The applicability of vortex identification methods for complex vortex structures in axial turbine rotor passages ＊

Yu-fan Wang, Wei-hao Zhang, , Xia Cao, Hong-kai Yang

1. National Key Laboratory of Science and Technology on Aero-Engine Aero-thermodynamics, School of Energy and Power Engineering, Beihang University, Beijing 100191, China 2. Collaborative Innovation Center of Advanced Aero-Engine, Beijing 100191, China

Abstract: The complex vortex structures in the flow around turbine rotor passages, with weak or strong, large or small vortices,interacting with each other, often generate most of aerodynamic loss in turbomachines. Therefore, it is important to identify the vortex structures accurately for the flow field analysis and the aerodynamic performance optimization for turbomachines. In this paper, by using 4 vortex identification methods (the Q criterion, the Ω method, the Liutex method and the Ω-Liutex method), the vortices are identified in turbine rotor passages. In terms of the threshold selection, the results show that the Ω method and the Q-Liutex method are more robust, by which strong and weak vortices can be visualized simultaneously over a wide range of thresholds. As for the display consistency of the vortex identification methods and the streamlines, it is shown that the Liutex method gives results coinciding best with the streamlines in identifying strong vortices, while the Ω-Liutex method gives results the most consistent with the streamlines in identifying weak vortices. As to the relationship among the loss, the vortices and the shear, except for the Q criterion, the other three methods can distinguish the vortical regions from the high shear regions. And the flow losses in turbine rotor passages are often related to high shear zones, while there is a small loss within the core of the vortex. In order to obtain the variation of vortices in the turbine rotor passages at different working points, the Liutex method is applied in 2 cases of a turbine with different angles of attack. The identification results show that the strengths of the tip leakage vortex and the upper passage vortex are weaker and the distance between them is closer at a negative angle of attack. This indicates that the Liutex method is an effective method, and can be used to analyze the vortex structures and their evolution in turbine rotor passages.

Key words: Vortex identification, tip leakage vortex, turbine rotor passage, Liutex method, Ω-Liutex method

Introduction

The flow field in the turbine passage is extremely complex, with multi-scale and multi-intensity vortices,and the interaction among these vortices often generate loss in the flow field. In addition, the evolution mechanism of each vortex is different, and is affected by the end wall boundary layer and the rotor blade rotation, resulting in a strongly three-dimensional flow field, strongly sheared and strongly unsteady[1]. The most vortex structures are located in the end wall with a great end wall losses in turbine passages. Denton[2]estimated that the end wall losses can account for 2/3 of the total flow losses. In order to reduce the flow losses in the turbine, a substantial number of researches were carried out based on the blade three-dimensional modeling[3], the end-wall modeling[4], the leading edge modification[5],and the tip modeling[6-7]. In this kind of studies, the secondary flow structure and its evolution are the inevitable topics, a

nd the accurate identification of the vortex structure is extremely important.

Since Helmholtz[8]proposed the “vortex motion”in his three vorticity theorems, the definition and the identification of vortices have attracted much attention.Brachet et al.[9]defined vortices as the region dominated by a negative velocity gradient. Babiano et al.[10]proposed that the vortices are any region where the vorticity is greater than a certain threshold. For a long time thereafter, the vorticity was regarded as an indicator for reliably identifying vortices. In 1989,Robinson et al.[11]pointed out that in the turbulent boundary layer, especially in the region near the wall,the correlation between the large vorticity region and the actual vortices is rather weak. In recent studies,Dong et al.[12-13]also pointed out and verified that there is no necessary relationship between the vortices and the vorticity through the study of the transition section of the boundary layer, and that the vorticity cannot be applied to identify vortices or rotation regions. With the recognition of this key issue, it becomes a controversial issue how to identify the vortices, and various vortex identification methods were proposed. Since 1988, a series of criteria based on the velocity gradient tensor were proposed for the vortex identification, theQcriterion[14], the Δ criterion[15], the2λcriterion[16]and theciλcriterion[17]and they are often used in the analysis of the flow field in turbine passages. However, there are some common defects in the above-mentioned vortex identification methods, such as the threshold selection and the misidentification of a strong shear zone, which makes it difficult to study the internal flow field in turbine passages.

