Vortex dynamics and entropy generation in separated transitional flow over a compressor blade at various incidence angles

Mingyang WANG, Ziliang LI, Ge HAN, Chengwu YANG,Shengfeng ZHAO, Yanfeng ZHANG, Xin’gen LU,*

a Key Laboratory of Light-Duty Gas-Turbine, Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing 100190, China

b University of Chinese Academy of Sciences, Beijing 100049, China

KEYWORDS Aerodynamic performance;Compressor airfoil;Entropy;Incidence angle;Separation bubble;Vortex dynamics

Abstract The transition process within a Laminar Separation Bubble(LSB)that formed on a compressor blade surface was investigated using Large Eddy Simulations(LESs)at a Reynolds number of 1.5×105 and incidence angles of 0°,+3°,and+5°.The vortex dynamics in the separated shear layers were compared at various incidence angles and its effects on the loss generation were clarified through entropy analysis. Results showed that transition onset, which was accurately identified by the Linear Stability Theory (LST), was significantly promoted at the increased incidence angle. As such, the development of LSB was suppressed and the relative role of viscous instability played in the transition process was weakened. At the incidence angle of 0°, two-dimensional spanwise vortices detached from the blade surface and rolled up periodically, which were further stretched and eventually evolved into large-scale hairpin vortices.As time passed,the fully developed hairpin vortices broke down into small-scale eddies.Meanwhile,the flow near the wall reversely ejected into the outer separated shear layers and a sweeping process happened subsequently, forcing the separated shear layers to reattach and accelerating the generation of turbulent fluctuations. By comparison,the strength of vortex rolling-up was weakened at higher incidence angles, and the vortex pairing and breakdown of large-scale vortices were less pronounced. Therefore, the level of turbulent fluctuations that generated in the separated shear layers was reduced.Detailed entropy analysis showed that the turbulent dissipation effect related to the Reynolds shear stresses determined the largest amount of positive entropy generation, which declined to a lower level as the incidence angle increased from 0° to +5°. Correspondingly, the profile loss was reduced by 50.4%.

1. Introduction

In recent years,reducing the weight and improving the aerodynamic performance of components have become common practices in turbomachinery design. In general, weight reduction inevitably increases the blade loading and results in a stronger adverse pressure gradient, which may trigger laminar separation on the blade surface, particularly for the components that operate at fairly low Reynolds numbers. As such,the unsteady separated shear layers undergo a transition to turbulence and reattach, and a Laminar Separation Bubble(LSB) appear.Unfortunately, the transition within an LSB will cause larger profile lossand seriously damage the adiabatic efficiency and operating range of compressors.Therefore, it is of crucial importance to investigate the transition process within an LSB and its effect on the loss generation.

Numerous experimental and numerical investigations have reported that the transition process in LSB depended on several parameters, such as the Reynolds number, free-stream turbulence intensity, pressure gradient, and blade surface roughness.Most of these investigations addressed the vortex dynamics and instability mechanisms in the separated shear layers,and it has been established that the transition process is highly dependent of the unsteady vortex behavior.A scanning Particle Image Velocimetry (PIV) measurement was conducted by Burgmann et al.to detect the dynamics of the vortex rolling-up on the surface of a SD7003 airfoil.The results showed that quasi-periodic large-scale vortices rolled up downstream of the LSB and abruptly burst with a strong ejection of fluid into the mean flow. Li and Yanginvestigated the effects of freestream turbulence on the transition process.They found that the streamwise streaks caused by freestream turbulence accelerated the distortion of spanwise vortices and promoted the transition process. Li et al.numerically and experimentally investigated the threedimensional vortex structures that formed on a NACA 0012 airfoil in the dynamic stall stage.Results showed that a leading edge vortex detached from the blade surface, which interacted with the streamwise vortices and accelerated the laminar-toturbulent transition.

