Flow loss and structure of circular arc blades with different leading edges

Abstract

Due to much lower cost of development and production, circular arc blades are often applied to axial flow fans. However, compared with NACA 65 profile, circular arc blades have higher losses, which are caused by the formation of separation bubbles or complete separation at the leading edge. Therefore, it is necessary to identify a specific geometry of leading edge, which can reduce the separation bubbles or complete separation. In this article, circular arc blades with different leading edges are examined by numerical method. The numerical investigations were performed with Reynolds numbers of $2 \times 10^{5}$ and $4 \times 10^{5}$ . All examinations were performed for different incidence angles. The influence of the sidewall on the flow loss and behavior is taken into account. In this article, the influence of leading edge geometry and Reynolds number on the flow losses is shown. The occurrence of flow behavior such as separation bubbles at the leading edge, secondary flow, and corner stall is shown and discussed. The flow structure on the blade and the corresponding sidewall region is illustrated with numerical streamline figures.

Keywords

Circular arc blade leading edge geometry separation bubbles flow structure in the sidewall region

Introduction

Due to the resource shortage, many blade profiles have been designed and tested to develop a blade with high efficiency, for example, the well proven and explored NACA profiles (National Advisory Committee for Aeronautics).¹ NACA profiles can provide good functionality and performance, but their development and production are also very expensive. In order to reduce the manufacturing costs, an alternative approach is circular arc blade. However, the flow losses of circular arc blade is higher than NACA 65 profile. Therefore, it is necessary to find a way to reduce the flow losses of circular arc blade by investigating the flow structure, aerodynamic quality values, and different profile geometries.

Generally, the losses of a turbine blade can be divided into profile loss and secondary flow loss. The profile loss is caused by the growth of the boundary layer on the blade and the formation of separation bubbles. The secondary flow loss is caused by the friction of the blade and sidewall. Therefore, in this article, the flow behavior such as secondary flow and separation bubbles are investigated.

Secondary flow

In 1980, Langston² was the first to illustrate a secondary flow model for two-dimensional (2D) boundary layer along the sidewall. In the present days, the researches about secondary flow are focusing on how to reduce secondary flow and the corresponding influence of the induced conditions on the secondary flow. Zess and Thole³ tried to use a leading edge fillet to eliminate the horseshoe vortex at the leading edge of a turbine cascade. Liu et al.⁴ and Zander and Nitsche⁵ equipped synthetic jet actuators on the suction surface of a compressor cascade and investigated their influence on the secondary flow. Hermanson and Thole⁶ and Chen et al.⁷ studied the inflow variations (including inflow velocity, inlet boundary layer thickness, and inflow turbulence intensity) which affect the secondary flow behavior of a turbine cascade. Additionally, secondary flow is also normal in turbomachine, which reduces the energy performance and induces operation instability.^8,9

K Liesner et al.¹⁰ studied four different types of sidewall with suction slots and found out that with the reduction of the secondary flow, the total loss of compressor cascade can be significantly decreased. YC Nho et al.¹¹ determined that the cascade geometry can affect the secondary flow. Based on experimental and numerical investigations, HM El-Batsh¹² studied the secondary flow of an annular turbine cascade, and his study shows that the secondary flow is affected by the pressure gradient along blade span. In his case, the secondary flow near the hub was stronger than near the shroud, and the losses were concentrated near the hub.

In conclusion, the results about secondary flow can be written as follows:

Secondary flow is concentrated in blade–sidewall juncture. So, attempts of reducing secondary flow can take place in this region.

Secondary flow can be effected by inlet variations such as blade geometry, incidence, and Reynolds number.

Separation bubbles

Separation bubbles at the leading edge of blades are a challenging task of aerodynamics, and the behavior of separation bubbles have been examined by several authors with both numerical and experimental investigations. In 1963, Gaster¹³ researched systematically the behavior of separation bubbles. Using a low-speed water tunnel, Courtine and Spohn¹⁴ examined the separation bubbles for a constant flow over a generic bluff body with different front edge radii. Lamballais et al.¹⁵ studied the formation of the separation bubbles over a generic half body with a rounded edge based on Reynolds number of Re = 1250. His work determined the combined effects of curvature and aspect ratio on the separation bubbles.

