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# International Journal of Turbo & Jet-Engines

Ed. by Sherbaum, Valery / Erenburg, Vladimir

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Volume 33, Issue 1

# Numerical Study of the Effect of Secondary Vortex on Three-Dimensional Corner Separation in a Compressor Cascade

Yangwei Liu
• School of Energy and Power Engineering, National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, Beihang University, Beijing 100191, China
• Collaborative Innovation Center of Advanced Aero-Engine, Beijing 100191, China
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• Other articles by this author:
/ Hao Yan
• Corresponding author
• School of Energy and Power Engineering, National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, Beihang University, Beijing 100191, China
• Collaborative Innovation Center of Advanced Aero-Engine, Beijing 100191, China
• Email
• Other articles by this author:
/ Lipeng Lu
• School of Energy and Power Engineering, National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, Beihang University, Beijing 100191, China
• Collaborative Innovation Center of Advanced Aero-Engine, Beijing 100191, China
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• Other articles by this author:
Published Online: 2015-02-06 | DOI: https://doi.org/10.1515/tjj-2014-0039

## Abstract

The complex flow structures in a linear compressor cascade have been investigated under different incidences using both the Reynolds-averaged Navier–Stokes (RANS) and delayed detached eddy simulation (DDES) methods. The current study analyzes the development of horseshoe vortex and passage vortex in a compressor cascade based on DDES results and explores the effect of the passage vortex on corner separation using the RANS method. Results show that the effect of horseshoe vortex on three-dimensional corner separation is weak, whereas the effect of passage vortex is dominant. A large vortex breaks into many small vortices in the corner separation region, thereby resulting in strong turbulence fluctuation. The passage vortex transports the low-energetic flow near the endwall to the blade suction surface and enlarges corner separation in the cascade. Hence, total pressure loss increases in the cascade.

Keywords: cascade; corner separation; secondary flow; vortex

PACS: 47.11.-j

## 1 Introduction

Three-dimensional (3D) separations, which are also referred to as corner separation, have been identified as an inherent flow feature of the corner formed by the blade suction surface and endwall of axial compressors [1]. In compressors, the high loading needs more turning angle achieved in blade passage, thus the flow is much more susceptible to separate [2]. The 3D corner separation may lead to deleterious consequences, such as passage blockage, limiting on static pressure rise, a considerable total pressure loss and reduction in compressor efficiency, and eventually stall and surge especially for highly loaded compressor [3]. Hence, the corner separation should be predicted accurately through computational fluid dynamics (CFD) and controlled effectively to enhance compressor performance in the routine design. However, this issue is difficult to manage and control. This is perhaps primarily because the nature and characteristics of 3D separations in compressors are not clearly understood, nor are the mechanisms and factors that influence their growth and ultimate size [4].

The corner separation that appears reverse flow on both endwall and suction surface in compressor is typically 3D; sometimes if the reverse flow is enlarged to some extent that the separation deteriorates compressor performance seriously, the corner stall appears. The principal 3D effect is the secondary flow, caused by the cross-passage pressure gradient, which lowers stagnation pressure and introduces low-momentum fluid into the hub-corner region. The types of secondary flow structures in compressor cascades include horseshoe vortex, passage vortex, shedding vortex and leakage vortex. These structures significantly influence evolution of corner flow [5, 6].

Several early studies have discussed the relation between the effects of secondary flow and the mechanism of 3D corner separation. Based on the flow topology on blade suction surface and endwall, Schulz [7] anticipated a vortex ring structure in corner separation region, which is formed by joining the blade suction surface and the hub somewhere near the blade trailing edge. Hah and Loellbach [8] noted that a strong twister-like vortex was formed near the rear part of the blade suction surface. The low-momentum fluid within the hub boundary layer was transported toward the suction side of the blade by this vortex. Gbadebo et al. [4] once considered the leading-edge horseshoe vortex plays a major role in mechanism of 3D separation. These studies indicate that 3D corner separation is closely related to the secondary flow in compressor cascades. However, the opinions regarding vortex structures in the separation region differ. Moreover the effect of secondary flow on the inducement of 3D corner separation is controversial.