The vortex dynamics in separated shear layers also behaved differently at various incidence angles. Lambert and Yarusevychexperimentally compared the complicated vortex evolution process over a NACA 0018 airfoil at the incidence angles of 0°, 5°, 8°, and 10°. It was reported that the vortex shedding frequency increased and the vortex merging was more pronounced at larger incidence angles. The PIV technique was applied to investigate the unsteady vortex structures on a compressor blade by Shi and Ma. They found that the separated shear layers remained coherent near the trailing edge at a lower incidence angle,and the separation occurred earlier with the increase of incidence angle. Wang and Zhaoalso conducted a detailed PIV measurement to study the dynamic stall phenomenon of two different typical rotor airfoils. The results revealed that the strength of the leading edge vortex behaved differently at various incidence angles,which also significantly altered the dynamic stall characteristics.

Recently,several researchers have paid attention to the loss mechanisms in the LSB.For example,the concept of deformation work terms was used by Simoni et al., who reported that the vortex rolling-up and subsequent breakdown were responsible for the largest amount of loss production. Similarly, the entropy analysis was also conducted to quantify the loss source in the transition process,it was demonstrated that the entropy production rate in the boundary layer increased as the laminar separation moved upstream. A PIV measurement was conducted by Skifton et al.to investigate the entropy generation in a bypass transition process. They found that the viscous effect became less dominant in the transition process, while the contribution of turbulent dissipation to the entropy generation increased. Furthermore, to reduce the flow loss in the boundary layer separation or transition process,some passive or active flow control methods have also been developed.

Although significant progress has been made towards a deep insight into the transition process in the LSB,the detailed vortex dynamics and its effects on the loss generation process at various incidence angles are still far from being completely understood.In the present study,the transition process within an LSB formed on a compressor blade at the incidence angles of 0°,+3°,and+5°(the Reynolds number based on the blade chord length and the inlet velocity was 1.5 × 10) was investigated using Large Eddy Simulations(LESs).The amplification of turbulent fluctuations in LSB was analyzed, and the transition onset was clearly identified by the Linear Stability Theory(LST).Then we compared the vortex evolution process at various incidence angles and analyzed its effects on the generation of turbulent fluctuations. Finally, a detailed analysis was conducted to evaluate the contribution of viscous dissipation and turbulent dissipation effects to the entropy generation,and the underlying relationship between the complicated vortex dynamics and the loss level was clarified.

2. Compressor airfoil

A compressor airfoil named V103was selected in the present study (Fig. 1). The inlet Mach number (Ma) is 0.67 and the inlet Reynolds number (Re) is 1.5 × 10at the design condition. The parameters of the V103 cascade are shown in Table 1.

Fig. 1 V103 cascade.

Table 1 Aerodynamic and geometry parameters of V103 cascade13.

3. Numerical methods

3.1. Solution method

The flow characteristics in the LSB were resolved by LES with the commercial software ANSYS-CFX. In LES, the scale of vortices is compared with the filter width to identify the socalled large- and small-scale eddies. Then, the large-scale eddies can be directly resolved, while the energy transfer between the large- and small-scale eddies is modeled by the Subgrid-Scale(SGS)model(LES WALE model).The spatial discretization was realized through a second-order central differencing scheme, and the time marching was based on a second-order backward Euler scheme.

3.2. Boundary conditions

The inflow plane in the computational domain was located at 1.5Cupstream of the leading edge. The total pressure P,total temperature T, and inlet flow angle βwere specified,and no turbulent fluctuations were generated in the inlet plane.The outlet boundary was extended at 2Cdownstream of the trailing edge, and the average static pressure Pwas given.The spanwise size was set as 0.15C(0.162 C),which was adequate in revealing the evolution process of three-dimensional vortices by LES.The periodic conditions were imposed on the spanwise and pitchwise directions, and the blade surfaces were considered as no-slip adiabatic walls.

3.3. Spatial grid size

The grid size and topology were identical for all numerical cases(Fig.2),and the number of total grid nodes was approximately 9.6×10.The normalized grid size in the wall-normal direction satisfied Δy<1, and the maximum streamwise and spanwise grid space(Δx, Δz) was lower than 15 in the separated transitional region, as shown in Fig. 3. It has been demonstrated that the grid resolution was sufficient in LES.The time step in all calculations was set as 1.5 × 10s to ensure that the Courant-Friedrichs-Lewy(CFL)number was lower than 1.Each calculation run approximately 40 flow-through times to establish a statistical convergence state, and the turbulent statistics were collected in the subsequent 8 flow-through times.