Shyy et al.¹⁶ and Diwan and Ramesh¹⁷ found out that the variation of Reynolds number and attack angle has a significant effect on the length and height of separation bubbles. Using advanced laser anemometry techniques, Crompton and Barrett¹⁸ investigated the structure of the separation bubbles behind the sharp leading edge of a flat plate at Reynolds number from $Re = 0.1 \times 10^{5}$ to $Re = 5.5 \times 10^{5}$ . Crompton’s work shows that the length of the separation bubbles was first determined by the plate incidence and second affected due to an increased shear layer caused by the Reynolds number.

Hamakhan and Korakianitis¹⁹ and Zhang et al.²⁰ have changed the leading edge shape of Hodson–Dominy blade to investigate the influence of leading edge on the flow behavior and find a leading edge shape which can reduce the separation bubbles. SCT Perkins and AD Henderson²¹ tested the influence of Reynolds number and incidence on boundary layer development at the leading edge of a blade. His work shows that the length of separation bubbles seems to reduce with the increasing Reynolds number. By the investigation of the cascades with circular leading edge and elliptic leading edge, Liu et al.²² and APS Wheeler et al.²³ studied the influence of leading edge geometry on separation bubbles behavior. Their works show that the leading edge shape has a large influence on the formation of separation bubbles.

The results of the investigations about separation bubbles so far can be stated as follows:

The leading edge geometry, attack angle, and Reynolds number can affect the separation bubbles structure.

Usually, the length of separation bubbles decreases with the increasing Reynolds number.

Separation bubbles have a significant influence on the blade aerodynamic performance.

Circular arc blade

Most researches about circular arc blade are based on theory of Weinig,²⁴ who has developed an analytical theory based on potential flow for cascades consisting of thin-cambered blades. JB McDevitt et al.²⁵ have examined the transonic flow around a circular arc blade with Reynolds number from $1 \times 10^{6}$ to $17 \times 10^{6}$ . A design method for cascades consisting of circular arc blades with constant thickness has been worked out,²⁶ and it is based on a singularity method combined with a computational fluid dynamics (CFD)-data-based flow loss model. Suzuki et al.²⁷ have investigated the total pressure loss coefficient of thin circular arc blade of different camber angles. Suzuki’s work shows that the result of CFD simulation is sufficiently useful when the camber angle of the circular arc blade is below about $49.5 °$ . According to his result, the camber angle of all investigated blades in this article is chosen as $φ = 20 °$ .

A comparison between NACA 65 profile and circular arc blade for different spacing ratios, Reynolds numbers and incidences has already been made, and it can be stated that the main reason of the higher losses of circular arc blade is the separation bubbles induced by the leading edge of the blade. Therefore, the further research intends to find a leading edge geometry of circular arc blade, which can reduce separation bubbles.

In order to achieve this purpose, in this article, circular arc blade with different leading edges is examined by numerical method. The aims of this article are the following:

Investigate and compare the flow loss of circular arc blade with different leading edges based on 3D CFD simulations.

Identify a suitable leading edge geometry of circular arc blade, which can reduce separation bubbles.

Compare the flow structure and streamline of circular arc blade with different leading edges.

Show the flow loss in the sidewall region of the investigated blades.

The investigated blades

In this article, the investigated blades are circular arc blade with constant thickness, that is, $e = 3 mm$ , each blade possesses a camber angle $φ = 20 °$ and stagger angle $λ = 20 °$ (see Figure 3). The chord length l = 100 mm. The investigated spacing ratios is t/l = 1.0. The flow loss and behavior of all blades are examined in dependence of incidence and Reynolds number. The geometry of new leading edge is designed in use of the shape of ellipse (see Figure 1).

Figure 1.

Geometry variation of leading edge.

In order to keep the thickness of all blades constant, the length of minor axis of the ellipse a is fixed as a = 1.5 mm. The major axis of the ellipse b is changed from the original profile of b = 1.5 mm to three different leading edge geometries (3, 6, and 10 mm) with the corresponding names blade-ori, blade-3, blade-6, and blade-10 (see Figure 2). The configuration of the circular arc blade is shown in Figure 3, and the parameters are represented in Table 1.

Figure 2.

The investigated blades: (a) blade-ori (b = 1.5 mm), (b) blade-3 (b = 3 mm), (c) blade-6 (b = 6 mm), and (d) blade-10 (b = 10 mm).

Figure 3.

Nomenclature of investigated blades.

Table 1.

Parameters of used blades.

