Volumetric entropy generation rate associated with tip clearance flow in linear cascade

The tip clearance flow structure and its corresponding entropy generation rate of a linear cascade with 1.85mm radial tip clearance operating under three different incidence angle are investigated. The incidence angle influences the tip leakage vortex position as well as the size of the tip leakage vortex. Larger incidence angle results in larger tip leakage vortex and pushes the tip leakage vortex further away from the suction side. The entropy generation rate related to the tip clearance flow structure is calculated and it is confirmed that tip clearance flow is the main major source of aerodynamic loss in linear cascade. Larger incidence angle leading to higher entropy generation rate. Results also show that it’s not the core region of tip leakage vortex but the edge vortex and shear layer of the tip clearance flow which has the highest entropy generation rate. When operating under an incidence angle of 4°, there exist an extra high entropy generation rate region in main flow. It is found that the compression and expansion of fluid due to the existence of tip clearance vortex was the main reason for this extra high entropy generation.


Introduction
The radial clearance between stator-vane to end-wall and between rotor-blade to end-wall is indispensable for turbomachine operation and adjusting. The clearance flow, including tip clearance flow and hub clearance flow, due to the existence of radial clearance, however, has been a major source for loss, blockage, instability and noise, and is detrimental to the aerodynamic, aero-acoustic and structural performance of turbomachine. Since clearance flow has significant influences on turbomachinery performance, it attracts numerous researchers' attention. A number of numerical and experimental investigations have been carried out across the world for the purpose of getting deep insight into the detailed structure of the clearance flow. The investigation of Kang and Hirsch shows that horseshoe vortex and multiple vortex exist in tip region and the existence of tip clearance vortex results in passage vortex close to end wall and suction side, Kang and Hirsch also find that the vorticity of tip clearance vortex increase right after it formed in the leading edge region and then decrease gradually [1] [2] . Kang and Hirsch also indentified tip separation vortex and pointed out that leakage flow mixing has a significant influence on the internal loss of compressor cascade [3] . Zhong studied the unsteadiness of tip clearance flow using DDES method and found that smaller tip gap results in weaker anisotropy [4] . The investigation of Kunz shows that Navier-Stokes procedures with k-epsilon turbulence model has satisfactory performance in capturing important physical phenomena of tip clearance flow [5] . Tian and Li found that the intensity and influence area of IOP Publishing doi:10.1088/1757-899X/1081/1/012045 2 tip leakage vortex increase with increasing tip leakage clearance [6] . The investigation of Muthanna and Devenport reveals that the dynamics of tip clearance vortex are dominated by streamwise meanvelocity deficit the vortex produces [7] . Alexej studied the tip clearance flow of axial-flow fan using LES method and found that tip clearance flow results in lower turbulence kinetic energy [8] . Some passive methods has been utilized to control the tip clearance flow. Mehdi pointed out that porous treatment can reduce the intensity of tip vortex as well as the shear layer rollup and boundary layer separation [9] . The investigation of Chen demonstrated that circumferential groove can suck low energy tip clearance flow and thus has a significant importance in extending stall margin of centrifugal compressor [10] .Wang studied the effects of blade tip profile and found that optimized blade tip profile can reduce tip leakage flow velocity, weakens the strength of mixing, and therefore results in loss reduction [11] . The loss induced by clearance flow is a dominant part of turbomachinery total loss. Clarifying the loss mechanism associated with the complex tip clearance flow structure is beneficial for developing design method to improve the turbomachinery efficiency as high as possible. According to the theory of the second law analysis, the volumetric entropy generation rate is capable to present the local loss intensity and thus providing researchers the possibility of clarifying the loss mechanism [12] . Up to date, three main approaches has been proposed for volumetric entropy generation rate calculation. By adding a turbulent conductivity and turbulent viscosity to the molecular conductivity and viscosity, Moore and Moore proposed the first method to close the volumetric entropy equation [13] . The Moore model has been utilized for studying end-wall sealing flows [14] and tip cavity flows [15] in turbines based on RANS method. Kramer-Bevan [16] proposed a new model for volumetric entropy equation closing and has been used by Kock [17] and Takakura [18] to investigate the entropy generation rate in the tip region of a high-pressure turbine using RANS simulation results. Adeyinka [19] extended past Reynolds averaging techniques for the momentum and scalar transport equations to the second law for turbulent and has been used by Orhan [20] to study loss mechanism of an axial turbine cascade.The applicability of the model proposed by Kramer Bevan in loss mechanism study has been validated and is used in this paper. Direct and quantitative study of clearance flow in actual turbomachines is challenge due to some safety issues and some technique limitation. Linear cascade with gap can reproduce flow structures quite similar to the tip clearance flow in actual turbomachines and providing researchers an efficient and effective way of studying clearance flow phenomenon. In this paper, a numerical research using RANS is carried out to study the tip clearance flow structure and the loss mechanism associated with tip clearance flow under three different incidence angle (α). The paper is organized as follows: After the validation of the RANS method, the numerical model is introduced and the numerical results are presented. The tip clearance flow structure and the loss production associated with these flow structures under three different incidence angle is then analyzed in detail to understand the tip clearance flow structure, the corresponding loss-generating mechanism and the influence of incidence angle on clearance flow structure and the corresponding loss mechanism.