4. Results and discussion

In the present study, the structure of LSB and associated turbulent fluctuations were first compared at various incidence angles.Next,the transition onset location was accurately identified by the LST, and the instability mechanisms dominating the transition process were clarified. Then, the complicated vortex dynamics at various incidence angles were compared and its effects on the generation of turbulent fluctuations were discussed. Finally, a detailed entropy analysis was conducted to highlight the loss mechanism in the separated transitional flow.

Fig. 2 Computational grid of V103 cascade.

Fig.3 Distribution of normalized grid size in streamwise(Δx+),wall-normal (Δy+), and spanwise (Δz+) directions.

Fig. 4 Comparison of isentropic Mach number at various incidence angles.

4.1. Time-averaged flow characteristics at various incidence angles

The predicted isentropic Mach number (Ma) at various incidence angles was compared in Fig. 4.The numerical result on the suction surface generally coincided with the experimental data at the incidence of 0°.However, a little discrepancy was observed from x/C=30%to x/C=45%,which may be caused by the difference between the level of freestream turbulence intensity imposed on the inlet plane in the experimental and numerical studies.At the incidence angle of 0°(i=0°),a constant pressure region was observed from the Madistribution, indicating that the laminar separation and the subsequent transition appeared. From the skin friction coefficient(C) shown in Fig. 5, the laminar separation occurred at x/C= 32.5%, which underwent a transition to turbulence and reattached in the vicinity of x/C= 75.0%. Therefore,a closed LSB that covered 42.5% of the axial chord length formed. At the incidence angle of +3° (i=+3°), the airflow more rapidly accelerated near the leading edge and subsequently diffused. The location of laminar separation moved to x/C= 27.2%, and turbulent reattachment occurred at x/C=64.2%.Compared with the case of i=0°,the bubble length was decreased by approximately 5.5% of the axial chord. With the incidence angle increasing to + 5°, the laminar separation and turbulent reattachment occurred much upstream. As such, the development of LSB was further suppressed(the length was decreased by 15.6%of the axial chord compared with the case of i = 0°), which was also evidenced by the shorter plateau region in the Madistribution.

Fig.5 Comparison of skin friction coefficient on suction surface at various incidence angles.

Fig. 6 Time-averaged streamwise velocity on suction surface at various incidence angles.

Fig. 7 Root-mean-square velocity fluctuations on suction surface at various incidence angles.

4.2. Instability mechanisms in transition process at various incidence angles

The development of LSB was directly related to the transition process.The amplification processes of velocity fluctuations at three incidence angles were compared in Fig. 8 (a), and the amplitudes located in the separated shear layer were presented in a logarithmic form (Fig. 8(b)). At each incidence angle,exponential growth was observed in a specific streamwise region, as marked by the inclined line, which matched well with the LST.Based on the LST,the amplification of fluctuations in separated shear layers first followed an exponential path (similar to the amplification process that occurred in the attached boundary layer).However,with the further amplification in the fluctuations, three-dimensional nonlinear effects appeared.As such,the growth began to deviate from the exponential path (cannot be predicted by the LST) and the transition onset occurred.Therefore, one can see that the transition onset took place at x/C= 57.5% at the incidence angle of 0°.With the increase in incidence angle,the transition onset occurred much earlier at x/C= 52.5% and x/C= 45% at the incidence angles of +3° and +5°, respectively. It is of interest to note that the amplification rate of velocity fluctuations in the separated shear layer was nearly independent of the incidence angle, but the location of initial growth in velocity fluctuations shifted upstream at higher incidence angles, which contributed to an earlier transition onset.The transition completion was estimated to be located at the maximum velocity fluctuation, it can be seen that the transition also completed more rapidly at larger incidence angles,which enhanced the turbulent momentum exchange and accelerated the separated shears to reattach. Therefore, the turbulent reattachment also moved upstream.

