Aerodynamic design of a high-efficiency two-stage axial turbine, using streamline curvature method and performance optimisation by clocking of stator blades

Abstract

Aerodynamic design of a high-efficiency two-stage axial turbine is carried out using a hybrid method through implantation of a two-step design procedure. In the first step, the well-known streamline curvature (SLC) and free vortex (FV) methods are properly combined to establish three-dimensional geometries of the blades at each row and to obtain the flow field properties. The second step is provided to obtain the highest aerodynamic efficiency by optimum clocking of the second stator blades relative to the first ones through executing steady and unsteady computational fluid dynamics (CFD) of three-dimensional viscous flow. Slight discrepancies were observed between gas dynamics results of the SLC and those of CFD. Total pressure and temperature at the turbine outlet, obtained from SLC method, differed from those obtained by 3D-CFD technique by 13.06% and 1.88% respectively. Aerodynamic efficiency of the turbine is obtained about 91.83%, based on 3D-CFD. Time-averaged results showed that under the optimum clocking of the second row stator blades, inlet total pressure and output power of the second rotor increase by 0.23%, and 0.93%, respectively, in comparison to the worst clocking case. These augmentations resulted in increased total to total efficiency of the second stage by 0.444%. Additionally, the total output power of the two stages increased by 0.71% through the optimum clocking. Modeling the unsteady wake flow trajectory within the blades passages confirmed that all of these beneficial effects happen if the upstream wake impinges on the leading edge region of the second stator blades.

Nomenclature

C

blade real chord (mm)

C _p

pressure coefficient

E

specific energy of a fluid particle (J/kg)

H

specific enthalpy (J/kg)

I

specific internal energy (J/kg) or rothalpy (J/kg)

K

specific turbulent kinetic energy (J/kg)

K

ratio of specific heats= ${{\rm{c}}_{\rm{p}}}/{{\rm{c}}_{\rm{v}}}$

$\dot{\boldsymbol{m}}$

mass flow rate (kg/s)

M

mach number

N

rotational speed (rpm)

P

pressure (Pa)

Pr

pressure ratio

r

radius (mm)

Re $_{\boldsymbol\theta}$

momentum thickness Reynolds number

s

static entropy (J/kg.K)

T

temperature (K)

t

time (s)

Tu

turbulent intensity (%)

u,v, w

velocity components (m/s)

V or C

absolute velocity (m/s)

W

relative velocity (m/s)

Y ⁺

dimensionless wall distance

Greek symbol

$\boldsymbol\alpha$

flow absolute angle (deg)

$\boldsymbol\beta$

flow relative angle (deg)

$\boldsymbol\gamma$

intermittency

$\boldsymbol\varepsilon$

turbulent kinetic energy dissipation rate (m²/s³]

${\boldsymbol\varepsilon _{\boldsymbol{F}}}$

flow turn angle (deg)

$\boldsymbol\eta$

efficiency

$\boldsymbol\lambda$

pressure loss coefficient

$\boldsymbol\mu$ _t

dynamic turbulent viscosity (N.s/m²)

$\boldsymbol\nu$ _t

kinematic turbulent viscosity (N.m/kg)

$\boldsymbol\rho$

density (kg/m³)

$\boldsymbol\tau$

shear stress (Pa)

$\boldsymbol\varOmega$ _s

specific speed

$\boldsymbol\varOmega$ or $\boldsymbol\omega$

angular velocity (rad/s)

Superscript

$ \to $

vector

Superscript

0 or t

stagnation condition

1 or in

inlet

2 or out

outlet

1

stator inlet

2

stator outlet or rotor inlet

3

rotor outlet

abs

absolute

Acronyms

3D

three-dimensional

Des.

design condition

rpm

revolution per minute

s

static condition

tt

total to total

URANS

Unsteady Reynolds-Averaged Navier-Stokes

1.0 Introduction

Aerodynamic design of an axial turbine, as the main component of numerous gas turbine engines, is a complex process. Viscous three-dimensional computational fluid dynamics (CFD) methods can be useful in development and optimisation of flow passage and blades geometries to suitable accuracies. These methods, at least at the initial steps of design process are not commonly recommended, since they are usually time consuming and expensive. Design process can be started by fairly primitive methods to lay out the overall design, and then, terminate with sophisticated 3D-CFD simulations. The well-known streamline curvature (SLC) method can be used as an initial step of design process. In this method, two surfaces of blade to blade (S1) and hub to tip (S2), which are originally proposed by Wu [1], are utilised to generate computational domains. Hub to tip axisymmetric computations on S2 surfaces have turned a main element in turbomachines design, and blade to blade analysis can be used for preliminary design of the blades profiles.

Novak used the differential form of continuity equation within SLC method to establish an expression for estimation of meridional velocity gradient [2]. His attempt led to eliminate uncertainty in numerical estimation of this parameter as a result of exerting quasi-normal lines between the those representing blade leading edge and trailing edge. Denton used SLC calculations to predict the flow field in a transonic axial flow turbine [3]. By comparison of numerical results with those of experimental measurements, he concluded that the errors in numerical method are mainly due to the empirical correlations for modeling the various types of losses. Aungier introduced a SLC technique, known as approximate normal equilibrium model [4]. This technique aims in elimination of the major iteration loop, which is the primary source of convergence problems, and also reduces the computational time in comparison to the full normal equilibrium model.

Korakianitis and Zou [5] introduced a specialised variant of SLC method that incorporates the axial slope of streamlines. This approach considers an iteration loop for mass-flow balance, while the radial momentum equation at each flow station is solved using a one-pass numerical predictor-corrector technique, resulting in reduced computational time. Wennerstrom [6, 7], Boyer [8], Templalexis [9], Zhu [10], and Pachidis [11] have contributed to the advancement of SLC methodologies, each proposing their own procedures and enhancements. Abbasi et al. conducted an investigation on effects of inlet distortion on axial compressor performance using SLC method, considering both design and off-design conditions [12]. Their study demonstrated good agreement between numerical results and available experimental data.

Kim utilised the SLC method for predicting the performance of a 5-stage axial turbine and discussed its advantages and limitations compared to CFD technique [13]. However, it should be noted that in Kim’s study, the quasi-normal lines were not distributed on the whole surfaces of the rotor and stator blades. Instead, they were only implemented at the leading and trailing edges of the blades and within their axial gaps. Ainley and Mathieson have used a 2-step procedure to estimate the performance of multi-stage axial turbines [14]. In the first step the pressure loss and flow angles for each blades row were determined. The second step was denoted to use of SLC method based on a fixed set of boundary conditions and a given rotational speed.

