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This paper introduces an investigation of the effect of radial pressure gradient on the secondary flow generated in turbine cascades. Laboratory measurements were performed using an annular sector cascade which allowed the investigation using relatively small number of blades. The flow was measured upstream and downstream of the cascade using a calibrated five-hole pressure probe. The three-dimensional Reynolds Averaged Navier Stokes equations were solved to understand flow physics. Turbulence was modeled using eddy-viscosity assumption and the two-equation Shear Stress Transport (SST)

Large-scale steam and gas turbines are always used in power generation and industrial applications. Therefore, turbine efficiency and performance have major concern. The losses in a turbine can be divided into profile loss, secondary flow loss, and tip clearance loss. The profile loss is caused by the growth of the boundary layer on the blades. Secondary flow loss is generated due to the deflection through blade channel. Tip leakage loss is induced due to pressure difference between blade pressure side and blade suction side when the tip clearance gap exists. There are many factors which influence turbine losses. The pressure gradient, turbulence level, blade geometry, incoming velocity, and inlet boundary layer thickness represent important parameters affecting turbine efficiency.

It is practically very difficult to perform detailed flow field measurements in an engine at operating conditions. Understanding the physics that governs the flow and the associated turbine cascade losses has been obtained through wind tunnel experiments. These laboratory tests not only allow detailed flow field measurements but also give the experimenter the possibility to investigate the effect of several parameters separately.

Experimental studies using linear turbine cascades introduce the aspect of flow periodicity by arranging a number of blades of constant cross-sections separated by a constant pitch. Linear cascade experiments provide several advantages such as geometric simplicity, simple adjustment, large blade sections, and simply changing incidence angle. They are used as a tool to provide quasi-three-dimensional blade-to-blade data for the simulation of the flow. Linear cascades have been used extensively and have succeeded in providing a better understanding of the physics involved. They simplified the problem and allowed the usage of large-scale cascades to provide better detailed measurements in different regions of the turbine. Linear turbine cascades have been used very extensively for basic investigations of secondary flows through turbine cascades [

The reliability of the experimental data is improved by using numerical calculations for the interpretation of the data. Linear turbine cascade measurements are also commonly used in defining the proper boundary conditions for numerical calculations and the selection of appropriate turbulence models and validation of the numerical techniques.

Although the experimental data of the secondary flows obtained in linear cascades are very valuable for numerical validation, they cannot be used directly for calculating turbine flows where the radial static pressure gradient field plays a particularly important role with respect to the spanwise distribution of losses and outlet flow angles. Figure

Pressure gradient in linear and in annular cascades.

The annular turbine cascade consists of an annular space between two concentric cylinders which contains a turbine blade row. The main advantage of annular cascades over linear cascades is the possibility of simulating the radial static pressure gradient and, therefore, simulating turbine flow conditions more closely than the linear cascade. However, the increased number of blades in annular cascades makes the blades have smaller scale than that of linear cascades causing increased probe blockage effects and higher measurement errors due to stronger pressure gradients. Recently, El-Batsh and Bassily Hanna [^{4} to 26 × 10^{4}. A very comprehensive review of advanced applicable techniques to both linear and annular cascade testing has been published by Hirsch [

Annular large-scale sector cascades are used as a compromise between the advantages of linear and annular cascades simulating properly the radial pressure gradient field and increasing the blade size which enables accurate measurements for laboratory investigation using a relatively low number of blades. In addition, blades with large aspect ratios including blade twist and changes of section area along blade span can be examined using annular sector cascades. Furthermore, existing engine parts can be examined, and the blade profiles may be obtained from the engine. The established radial pressure gradients ensure that secondary flows develop as they would exist in an operating engine. Model and testing costs for annular sector cascades are considerably lower than those for their fully annular equivalents.

This study aims to investigate the effect of radial pressure gradient on the three-dimensional flow through radial turbine cascades without tip clearance. The study aims also to predict secondary flow generated in real turbines. This is achieved by using an annular turbine cascade sector with large-scale turbine blades. The effects of the radial pressure gradient on the secondary flow and loss mechanism through the cascade are examined. To interpret the data and to understand the flow physics, numerical calculations are performed as well.

Limited information is available concerning experiment design using annular sector cascades. Vogt and Fransson [

The annular sector cascade constructed in this study was equipped with 5 blades representing blade profile exact replicas of the first stage rotor of the gas turbine engine from General Electric working in electric power generation. The rotor contains 92 blades and has hub and tip diameters of 1946 mm and 2366 mm, respectively. The blade profile is tapered to represent actual turbine blade with different cross-sections along blade span. Radial pressure gradient exists in the rotor and in the stator. In the stator, flow deflection caused by the hub and the casing produces radial pressure gradient which changes secondary flow pattern between blade tip and blade root. In the rotor, the problem is more complicated due to the rotation of the blades and the associated centrifugal force. This effect makes the design usually based on radial equilibrium. However, experimental measurements for the rotor during rotation are rather complicated and in most cases are not possible if we have to consider the actual speed of rotation. Therefore, the measurements are performed in the present study while the blades are fixed. This would give the flow field in the absence of the centrifugal forces. The measurements also allow numerical model verification. Further study will be performed which allows blade rotation using the adopted numerical technique which would simulate the actual case in the machine with the same rotating speed. Blade coordinates were obtained using a scanning system which is used in reverse engineering. It provided the blade coordinates over eight cross-sections along blade span. Figure

Cascade parameters.