To enhance our conventional understanding of corner separation and to investigate the effect of secondary vortices on 3D corner separation, numerical simulations are conducted under different incidences in a linear Prescribed Velocity Distribution (PVD) compressor cascade, in which cascade the flow mechanisms and some control methods for 3D corner separation have been investigated at Cambridge University [911]. In this paper, the detailed vortex structures in corner separation are analyzed using the delayed detached eddy simulation (DDES) method. The effect of secondary flow on 3D corner separation is investigated with the Reynolds-averaged Navier–Stokes (RANS) method.

## 2.1 Description of PVD cascade

A linear compressor cascade obtained from Cambridge University is used as computational geometrical model to investigate the corner separation. The cascade parameters are summarized in Table 1.

## 2.2 RANS numerical method

In the simulation, one-blade passage is selected to investigate the separation. Periodic boundary conditions are set on the two sides of flow passage. As the blade is symmetric between the hub and tip without clearance, in order to reduce computation quantity, the computational domain is reduced to half span with symmetric condition set on the passage top surface.

Hexahedral structural meshes are generated in O4H topology by using NUMECA, Autogrid, shown in Figure 1. The y+ of first grid adjacent to the wall is approximately 1.0. A series of grids have been generated with different grid densities and distributions to check the grid independence of the solution. Cases with 0.41 million, 0.79 million, 1.2 million, 1.58 million, 1.97 million grid numbers are calculated, respectively, and the velocity distribution near the trailing edge are extracted, as shown in Figure 2. The results show there is little difference when the grid number exceeds 1.58 million. In order to ensure calculating accuracy, the case with 1.97 million is selected in this paper.

Figure 1:

Figure 2:

Velocity distribution at the trailing edge

Table 1:

The commercial CFD software FLUENT 6.3.26 is used to conduct the numerical simulations. The pressure-based implicit solver is applied. A second-order upwind scheme and the central-differencing scheme are used for the convection terms in RANS and DDES, respectively. In cases for RANS calculation, turbulence model is one of the key elements and a weakness in CFD for engineering [12, 13]; therefore, the Reynolds stress model (RSM) is used according to our previous studies [12, 14].

According to the experiment, the velocity of the main flow is set as 23.0 m/s given the profile of boundary layer at inlet, as shown in Figure 3. The turbulent intensity is 1.5% at inlet. Meanwhile, the endwall before the leading edge of blade is set as slip wall in some computational cases to generate different strength of secondary flow in cascade.

Figure 3:

Velocity profile at inlet

## 2.3 Validation of RANS numerical predictions

The oil flow visualization and the corresponding numerical limiting streamlines at 0° incidence are presented in Figure 4. The limiting streamlines on the blade suction surface and the endwall accurately indicate the starting point of the corner separation and the radial separation scale. The separation region is approximately similar in size to that in the experiment.

Figure 4:

Experimental oil flow visualization and numerical limiting streamlines

Furthermore, the validation is carried out on pressure distribution along blade surface to confirm that the simulation could reflect the real flows measured in the experiment. The static pressure coefficient ${C}_{p}$ is defined as follows: ${C}_{p}=\left(P-{P}_{1}\right)/\left({P}_{01}-{P}_{1}\right)$(1)

where ${P}_{1}$ and ${P}_{01}$ are the reference static and total pressure at inlet, while $P$ is static pressure at the local point.

Figure 5 displays the distribution of static pressure on the blade surface at 54.0% and 89.0% span, respectively. The numerical results agree with the experiment data well, regardless of whether the results are obtained from the middle region (54.0% span) or hub region (89.0% span) of the blade.

Figure 5:

Static pressure coefficient on blade surface at 0° incidence: (a) 54% span and (b) 89% span

By comparing with experimental data, the numerical simulation provides a solid foundation for separation physics analysis. Therefore, this RANS method is appropriate for analyzing the aerodynamic performance of the PVD cascade in the following sections.

## 2.4 DDES simulation method

The detached eddy simulation (DES) method was developed to address the challenge of high-Reynolds number and massively separated flows, which must be addressed in such fields as aerospace and ground transportation, as well as in atmospheric studies [15]. This method can accurately predict the complexity and unsteadiness naturally associated with the compressor flow. In 2006, Spalart et al. proposed DDES method, which incorporates a simple modification into the initial DES. DDES introduces kinematic eddy viscosity into Spalart-Allmaras (SA) turbulence model to take both effects of grid spacing and eddy-viscosity field into considerations [16]. In order to study the evolvement of secondary vortex in compressor cascade, DDES method based on SA model is used in some cases. Considering computing consumption and difficulties of making huge numbers of transient flow field time averaged, only part of results in the paper are analyzed using DDES method.