The aerodynamic efficiency of a low-pressure turbine (LPT) strongly affects the specific fuel consumption of an engine, where 1% increase in its polytropic efficiency improves fuel consumption by 0.5–1.0% [15]. LPTs are also expensive due to their relatively high number of expensive components. There would appear to be little chance for improving the aerodynamic performance via optimisation of blade profiles when the isentropic efficiency exceeds 90% [16]. However, ‘blade-wake interaction’ and its effects on flow structure are taken into account for designing high-efficiency axial turbine stages, especially in LPTs [17, 18]. Understanding the trajectory of this wake flow is crucial for optimising the aerodynamic performance of the turbine. Impingement of the upstream wake with the subsequent blades can cause transition from laminar to turbulent flow to occur on their surfaces [17, 19, 20]. Proper impingement of the wake flow on the subsequent blades can increase the lift force, which is obviously desirable.

Clocking of the blades (i.e. adjusting the circumferential positions of the blades relative to each other) is a passive control method for guiding the trajectory of the wake flow originated from the upstream blades. As a result, it would be possible to enhance the aerodynamic performance of a turbine by optimising the clocking positions (CLP) of the blades. Figure 1 illustrates five different clocking positions (CLP0 to CLP5) of the second stator blades relative to the first stator. Maximum efficiency will be achieved if the first stator wake impinges on the leading edge of the second stator blades row [21–23].

Figure 1. Clocking of second stage stator blades relative to first stage stator.

Huber experimentally obtained an increase of 0.8% in a 2-stage axial turbine efficiency as a result of optimum clocking of the second stator blades [24]. Arnone numerically concluded that the efficiency of a 3-stage low-pressure turbine (LPT) can be increased by 0.7% under the optimum clocking [25]. Clocking analyses of a 1.5-stage axial turbine, conducted by Reinmoller, led to 1% improvement in efficiency [26]. Jouini, based on his studies on a high-pressure turbine (HPT), obtained an increase of 4% in aerodynamic efficiency via optimum clocking [27]. Arnone numerically concluded that the optimum clocking would be accompanied by higher pressure difference between the suction and pressure sides of the blades, which leads to higher outlet total pressure [21].

Comparing results of steady and unsteady analyses, performed by frozen rotor and sliding mesh approaches respectively, Bohn showed that steady results could be effective in identifying optimum CLP during preliminary design level [28]. Examining rotor-stator interactions, Gaetani concluded that clocking mechanism in LPTs is more related to the wake flow behaviour rather than the secondary flows structure [17]. He detected that under optimum clocking, wake flow can cause high lift forces to be exerted on the blades surfaces. Sobczak has simulated flow field in a 2-stage low aspect ratio turbine utilising frozen rotor technique and shear stress transport (SST) turbulence model in the CFX flow solver [29]. His findings indicate that maximum efficiency can be achieved if the upstream rotor wake impinges on the leading edge of the next stator blade.

Konig experimentally investigated effects of the first stator clocking on aerodynamic performance of a 1.5-stage LPT [20]. He found that minimum pressure loss is accompanied by impinging the first stator wake on the second stator blade leading edge. The same result is achieved by Zhu, through his numerical investigations on the second stator clocking effects in a 1.5-stage axial turbine along its mid-span [30]. Touil and Ghenaiet analysed five different clocking positions for a 2-stage axial turbine [31, 32]. They concluded that under optimum clocking, the isentropic efficiency of the whole two stages and the second stage increase by 0.08% and 0.14%, respectively.

In contrast to the axial/centrifugal compressors, there are unfortunately lack of enough references in the literature about designing axial/radial turbines utilising SLC method. Of the most important reasons behind it seems to be divergency occurred in the results during the numerical operations. This divergency may occur due to unpredictable load distribution along each streamline over the blade surfaces. In contrast to the axial compressors, the flow turning angle of the turbine blades are usually high. This makes it difficult to easily find a suitable streamwise load distribution. In the current research work, the well-known free vortex method is utilised for initialising the flow properties along the leading and trailing edges of the HPT and LPT blades. As a result, it provided to overcome the divergence problem usually happen during the mathematical operations of SLC method. In addition, efforts are made to overcome the lack of comprehensive phenomenology on blade-wake interaction (i.e. transient effects of wake on the blades loadings and unsteady aerodynamic performance of the turbine) under various clocking positions, in the current research work.

The main objective of the current research work is to design a high-efficiency two-stage axial turbine for a gas turbine engine. An in-house computerised code is developed in this respect, which utilises a hybrid method as a combination of SLC and free vortex (FV) methods. Flow field on the turbine meridional plane and geometries of the blades profiles are the ultimate outcomes of the above-mentioned code. Clocking analyses of the proposed turbine have been carried out to approach the maximum aerodynamic efficiency through 3D steady and unsteady numerical flow simulations. Various CLPs of the second stator blades have been examined to optimise the aerodynamic performance of the second stage and the whole turbine. In addition, clocking effects on the steady and time-dependent flow characteristics, with emphasis on the wake flow trajectory under different conditions, are studied in detail.

2.0 Hybrid method

A computerised code is developed for designing of axial turbines, which is based on combination of streamline curvature (SLC) method and free vortex (FV) flow postulation. The SLC method is implemented to analyse the flow field on a set of streamlines distributed from hub to tip on a surface, known as the S2 surface. Nonetheless, convergence of SLC method depends typically on some input parameters as the initial conditions or primary requirements, among which proper estimation of spanwise distribution of the axial velocity is the most important one. The free vortex method is used to properly estimate input parameters along the leading and trailing edges of the blades at each row.

2.1 Free vortex method

Figure 2 illustrates the process undertaken prior to execution of the free vortex part of the computerised code. This algorithm iteratively adjusts the blade circumferential velocity (U) and each stage loading coefficient ( $\varPsi$ ) in order to attain the desired total pressure ratio of the turbine.

Figure 2. Algorithm for determination of stage loading and rotational speed.

Regarding output power of each stage, tangential component of absolute velocity at the rotor blade trailing edge can be computed using the well-known Euler equation as follows.

(1) \begin{align}{W_{Rotor}} = \omega \tau = \dot m\omega \left[ {\left( {{r_2}{C_{\theta 2}}} \right) - \left( {{r_3}{C_{\theta 3}}} \right)} \right]\;\; \to \;\;{C_{\theta 3}} = \frac{{\left[ {{r_2}{C_{\theta 2}} - \frac{\tau }{{\dot m}}} \right]}}{{{r_3}}}\end{align}

Figure 3 illustrates the algorithm employed to calculate flow properties in both the absolute and relative frames. Velocity triangles at the inlet and outlet of a rotor blade are schematically depicted in Fig. 4.

Figure 3. Algorithm for flow properties calculations.

Figure 4. Velocity triangles for rotor blade.