In 2016, a vortex identification method called theΩmethod, with a high robustness, was first proposed by Liu et al.[18]. Later, Zhang et al.[19-20]applied this method for the flow field analysis of a reversible pump turbine, and verified the robustness of theΩmethod and the ability of identifying simultaneously strong and weak vortices. Recently, Liu and his team[21-24]further proposed the third generation vortex identification method[25]-the Liutex method with the identification parameter being a vector. Gao and Liu[26]compared the Liutex method with the vortex identification methods based on the eigenvalue of the velocity gradient tensor in the mathematical and physical sense. The conclusion is that the Liutex method eliminates the contamination by the shear compared with other methods, so it can quantify accurately the local rotational strength. Dong et al.[27]put forward a new normalized vortex identification method named theΩ-Liutex method, based on the Liutex andΩmethods.

The Liutex andΩ-Liutex methods were used in the vortex identification in the cases of the flow transition and the swirling jet, while they were not applied to the analysis of complex flow fields such as the flow fields in turbines. The applicability of the Liutex method in the analysis of complex flow fields with multi-scale and multi-intensity vortices should be verified. In this paper, theQcriterion and the latest three kinds of vortex identification methods are applied to the identification of the vortices in turbine rotor passages. The identification results are compared and analyzed, and the advantages and disadvantages of these methods are summarized. Then, the Liutex method is applied under different working conditions,and the applicability of this method in turbine rotor passages is verified by the analysis of identification results and the flow field.

1. Numerical method

Steady-state RANS computations are performed by using ANSYS CFX 12.0 with the shear stress transport (SST) turbulence model. The N-S equation is discretized by the finite volume method and the“high precision” scheme in the CFX is adopted. The wall boundary is set to be adiabatic, ignoring the effect of the

wall heat transfer.

In the numerical simulation, a single blade passage is modeled, including the rotor and stator domains. Periodic conditions are assumed at the boundaries with the adjoining passages, while the mixing plane approach is employed at the stator-rotor interface. The commercial software NUMECA Autogrid5 is used to generate the static sub-domain computational grid, while the ICEM is used to generate the rotor domain computational grids. The mesh elements in both domains are hexahedrons. The number of grids in the rotor domain is about 1.96×106,and that in the stator domain is about 6.4×105. The grids are shown in Fig. 1. In order to obtain the flow details of the tip clearance region, the mesh of the clearance region is specially refined in the ICEM, and 32 cells are used in the radial direction of the tip clearance.

The mass flow-averaged total pressure and the circumferential average total temperature are imposed in the inlet plane, and the turbulence intensity is set to 5%. The exit static pressure is adjusted for the average total-to-total pressure ratio of 4.

2. Results and discussions

2.1 Identification of vortex structures in turbine rotor passages

In order to accurately identify vortex structures in the turbine rotor passages, 4 methods will be adopted i.e. theQcriterion, theΩmethod, the Liutex method and theΩ-Liutex method.

TheQcriterion, proposed by Hunt et al.[14]in 1988, is a common method for the vortex identification, based on the second invariant of the eigenvalue equation of the velocity gradient tensor.TheQcriterion represents the extent that the amplitude of the local vorticity term exceeds that of the local strain term, which can be expressed as whererepresents the Frobenius norm andAis the symmetric part of the velocity gradient tensor andBis the anti-symmetric part.

TheΩmethod was first proposed by Liu et al.[18]in 2016, and was proved to be Galilean invariant[28]. This method was applied in the boundary layer transition[12-13,29], and the applicability of this method in the flow transition section was verified. In theΩmethod, the vortices are defined as the region where the vorticity overtakes the deformation and can be calculated by the following formula

Fig.1 (Color online) Computational mesh

In a practical calculation,εis usually introduced into the denominator to remove the noises caused by the division by zero or very small numbers. In order to restrain the influence ofεon the calculation results, Dong et al.[30]suggested thatε=is a good choice empirically.