In cases for DDES calculation, the most settings are the same as RANS case. The differences are: (1) the grid number reaches 10 million in the whole passage which meets the requirement of DDES; (2) SA turbulence model is employed; (3) dual time step method is applied with an outer iteration physical time step of 1.0 × 10−5 s and 50 inner iterations per physical time step.

## 3 Vortex structure in PVD cascade with DDES results

DDES method can help to clarify flow process in the cascade because detailed vortex structures and turbulent fluctuation can be captured. According to Figure 6, a large vortex is generated at the corner separation region. From the velocity contour in Figure 7, the velocity at the rear region of blade suction surface is obviously lower than that in the other parts. A large vortex breaks into many small stripe vortices at the trailing edge of the blade and these stripe vortices mix with the free stream flow exhibited in Figure 6. Hence, there are many small areas in the low-velocity region instead of a whole area in Figure 7. As per velocity contour presented in this figure, the corner separation starts approximately at 40% chord and enlarged gradually along the suction surface.

Figure 6:

Iso-surface of vorticity magnitude (Ω = 3,000)

Figure 7:

Velocity contour in transient flow filed

Figure 8 shows the slice cut at the trailing edge of the blade. Many small vortices are generated in the corner separation region, and their rotating direction is opposite to that of adjacent vortices. Part of the area in Figure 8 is magnified in Figure 9; thus, the location and rotating direction of the vortex can be clearly observed. These vortices, shedding from the wall or induced by other vortices, lead to strong turbulence fluctuation in the corner region, as the pressure fluctuation of one monitor point in the corner separation region shown in Figure 10.

Figure 8:

Contour of streamwise vorticity magnitude

Figure 9:

Streamlines at the blade trailing edge

Figure 10:

Static pressure of a monitor point in corner separation region

The vorticity criterion $Q$ is used to identify the major vortex structures in the corner separation region. $Q$ is defined as: $Q=\frac{1}{2}\left({\mathrm{\Omega }}_{ij}{\mathrm{\Omega }}_{ij}-{S}_{ij}{S}_{ij}\right)$(2)where ${\mathrm{\Omega }}_{ij}$ is the vorticity tensor and ${S}_{ij}$ is the shear strain tensor.$Q$ represents the local balance between the shear strain rate and vorticity magnitude [17, 18]. Figure 11 depicts the iso-surface of $Q$=100,000 at 0° and 5° incidences, respectively. The horseshoe vortex that formed at the blade leading edge bifurcated into suction side leg and pressure side leg. The pressure side leg vortex runs toward the suction surface of adjacent blade because of pitchwise pressure gradient and merges with the passage vortex gradually. The suction side one dissipates quickly near the blade suction surface as the vortex is counterclockwise (looking from upstream) whose rotation is opposite to the main stream. The strength of horseshoe vortex enhances as the incidence increased, especially the pressure side leg vortex. The passage vortex that starts at the mid-chord position evolves quickly in the cascade and becomes the dominant vortex structure in the cascade passage. The passage vortex induces the corner vortex and its strength greatly affects the scale of the corner separation region.

Figure 11:

Iso-surface of Q = 100,000: (a) at 0° incidence and (b) at 5° incidence

## 4 Effects of secondary vortex on 3D corner separation in compressor cascade with RANS results

The basic flow mechanism of 3D corner separation has long been universally accepted. According to Mertz [19] and Horlock et al. [20], two basic mechanisms could affect the formation of 3D separations in compressor blade passages: the adverse pressure gradient in the streamwise direction and the secondary flow effects (circumferential and radial flow migration in the near-wall regions and the skew of the endwall boundary layer) in cascade passages. As the major secondary flow in compressor stator cascade, the horseshoe vortex and passage vortex develops differently in compressor cascade, so their effects on 3D corner separation is quite different.