Regarding Fig. 4, at each operational point between fixed and moving components (i.e. stator and rotor blades) the relative velocity can be calculated by Equation (2).

(2) \begin{align}\vec W = \vec C - \vec U\end{align}

Following equations are utilised to compute the loading factor $\varPsi$ , flow coefficient Φ and degree of reaction R.

(3) \begin{align}\varPsi = \frac{{\overrightarrow {{V_{2\theta }}} - \overrightarrow {{V_{3\theta }}} }}{U}\end{align}

(4) \begin{align}\Phi = \frac{{{V_x}}}{U}\end{align}

(5) \begin{align}R = 1 - \frac{{\overrightarrow {{V_{2\theta }}} + \overrightarrow {{V_{3\theta }}} }}{2}\end{align}

After finishing with the above calculations along the turbine passage mean-line, the free vortex (FV) part of the in-house computerised code is activated. Using FV method, as a subset of the well-known radial equilibrium law, one can compute the flow properties at the other radial positions.

Axial component of the vorticity for an axisymmetric flow in the cylindrical coordinate system is defined as follows [33].

(6) \begin{align}\xi = \frac{{\partial {c_\theta }}}{{\partial r}} + \frac{{{c_\theta }}}{r} = \frac{1}{r}\frac{\partial }{{\partial r}}\left( {r{c_\theta }} \right)\end{align}

The above equation, considering free vortex assumption (i.e. zero vorticity), leads to: rc _θ = const, and consequently to the following equations for calculations of tangential and axial components of velocity at the leading and trailing edges of the rotor blades at each turbine stage.

(7) \begin{align}{r_h}{c_{\theta 2h}} = {r_t}{c_{\theta 2t}} = {r_m}{c_{\theta 2m}} = {C_2}\end{align}

(8) \begin{align}{r_h}{c_{\theta 3h}} = {r_t}{c_{\theta 3t}} = {r_m}{c_{\theta 3m}} = {C_3}\end{align}

(9) \begin{align}{c_{x2h}} = {c_{x2t}} = {c_{x2m}} = {C_{x2}}\end{align}

(10) \begin{align}{c_{x3h}} = {c_{x3t}} = {c_{x3m}} = {C_{x3}}\end{align}

As already mentioned, the FV method is implemented to estimate the flow properties along the leading and trailing edges of the blades at each row. Then, they are linearly distributed along each streamline in order to provide initial conditions for analysing the turbine flow field by SLC method. Finally, all the necessary data would be extracted from the SLC calculations to characterise the blades profile at each radial position. More details are presented in the following section.

2.2 Streamline curvature method

As already mentioned, in streamline curvature (SLC) method two surfaces of blade to blade (S1) and hub to tip (S2) are initially introduced. These two surfaces are drawn and shown in Fig. 5. Approximate normal equilibrium model, introduced by Aungier [34], has been utilised to develop the SLC method equations on the S2 surface.

Figure 5. Definition of S1 and S2 surfaces.

2.2.1 Meridional coordinate system

In SLC method the turbine flow passage is initially divided into a group of annular stream-tubes which are bounded by stream-surfaces. Figure 6 illustrates schematic drawing of one stream surface and the unit vectors used in meridional coordinates (denoted by e _m and ${e_\theta }).$ Quasi-normal lines are also drawn in this figure.

Figure 6. Definition of stream surface and quasi-normals.

The quasi-normal angle, designated by λ, represents the angle between any quasi-normal and the vertical direction. The slope of a streamline at any arbitrary point is indicated by $\varphi$ . Following relations can be obtained from Fig. 6.

(11) \begin{align}tan\lambda = \frac{{\Delta z}}{{\Delta r}} = \frac{{{z_h} - {z_s}}}{{{r_s} - {r_h}}}\end{align}

(12) \begin{align}\sin \varphi = \frac{{\partial r}}{{\partial m}}\end{align}

The local angle between the lines of quasi-normal and real normal on a streamline ( $\varepsilon$ ) can be obtained through Equation (13).

(13) \begin{align}\varepsilon = \varphi - \lambda \end{align}

By considering the real normal direction (n), as the third direction in Fig. 6, the tangential component of the relative velocity in the natural coordinate system (θ, m, n) can be obtained by Equation (14).

(14) \begin{align}{W_\theta } = {C_\theta } - \omega r\end{align}

The meridional component of the fluid velocity (C _m ) can be calculated by Equation (15).

(15) \begin{align}{W_m} = \sqrt {W_z^2 + W_r^2} = {C_m}\end{align}

2.2.2 Approximate quasi-normal equilibrium model

Flow properties profiles along a quasi-normal are modeled as a non-viscous and adiabatic flow problem. Assuming axisymmetric flow, a solution on the meridional plane can effectively simulate the flow field. The mass flow rate along one quasi-normal line, extended from hub to tip, in its integral form can be obtained by Equation (16) (see Fig. 6).

(16) \begin{align}\dot m = 2\pi \mathop \int \nolimits_0^{{y_s}} r\rho {C_m}cos\varepsilon \;dy\end{align}

The momentum equations in the tangential and normal directions, in the steady and axisymmetric form, can be written as follows.

(17) \begin{align}\frac{{\partial \left( {r{W_\theta } + {r^2}\omega } \right)}}{{\partial m}} = \frac{{\partial r{C_\theta }}}{{\partial m}} = 0\end{align}

(18) \begin{align}{\kappa _m}C_m^2 + \frac{{{W_\theta }}}{r}\frac{{\partial \left( {r{W_\theta } + {r^2}\omega } \right)}}{{\partial n}} + {C_m}\frac{{\partial {C_m}}}{{\partial n}} = \frac{{\partial I}}{{\partial n}} - T\frac{{\partial s}}{{\partial n}}\end{align}

Streamline curvature ( $\kappa$ _m ) is defined as the following relation.

(19)

The energy equation, in its unsteady and axisymmetric form, can be expressed as follows.

(20) \begin{align}\frac{{\partial I}}{{\partial t}} - \frac{1}{\rho }\frac{{\partial P}}{{\partial t}} + {W_m}\frac{{\partial I}}{{\partial m}} + \frac{{{W_\theta }}}{r}\frac{{\partial I}}{{\partial \theta }}\mathop \to \limits^{steady\;form} \;\;\frac{{\partial I}}{{\partial m}} = 0\;\;\;\;\end{align}

(21) \begin{align}I = H - \omega r{C_\theta }\end{align}

Substituting Equations (17) and (18) into the energy equation, one can conclude that the entropy term remains constant along one streamline (i.e. $\partial s/\partial m = 0$ ). Consequently, the meridional gradient of the entropy and angular momentum (rC _θ ) would be locally zero along one quasi-normal line. Finally, the equation of normal equilibrium in the meridional coordinate system can be derived as Equation (22).