Root | tip | |
---|---|---|

Chord length, | 107 | 102 |

Axial chord, | 105 | 78 |

Inlet angle, | 39 | |

Exit angle, | 26 | |

Blade span, | 210 | |

Blade spacing (pitch), | 66.45 | 80.8 |

Aspect ratio, | 1.96 | 2.06 |

Inner diameter, | 1946 | |

Outer diameter, | 2366 | |

Number of blades, | 92 |

Diagram illustrating cascade parameters at the tip and the three-dimensional blade profile.

The sector annular cascade was constructed from two concentric cylinder sectors representing the hub and the casing. The diameters of the cylinders are the hub and tip diameters as given in the real machine. The measurements were obtained upstream and downstream of the middle blade which was obtained from the real gas turbine engine and was not manufactured. The other blades were manufactured by casting and finished to obtain smooth surfaces. The blades were fixed to the hub and the casing, and, therefore, there was no tip clearance gap. The angle between two adjacent blades is 360/92 = 3.913°.

The annular sector cascade (Figure

Annular sector cascade.

The upstream flow measurements were obtained at a distance of 105 mm upstream of the blade leading edge while the downstream measurements were obtained at a distance of 52.5 mm downstream of the blade trailing edge. The flow was measured downstream of the annular sector cascade using a mesh with 8 equally spaced points in the pitchwise direction and 25 points in the spanwise direction. A traverse mechanism was used to allow probe rotation in the pitchwise direction and translation in the spanwise direction. The upstream and the downstream measurement meshes were vertical. The probe was moved with a step of 5 mm in the region close to the hub and to the casing and 10 mm elsewhere. Figure

Measurements mesh.

Five-hole pressure probes are widely used in turbomachinery for the measurement of fluid flow because they are relatively accurate and robust. Although more advanced techniques are available today such as laser and hot wire, the advantages of the five-hole probe make it superior for turbomachinery flow measurements. They are recently used for turbomachinery applications by many researchers [

The five-hole probes can be used to measure total and static pressures as well as gas velocity and flow angle. This is achieved by extensive calibrations for the probe to cover all of the expected flow conditions that will be encountered by the probe during wind tunnel measurements. A five-hole probe has five pressure ports which are distributed on a conical tip. The five-hole probe used in this study has L-shape with tip diameter of 3 mm as shown in Figure

L-shaped five-hole probe with a conical tip.

Photo for the five-hole probe

Probe coordinate system

Probe calibration was performed here by placing the probe into a calibration rig with uniform and one-dimensional flow. The total pressure (

The pitch angle coefficient

During wind tunnel measurements, the probe was inserted into the unknown flow field, and the pressures at the tip of the probe (

Calibration map for the yaw and pitch dimensionless coefficients.

Calibration map for the total pressure coefficient.

Calibration map for the static pressure coefficient.

The three-dimensional flow through the cascade was obtained by solving the flow governing equations. In all calculations performed here, the Mach number was small, and therefore the flow was considered incompressible, and as a consequence solving energy equation was not required. Since the flow through turbine cascades is almost turbulent, an appropriate turbulence model was required. The selected turbulence model is able to predict the losses with reasonable accuracy. A commercial CFD code was employed to solve the flow-governing equations.

Fluid flow characteristics are described by the conservation of mass (continuity equation) and momentum (Navier-Stokes equations). For turbulent flows, Reynolds averaging procedure is commonly used, and the governing equations are called Reynolds Averaged Navier Stokes equations. For incompressible turbulent flow neglecting external forces they are given by [

Bardina et al. [

The SST

A control-volume technique was employed to convert the differential equations to algebraic equations which can be solved numerically. The upwind scheme was used to represent the convection terms of the governing equations. The semi-implicit method for pressure-linked equation (SIMPLE) was used to solve the discretized equations. A CFD commercial code was employed here to solve the equations. All computations have been carried out using a personal computer with a single processor of Intel I5 with frequency 2.4.

The flow was solved around a single blade considering periodic flow. The inlet boundary was considered at a distance of axial chord upstream of the blade leading edge while the outlet boundary was considered at a distance of axial chord downstream of the blade trailing edge. The grid was generated for the entire blade span since there is no symmetry plane at midspan which exists in linear cascades. Fixed wall boundaries were considered at the blade surface and at the hub and at the casing. A two-dimensional grid was generated at the hub and was copied in the spanwise direction to form the three-dimensional grid. The grid has a total number of about 996000 grid points. The grid size was based on the grid-independent results obtained by Hildebrandt and Fottner [

Computational grid.