As depicted in Figure 12, the streamlines near the endwall are slightly twisted. Moreover, no strong vortex flow is generated by horseshoe vortex at the leading edge of the blade. Further studied from Figure 13, one strand of streamlines (the pink ones) beside the suction surface begins to separate from the leading edge at 40% chord. This strand rolls up along the suction side of the blade and finally flows out of the cascade. The other strand of streamlines separates at nearly the same position and twists in the space downstream of the cascade to form the passage vortex. The analysis results of the streamlines in cascade indicate the effect of horseshoe vortex on corner separation is weak. Hence, we focus on the effect of passage vortex on the corner separation, which is the dominant secondary vortex, in the following sections of this paper.

Figure 12:

Streamlines near the endwall

Figure 13:

Streamlines near the endwall and blade

To study the effect of passage vortex on corner separation, the endwall before the leading edge part is set as slip wall in some computation cases, labeled as slip case, as shown in Figure 14. When air approaches the leading edge of the blade, the boundary layer in slip case is not fully developed at the leading edge, so the passage vortex is weaker compared with that in the non-slip case as shown in Figure 15.

Figure 14:

Endwall boundary condition in slip case

Figure 15:

Streamlines at different streamwise sections at 0° incidence. (a) non-slip case. (b) slip case

Figure 16 shows the surface limiting streamlines of the non-slip case and slip case at 0° incidence. The extent of separation area reaches at nearly 40% span near the trailing edge in non-slip case, whereas it only reaches 25% in the slip case. The comparison shows the separation area in the non-slip case is larger than that in slip case. Thus the passage vortex can lead to severer separation in the cascade.

Figure 16:

Limiting streamlines at 0° incidence: (a) non-slip case and (b) slip case

The loss in compressor cascade is defined as Yp = (P01P0)/(0.5 × ρ × V12). The total pressure loss Yp is a key value to access the compressor performance. High-valued Yp indicates significant loss in the compressor cascade. Figure 17 compares the total pressure loss near the blade trailing edge at incidences of 0° and 5°, respectively. When the blade operates at design condition (0°), the loss core (high-value loss region) is larger and the boundary layer thickness is thicker in non-slip case than that in the slip case. The loss core in non-slip case is close to the main stream. The trend at incidence of 5° is similar, and the loss core area is significantly larger than that at 0° incidence. When the passage vortex exists, the corner separation region is enlarged and its total pressure loss increases. From all the comparison, it can be postulated that the passage vortex transports the low-energetic flow near the endwall to the suction surface, which forms the corner region in the cascade. Because the passage vortex is weak in slip case, the flow excursion near the endwall is insignificant. Furthermore, the size of 3D corner separation is small.

Figure 17:

Contours of exit total pressure loss coefficient at 50% chord from the trailing edge: (a) non-slip case at 0°; (b) slip case at 0°; (c) non-slip case at 5° and (d) slip case at 5°

Figure 18 depicts the spanwise distribution of pressure on suction surface. The C-shaped pressure distribution can be observed before 50% chord position. The static pressure near the hub is higher than that at middle span. This result is mainly attributed to the following reasons: the cross-pressure gradient from the pressure surface to the suction surface and the accumulation of low-energetic fluid from the boundary layers of both hub and suction surface. The C-shaped pressure distribution gradually weakens after the 50% chord position because corner separation is initiated and the boundary layer separates on blade suction surface. Figure 19 compares pressure distribution on the suction surface in the non-slip case and slip case. The results suggest the static pressure rise considerably more near the endwall in the non-slip case than in the slip case. Therefore, the extent of low-energetic fluid accumulation in the non-slip case is greater than that in the slip case, which indicates the secondary flow is important in this phenomenon.

Figure 18:

Static pressure distribution on suction surface at 0° incidence

Figure 19:

Comparison of static pressure distribution on suction surface at 0° incidence

As suggested by Gbadebo et al. [4], the thickness of 3D separated layer can be simply denoted using the concept of relative displacement thickness ${\mathrm{\delta }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\ast }$, which is defined as follows: ${\mathrm{\delta }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\ast }=\frac{2}{R}{\int }_{{r}_{\mathrm{h}\mathrm{u}\mathrm{b}}}^{{r}_{\mathrm{m}\mathrm{i}\mathrm{d}}}\left[{\mathrm{\delta }}^{\ast }\left(r\right)-{{\mathrm{\delta }}^{\ast }}_{\mathrm{m}\mathrm{i}\mathrm{d}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}}\right]dr/c$(3)where displacement thickness is given as: ${\mathrm{\delta }}^{\ast }\left(r\right)={\int }_{0}^{\mathrm{\delta }}\left[1-\frac{\mathrm{\rho }v\left(r,s\right)}{{\mathrm{\rho }}_{\mathrm{f}\mathrm{s}}{V}_{\mathrm{f}\mathrm{s}}}\right]dy$(4)The relative displacement thickness in non-slip case is thicker than that in slip case. It indicates the boundary layer of the blade suction surface in the non-slip case is thicker than in the slip case and the corner separation in the former is severer, as illustrated in Figure 20.

Figure 21 displays the outflow angle at the cascade exit. As a result of 3D corner separation, the outflow angle increases with the increase in span fraction from 0.5 to 0.9. This angle decreases sharply at the endwall region because of the occurrence of overturning in both cases. The corner separation is larger in the non-slip case than in the slip case; thus, the boundary layer is thicker on the surface and the outflow angle is larger. In particular, the outflow angle in the non-slip case decreases significantly at the span fraction ranges from 0.9 to 1.0 as a result of the considerable overturning in the cascade that is caused by the secondary flow.

Figure 20:

Relative displacement thickness at cascade exit at 0°

Figure 21:

Outflow angle at 0° incidence

Based on the incidence characteristic presented in Figure 22, the total pressure loss increases suddenly when the incidence exceeds 3°, which indicates the corner stall happens in cascade. The total pressure loss is larger in the non-slip case than in the slip case regardless of whether or not the corner stall appears. Therefore, the passage vortex enlarges separation and aggravates the aerodynamic performance in compressor cascade.

Figure 22:

Loss of total pressure coefficient at different incidences

## 5 Conclusion

In this study, both DDES and RANS methods are applied to investigate the development of secondary vortices and their effects on corner separation in a linear compressor PVD cascade. The detailed secondary flow structures are analyzed using DDES results. The effects of secondary vortices are explored by RANS method with different endwall boundary conditions before blade leading edge. The conclusions are obtained as follows:

In the corner separation region, a large vortex breaks into many small vortices. These vortices result in strong turbulence fluctuation in the corner separation region. Hence, 3D corner separation is an unsteady phenomenon in compressor cascade.

Horseshoe vortex and passage vortex are generated in the compressor cascade. The horseshoe vortex which is formed before the leading edge is bifurcated into suction side leg and pressure side leg. The suction side leg disappears quickly when the flow passes through the cascade. The pressure side leg merges with passage vortex and moves to the suction surface of the adjacent blade. The horseshoe vortex has a very weak effect on 3D corner separation.

The passage vortex starting at mid-chord position in cascade passage develops into a dominant secondary vortex and significantly enhances corner separation in the PVD cascade. The passage vortex transports the low-energetic fluid within the endwall boundary layer toward the blade suction surface and increases total pressure loss.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 51376001, No. 51420105008, No. 51136003, No. 51006006), the National Basic Research Program of China (2012CB720205, 2014CB046405), the Aeronautical Science Foundation of China (2012ZB51014), the Beijing Higher Education Young Elite Teacher Project and the Fundamental Research Funds for the Central Universities. The authors would like to thank Whittle Laboratory and Rolls-Royce Plc for providing their experimental results. The authors also thank Aeroengine Simulation Research Center of Beihang University for the permission of using FLUENT.

## Nomenclature

C

Chord

Cp

Static pressure coefficient

Yp

Loss of total pressure coefficient

P1

Reference static pressure at inlet

P01

Reference total pressure at inlet

P

Static pressure at the desired point

P0

Total pressure at the desired point

R

Gas constant

${\mathrm{\delta }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\ast }$

Relative displacement thickness

${\mathrm{\rho }}_{\mathrm{f}\mathrm{s}}$

Free stream density

Vfs

Free stream velocity

V0

Mass-averaged velocity at inlet

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Accepted: 2015-01-15

Published Online: 2015-02-06

Published in Print: 2016-04-01

Citation Information: International Journal of Turbo & Jet-Engines, Volume 33, Issue 1, Pages 9–18, ISSN (Online) 2191-0332, ISSN (Print) 0334-0082,

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