(22) \begin{align}{C_m}\frac{{\partial {C_m}}}{{\partial y}} + {\kappa _m}C_m^2cos\varepsilon - {C_m}sin\varepsilon \frac{{\partial {C_m}}}{{\partial m}} + \frac{{{W_\theta }}}{r}\frac{{\partial r{C_\theta }}}{{\partial y}} = \frac{{\partial I}}{{\partial y}} - T\frac{{\partial s}}{{\partial y}}\end{align}

After performing mathematical operations for an adiabatic, inviscid and compressible flow, Equation (22) can be rearranged as Equation (23).

(23)

Here, M _m and M _θ represent the meridional and tangential components of Mach number (i.e. M _m = C _m /a and M _θ = C _θ /a, respectively). Equation (23) can be used to calculate C _m distribution along one quasi-normal line. This calculation starts from the first quasi-normal, and then, marches downstream along the subsequent lines. The flow field can be analysed as a non-viscous problem by solution of the SLC equations, while the viscous dissipative effect is modeled via empirical loss models. In the present research work, six different types of losses are considered by the present authors in their own in-house code. These losses are: profile loss [34], secondary flow loss [35], trailing edge loss [34], shock wave loss [34], supersonic expansion loss [34] and blade tip clearance loss [34]. For the sake of brevity, the relevant equations are not included in this paper. For more comprehensive information one can refer to Ref. (34). Figure 7 shows the computational algorithm utilised in the present research work.

Figure 7. Computational algorithm of SLC method.

3.0 Turbine characteristics

Table 1 summarises some of the constant input parameters which are used for designing the turbine. The proposed turbine consists of a one-stage high-pressure turbine (HPT) and a one-stage low-pressure turbine (LPT).

The in-house code offers great flexibility in mesh generation in terms of grid size within the computational domain. This flexibility extends to the entire turbine passage, including both the stator and rotor blades, as well as the axial gap between the blades rows. Initial location of streamlines on the meridional plane is specified by equal radial spacing between each two subsequent streamlines. SLC calculations continue through an iterative process and terminate while the same mass flow rate passes through each two adjacent streamlines. During the solution process the location of the streamlines are being continuously modified to their final positions. Figure 8 displays the initial and final positions of the streamlines within the turbine flow passage. Final mesh structure including the quasi-normals and streamlines are shown in Fig. 9.

Table 2 summarises turbine blades row characteristics at the mid-span, extracted from executing the in-house computerised code.

Aerofoil terminology defined for axial turbine blades, necessary for design process, is introduced in Fig. 10 [34]. At each radial position the blades spacing, stagger angle, inlet and outlet angles, real and axial chord lengths are obtained from executing the in-house computerised code. While the other parameters, including flow passage throat diameter, trailing and leading edges radii and wedges angles are obtained based on the empirical correlations [34]. For the sake of brevity these correlations are not included in this paper. The full geometry of the turbine, designed in this research work, is illustrated in Fig. 11 along with the top views of the HPT and the LPT rotor blades.

Gas dynamic data within the computaion domain is presented in the subsequent sections.

4.0 Three-dimensional viscous flow simulations

Both the steady and unsteady analyses of the 3D viscous flow field within the proposed turbine are conducted based on the frozen rotor and sliding mesh approaches, respectively. Governing equations consist of three-dimensional compressible U-RANS equations together with the energy equation. These equations are being solved based on the finite volume scheme using the well-known commercial CFX software. Coupled equations of the pressure-velocity for co-located grid arrangement are solved using interpolation technique introduced by Rhie and Chow [36]. High-resolution scheme is used for the advection terms in momentum and turbulence model equations. For turbulence modeling, a combination of SST k-ω turbulence model and γ-Re _θ transition equation [37] is utilised to effectively capture the probable transition phenomenon.

Table 1. Constant input parameters

Figure 8. Initial and final streamlines.

Figure 9. Final mesh structure.

Table 2. Some turbine characteristics at mid-span

Figure 10. Axial turbine aerofoil terminology.

Figure 11. Full geometry of LPT and HPT.

4.1 Equations of conservation laws

The governing main equations are introduced as follows:

1. – Continuity equation

   (24) \begin{align}\frac{{\partial \rho }}{{\partial t}} + \nabla .\left( {\rho .\vec u} \right) = 0\end{align}

2. – Momentum equation

Since fully coupled solution procedure of the velocity and pressure fields is used through the calculations, the momentum equations (the Navier-Stokes equations) are employed in the rotating frame based on the absolute velocity. The vector form of the momentum equation is presented by Equation (25).

(25) \begin{align}\frac{{\partial \vec u}}{{\partial t}} + \vec u.\nabla \vec u = \frac{{ - 1}}{\rho }\nabla P - \vec \Omega \times \left( {\vec \Omega \times \vec r} \right) - 2\vec \Omega \times \vec u + \nu {\nabla ^2}\vec u\end{align}

The second and third terms on the right hand side of the above equation represent centrifugal and Coriolis accelerations, respectively.

1. – Energy equation

The energy conservation of a fluid particle is ensured by equating the rate of change of the fluid particle energy to the sum of the net rates of the work and heat transfer to the fluid element and the rate of increase of energy due to the other sources. For compressible flows the energy equation can be derived based on the specific total enthalpy (h ₀ ). The general form of this equation is introduced as Equation (26).

\begin{align*}\frac{{\partial \left( {\rho {h_0}} \right)}}{{\partial t\;}} + div\left( {\rho {h_0}u} \right) = iv\left( {k\;grad\;T} \right) + \end{align*}

(26) \begin{align}\frac{{\partial p}}{{\partial t}} + \left[ {\frac{{\partial \left( {u{\tau _{xx}}} \right)}}{{\partial x}} + \frac{{\partial \left( {u{\tau _{yx}}} \right)}}{{\partial y}} + \frac{{\partial \left( {u{\tau _{zx}}} \right)}}{{\partial z}} + \frac{{\partial \left( {v{\tau _{xy}}} \right)}}{{\partial x}} + \frac{{\partial \left( {v{\tau _{yy}}} \right)}}{{\partial y}} + \frac{{\partial \left( {v{\tau _{zy}}} \right)}}{{\partial z}} + \frac{{\partial \left( {w{\tau _{xz}}} \right)}}{{\partial x}} + \frac{{\partial \left( {w{\tau _{yz}}} \right)}}{{\partial y}} + \frac{{\partial \left( {w{\tau _{zz}}} \right)}}{{\partial z}}} \right] + {S_h}\;\end{align}

4.2 Turbulence modeling

The SST k-ω turbulence model includes two transfer equations for k and ω quantities [38] as follows.