Blade type	Circular arc blade
Chord long, l (mm)	100
Camber angle, $φ (°)$	20
Stagger angle, $λ (°)$	20
Thickness/chord long (e/l)	0.03
Height of the blade/chord long (b/l)	2
Spacing ratio (t/l)	1.0

Numerical investigation

Three-dimensional (3D) flow domains were created for all blades with different leading edges. Using the software package Icem CFD from Ansys 15.0, the flow domains were discretized with block-structured grids. In order to determine the flow behavior of the boundary layers, the grids have been refined continuously toward the walls. The maximum spacing of the meshes to the walls is smaller than 0.02 mm, so that the $y^{+}$ is smaller than 2. A 3D mesh consists of about 4 million cells. Details of the grid are shown in Figure 4. The plane for computing the results is 50 mm downstream from the trailing edge of the blade (Figure 5).

Figure 4.

Grid detail: (a) grid detail on the leading edge and (b) grid detail on the trailing edge.

Figure 5.

Flow domain with boundary condition.

The simulation were conducted with software Fluent 15.0. Reynolds-averaged Navier–Stokes (RANS) equations were solved on an unstructured grid. A second-order finite-volume-discretization was applied to a collocated grid. In this article, the flow behavior such as transition from laminar to turbulent boundary and bubbles separations must be considered, so the transition shear stress transport (SST) model was selected as turbulence model. The transition SST turbulence model is based on the coupling of the SST transport equations with the work of Menter et al.,²⁸ who combines the SST-k-ω-turbulence model with empirical correlations for the calculation of the turbulent onset and the length of the transition region.

The numerical investigations are performed with Reynolds Numbers of $2 \times 10^{5}$ and $4 \times 10^{5}$ . The Reynolds number is defined by

Re = \frac{ϱ \cdot l \cdot C_{1}}{μ}

(1)

with $ϱ$ is the density of fluid, l is the chord length of blade, $C_{1}$ is inflow velocity, and $μ$ is dynamical viscosity.

In order to consider the influence of boundary layer on the sidewall, the velocity profile of incoming flow has been investigated based on the free-flow measurement in the wind tunnel, which is used in the laboratory. These velocity profiles have been implemented in the inlet boundary conditions with a user-defined function (UDF). The used velocity profiles are depicted in Figures 6 and 7. Here, $δ$ is the thickness of the boundary layer of the sidewall on the inlet, which is measured by the wind tunnel in our lab. x is the distance between the current position and the sidewall in the blade height direction (in this work is in x-direction). Therefore, when x/δ is 1, the current position is on the edge of the boundary layer. When x/δ is 0, the current position is on the sidewall. $C_{x}$ is the velocity at the current position x. $C_{\max}$ is the max velocity in the boundary layer of the sidewall on the inlet, which is also measured by the wind tunnel in our lab.

Figure 6.

Velocity profile on inlet, $Re = 2 \times 10^{5}$ .

Figure 7.

Velocity profile on inlet, $Re = 4 \times 10^{5}$ .

The turbulent intensity $T_{u}$ was 2% and the turbulent length scale was $0.07 \times l_{eff} = 0.014 m$ . All examinations were performed for the incidence angles from $i = - 15 °$ to $i = 10 °$ with a stepwise movement of $5 °$ . Wall functions were not used so that the RANS equations were solved for all cells. The Mach number is $Ma < 0.3$ , which characterizes an incompressible inflow. Furthermore, the assumptions of isothermal and steady state conditions were made. So, the energy equation was not needed, and no boundary condition for the temperature was set. For the other boundary conditions, the respective default settings were used. The boundary conditions at each surface are listed in Table 2.

Table 2.

Boundary conditions at each surface.

Surface	Boundary conditions
Inlet	Velocity inlet with UDF
Outlet	Pressure outlet
Periodic boundary	Periodic
Symmetry plane	Symmetry
Sidewall	Non-slip wall

UDF: user-defined function.

In order to check whether the mesh resolution was sufficient, the grid was coarsened by doubling the mesh spacing within the wall regions. Three different meshes have been tested, and the results are depicted in Figure 8. It was determined that the coarsening of the mesh had no significant influence on the results.

Figure 8.

Independence test of mesh number of blade-ori, $Re = 2 \times 10^{5}$ .

Experimental investigation

In this article, the experimental investigation for the original circular arc blade was conducted to guarantee the validation of the numerical results with the experimental measurements.