The main idea of the Liutex method is to define the vortex as a new vector with a direction and a strength. The direction is the local rotation axis, which is the real eigenvector of the velocity gradient tensor,and its strength is determined by the sizes of the in-plane vorticity term and the deformation term perpendicular to the rotation axis with two coordinate transformations. The explicit expression of the Liutex strengthR[31]is

whereϖis the vorticity,ris the real eigenvector of the velocity gradient tensor,λciis the imaginary part of the conjugate complex eigenvalue andrepresents the vector dot product. The Liutex method is unique, Galileo invariant[24]and systematic.

TheΩ-Liutex method was proposed by Dong et al.[27], based on the idea of theΩmethod and the Liutex method. It is a normalized Liutex identification method with the threshold range of 0-1. According to the definition, theΩ-Liutex method can be represented by the following formula

whereαandβcan be calculated by the following formulae:

In the practical calculation by theΩ-Liutex method,ε=b(β2-α2) will also be introduced

maxon the denominator. It is more appropriate to take 0.001-0.002 forb.

Fig. 2 (Color online) Vortex structures in rotor passage identified by different vortex identification methods and streamwise vorticity distribution (UHV: upper horseshoe vortex,LHV: lower horseshoe vortex)

Figure 2 shows the vortex structures in the rotor passage identified by the above 4 vortex identification methods, and the iso-surfaces are colored according to the streamwise vorticity. In order to better identify the main vortex structures in the passage, it is necessary to select appropriate thresholds based on a professional experience. As can be seen from Fig. 2, the four vortex identification methods can identify the tip leakage vortex (TLV), the upper passage vortex(UPV), the lower passage vortex (LPV) and the horseshoe vortices (HV), but with different relative sizes of the vortices. The suction side corner vortex(SSCV) is relatively weaker as compared with the tip leakage vortex and the passage vortices. The identification results of the SSCV with the above four methods are quite different. There is no SSCV in the identification results of theQcriterion and the Liutex method, while theΩmethod and theΩ-Liutex method can identify the SSCV extremely clearly. This indicates that theΩmethod and theΩ-Liutex method have better performance in identifying and visualizing weak vortices.

Fig. 3 (Color online) Vortex structures in rotor passage identified bydifferent vortex identification methods with small threshold and streamwise vorticity distribution

Figure 3 shows the identification results of the 4 kinds of vortex identification methods with small thresholds. Comparing with Fig. 2, it can be found that the identification results of theQcriterion and the Liutex method are greatly affected by the threshold value. Some vortices may not be recognized with a higher threshold. And the most obvious change of the vortices in the passage is seen from the SSCV,followed by the scraping vortices outside the tip clearance. The horseshoe vortex structure near the casing identified by the Liutex method also changes significantly. At a higher threshold, the pressure side branch of the horseshoe vortex cannot be identified by the Liutex method. Although the identification results of theΩandΩ-Liutex methods vary with the threshold, the key vortex structures can still be observed. Regardless of the threshold value, in Figs.2(b), 2(d) and Figs.3(b), 3(d), the SSCV can be clearly observed, and the evolution process of the UPV can still be recognized. These two methods are less sensitive to the thresholds, which reduces the difficulty of the threshold selection. Even if the threshold selected is not the most appropriate, some important vortex structures can still be identified. On the contrary, with theQcriterion and the Liutex method, if the threshold selected is not appropriate, some vortices will be missed or misidentified. Therefore, the threshold selection of theQcriterion and the Liutex method often requires some professional experience.