Transport equation for turbulent kinetic energy term (k):

(27) \begin{align}\frac{{D\rho k}}{{Dt}} = {P_k} - {D_k} + \frac{\partial }{{\partial {x_j}}}\left[ {\left( {\mu + {\sigma _k}{\mu _t}} \right)\frac{{\partial k}}{{\partial {x_j}}}} \right]\;\;\end{align}

${\rm{and}}$ the transport equation for dissipation term (ω), in the form of the following equation.

(28) \begin{align}\frac{{D\rho \omega }}{{Dt}} = \frac{\gamma }{{{\nu _t}}}{P_k} - \beta \rho {\omega ^2} + \frac{\partial }{{\partial {x_j}}}\left[ {\left( {\mu + {\sigma _\omega }{\mu _t}} \right)\frac{{\partial \omega }}{{\partial {x_j}}}} \right] + 2\left( {1 - {F_1}} \right)\frac{{\rho {\sigma _{\omega 2}}}}{\omega }\frac{{\partial k}}{{\partial {x_j}}}\frac{{\partial \omega }}{{\partial {x_j}}}\end{align}

In the above equations D _k and P _k are dissipation and production terms, respectively. See Ref. [38) for more details.

4.3 Transition modeling

Non-gradual development of intermittency weighting factor inside the boundary layer is a problem in transition modeling, triggered at a defined streamwise location. To address this issue, a technique known as $\gamma - \tilde{R}e_{\theta t}$ model, introduced by Langtry and Menter [37], is employed. This model defines a transport equation for intermittency weighting factor to solve the problem and to survey the flow history. Central idea behind this model is based on the concept of vorticity Reynolds number (Re _v ) [39]. Consequently, transition-onset momentum thickness Reynolds number would be linked to the local boundary layer quantities. This concept is described by the following equations.

(29) \begin{align}R{e_\nu } = \frac{{\rho {y^2}}}{\mu }\left| {\frac{{\partial u}}{{\partial y}}} \right|\end{align}

(30) \begin{align}R{e_\theta } = \frac{{max\left( {R{e_\nu }} \right)}}{{2.193}}\end{align}

This model uses a correlation, such as that offered by Abu-Ghannam and Shaw [40], to determine the onset of transition based on the turbulence intensity and pressure gradient parameter evaluated at the edge of the boundary layer. Transport equations for numerical intermittency (γ) and transition-onset momentum thickness Reynolds number ( $\tilde R{e_{\theta t}}$ ) are introduced as follows, respectively:

(31) \begin{align}\frac{{\partial \left( {\rho \gamma } \right)}}{{\partial t}} + \frac{{\partial \left( {\rho {U_j}\gamma } \right)}}{{\partial {x_j}}} = {P_\gamma } - {D_\gamma } + \frac{\partial }{{\partial {x_j}}}\left[ {\left( {\mu + \frac{{{\mu _t}}}{{{\sigma _f}}}} \right)\frac{{\partial \gamma }}{{\partial {x_j}}}} \right]\end{align}

(32) \begin{align}\frac{{\partial \left( {\rho \tilde R{e_{\theta t}}} \right)}}{{\partial t}} + \frac{{\partial \left( {\rho {U_j}\tilde R{e_{\theta t}}} \right)}}{{\partial {x_j}}} = {P_{\theta t}} + \frac{\partial }{{\partial {x_j}}}\left[ {{\sigma _{\theta t}}\left( {\mu + {\mu _t}} \right)\frac{{\partial \tilde R{e_{\theta t}}}}{{\partial {x_j}}}} \right]\end{align}

The role of Equation (32) is to transform the non-local empirical correlation into a local quantity to compute the transition length function (F_length) and critical momentum thickness Reynolds number (Re _θc ); which are needed for the numerical intermittency equation. If the Re _θ , computed by Equation (30), becomes greater than Re _θc (which is a local parameter as a function of $\tilde R{e_{\theta t}}$ ), γ would be activated to promote the production term (P _γ ). The Final modified version of the transport equation for (k) term in SST k-ω turbulence model is given by Equation (33).

(33) \begin{align}\frac{{\partial \left( {\rho k} \right)}}{{\partial t}} + \frac{{\partial \left( {\rho {u_j}k} \right)}}{{\partial {x_j}}} = {\tilde P_k} - {\tilde D_k} + \frac{\partial }{{\partial {x_j}}}\left[ {\left( {\mu + {\sigma _k}{\mu _t}} \right)\frac{{\partial k}}{{\partial {x_j}}}} \right]\end{align}

See Ref. [37] for more details.

4.3.1 Necessity on transition modeling

A key factor in clocking studies is related to proper modeling of transition, which leads to more realistic and precise results [25]. In this type of study, the fraction of laminar flow based on an empirical correlation suggested by Mayle [41] can be estimated by the following relation:

(34) \begin{align}\frac{{R{e_{xt}}}}{{R{e_L}}} = \frac{{\left( {3.8 \times {{10}^6}} \right){{\left( {100\;Tu} \right)}^{ - \frac{5}{4}}}}}{{R{e_L}}}\end{align}

In the above equation, Re _xt represents transition-onset Reynolds number. Also, turbulence intensity is calculated using turbulent kinetic energy (k) and flow velocity (V) as follows.

(35) \begin{align}Tu = \sqrt {\left( {\frac{{2k}}{3}} \right)} /V\end{align}

During primary flow simulations, the fraction of laminar flow on the blades surfaces has been estimated at mid-span. Results are presented in Table 3. Concerning this table, one can deduce that simulation with the assumption of only fully turbulent flow is not acceptable, since the fraction of laminar flow is not negligible on the blades surfaces. Therefore, modelling the transition phenomenon would be required to accurately capture the flow behaviour and to obtain reliable final results.

Table 3. Fraction of laminar flow of different components at mid-span

4.4 Computational domain

Figure 12 represents the computational domain including the number of blades at each row. For data transfer between the sub-domains, an interface is needed for the flow simulation studies. ‘Frozen rotor’ and ‘sliding mesh’ approaches are employed for steady-state and unsteady simulations, respectively. In addition, periodic boundary condition is imposed on the lateral surfaces of the computational domain.

Figure 12. Computational domain and boundary conditions.

Full structured mesh system has been applied for the computational domain using Turbogrid software. Grid topology around the blade profiles is of O-grid type. Mesh structure on the blades surfaces, inlet boundary and hub is shown in Fig. 13 for each sub-domain. Fine meshes are distributed for the regions of high sensitivity, such as the blades leading and trailing edges regions.

Figure 13. Surface grids structure.

Table 4 introduces number of grids generated within each sub-domain.

