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 [Reference Yahya33].

(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 [Reference Aungier34], 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 [Reference Aungier34], secondary flow loss [Reference Kacker and Okapuu35], trailing edge loss [Reference Aungier34], shock wave loss [Reference Aungier34], supersonic expansion loss [Reference Aungier34] and blade tip clearance loss [Reference Aungier34]. For the sake of brevity, the relevant equations are not included in this paper. For more comprehensive information one can refer to Ref. (Reference Aungier34). Figure 7 shows the computational algorithm utilised in the present research work.

Figure 7. Computational algorithm of SLC method.

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 [Reference Wilcox38] 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. [Reference Wilcox38) 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 [Reference Langtry and Menter37], 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 ) [Reference Van Driest and Blumer39]. 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 [Reference Abu-Ghannam and Shaw40], 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. [Reference Langtry and Menter37] 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 [Reference Arnone, Marconcini, Pacciani, Schipani and Spano25]. In this type of study, the fraction of laminar flow based on an empirical correlation suggested by Mayle [Reference Mayle41] 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 [Reference Menter, Smirnov, Liu and Avancha42].

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. [Reference Hodson and Howell22] and [Reference Stieger and Hodson45], 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.