To further detect the instability mechanisms that determined the transition process at various incidence angles, the Power Spectrum Density (PSD) was calculated based on the instantaneous pressure data, as shown in Fig. 9. At the incidence angle of 0°, a distinct peak was observed at a frequency of 7731 Hz (the dominant frequency f), which was related to the periodic vortex shedding.At higher incidences of+3°and+5°, a dominant frequency also appeared, which occurred at f=7499 Hz and f=6309 Hz,respectively.This result indicated that the incidence angle significantly affected the unsteady vortex behavior in the transition process.

Numerous investigations have demonstrated that the Tollmien-Schlichting (T-S) mechanism was responsible for the breakdown into turbulence.However, some studies reported that the Kelvin-Helmholtz(K-H)instability was decisive.It was also proposed that the two instability modes might coexist in LSB and that the relative role was determined by the boundary layer velocity profile and strength of the reverse flow.The dominant frequencies for the three cases were scaled by the boundary layer velocity (U) and momentum thickness (θ) at the separation point to obtain Strouhal number St(St=fθ/U), as listed in Table 2. One can see that the Strouhal numbers were in the ranges of K-H instability reported in previous studies.Therefore, the inviscid K-H mechanism showed a great impact on the transition process.

Fig. 8 Velocity fluctuations amplification along streamwise direction at various incidence angles.

Fig. 9 Comparison of PSD at various incidence angles.

Table 2 Comparison of Strouhal numbers in several studies.

Rist and Maucheralso documented that the instability mechanism showed a large difference in various regions. For the outer separated shear layer, the inviscid K-H mechanism caused by the inflectional velocity profile was dominant, and the effect of Reynolds number is marginal. However, for the region in the reverse-flow,the viscous T-S instability was dominant and was susceptible to the Reynolds number.Moreover,the near-wall fluctuations may propagate upstream and can be characterized as an absolute instability.Increasing the distance from the wall enhanced the inviscid K-H instability in the outer separated shear layers,while the inner viscous instability would be promoted in a stronger reverse-flow region.Although the K-H mechanism is active in the transition process (Fig. 9), the inner fluctuation peak near the wall (Fig. 7)suggested the coexistence of the viscous instability.With the increase in incidence angle, the development of LSB was suppressed. Correspondingly, the strength of reverse flow was reduced and the inner peak in the velocity fluctuations was weaker(Fig.7).Therefore,it was assumed that the relative role of viscous instability in the near-wall region became less important.

4.3. Effects of incidence angle on vortex dynamics in transition process

As shown in Fig. 9, the vortex shedding frequency decreased with the increase in incidence angle, implying that the vortex dynamics behaved differently at various incidence angles.The transient vortex structures were revealed by an isosurface,which was identified by the Q-criterionand colored by the Mach number (Ma), as shown in Fig. 10. At the incidence angle of 0°, two-dimensional spanwise vortices periodically rolled up at x/C= 52.5% (t = t). Then the spanwise vortices (such as vortex B) gradually deformed (t = t+ 2Δt)and were stretched toward the streamwise direction in the vortex pairing process (t = t+ 3Δt). As such, the coherency of distorted vortices degraded, which broke down locally and generated small-scale vortices(t=t+4Δt).The appearance of small vortices in the vortex pairing indicated that a secondary instability was triggered and the transition onset occurred.As time passed, these fully-developed vortices eventually evolved into hairpin vortex structures, such as vortex Aat t = t+ 4Δt and t = t+ 5Δt. Due to the strong shear stress in the near-wall region,the head of the hairpin vortices gradually evolved into an elongated neck (marked by the red circles at t = t+ 4Δt and t = t+ 5Δt) and completely broke down into small turbulent eddies near the reattachment,which also indicated the completion of the transition process.