Total number of the elements is about 12.55 million. This number of elements has been selected as a result of examining independency of the results to the number of grids. Figure 14 shows this latter result, indicating variations of the total-to-total efficiency against the number of grids.

Y ⁺ values for the LPT stator and rotor blades are shown in Fig. 15. Due to implementation of γ-Re _θ transition model, the Y ⁺ must be kept below 5, otherwise the onset of transition will be estimated earlier than its real location [42].

5.0 Comparison between slc and 3d-cfd results

Figure 16 presents spanwise variations of the performance parameters of loading factor ( $\varPsi$ ), flow coefficient (Φ) and degree of reaction (R) for both the HPT and LPT. Average value of each parameter is also superimposed in each figure. Each turbine shows nearly a constant loading factor along the blades row span, which is obviously desirable. The degree of reaction exhibits as cumulative trend from the hub towards the casing.

It should be noted that continuous monitoring of control parameters has been executed during the design process. These control parameters are introduced as follows.

• Degree of reaction (R); Based on some references (for example: [33] it is recommended to set the R value about 0.55∼0.6 in order to reach the maximum efficiency. Because the flow field behaves better in the rotor blades in comparison to the stator blades. In the rotor blades the flow separation is getting delayed due to existence of the centrifugal and Coriolis forces, resulting in more stable boundary layer [43].

• Local loading factor ( $\varPsi$ _loc ); It is recommended to avoid exceeding $\varPsi$ _loc = 2.4 at hub region of LPT’s [44]. This value is equal to 1.42 for the present design case. In addition, the loading factor of the present HPT varies from 1.71 to 1.68 while moving from the hub to the tip, which is laid within its acceptable range of 1.4 to 2 [44].

• Hub relative Mach No. (M _{rel, hub} ); It is recommended that do not exceed M _rel above 0.7 for both the HPT and LPT at the hub region [21]. In the present turbine, M _{rel, hub} is equal to 0.66 and 0.59 for the HPT and LPT, respectively.

• Specific speed (Ω _s ); This parameter is obtained about 0.366 and 0.786 for the HPT and LPT, respectively, which are located within their limits [33].

Flow field results obtained from the SLC method are compared with those extracted through the 3D-CFD simulations. These results are presented in Fig. 17 in terms of contours of the main gas dynamic properties on the turbine meridional plane.

It is logically expected that the results obtained via the CFD approach do not exactly match with those extracted from the SLC method. Referring to Fig. 17, the most severe pressure gradients occur in the rotor blades. The difference in the absolute total pressure between SLC method and the 3D-CFD at the turbine outlet is 13.06%. Additionally, the difference in the absolute total temperaure between these two methods at the outlet of the baldes rows, including the first rotor, the second stator and the second rotor is 5.83%, 5.26% and 1.88%, respectively. It can be concluded that, in spite of some simplifications considered in SLC method (on the S2-surface), it is still as an appropriate technique in initial steps of design process of axial turbines (or axial compressors).

Table 4. Number of elements and sub-domains specification

6.0 Optimum clocked position of lpt stator blades

Five clocking positions of the LPT stator blades relative to the HPT ones, as shown in Fig. 1, are examined. To define these positions, the blades row is revolved circumferentially with equal steps of 2°, which are designated by CLP1 to CLP5. Also, CLP0 is attributed to the initial situation.

Figure 14. Variations of total-to-total efficiency with number of meshes.

Figure 15. Y-plus contour on LPT blades.

6.1 Optimum clocking position

Results of the LPT total-to-total efficiency, inlet absolute total pressure and its output power are shown in Fig. 18 for various CLPs. In addition, output power for the HPT and the whole turbine (HPT+LPT) are presented in this figure.

Figure 16. Spanwise variations of performance parameters.

Figure 17. Comparison between SLC and 3D-CFD results on turbine meridional plane.

Figure 18. Turbine specifications.

As can be detected from Fig. 18, the maximum efficiency of the LPT is obtained in CLP1 case (i.e. the LPT stator blades are clocked circumferentially 2° relative to its initial position). The worst case belongs to CLP4 (i.e. the LPT stator blades are clocked circumferentially 8° relative to its initial position). The efficiency in the CLP1 is increased by 0.415% in comparison to CLP4. As can be observed in these figures, the maximum inlet total pressure and output power of the LPT rotor belong to CLP1, which in comparison to CLP4, are increased by 0.097% and 0.547%, respectively.

Although the HPT stator is fixed and clocking is imposed on the LPT stator, characteristics of the upstream flow are affected by the downstream flow conditions. Consequently, the performance of the HPT rotor blades row can be changed. Regarding Fig. 18, the general trend of the total power of the whole turbine (HPT+LPT) is the same as that already introduced for the LPT rotor. It can be concluded that under the clocking process the total power has been mostly affected by performance of the LPT. The total output power of the whole turbine in CLP1 case has increased by 0.4% in comparison to CLP4.

Total-to-total efficiency of the whole turbine, HPT and LPT are separately presented in Fig. 19 for each clocking position. These latter results quantitatively support all the discussion presented in this section. Fidelity of the executed hybrid method in designing a high efficiency axial turbine is obvious by referring to Fig. 19. As shown in this figure the total-to-total efficiency of the whole turbine in its initial condition (i.e. CLP0) is more than 90% which is perfectly satisfactory.

Figure 19. Total-to-total efficiency of LPT, HPT and the whole turbine.

6.2 Aerodynamic performance of clocked stator

Steady aerodynamic performance of the LPT stator is investigated in terms of the pressure loss coefficient and outlet velocity entering the subsequent rotor. These results are presented in Fig. 20 at various CLPs. As can be detected, the highest velocity (i.e. the highest dynamic pressure) belongs to CLP1 case. As a result, the highest total pressure will occur in this case (see Fig. 18). Therefore, it can be deduced that CLP1 is associated with the lowest pressure loss coefficient. It is worth mentioning that in CLP1, pressure loss coefficient is decreased by 15.1% in comparison to CLP4.

Figure 20. LPT stator total pressure loss coefficient and outlet velocity at various CLPs.

6.2.1 Blade-to-blade wake flow trajectories

Trajectory of the wake flow, originating from the HPT rotor blades row, is presented and discussed in this section for CLP1 and CLP4 cases on different spanwise blade-to-blade surfaces, in terms of static entropy contours. These results are presented in Fig. 21 for 10, 50 and 90% of the blades span.

Figure 21. Entropy contours at different blades span.

Additionally, Table 5 summarises circumferential-averaged flow properties at the LPT stator outlet for 10, 50 and 90% of the blades span which support the following discussion.