The measurement was conducted in a wind tunnel, which consists of an inlet nozzle, an axial compressor, a honeycomb rectifier, an acceleration nozzle, and a measuring section. The inlet nozzle has an inner diameter of 710 mm and a length of 1575 mm. Next to the inlet nozzle is an axial compressor, which is controlled by a frequency converter and has the power up to 37 kW. The honeycomb rectifier is used to reduce the swirled flow from the axial compressor by transferring it into a one-directional flow stream. After the honeycomb, the flow goes through the acceleration nozzle to reduce velocity differences and forms an uniform velocity distribution on the measuring section, which is located at the end of the wind tunnel (see Figure 9).

Figure 9.

Wind tunnel.

The total and dynamic pressures of the incoming flow are measured by a Prandtl probe, which is located in the center of inflow section (see Figure 10). The pressures on the measuring plane are detected by the five-hole probe, which is mounted in the traverse system. This system consists of the step motors, which can move the five-hole probe in both x and y directions, thus with the computer control the five-hole sonde can measure the pressure of any point on the measuring plane.

Figure 10.

Wind tunnel and measurement section.

Two side channel compressors are mounted on the top and bottom of the measuring section. The speed of the side channel compressors is controlled by a frequency converter. The task of both side channel compressors is to ensure that the boundary layers on the upper and lower wall are sucked away so that the flow in y-direction becomes periodic.

Results

Loss coefficient and turning angle

The dimensionless loss coefficient $ζ$ and the turning angle $Δ β$ of the original blades are shown in Figures 11 and 12. The loss coefficient $ζ$ is defined by

ζ = \frac{\bar{p_{tot 1}} - \bar{p_{tot 2}}}{\frac{ϱ}{2} \cdot C_{1}^{2}}

(2)

with $\bar{p_{tot 1}}$ and $\bar{p_{tot 2}}$ are the mass-averaged total pressure on inlet and measuring plane. $C_{1}$ is the velocity on the inlet. The outflow angle $β_{2}$ and the resulting turning angle $Δ β$ are defined as

\tan β_{2} = \frac{\bar{C_{2 y}}}{\bar{C_{2 z}}}, Δ β = β_{1} - β_{2}

(3)

with $\bar{C_{2 y}}$ and $\bar{C_{2 z}}$ are the mass-averaged velocity on the measuring plane in y- and z-directions, respectively. $β_{1}$ is the inflow angle, $β_{2}$ is flow angle on measuring plane.

Figure 11.

Loss coefficient of original circular arc blade.

Figure 12.

Turning angle of original circular arc blade.

As the results show, the $ζ$ and $Δ β$ of original circular arc blade have a good agreement between numerical and experimental methods.

The dimensionless loss coefficient $ζ$ and the turning angle $Δ β$ of the blades with different leading edges in numerical method are shown in Figures 13 and 14. The loss coefficient $ζ$ of blade-ori (b = 1.5 mm), blade-3 (b = 3 mm), blade-6 (b = 6 mm), and blade-10 (b = 10 mm) show a similar distribution on inflow angle $β_{1}$ . The flow losses increase with the decreasing Reynolds number. The inflow angle $β_{1} = 25 °$ characterizes the operational area with the lowest loss value. In general, the loss values of blade-ori are higher for every inflow angle than the other investigated blades. When $β_{1} > 35 °$ or $β_{1} < 20 °$ , the losses of blade-6 are the lowest flow losses. The turning angle $Δ β$ of all blades are almost the same in both Reynolds numbers.

Figure 13.

Loss coefficient of the blades with different leading edges.

Figure 14.

Turning angle of the blades with different leading edges.

Separation bubbles

Due to an adverse pressure gradient, laminar boundary layer separates from the surface of the blade, then separation bubbles appear and the laminar flow transforms into turbulent flow. The separation bubbles occur usually at the leading edge of thin blades and can modify the flow structure of the blade and consequently have an influence on the blade aerodynamic performance.

The CFD simulation results of the separation bubbles at the leading edge of each blades are depicted in Figures 15 –22. With an inflow angle of $β_{1} = 15 °$ , blade-ori, blade-3, and blade-10 have large separation bubbles or complete separation at the leading edge, which can cause high losses. Additionally, it is noticeable that there are only small separation bubbles at the leading edge of blade-6, which leads to lower flow losses compared to the other three blades. These findings have a good agreement with the results of loss coefficient in Figure 13.

Figure 15.

Separation bubbles of blade-ori, $β_{1} = 15 °$ : (a) Reynolds number = 2 × 10⁵ and (b) Reynolds number = 4 × 10⁵.

