Flow Mechanisms in Low-Pressure Turbines

Abstract

In aircraft engine, the main task for low pressure turbine (LP turbine, LPT) is to drive rotational components, for example the fan or booster stages. It also can be used as direct power output apparatus, which provides shaft power to drive a propeller, fan, or other lift or thrust equipment. In turboprop and turboshaft engine, LP turbine is also known as power turbine or free turbine. In the spatial position of the flow path, LP turbine locates behind the high pressure turbine (HP turbine, HPT). Between the HP turbine and LP turbine in the high bypass ratio (BPR) turbofan engines, there usually arranges the bearing freamework known as the inter-turbine ducts, and turbine rear frame (TRF) ducts which is connecting to outlet nozzle.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

4.1 Geometrical and Aero-Thermal Characteristics of Low-Pressure Turbines and their Development Trends

4.1.1 The Geometrical and Aero-Thermal Characteristics of LP Turbines

In aircraft engine, the main task for low-pressure turbine (LP turbine, LPT) is to drive rotational components, for example the fan or booster stages. It also can be used as direct power output apparatus, which provides shaft power to drive a propeller, fan, or other lift or thrust equipment. In turboprop and turboshaft engine, LP turbine is also known as power turbine or free turbine. For the spatial position in the flow path, LP turbine locates behind the high-pressure turbine (HP turbine, HPT). Between the HP turbine and LP turbine , in high bypass ratio(BPR) turbofan engines, there usually arranges the bearing framework known as the inter-turbine duct. And the LP turbine is connected with outlet nozzle using the turbine rear frame (TRF) .

The mission of the aircraft and its operating environment determine that LP turbine has its own characteristic geometrical and aero-thermal parameters. Comparing with the HP turbine geometry, due to the lower rotational speed and different aero-thermo parameters, LP turbine is with larger size, more stages, and heavier weight. Take the LP turbine that directly drives the fan in high BPR turbofan engine as an example. Since the LP turbine and the fan are connected via the same shaft, they must be in the same rotational speed. Considering the fan performance and noise restrictions, tip velocity of fan can be generally established. With increasing the fan diameter, it results in a low level rotational speed of the LP shaft, which limits single-stage LP turbine power capability. Also because the LP turbine has relatively low inlet temperature and pressure, it further limits the work capability of units mass flow rate. Additionally, though the gas mass flow rate through the LP turbine is less than that through the fan, the power output from LPT should not be low. Eventually it leads to the design of LP turbine has larger outer diameter and more stages to avoid lack of power capacity. For the LP turbine that drives BPR turbofan, it can be 5–7 stages and contributes 25–30% of the whole engine weight. In addition, since the hot gas already expands in the HP turbine, the gas density reduced significantly, which leads to increased volume flow rate. Meanwhile, due to the Mach number limitation at LP turbine outlet to be in the range of 0.4–0.6, it ultimately leads to an increasing cross section area of the flow passage. Therefore, geometrically LP turbine has longer blade with larger aspect ratio and smaller hub diameter. A typical LP turbine blade aspect ratio is between 3:1 and 7:1. In high-loaded low pressure turbine cascade, the flow turning angle is large, typically 100°–110° or even higher, so the cascade has a relatively smaller convergence. To summary, the main geometric characteristics of the LP turbine cascade are, low hub-to-tip ratio, high aspect ratio, large turning angle and small convergence passage.

In terms of velocity triangle , the low shaft speed means a lower tangential speed. Generally, it results in significantly larger flow coefficients and blade loading factors when compared to the HP turbine, and thus is not conducive to improving the aerodynamic performance of LP turbines [1].