1. – 10% span: At 10% span in CLP1, a wake impinges on S2-1 leading edge leaning toward the suction side and the S2-2 and S2-3 blades are completely covered by the upstream wakes. Also, a wake passing near the S2-3 suction side gradually merges with the wake which completely covers its suction and pressure sides. In contrast, in CLP4, the wakes originated from R1-5 and R1-6 blades and the wake passing through the middle space of the passage between these two blades, altogether enter the passage between S2-1 and S2-2. Also, two wakes enter the middle space of the passage between S2-3 and the corresponding blade of S2-1 without hitting their leading edges. Such entry, without wake impinging on the leading edge, increases the pressure loss. Regarding Table 5, at 10% span under CLP1 in comparison to CLP4, the outlet total pressure of the LPT stator is increased by 0.905% due to increasing the absolute velocity. In addition, it can be detected that increasing the absolute velocity results in higher absolute total temperature at the LPT stator outlet in CLP1.

2. – 50% span: As can be detected from Fig. 21 at 50% span in CLP1, the wake flow of R1-3 blade, impinges on the S2-1 leading edge and sweeps its suction side surface. Also, a wake hits the S2-3 leading edge and sweeps its pressure side. In CLP4, only one wake flow impinges on S2-2 leading edge and covers its surfaces. Also, two strong wakes, originated from the upstream rotor blades row, enter the middle space of the passage between S2-1 and S2-2 blades. Two other wakes enter the middle space of the passage between S2-2 and S2-3 without hitting the blades leading edge. Better wake entry situations in CLP1, in comparison to CLP4, causes the total pressure at the LPT stator outlet to increase by 0.123% (see Table 5).

3. – 90% span: The events at 90% span in both CLPs are almost identical to those at 50% span. Regarding Table 5 data at 90% span, the total pressure at the LPT stator outlet in CLP1 is increased by 0.495% in comparison to CLP4.

6.3 Clocking effects on unsteady performance

Investigation of clocking effects on unsteady performance of the turbine has been performed for the CLP1 and CLP4 cases, i.e. the optimum and the worst CLPs, respectively. Time-averaged results for some selected flow properties are calculated and results are presented in Table 6.

It can be deduced that in comparison to CLP4, the absolute total pressure at the LPT rotor inlet and its output power in CLP1 are increased by 0.23% and 0.93%, respectively. These augmentations in CLP1 result in an increase of 0.444% in aerodynamic efficiency of the LPT in comparison to CLP4. All these latter augmentations are as a result of 20.3% decrease in the total pressure loss of the LPT stator. Absolute velocity at the LPT rotor inlet in CLP1 is increased by 0.607% in comparison to CLP4. The outlet velocity of the LPT stator increases under the optimum clocking, which in turn, causes the total pressure at the LPT rotor inlet region to increase. In the CLP1 case, the HPT rotor output power is increased by 0.54% in comparison to the CLP4 case. Finally, it can be deduced that the output power of the whole two HPT+LPT in CLP1 is increased by 0.71% in comparison to CLP4.

6.3.1 Unsteady performance of LPT stator

Unsteady aerodynamic performance of the LPT stator blades is investigated and time dependent results are shown in Fig. 22. The horizontal axis of these plots indicates the normalised time, which is shown by t/ $\tau$ . Here, $\tau$ is attributed to the time taken by a wake flow while it travels from the stator blade entry to its exit section.

Table 5. Performance parameters at different blades span at LPT stator outlet

Table 6. Time-averaged results of rotor blades rows

Figure 22. Spanwise-averaged instantaneous results of LPT stator.

Regarding Fig. 22, outlet velocity of the LPT stator in CLP1 is always higher than that in CLP4, which results in higher outlet total pressure. As expected, CLP1 is accompanied by the lowest pressure losses at any time.

Instantaneous entropy contours at the LPT stator inlet are presented in Fig. 23 for three subsequent normalised times. In CLP1, at t/ $\tau$ = 0, a continuous wake segment extended from hub to tip impinges on the S2-2 leading edge. However, in CLP4, at the same time, two wakes enter the passage between S2-2 and S2-3 blades without impinging the blade leading edge and the other two ones enter the passage between S2-1 and S2-2 blades. Comparison of wake entry situation between both the CLPs at t/ $\tau$ = 0.5 reveals the fact that in CLP1 upstream wakes impinge on the leading edges of S2-1 and S2-3 blades, which results in an increase in velocity at the LPT stator outlet in comparison to CLP4 (see Fig. 22, too). As the time advances up to t/ $\tau$ = 1, once again, a wake impinges on S2-l leading edge. Meanwhile, in CLP4, the wake segments enter the passage between the blades without any collision on the blades surfaces.

Figure 23. Instantaneous entropy contours at LPT stator inlet boundary.

Referring to Fig. 22 one can observe that at all instances, the outlet total pressure of the LPT stator in CLP1 is higher than that at CLP4. This superiority is a direct result of more frequent impingement of the upstream wakes on the leading edges of the LPT stator blades in CLP1 in comparison to CLP4. Instantaneous total pressure contours at the LPT stator outlet, presented in Fig. 24, qualitatively support the above conclusion.

Figure 24. Instantaneous contours of total pressure at LPT stator outlet boundary.

6.3.1.1 Blade-wake interaction under clocking process

Trajectory of the wake flow within the blades passages of the LPT stage, basically originated from the HPT stator blades row, is presented and discussed in this section in terms of the instantaneous entropy contours. These results are presented in Fig. 25 for the subsequent five normalised times, at the blades mid-span. In addition, no significant event has been observed within the first stator blades while imposing the clocking on the second stator. Therefore, the contours shown in Fig. 25 run from the first rotor. In this figure the wakes are labeled by ‘W’ followed by two numbers. The first number corresponds to the specific normalised time, while the second number, starting from zero, indicates the sequence of the wakes at that particular time.

Figure 25. Instantaneous entropy contour at blades mid-pan.

As can be detected from Fig. 25, in CLP1 case, the W1-1 wake enters the passage between S2-1 and S2-2 blades after hitting the S2-2 leading edge, and then, the deformed wake is convected through the passage. Continuously, subsequent wakes follow a similar pattern, entering the passage in the same manner. This behaviour is also observed for the passage between the S2-2 and S2-3 blades. Stretching and re-orientation processes experienced by the deformed wakes lead to a negative jet towards the suction side. This phenomenon, explained more detail in Refs. [22] and [45], results in acceleration of the flow on the suction side of the flow passage. Consequently, the lift force acting on the blades and the outlet dynamic pressure are increased.