At the higher incidence angle of +3°, the unsteady vortex behavior,such as the deformation of two-dimensional vortices,vortex pairing or merging, and breakdown of large-scale vortices were similar to those of i = 0° case. However, there are also some differences. For example, the initial shedding of the two-dimensional spanwise vortices occurred earlier at x/C= 45.0%. Moreover, the distance between the two neighboring spanwise vortices was slightly larger. As the incidence angle increased to +5°, the development of the LSB was further suppressed. Correspondingly, the strength of the vortex rolling up was weakened and the breakdown of large-scale hairpin vortex structures was less pronounced.

The near-wall streamlines at several time steps were presented in Fig.11.At three incidence angles,a clockwise vortex was induced by the K-H instability, and strong reverse flow appeared inside it. Furthermore, a secondary vortex (marked by the red rectangle) was also observed downstream of the main vortex. As such, the near-wall flow from downstream ejected reversely into the separated shear layer,and a sweeping process happened subsequently, which forced the separated shear layers to reattach and accelerated the generation of turbulent fluctuations. By comparison, the level of reverse flow was reduced at higher incidence angles. Also, the ejectionsweeping process was suppressed.

Fig. 10 Spanwise vortex evolution process at various incidence angles.

It is of interest to note that the reverse flow mixing inside the bubble, the breakdown of three-dimensional hairpin vortices, and the ejection-sweeping of the near-wall flow directly determined the generation of turbulent fluctuations in the separated shear layers. With increasing the incidence angle, the development of LSB was suppressed. For such a condition,the level of reverse flow mixing was reduced. Meanwhile, the strength of rolling-up and breakdown of large-scale vortices were also weakened. Therefore, the resulting turbulent fluctuations declined to a lower level, which was likely to have a great impact on the loss generation. More details will be discussed in the following section.

Fig. 11 Transient streamlines near suction wall at various incidence angles.

4.4. Entropy generation mechanism in transition process at various incidence angles

As mentioned above, the incidence angle significantly altered the vortex behavior in the transition process, which directly determined the loss level and the aerodynamic performance of the compressor airfoil.Fig. 12 compared the profile loss at various incidence angles (the total pressure loss coefficient ω was defined as the difference between the time-averaged total pressure at the inlet plane (P) and the outlet plane(P), which was further normalized by the dynamic pressure at the inlet plane (P- P), ω = (P- P)/(P- P)).One can see that the profile loss was decreased by 50.4% with the incidence angle increasing from 0° to +5°. In the next,entropy analysis will be carried out to clarify the loss mechanism at various incidence angles.

In the separated transitional flow,the entropy will be inevitably generated by viscous shear stresses and turbulent dissipation effect. To clarify the entropy generation mechanism at various incidence angles,a detailed analysis of how and where the local entropy was generated was carried out. The timeaveraged entropy generation can be expanded to two groups of dissipation terms; the first one represented the viscous dissipation (μΦ) and the other was the turbulent dissipation(ρε). The Entropy Generation Rate (EGR) per unit volume(S) can be expressed as Eq. (1):

where T was the temperature,and μ was the dynamic viscosity.

The viscous dissipation term (μΦ) was determined by the mean velocity gradient, and the turbulent dissipation term(ρε) was associated with the velocity fluctuation gradient.For the two-dimensional flow (x: streamwise direction, and y: wall-normal direction), μΦ and ρε can be expressed as Eqs. (2) and (3):

Fig.13 compared the viscous dissipation and turbulent dissipation that contributed to the total entropy generation rate at various incidence angles. At the incidence angle of 0°, the high-level viscous dissipation term μΦ was observed in the attached boundary layer(before laminar separation and downstream of the turbulent reattachment).However,the relatively high-level μΦ was also maintained along the external boundary of the separated shear layer due to strong shear in the inflectional velocity profile, but it rapidly decreased to a lower level near the maximum thickness of the bubble. For the entropy generation related to the turbulent dissipation ρε, the value before laminar separation was almost negligible, which was amplified just downstream of laminar separation and achieved its maximum near the region where the large-scale vortices completely broke down. It is noteworthy that the ρε also remained at a high level in the near-wall region downstream of the reattachment, which might be caused by the random fluctuations in the turbulent boundary layers. With the increase in incidence angle,the thickness of the separated shear layer was reduced. Therefore, the region of high-level μΦ moved closer to the wall.Meanwhile,as the strength of vortex rolling-up and breakdown was weakened, the region of highlevel ρε shrank and the total entropy generation rate declined to a lower level.