Fig. 4 (Color online) The distribution of identification parameters by four methods and surface streamlines at a distance of 50% of rotor chord (SS: suction side, PS: pressure side,A: tip leakage vortex, B: upper passage vortex, C: scraping vortex)

A vortex is intuitively recognized as a part of the fluid in the rotational/swirling motion[23]. The most intuitive and convenient method to judge the existence and the location of vortices is by observing the streamlines. For the display consistency of the vortex identification methods and the streamlines, Fig.4 focuses on the difference between the vortex structures reflected intuitively by the surface streamlines and the vortex structures identified by the 4 vortex identification methods. In Fig. 4(a), there are significant differences between the vortices (e.g., zone B,zone C) identified by theQcriterion and the surface streamlines. TheΩmethod is more accurate if the location of the vortices identified by streamlines is taken as the standard, and the Liutex method is the most accurate in identifying the UPV (in zone B of Figs. 4(b), 4(c)). However, the identified vortex structures of strong vortices (e.g., zone A, zone B) by theΩ-Liutex method departs from those visualized by streamlines. It is noteworthy that theΩ-Liutex method gives results with a high consistency with streamlines in the identification of weak vortices (e.g.,zone C of Fig. 4(d)). From the above observations, it is easy to find that the Liutex method is the most consistent with streamlines in the identification of strong vortices, while theΩ-Liutex method is the most consistent with streamlines in identifying weak vortices.

Fig. 5 (Color online) The distribution of identification parameters by four methods and the contour distribution of streamwise vorticity at a distance of 50% rotor chord

Some vortex identification methods are contaminated by shears, with inaccurate identification of vortices. For example, theQcriterion in Fig. 2(a)misidentifies the end wall boundary layer as vortices.Figure 5 shows the distribution of identification parameters of four methods and the contour distribution of the streamwise vorticity at a distance of 50% of the rotor chord. In Fig. 5, zone A corresponds to the tip leakage vortex, and zone B corresponds to the upper passage vortex. Combining with the contour of the streamwise vorticity, the locations of the vortices identified by theQcriterion coincides basically with the high vorticity regions. There is a deviation between the vortices identified by the Liutex method and the high vorticity regions, while the vortices identified by theΩmethod and theΩ-Liutex method obviously deviate from the high vorticity regions. In the identification results of different methods, the difference of the coincidence between the vortical regions and the high shear regions is due to the different levels of contamination caused by shear. Except for theQcriterion, the other three methods perform well in distinguishing the vortical regions from the high shear regions, which indicates that there may be only a little contamination by shear in these three methods.

The tip clearance loss accounts for a large proportion of flow losses in the rotor passage. In order to further understand the relationship among the losses,the vortices and the shear, the local turbulent dissipation terms at a distance of 50% of chord and 90%of chord are shown in Fig. 6. And the contour of the streamwise vorticity is shown in Fig. 6. The turbulent dissipation term is defined as

Fig. 6 (Color online) Turbulent dissipation and streamwise vorticity distribution

whereμeff=μ+μτ,μτis the eddy viscosity andμis the dynamic viscosity. The turbulent dissipation causes the loss of the mechanical energy irreversibly converted to the internal energy, which will inevitably lead to the entropy production and can be used to measure the local loss. It is found that the high loss distributions at a distance of 50% and 90% of the rotor chord coincide well with the high vorticity regions(zone B in Fig. 6). As shown in Fig. 4(c), Fig. 6(a),zone A corresponds to the core of the tip leakage vortex, and there is only low-intensity turbulent dissipation in the core region of the tip leakage vortex,which indicates that the fluid rotation is not the essential cause of the loss. The high loss region near the tip clearance is mainly caused by the shear layer on the casing and the shear between the tip leakage flow and the main stream flow.

2.2 Analysis of vortices in turbine rotor passages under different conditions

With the adjustment of the rotor speed, the angle of attack at the inlet of the rotor blade can be changed.According to the analysis of our simulation results, the inlet angles of attack in case 1, case 2 are adjusted to 1.74° and -11.69°, respectively.