In the aspect of aero-thermal process, since the LP turbine locates after the HP turbine or ITD, it faces the upstream blade wake , leakage flow , secondary passage vortices, and also the existing of strong shock wave near high loaded HP turbine trailing edge. All these lead to a complicated inflow condition for LP turbine, and aggravate the uneven distribution of the inflow aero-thermal parameters. It means that design should consider the matching between LP and HP turbine within all operating points. In addition, the LP turbine outlet normally requires axial out flow. With small convergence of the blade passage, it is not easy to achieve in the final stage of a LPT. However, if the outlet guide vane in the turbine rear frame (TRF) is designed with aero-loading, the LP turbine outlet flow angle may deviate from the axial for 20°–30°. This is beneficial to reduce the number of LP turbine rows. Meanwhile, the LP turbine passage has the lowest hot gas density and the lowest Reynolds number level [2] in the engine. As shown in Fig. 4.1, during high altitude cruise, the LP turbine Reynolds number can be as low as 3 × 10⁴ to 5 × 10⁴ [3], especially for the final stage. In this case, low Reynolds number effect is one of the key factors on LP turbine aerodynamics performance, for example the efficiency and flow capacity, which must be considered specific [4]. To summarize, for conventional LP turbine, the main aero-thermo characteristics include low Mach number, low Reynolds number , high loaded, and complex inflow boundary conditions.

Fig. 4.1

Reynolds number levels of components in a mid-size engine [2]

4.1.2 Flow Characteristics and Losses in LP Turbine Passages

Figure 4.2 shows the typical flow visualization results on a LP turbine blade [5]. As shown in the figure that the flow field near blade tip and hub has been strongly impacted by the secondary flow with three dimensional characteristics. Secondary flow loss also dominates these regions. However, due to the large aspect ratio of the blade, the region impacted by the secondary flow is limited, thus its loss contributes small portion of the overall losses. In most of the blade span, flow shows quasi-two-dimensional characteristics that boundary layer associated profile loss dominates. Since the LP turbine rotor blade normally has shroud , the tip leakage flow is relatively weak and its loss is not significant. So in general, the profile loss is the most important source of aero-loss, the main factor which affects the performance, and also the major point should be considered in the design.

Fig. 4.2

Flow visualization on a LP turbine blade [5]

The profile loss is mainly determined by the status of the boundary layer development on the blade, base region formed due to boundary layer shedding at the trailing edge, and mixing between wake flow and main flow [6], etc. As shown in Fig. 4.3, due to the small radius of the trailing edge, the base region loss and mixing loss is less. Relatively, the flow condition of boundary layer contributes the major portion of aero loss, which directly affects the friction loss and separation loss, etc. Here, the blade surface friction loss is proportional to the cubic of hot gas velocity. Due to the velocity difference between pressure side and suction side, the pressure side friction loss is significant less comparing to that on suction side. In addition, due to the adverse pressure gradient near the trailing edge, flow separation often happens on the suction side, which causes more serious damage [5], resulting in a sharp increase of loss. Studies showed that the suction side total loss raised about 60% while about 20% on the pressure side. Mixing loss caused by wake is about 16% and the rest is base region loss, which accounting for about 4% [7]. Therefore, effective control of the boundary layer state, especially on suction side, is particularly important for LP turbine profile design.

Fig. 4.3

Estimated loss breakdown for datum blade [7]

Although the secondary flow loss contributes small proportion of the overall loss, it cannot be simply ignored in detail design process. The shrouded LP turbine blades can suppress leakage flow and vortices, but the flow in the shroud region is complicated. If the shroud is not designed properly, leakage flow will interact with main flow and leads to the influence that may not smaller than the unshrouded blades [8]. In addition with ultra-high-lift LP turbine profile, the corresponding secondary flow changes significantly. With the interaction between secondary flow and separation bubble near the leading edge of the pressure side, it enhances secondary flow intensity and further distorts the boundary layer [9], the secondary flow loss increases. Therefore, secondary flow loss cannot easily be ignored in high performance and high-loaded LP turbine design.

4.1.3 Development Trends of LP Turbine Design

In the past two decades, research of fluid mechanism and high-performance high-loaded LP turbine aero-design has seek significant attention from engine companies and universities. With the support from many research programs, the understanding of flow mechanism and the capability of component design have been improved rapidly. The new technologies are more and more used in aero-engine design, which effectively improves its performance. Technically, some major trends are:

• 1. Through the careful study of aerodynamics in real geometry and thermo environment, establish the mathematical-physical model; develop detail design methods, explore the potential of improving efficiency, provide support to enhance performance.