Figure 16.

Separation bubbles of blade-3, $β_{1} = 15 °$ : (a) Reynolds number = 2 × 10⁵ and (b) Reynolds number = 4 × 10⁵.

Figure 17.

Separation bubbles of blade-6, $β_{1} = 15 °$ : (a) Reynolds number = 2 × 10⁵ and (b) Reynolds number = 4 × 10⁵.

Figure 18.

Separation bubbles of blade-10, $β_{1} = 15 °$ : (a) Reynolds number = 2 × 10⁵ and (b) Reynolds number = 4 × 10⁵.

Figure 19.

Separation bubbles of blade-ori, $β_{1} = 30 °$ : (a) Reynolds number = 2 × 10⁵ and (b) Reynolds number = 4 × 10⁵.

Figure 20.

Separation bubbles of blade-3, $β_{1} = 30 °$ : (a) Reynolds number = 2 × 10⁵ and (b) Reynolds number = 4 × 10⁵.

Figure 21.

Separation bubbles of blade-6, $β_{1} = 30 °$ : (a) Reynolds number = 2 × 10⁵ and (b) Reynolds number = 4 × 10⁵.

Figure 22.

Separation bubbles of blade-10, $β_{1} = 30 °$ : (a) Reynolds number = 2 × 10⁵ and (b) Reynolds number = 4 × 10⁵.

When $β_{1} = 30 °$ , due to the suitable leading edge geometry of blade-6 and blade-10, the inflow can flow on the surface of the blade without detachment, thus no separation bubbles occur at the leading edge. Therefore, blade-6 and blade-10 have relative lower losses than blade-ori and blade-3 in this region. However, the separation bubble length of blade-ori is larger than blade-3, so blade-ori has the highest losses in this region. In conclusion, by comparing the separation bubbles in dependence of incidence and Reynolds number, blade-6 has the best performance of reducing separation bubbles.

The length of separation bubbles decreases when Reynolds number is increasing. This result has a good agreement with SCT Perkins and AD Henderson,²¹ who conducted the measurement of a blade with circular arc leading edge at Reynolds numbers of $2.6 \times 10^{5}$ and $4 \times 10^{5}$ . Thier work states that the length of the separation bubbles seems to reduce with increasing Reynolds number.

Streamline on the sidewall

Besides the separation bubbles, secondary flow plays also an important role in the flow behavior of the blade. In Figures 23 –26, the flow structure (near the sidewall) of each blade are depicted.

Figure 23.

Streamline of blade-ori, $β_{1} = 30 °$ and $Re = 2 \times 10^{5}$ .

Figure 24.

Streamline of blade-3, $β_{1} = 30 °$ and $Re = 2 \times 10^{5}$ .

Figure 25.

Streamline of blade-6, $β_{1} = 30 °$ and $Re = 2 \times 10^{5}$ .

Figure 26.

Streamline of blade-10, $β_{1} = 30 °$ and $Re = 2 \times 10^{5}$ .

Due to the viscous effect of the flow and the friction on the sidewall, the flow velocity close to the wall is very small, thus the centrifugal force of the flow is reduced in this region. Since the static pressure is essentially the same in the boundary layer as it is in the adjacent outer flow, it is typical that streamline is curved within the boundary layer of the sidewall (on the sidewall).

The flow structure near the sidewall of all investigated blades is similar. At the inlet area, the streamline flows parallel toward to the blade with the given incidence. When the inflow hits the saddle point near the leading edge of the blade, then the streamline separates into two parts. One part moves toward to the suction side of the blade under the influence of the secondary flow. The other part of the streamline turns and transforms into the secondary flow.

Loss distribution in the sidewall region

The ζ-distributions of all investigated blades are depicted in Figures 27 –36. The ζ-distribution is investigated on the plane, which is parallel to the inflow plane and 50 mm downstream behind the trailing edge of the blade (see Figure 5). On the left side of the figures is the symmetry plane (which is marked as 1), and on the right side of the figures is the sidewall (which is marked as 2); the upper and lower sides of the figures are periodic boundary. Again the friction of the sidewall and the blade affects the inflow velocity, which leads to a lower velocity and therefore to a decrease in the total pressure. Furthermore, this leads to higher flow losses in the corner region between the blade and the corresponding sidewall.

Figure 27.

Loss distribution on the measuring plan of blade-ori (original blade), $β_{1} = 15 °$ and Reynolds number = $2 \times 10^{5}$ .