Model and Numerical Method
2.1. The studied cascade the linear compressor cascade of LMFA at Ecole Centrale Lyon is selected for the purpose of evaluating the accuracy of numerical methods as there are many detailed and accurate experiments of this cascade were carried out in previous researches. The geometric parameters of the cascade are shown in Table 1. The inlet velocity is 40m/s so that the flow in the cascade can be treated as incompressible. The incident angle is set to 4 degrees, so that the corner separation exist. More detailed information can be found in Reference [21] . In order to study the effects of incidence angle on clearance flow structure and entropy generation rate, a 1.85mm tip clearance gap is added to the LMFA cascade and three different operating condition, which corresponding to an incidence angle of -4 o , 0 o and 4 o , were set in this study.

Numerical setup
The HOH mesh topology is employed for meshing the studied cascade. To guarantee the accuracy of the numerical simulation, the Y-plus of the first layer near-wall of computational mesh is less than 1.
The length of the inlet and outlet region are 1.5 times of the chord length (1.5C) and 2C in order to make these two boundaries less reflective. Half of blade span is used to carry out this research for the purpose of reducing the computational consumption. The mesh dependence was studied and the grid number chosen for this study is about 3.0 million. Mesh information is show in Figure 1.

Numerical validation.
Static pressure coefficient is a common parameter to quantify the blade load distribution and Total pressure loss coefficient is a parameter corresponding to loss generation in turbomachine and they are defined as eqn.(1) and eqn.(2) respectively.   Figure 2, it is clear that RANS can predict the separation vortex and the total pressure loss corresponding to the separation vortex with satisfactory accuracy.  Figure 3 also presents that the tip leakage vortex size expanded as the tip leakage vortex being convected downstream. The generation and development process of tip leakage vortex revealed is consistent with other previous researches. Figure 3 also reveals the variation of tip leakage vortex structure with the change of incidence angle. The origin of the tip leakage vortex moves downstream along the blade suction side with decreasing incidence angle. The origin of the tip leakage vortex located around 3%,10% and 30% chord length for incidence angle of 4 o , 0 o and -4 o . Increasing incidence angle pushes the tip leakage vortex further faraway from the suction side. From Figure 3, it is also clear that larger incidence angle results in larger tip leakage vortex.  Figure 4 also reveals that tip leakage vortex size and its distance to suction side increase with incidence angle increase. Increasing incidence angle results in large static pressure difference between blade suction side and blade pressure side as well as rapid flow acceleration on the suction side, which is the fundamental reason for tip leakage flow structure and vortex strength variation with incidence angle.  Figure 5 presents the variation of total pressure loss coefficient with incidence angle. Data is extracted from a plane which is 0.363 times of axial chord length away from the blade trailing edge. Total pressure loss coefficient is calculated according to Eq. (2). From Figure 5, it is clear that the loss magnitude associated with the tip clearance flow increase with incidence angle. It can also be seen from Figure 5 that the region which is influenced by tip clearance flow expanded as incidence angle increases. For operating conditions with an incidence angle of -4 o , 0 o and 4 o , the origin of influence gradually increase from about 0.025m to around 0.035m. In the incidence angle range from -4 o to 4 o , the influence of incidence angle on the profile loss is relatively small. (4) Figure 6 Entropy generation rate around leading edge Figure 6 and Figure 7 clearly demonstrate the volumetric entropy generation rate associated with tip clearance flow around the leading edge and trailing edge region. Around the leading edge, when incidence angle is -4 o , the aerodynamic loss source is confined in the tip clearance region and it is obvious that the highest entropy generation rate occurs around the suction side, which means when cascade operating under incidence angle of -4 o , around the leading edge region, part of main flow fluid moves from the suction side to the pressure side, which results in edge vortex exist around the suction side. Increasing incidence angle, entropy generation rate associated with edge vortex located near the pressure side increase and the loss source extend to the main flow close to the suction side. Figure 6 also reveals that, around the leading edge, the magnitude of entropy generation rate of boundary layer is relatively small compared to that of the tip clearance flow. generation rate in the vortex core region, however, is relatively small compared to that of edge vortex and shear layer. From Figure 7, it is obvious that on the pressure side, there exist a region with high entropy generation.

Conclusions
Incidence angle plays an important role in deciding the origin, the size and the position of tip leakage vortex. The tip leakage vortex origin moves downstream with the increase of incidence angle, the vortex size increase with the increase of the incidence angle. Increasing incidence angle moves the tip leakage vortex further faraway from the blade suction side. Tip clearance flow is the main major source of aerodynamic loss in linear cascade. Entropy generation rate associated with tip clearance flow increases with the increase incidence angle. Edge vortex and shear layer corresponding to the tip clearance flow has the highest entropy generation rate. The entropy generation rate associated with the tip leakage vortex, however, is relatively small when compared with the entropy generation rate of edge vortex and shear layer. When operating under an incidence angle of 4 o , there exist an another high entropy generation rate region in main flow near trailing edge. It is found that the compression and expansion of fluid due to the existence of tip clearance vortex was the main reason for this extra high entropy generation.