In aero-engine, different components have different impacts on the specific fuel consumption (SFC). Table 4.1 shows the components performance exchange to SFC for a civil aircraft cruise at 0.8 Mach number in 10,000 m altitude. It shows that LP turbine efficiency has significant impact on SFC [10] that 1% efficiency change in LP turbine causes 0.9% or even more SFC change. With the incense of bypass ratio (BPR) in civil turbofan engine, this sensitivity would be even larger. Thus, to increase LP turbine efficiency is important for reducing the cost of the engine or even the operation. But with matures of design technology, the development of individual technology has difficulty to significantly enhance the LP turbine efficiency. So, people gradually turned their attentions to some design details which had been overlooked in the past, such as shroud , hub seal plate, blade leading edge shape, blade-hub junction shape, endwall non-axisymmetric surfaces, and so on. More detailed design methods and guidelines are developed accordingly [11]. Although each individual technology has limited contribution to enhance performance, their integration is very impressive, so that the LP turbine performance still rose considerably.

Table 4.1 Fuel consumption sensitives with components efficiency in one civil aviation turbofan [10]

• 2. Deep understand the unsteady flow mechanism, investigate the ways to apply unsteady effects within design purpose and develop corresponding design method, increase loading with similar or improved efficiency in order to reduce weight, also reduce the production and maintenance cost by reducing parts.

The weight of LP turbine accounts for a large proportion of the whole engine weight, which can be up to one-third. And the huge amount of LP turbine blades, which can be even thousands, increases the cost of manufacturing and maintenance. Survey data shows that with 17% engine weight reduction, the operation cost can reduce by 1%, which means cost deduction $200,000 per year [12]. Thus another important LP turbine technology trend is to increase load with current or improved performance, which means to reduce number of blades and turbine weight. The most successful example is the “calmed” effect application. This technology improved the LP turbine loading by 30–40%, and has already been applied to the UK Rolls-Royce BR71 * series, TRENT series, Engine Alliance’s GP7000 series and other high bypass ratio (BPR) turbofan engine [13,14,15].

• 3. Based on high-performance high-loaded design methods, further investigate the influences of small geometric deviations and assembly errors, perform parameter optimization, thus to develop robust aero design techniques, and reduce production and maintenance costs.

In modern aero engine technology, manufacturing is crucial to engine performance and reliability. There are defective events occurred due to manufacturing and assembly errors. Due to the complicated flow field in turbomachinery, subtle changes in the key parameters would directly affect detailed flow field and components matching, which would eventually leads to significant change in performance. Study from Cambridge University shows that the change of geometry or surface roughness at the leading edge of the compressor blade will significantly affect the flow separation near compressor hub. Study from MIT indicates that the typical manufacture errors can cause efficiency decreased by 1% in multiple stages compressor [16, 17]. It also happens to turbine. Study from Beihang University (BUAA) shows that subtle changes on profile can lead to local flow deterioration, which would be amplified in whole engine environment and reduce performance significantly [18, 19]. Although there is not much publication in this area, it is an inevitable trend to investigate geometric error or assembling error for robust design.

• 4. Considering LP turbine component-matching in full operation range, study the factors and mechanisms that impact performance degradation after long-operation time in real work environment, it is possible to abstract low-order coupled model, develop multi-disciplinary integrated design technologies for LP turbine so as to significantly improve the performance and reduce its life-cycle cost.

With the increased requirements for aero engine, efficiency at design point is no longer the only performance indicator. Instead, the LP turbine performance in each mission and even in the full lifetime is used for judgment. It means that an good LP turbine should provide good performance over wide operating range, like takeoff, climb, cruise, decent, landing and so on. The performance reduction should be less over time. Also, a longer life time with reliable mechanical integration, less requirement to manufacturing and material, and easy maintenance are also demanded. It means that not only the aero-parameters need to be considered during design, but also the listed above factors. Eventually to develop the integrated design technology with multidisciplinary coupling can fulfill the performance requirements and full-life time cost.