To further evaluate the effects of viscous and turbulent dissipation on the entropy generation, Swas integrated along the wall-normal direction and the resulting entropy generation rate per unit area (S) can be expressed as Eq.(4):

Terms I-Ⅵin Eq. (4) represented the contribution of viscous dissipation, Reynolds shear stress, Reynolds normal stress, turbulent energy flux, turbulent diffusion, and pressure diffusion to the entropy generation, respectively. As demonstrated by Walsh et al., the effects of the last two terms,namely the turbulent diffusion and pressure diffusion, were almost negligible on the entropy generation. Therefore, only the first four terms in Eq. (4) were considered.

The distributions of Ssubjected to various terms along the streamwise direction at three incidence angles were presented in Fig. 14. For the cases at all incidence angles, the variation tendency of Salong the streamwise direction was similar.One can see that the Reynolds shear stress term(S_)achieved its peak value near the maximum thickness of the bubble(approximately at the mean transition location),which contributed to the largest amount of positive entropy generation in the separated shear layers. By comparison, the effects of Reynolds normal stress(S_)and viscous dissipation (S) were less pronounced. It is also noteworthy that the term of turbulent energy flux Swas negative in most parts of the bubble region, which transported energy to the main flow.

Fig. 12 Comparison of profile loss at various incidence angles.

Fig. 13 Viscous and turbulent dissipation terms contributed to entropy generation rate at various incidence angles.

Fig. 14 Comparison of entropy generation rate per unit area S′′ at various incidence angles.

Fig. 15 Effect of incidence angle on entropy generation rate per unit area.

Fig. 16 Comparison of Reynolds shear stress on suction surface at various incidence angles.

5. Conclusions

The transition process within an LSB that formed on a compressor blade was investigated by LESs. The detailed vortex dynamics at the incidence angles of 0°, +3°, and +5° were compared and its effects on the loss generation were highlighted.

At the incidence angle of 0°, a wedge-shape LSB appeared on the blade surface, and strong reverse flow was induced inside it. As the incidence angle increased from 0° to +5°,the initial amplification rate of disturbances in the transition was nearly independent of the incidence angle, but the initial growth of fluctuations occurred much upstream at the higher incidence angle,which triggered an earlier transition.For such conditions, the length of LSB was reduced by 15.6% of the axial chord and the level of reverse flow was decreased.Therefore, the relative role of the viscous instability mechanism in the near-wall region was less important.

The vortex dynamics behaved differently at various incidence angles,which also directly determined the level of turbulent fluctuations in the separated shear layers.At the incidence angle of 0°, two-dimensional spanwise vortices detached from the blade surface and rolled up periodically, which were further stretched and showed three-dimensional characteristics in the vortex pairing process. With further amplification in the disturbances, the distorted vortices evolved into hairpin vortex structures and eventually broke down into small-scale turbulent structures.Meanwhile,the ejection-sweeping process was observed, which forced the separated shear layers to reattach and accelerated the generation of turbulent fluctuation.Due to the suppression of LSB at the enlarged incidence angles, the level of turbulent mixing in the reverse flow was reduced. Moreover, the vortex rolling-up and the breakdown of large-scale hairpin vortices were less pronounced. Therefore, the generation of turbulent fluctuations was weakened.

The contributions of the viscous and turbulent dissipation effects to the entropy generation were analyzed. Results showed that the turbulent dissipation term related to the Reynolds shear stress was responsible for the largest amount of entropy generation. As the vortex rolling-up and breakdown were weakened with the incidence angle increasing from 0°to +5°, the maximum Reynolds shear stress declined to a lower level. As such, the profile loss was decreased by 50.4%.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

This study was co-supported by the National Natural Science Foundation of China ( No. 51836008) and the National Science and Technology Major Project of China (No.2017-II-0010-0024).