σ ð! r Þ¼t ðpð! r ; i⋅ΔtÞ pð! r ÞÞ2 ¼ t r ; i⋅ΔtÞ Nt i¼1 Nt i¼1
(6)
KTR ¼
Based on the method shown in eq. (6), Fig. 13 and Fig. 14 give the contour of standard deviation of pressure fluctuations on the whole rotor blade surface colored with the standard deviation of fluctuating pres sures. The figures plot a detailed information of pressure variations on the rotor blade. It is obvious that the standard deviation σ generally increases with the increase of tip clearance sizes on most areas of the blade surface, which illustrates that despite the influence of tip clear ance on static pressure distributions is mainly focused on the blade tip region, the influence of tip clearance on pressure fluctuations works throughout the rotor blade surface. In general, for the four cases with advance ratio of J ¼ 1.0, the standard deviation σ on blade tip and blade leading edge is much larger than that on other areas, and the standard deviation σ is larger on the suction surface than that on the pressure surface. Moreover, the standard deviation σ on the blade root increase sharply with the increase of tip clearance, mainly because the increasing of tip clearance sizes reduces the effective velocities at the rotor plane (According to the calculation result of time-averaged pressure distri butions on the rotor blade, when the tip clearance size increases, the blade loads will decrease, and in turn leads to the decrease of induced velocities at the rotor plane) and thus promotes the evolution of blade root vortices. The integral of unsteady pressures along the blade surface is the blade thrust and torque of the rotor. The unsteady thrust and torque usually consist of two parts, namely the time-averaged part (eq. (7)) depending on the operating condition, and the fluctuating part (eq. (8)) changing periodically with the rotation of rotor blade. The engineering practice shows that the time-averaged parts of thrust and torque are presented as the propulsion performance of a pump-jet propeller, and the fluctuation parts of thrust and torque are usually the main factor inducing the hull vibration for modern underwater vehicles.
Nt Nt 1 X 1 X Fði ⋅ ΔtÞ ; KQ ¼ Mði ⋅ ΔtÞ Nt i¼1 Nt i¼1
~ ¼ FðtÞ Fx
~ ¼ MðtÞ KTR ; Mx
KQ
(7) (8)
where, F and M are the dimensionless unsteady thrust and torque of the rotor blade, respectively. According to eqs. (7) and (8), the fluctuation of thrust and torque for a single blade of rotor and their frequency spectra are shown in Figs. 13 and 14. Both the thrust fluctuation amplitudes and torque fluctuation amplitudes increase significantly with the increase the tip clearance sizes, and the stator blade passing frequency fs is the main frequency. For the case of Gap ¼ 0, the fluctuation of thrust and torque is relatively regular, and the amplitude of the third-order frequency is significantly less than that of the first-order and second-order frequencies. However, for pump-jet propulsor with tip clearance, the amplitude of the thirdorder frequency increases significantly with the increase of tip clear ance sizes, and the amplitude of the third-order frequency is approxi mately equal to the amplitude of the second-order frequency for case of Gap ¼ 2 mm and Gap ¼ 4 mm. It shows that the tip clearance effects do not change the main frequency of thrust and torque fluctuations, but it has a significant impact on the amplitudes of fluctuations, especially in the higher frequency band (f � 3fs). 5. Conclusions This study presents numerical investigation of tip clearance effects on propulsion performance and pressure fluctuations of a pump-jet propulsor. The SST k–omega turbulence model based on RANS method and the sliding mesh technique are employed to compute the unsteady flow at the design operating point. Propulsion performance experiment for pump-jet propulsor with tip clearance of 1 mm is also
(a) Gap=0
(b) Gap=1mm
(c) Gap=2mm
(d) Gap=4mm
Fig. 13. Comparison of pressure fluctuation contours for rotor pressure surface, J ¼ 1.0. 11
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Ocean Engineering 192 (2019) 106500
(a) Gap=0
(b) Gap=1mm
(c) Gap=2mm
(b) Gap=4mm
Fig. 14. Comparison of pressure fluctuation contours for rotor suction surface, J ¼ 1.0.
carried out to validate the numerical computation method. The propulsion performance of pump-jet propulsor with tip clear ance of 1 mm, 2 mm and 4 mm are computed and compared with that without tip clearance. The results reveal that although the tip clearance size is very small with respect to the rotor diameter (only 0.676–2.7 percent of the rotor radius), the reduction of rotor thrust and propulsion efficiency caused by the tip clearance is very significant, and especially when the tip clearance size is enlarged, these effects will be more sig nificant. For the case of Gap ¼ 1 mm, due to the influence of tip clear ance, the rotor thrust and torque dropped by 6%–9% and 3%–5% respectively compared with the case of Gap ¼ 0, and consequently, the propulsion efficiency reduced by about 5.8% at the design operating point. The decrease of thrust and torque of the pump-jet propulsor with tip clearance is mainly caused by the change of time-averaged pressure distributions on the rotor blade in the range of 0.8R–1.0R, and the larger the tip clearance size, the more significant decrease of pressure differ ence between the two sides of the rotor blade. The influence of tip clearance on pressure fluctuations can be observed throughout the whole rotor blade surface. The amplitudes of pressure fluctuations increase significantly on the rotor blade surface when the tip clearance is enlarged, especially on the blade tip, blade root and the leading edge, but the main frequency of the pressure fluctua tions is not affected. In addition, for thrust and torque fluctuations, the tip clearance has magnified greatly the amplitudes of force fluctuations, especially in the higher frequency band. The amplitudes of thrust and torque fluctuations increase along with the increase of tip clearance sizes, and the amplitudes corresponding to the third-order, fourth-order, and even the fifth-order frequencies are obviously increasing more sig nificant. If the tip clearance effects can be eliminated or suppressed effectively, the pressure and force fluctuations will decrease significantly.
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