Figure 28.

Loss distribution on the measuring plan of blade-ori, $β_{1} = 15 °$ and Reynolds number = $4 \times 10^{5}$ .

Figure 29.

Loss distribution on the measuring plan of blade-3, $β_{1} = 15 °$ .

Figure 30.

Loss distribution on the measuring plan of blade-6, $β_{1} = 15 °$ .

Figure 31.

Loss distribution on the measuring plan of blade-10, $β_{1} = 15 °$ .

Figure 32.

Loss distribution on the measuring plan of blade-ori (original blade), $β_{1} = 30 °$ and Reynolds number = $2 \times 10^{5}$ .

Figure 33.

Loss distribution on the measuring plan of blade-ori, $β_{1} = 30 °$ and Reynolds number = $4 \times 10^{5}$ .

Figure 34.

Loss distribution on the measuring plan of blade-3, $β_{1} = 30 °$ .

Figure 35.

Loss distribution on the measuring plan of blade-6, $β_{1} = 30 °$ .

Figure 36.

Loss distribution on the measuring plan of blade-10, $β_{1} = 30 °$ .

Figures 27 –33 show that the ζ-distribution of blade-ori of CFD simulation has a relative good agreement with the measurement results. With $β_{1} = 15 °$ , the inflow angle $β_{1}$ is smaller than stagger angle $λ$ , thus the high flow losses are generated in the corner between the sidewall and the pressure side of the blade. In this region, blade-ori has more flow losses than the other three blades. Additionally, due to only small separation bubbles occurring at the leading edge of blade-6, the wake behind the blade-6 is the smallest.

When $β_{1} = 30 °$ , the high flow losses are generated in the corner between the sidewall and the suction side of the blade. The flow loss distribution of all investigated blades is almost the same in this region.

At the higher Reynolds number, for example, Reynolds number = $4 \times 10^{5}$ , the boundary layer on the blade and the corresponding sidewall is thinner and the flow loss in the corner region between them is lower. Therefore, the loss coefficient $ζ$ is lower at the high Reynolds number, which has also a good agreement with the results of total pressure loss coefficient in Figure 13.

Conclusion

In this article, circular arc blades with different leading edges are investigated by numerical method. The investigated blades in this work are applied in guide vane, so the flow losses and turning angle are tested. The main purpose of this article is to find a leading edge geometry of circular arc blade which can reduce separation bubbles and finally decrease the flow losses.

By changing the leading edge geometry of circular arc blade, the separation bubbles can be reduced and flow losses can be decreased. The blade-6 has the best performance of reducing separation bubbles and the lowest flow losses. Regarding of inflow angle $β_{1} = 15 °$ and $β_{1} = 40 °$ , the losses of blade-6 is lower than blade-ori by about 40%. Additionally, the variations of leading edge geometry and Reynolds number have no significant influence on the turning angle $Δ β$ for the corresponding blades.

Except the formation of separation bubbles at the leading edge of the blade, the leading edge geometry has no significant influence on the flow structure near the sidewall. Generally, streamline turns in the region which is close to the sidewall. Due to the suitable leading edge, blade-6 has less separation bubbles than the other three blades. The length of separation bubbles increases with the decreasing Reynolds number.

The high flow losses are developed in the corner region between the blade and the corresponding sidewall. Because only small separation bubbles occur at the leading edge of blade-6, the wake behind the blade-6 is also smaller than the other three investigated blades. At the higher Reynolds number, the boundary layer on the blade and the corresponding sidewall is thinner and the flow loss in the corner region between them is lower. Therefore, the loss coefficient $ζ$ is smaller at high Reynolds number.

The investigated circular arc blades in this work have a constant thickness. When the thickness is not constant, the separation point of the flow is different, so the flow structure and flow loss are also different.

Footnotes

Handling Editor: Roslinda Nazar

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Jacobs

Ward

Pinkerton

RM.

The characteristics of 78 related airfoil sections from tests in the variable-density wind tunnel. Technical report NACA-TR-460, 1 January 1933. Washington, DC: National Advisory Committee for Aeronautics.

Langston

LS.

Crossflows in a turbine cascade passage. J Eng Power: T ASME 1980; 102: 866–874.

Zess

Thole

KA.

Computational design and experimental evaluation of using a leading edge fillet on a gas turbine vane. J Turbomach 2002; 124: 167–175.