Fig. 7 (Color online) Static pressure distribution on the rotor blade and 3-D streamline and Liutex distribution on 7 different slices (a-g)

Figure 7 shows the distribution of the Liutex values on seven different slices, the 3-D streamlines near the casing and the distribution of the static pressure on the rotor blade surface. In case 1, one sees an obvious leakage flow from the suction side near the rotor leading edge, but none in case 2. A local high static pressure region (zone I in Figs.7(a), 7(b))appears in case 1, with a strong adverse pressure gradient in the region between the rotor leading edge and zone I, while there is basically a favorable pressure gradient along the streamwise direction in case 2 (zone I in Fig. 7(b)). From the above description, with the change of the attack angle, the tip leakage flow near the rotor leading edge changes significantly. This change of the leakage flow further affects the evolution of the vortices near the rotor leading edge. The horseshoe vortex and the scraping vortex on the suction side of the rotor leading edge can be clearly observed in zone II of Fig. 7(a). These two vortices promote the development of the upper passage vortices to a certain extent. In zone II of Fig.7(b), it can be found that the flow near the casing enters the gap from the suction side, so there are no significant scraping vortex and suction side branch of the horseshoe vortex, which inhibits the expansion of the upper passage vortex. Near the “d” slice, the tip leakage vortex can be observed in zone III (in Fig.7(b)), and the blocking effect of the leakage flow on the transverse flow generates a strong scraping vortex.

Compared with the evolution of the vortices near the casing in case 1, case 2, the most significant difference is the relative position of the upper passage vortex and the tip leakage vortex. In Fig. 7(c), the upper passage vortex (zone B) in case 1 is close to the middle of the blade, and is far away from the tip leakage vortex (zone A). In Fig.7(d), the upper passage vortex in case 2 is closer to the tip leakage vortex.Compared with case 1, the intensity and the range of the two vortices in case 2 are obviously suppressed.The smaller the distance between the upper passage vortex and the tip leakage vortex, the stronger the interaction between them, which promotes the mixing between them. In addition, the approaching of the upper passage vortex and the tip leakage vortex reduces their ranges relatively, therefore, reduces the mixing of the low-energy fluids in these vortices with the main stream and reduces the mixing loss.According to the above analysis, the Liutex method is an effective method, and it can clearly show the structures and the evolution of the vortices in the turbine rotor passage.

3. Conclusions

In this paper, 4 vortex identification methods are applied to study the turbine rotor passages, and the applicability and the robustness of each method are analyzed. The Liutex method is also applied to the vortex analysis in 2 cases of a turbine rotor passage with different angles of attack. The following conclusions are drawn:

(1) TheΩmehod and theΩ-Liutex method can capture strong and weak vortices simultaneously in turbine rotor passages, with a high robustness. TheQcriterion and the Liutex method are greatly affected by the threshold. When the threshold is small,the weak vortices can be identified by theQcriterion and the Liutex method, however, with not very clear structures. Therefore, theΩmethod and theΩ-Liutex method come into being for capturing both strong and weak vortices and the vortex structures in turbine rotor passages.

(2) A vortex is intuitively recognized as a part of fluids in the rotational motion, and the streamlines are often used as an intuitive and rough method for judging the location of the vortices. There is a high consistency between the vortices visualized by the Liutex method and the vortices visualized by the streamlines in cases of strong vortices, while, the weak vortices visualized by theΩ-Liutex method have higher consistency with those visualized by the streamlines.

(3) TheΩmethod, the Liutex method and theΩ-Liutex method have obvious advantages in distinguishing the vortices from the high shear regions,as compared with theQcriterion, maybe due to the less shear contamination in these three methods. By using the Liutex method, it is found that, the high loss regions in the turbine rotor passages are concentrated in the high shear regions, while the losses within the core of the vortex are small.

(4) With the application of the Liutex method,the difference in the structure and the evolution of the vortices in the turbine rotor passages with different attack angles can be clearly observed. At a negative angle of attack, the tip leakage vortex and the upper passage vortex are effectively suppressed, and the interaction between the two vortices is enhanced,which reduces the mixing between the vortices and the main stream. It is shown that the Liutex method is an effective method and can be applied to the analysis of the vortex evolution in turbine rotor passages.

Acknowledgements

This work is accomplished by using the code RortexUTA and the code Omega-LiutexUTA which are released by Chaoqun Liu at University of Texas at Arlington.