In CLP4 in comparison to CLP1, it takes W1-1 a longer time to impinge on the S2-2 leading edge. As it can be seen at t/ $\tau$ = 0.75, in CLP1 there are almost four deformed wakes (labeled as W1-1, W1-0, W1-2 and W2-0) in the passage between S2-1 and S2-2 (see Fig. 26). These wakes pass through the passage which cause flow acceleration on the suction side of the blades. But in CLP4, along with W3-0 which advances at least to the half of the middle space of the passage without hitting S2-2 leading edge, there are only three deformed wakes (labeled as W1-1, W1-0 and W2-0) in the passage. In CLP1 in comparison to CLP4, wake convection without hitting S2-2 leading edge is taken shorter by W3-0.

Figure 26. Time-dependent flow properties at LPT stator outlet.

The unsteady effects of all the above-mentioned events on the flow properties at the LPT stator outlet are investigated quantitatively and the results are presented in Fig. 26 in terms of total pressure, velocity and total pressure loss coefficient versus the subsequent five normalised times for 10, 50 and 90% blades span. In CLP1 case the outlet velocity from the second stator (and consequently the total pressure) is increasing from t/ $\tau$ = 0 up to 0.75 (see also Fig. 25). This is primarily attributed to the transient effect of W1-1 impinging on S2-2 leading edge along with influences of the previous and subsequent wakes. By dissipation of W1-1 at t/ $\tau$ = 1 the velocity and consequently the total pressure are decreased. Referring to Fig. 26 one can clearly confirm the beneficial effects of CLP1 in comparison to CLP4 at each spanwise position and at any instant.

In the CLP4 case the outlet total pressure is increased from t/ $\tau$ = 0.5 to 1 at 50% span. Decrement of the total pressure in the interval between t/ $\tau$ = 0 to 0.5, is due to the non-collision entry situation of the wakes passing through the middle space of the passages between S2-1 and S2-2 and between S2-2 and S2-3 (see Fig. 25, too).

General trend of variations of the total pressure and velocity at 90% span follows the same manner as observed at the mid-span. Additionally, at all the three spans in CLP1, the outlet total pressure of the LPT stator is higher in comparison to CLP4 at any time. As can be detected from Fig. 26, the total pressure loss coefficient is always lower in CLP1 in comparison to CLP4 at all the three span positions. Notably, in both CLPs, the lowest losses happen at mid-span, while the highest ones occur at 90% span.

6.3.1.2 Clocking effects on blades loadings

To investigate clocking effects on the LPT stator blades loading, instantaneous pressure distribution at the blades mid-span is integrated to obtain the lift force coefficient (C _L ). Results are presented in Fig. 27.

Figure 27. Instantaneous C _L of LPT stator blades at mid-span.

As can be detected from this figure, in CLP1 case from t/ $\tau$ = 0 to 0.75, S2-1 and S2-2 blades loading is increased continuously due to their wake positive effects on the suction surface, as discussed earlier (refer to section 6.3.1.1, too). Subsequently, at t/ $\tau$ = 1, their loadings are decreased due to the wake dissipation. Also, in CLP1 these two latter blades experience higher loading at all times in comparison to CLP4.

6.3.2 Unsteady performance of LPT rotor

The unsteady effects of LPT stator wakes trajectories on the LPT rotor blades loading are investigated quantitatively and some selected results are presented in Fig. 28 only for three blades (i.e. R2-2, R2-4 and R2-6) in terms of C _L . In addition, time-averaged results for all the seven LPT rotor blades and instantaneous results for LPT rotor (i.e. averaged C _L of all the blades) are shown in this figure (see also Fig. 21 for rotor blades numbering).

Figure 28. Instantaneous C _L of LPT rotor blades at mid-span.

As can be detected from this figure, in CLP1 case from t/ $\tau$ = 0 to 0.25, R2-2 blade loading is decreased. At t/ $\tau$ = 0.25, an upstream wake impinges on the R2-3 leading edge and enters the passage between R2-2 and R2-3 (see Fig. 25). Then from t/ $\tau$ = 0.25 to 1, the wake sweeps the R2-2 suction side, which results in increasing R2-2 loading (see Fig. 28). From t/ $\tau$ = 0 to 0.5 in CLP4, R2-2 loading is increased due wake effect on its suction side in this time interval, and then, is decreased as the wake is dissipated (see Figs. 25 and 28).

Regarding Fig. 28, the time-averaged C _L of LPT rotor blades in CLP1 is higher in comparison to CLP4. It should be noted that, the higher loading of LPT rotor blades in CLP1 in comparison to CLP4 is mostly due to higher total pressure at the inlet of LPT rotor besides stator wake trajectories impinging on rotor blade leading edge. Also, instantaneous C _L of LPT rotor is increased from t/ $\tau$ = 0 to 0.75, and then, is decreased which is higher in comparison to CLP4 at all time.

7.0 Conclusion

In the present research work, initially HPT and LPT units of a gas turbine engine is aerodynamically designed utilising a hybrid method. Then, its performance is optimised through the best clocking of the LPT stator blades in order to reach the maximum efficiency of the turbine. The hybrid method comprises a combination of the SLC and FV methods. The main conclusions drawn from the current research work can be categorised as follows.

1. – Comparison between SLC results and those of 3D-CFD shows discrepancies between turbine outlet total pressure and temperature, about 13.06 and 1.88, respectively. Based on 3D-CFD, turbine total to total efficiency is about 91.83% which proves the proper design procedure and optimum clocking of LPT stator blades.

2. – Clocking analyses of the LPT stator blades row, by consideration of six CLPs, were conducted by steady 3D flow simulation to find the optimum CLP. The steady results demonstrated that under optimum CLP, the turbine efficiency is increased by 0.122% and 0.415% in comparison to the initial and the worst CLPs, respectively. Furthermore, time-averaged results obtained by unsteady 3D flow simulations, revealed that under optimum CLP, efficiency of the LPT stage increases by 0.444% in comparison to the worst CLP. This optimisation is as a result of increasing the outlet total pressure of the LPT stator by 0.23% or decreasing in its total pressure loss by 20.3%. The key factor contributing to this improvement is increasing of outlet velocity of the LPT stator (i.e. dynamic pressure) under the optimum CLP, which results in augmentation in outlet total pressure of LPT stator.

3. – Flow visualisation of blade-wake interaction at mid-span of the LPT stator blades shows that in a special time interval (i.e. wake convection time through blade passage) under the optimum CLP, the impingement of wakes on the blade leading edge is expedited. This phenomenon results in cumulative blade loading and consequently increased outlet total pressure. In contrast, under the worst CLP, wakes take longer to pass through the blades passage without hitting blades leading edge, which results in increased total pressure loss and decreased blades loading. Instantaneous results show that under optimum CLP, in comparison to the worst one, the LPT blades loading is higher as a result of wakes effects on the blade suction side which is accompanied by increases in outlet velocity and dynamic pressure.