In the present study, the inlet velocity boundary condition was defined at the inlet while the outlet boundary condition was considered at the exit. Since the inlet velocity profile is important for the development of the secondary flow field, the inlet velocity was measured upstream of the cascade at a distance of axial chord. The velocity was measured by using the calibrated five-hole probe along a line from the hub to tip using the same spanwise distance given in Figure

Inlet velocity distribution.

The results presented here can be divided into three groups. Firstly, the velocity distribution through the cascade was examined. Then the static pressure was investigated because it affects the secondary flow through the cascade. Finally, loss distribution was examined, and the locations of high losses are distinguished.

Figure

Dimensionless velocity in blade channel at different locations along blade span.

Figure

Dimensionless velocity on the downstream plane at blade midspan.

Counter plots for the dimensionless velocity downstream the cascade.

Experimental measurements

Numerical calculations

The static pressure was examined through the blade channel using the static pressure coefficient which was defined as

Figure

Static pressure coefficient through blade channel at different locations.

Static pressure coefficient calculated at two locations through blade span.

Numerically calculated pressure distribution through blade channel

Pressure distribution

Pitchwise mass-averaged pressure distribution

The flow was investigated downstream of the cascade using deviation angle which was defined as

Figure

Pitchwise mass-averaged deviation angle.

The losses through turbine cascade were examined by using the total pressure loss coefficient which was defined as

Figure

Total pressure loss coefficient through blade channel at different locations.

Figure

Total pressure loss coefficient on the downstream plane at blade midspan.

Total pressure loss coefficient downstream the cascade.

Experimental measurements

Numerical calculations

Figure

Total pressure loss coefficient along blade span.

In order to verify the grid-independent numerical solution, a denser mesh was generated with increased grid size by 60% to obtain as mesh with 1593000 grid points. The cells were mainly added in the high gradient flow regions. Figure

Effect of grid size on loss prediction.

Effect of inlet turbulence intensity on loss prediction.

Secondary flow predicted downstream of the cascade.

Experimental

Numerical

This study introduced experimental measurements using annular sector cascade verified by numerical calculations for the effect of the radial pressure gradient on the three-dimensional flow in turbine cascades. The results presented here showed that the numerical calculations and the experimental measurements provided the same flow features. The main conclusions obtained from the present study are as follows.

The annular sector cascade with five blades provided reasonable results and could be used instead of fully annular cascade to reduce experiment cost and to obtain laboratory measurements for large scale blade profiles. Reducing the annular cascade to a sector rather than a full annulus reduces the required mass flow rate considerably whilst maximizing the size of the test object.

The secondary flow near the hub is stronger than that near the shroud. For the blade aspect ratio considered in this study, the secondary flow was found to move across blade mid-span.

The measured and the calculated total pressure loss coefficient indicated that the losses were concentrated near the hub, and therefore attempts to reduce the secondary loss should emphasize the losses near the hub.

The increase in pressure difference between the blade pressure and suction surfaces with blade height increases flow deflection through blade height. Therefore, the flow is underturned near the hub and overturned near the shroud.

The radial pressure gradient would also affect the case with tip clearance gap, and the leakage flow induced from the blade pressure surface to the blade suction surface will be different from that predicted in linear cascade. In addition, the interaction between the passage vortex and the tip leakage vortex would be different from that obtained in linear cascades. Therefore, it is recommended for future work to extend the present study to the case with clearance gap.

Constant

Chord length

_{x}

Axial chord

Total pressure loss coefficient

Mass-averaged total pressure loss coefficient

_{1}, F

_{2}

Functions

Blade span;

Turbulent kinetic energy

_{α}, k

_{β}, k

_{t}, k

_{s}

Probe calibration coefficients

_{1}, p

_{2}, p

_{3}, p

_{4}, p

_{5}

Pressures obtained by five-hole probe

Static pressure

_{t}

Total pressure

Mean pressure

_{ω}

Production of

Blade spacing

Fluctuating velocity

Mean velocity

Mass-averaged velocity

_{i}:

Coordinate in

Distance through axial chord

Distance along blade span.

Pitch angle

Yaw angle

_{1}

Inlet blade angle

_{2}

Exit blade angle

Exit flow angle

Deviation angle

^{'},

^{'}

Model coefficients

Model constant

_{ij}

Kronecker second-order tensor

Molecular viscosity

_{t}

Turbulent eddy viscosity

Specific turbulent dissipation rate

Absolute value of vorticity

Fluid density

_{k},

_{ω}

Model coefficients.

inlet

exit

tip or turbulent

root.