Liu

Sun

Guan

et al . The influence of synthetic jet excitation on secondary flow in compressor cascade. J Therm Sci 2010; 19: 500–504.

Zander

Nitsche

Control of secondary flow structures on a highly loaded compressor cascade. Proc IMechE, Part A: J Power and Energy 2013; 227: 674–682.

Hermanson

Thole

KA.

Effect of inlet conditions on endwall secondary flows. J Propul Power 2000; 16: 286–296.

Chen

Lisener

Meyer

et al . Effect of inflow variations on compressor secondary flow behavior. In: Proceedings of the XX Polish fluid mechanics conference, Gliwice, 17–20 September 2012.

Tan

Zhu

Wang

et al . Numerical study on characteristics of unsteady flow in a centrifugal pump volute at partial load condition. Eng Computation 2015; 32: 1549–1566.

Tan

Cao

et al . Numerical investigation of clocking effect on a centrifugal pump with inlet guide vanes. Eng Computation 2016; 33: 465–481.

10.

Liesner

Meyer

Lemke

et al . On the efficiency of secondary flow suction in a compressor cascade. In: Proceeding of the ASME turbo expo 2010: power for land, sea and air, Glasgow, 14–18 June 2010. New York: ASME.

11.

Nho

Park

Lee

et al . Effects of turbine blade tip shape on total pressure loss and secondary flow of a linear turbine cascade. Int J Heat Fluid Fl 2012; 33: 92–100.

12.

El-Batsh

HM.

Effect of the radial pressure gradient on the secondary flow generated in an annular turbine cascade. Int J Rotat Mach 2012; 2012: 509209.

13.

Gaster

. On the stability of parallel flows and the behaviour of separation bubbles. Technical report, Queen Mary University of London, London, 1963.

14.

Courtine

Spohn

. Dynamics of separation bubbles formed on rounded edges. In: Proceedings of the 12th international symposium on applications of laser techniques to fluid mechanics, Lisbon, 12–15 July 2004.

15.

Lamballais

Silvestrini

Laizet

Direct numerical simulation of a separation bubble on a rounded finite-width leading edge. Int J Heat Fluid Fl 2008; 29: 612–625.

16.

Shyy

Lian

Jian

et al . Aerodynamics of low Reynolds number flyers. Cambridge: Cambridge University Press, 2008.

17.

Diwan

Ramesh

ON.

On the origin of the inflectional instability of a laminar separation bubble. J Fluid Mech 2009; 629: 263–298.

18.

Crompton

Barrett

RV.

Investigation of the separation bubble formed behind the sharp leading edge of a flat plate at incidence. Proc IMechE, Part G: J Aerospace Engineering 2000; 214: 157–176.

19.

Hamakhan

Korakianitis

Aerodynamic performance effects of leading-edge geometry in gas-turbine blades. Appl Energ 2010; 87: 1591–1601.

20.

Zhang

Zou

Leading-edge redesign of a turbomachinery blade and its effect on aerodynamic performance. Appl Energ 2012; 93: 655–667.

21.

Perkins

SCT

Henderson

. Separation and relaminarization at the circular arc leading edge of a controlled diffusion compressor stator. In: Proceedings of the ASME turbo expo 2012: turbine technical conference and exposition, Copenhagen, 11–15 June 2012. New York: ASME.

22.

Liu

et al . Effect of leading-edge geometry on separation bubble on a compressor blade. In: Proceedings of the ASME turbo expo 2003, collocated with the 2003 international joint power generation conference, Atlanta, GA, 16–19 June 2003. New York: ASME.

23.

Wheeler

APS

Sofia

Miller

RJ.

The effect of leading-edge geometry on wake interactions in compressors. J Turbomach 2009; 131: 041013.

24.

Weinig

. The flow around blades of turbomachines. Technical report, 1935.

25.

McDevitt

Levy

Jr Deiwert

GS.

Transonic flow about a thick circular-arc airfoil. AIAA J 1976; 14: 606–613.

26.

Bian

Han

Böhle

A design method for cascades consisting of circular arc blades with constant thickness. Int J Fluid Mach Syst 2017; 10: 63–75.

27.

Suzuki

Setoguchi

Kaneko

Prediction of cascade performance of circular-arc blades with CFD. Int J Fluid Mach Syst 2011; 4: 360–366.

28.

Menter

Langtry

Likki

et al . A correlation-based transition model using local variables—part I: model formulation. J Turbomach 2004; 128: 413–422.