4.2 Boundary Layer Spatial-Temporal Evolution Mechanism in LP Turbines

The boundary layer and its evolution have significant impact on LP turbine loss. The key phenomena to determine the boundary layer characteristics are flow separation, transition, etc. Therefore, fully understanding mechanisms of separation and transition has significant importance for further organizing and controling the boundary layer flow, which is essential to reduce LP turbine profile loss .

4.2.1 Flat Plate Boundary Layer Evolution and Flow Losses

4.2.1.1 Flat Plate Boundary Layer Evolution and Flow Loss with Zero Pressure Gradient

To discuss boundary layer evolution in cascade, it is essential to first understand laminar boundary layer transition mechanism on a flat plate in steady flow condition without pressure gradient . The laminar flow falls to different transition paths with different flow disturbances. For this, there are two classic transition modes: If the external disturbances are weak, and the plate wall is smooth, the natural transition happens; if the external disturbances are strong, the bypass transition happens. Recent studies show a more detailed transition road map. In Fig. 4.4, Morkovin shows transition modes with a variety external disturbances [20, 21]. According to this classification, between the natural and bypass transition modes, there also exists “transient growth” mode. Under this mode, different disturbances level corresponds to different transition paths (B–D, see Fig. 4.4) with not exactly the same mechanism.

Fig. 4.4

Boundary layer transition roadmap [20]

There are substantial studies on natural transition. Figure 4.5 shows the classic description of flat plate boundary layer natural transition [22]. Laminar boundary layer begins to form at the plate leading edge and develops downstream. Meanwhile it “feels” flow disturbances [23, 24]. When the boundary layer Reynolds number exceed critical value, laminar boundary layer becomes unstable with Tollmien-Schlichting waves (T-S waves) showing inside it. The amplitude of the waves enlarges gradually during propagation. With the secondary instability, $\lambda$ vortices appear. The $\lambda$ vortices further develop and deform due to nonlinear impacts, and form three-dimensional hair-pin vortices. The rapid growth of hair-pin vortices eventually leads to the breakdown of the laminar flow and form local turbulent spots. Turbulent spots, which are from different originations, convect and diffuse with main flow, eventually combine with each other, and form a fully turbulent boundary layer. This is typical natural transition.

Fig. 4.5

Natural transition of a flat plate boundary layer [22]

However, when there are strong flow disturbances, such as high level freestream turbulence, surface roughness , noise, vibration or other disturbances in tubormachinary, the first three stages in the natural transition mode may be skipped. Disturbances will penetrate through boundary layer and directly induce turbulent spots, thereby lead the transition process. This is the so-called bypass transition mode. The T-S waves cannot be observed in this mode. Without those three unstable stages, the bypass transition process become faster and its length become shorter. From the flow structure perspective view, the main characteristic of bypass transition is the appearance of streamwise streaks. Studies have shown that those streaks, which develop from the leading edge of plate, are related with low frequency disturbances. Though those disturbances have large scale in streamwise direction, their length scales in spanwise and vertical direction are in the same order with the boundary layer [25]. Figure 4.6 shows the bypass transition sketch with different freestream turbulence intensity (FSTI) and freestream velocity [26]. The results show the streaky structures in four conditions. With streaks developing along streamwise direction, they twist and swing, and ultimately become turbulence. Under low FSTI, the streaky structures last longer, while turbulent spots may appear randomly between them. Under high FSTI, these streaky structures are relatively more chaotic mainly due to the turbulent spots appear earlier. In addition, the turbulence spots show larger length scale in spanwise under the above two corresponding conditions. This is, on the one hand, because the turbulence length scales in those two conditions are larger, and on the other hand, due to the flow speed reduction and the resulting incensement of boundary layer thickness.

Fig. 4.6

Streamwise velocity streaks in by-pass transition [26]

Although the hydrodynamic stability theory gives some explanations of the streaks, until the end of last century, it was still unclear with the mechanism for the bypass transition , in particular the formation of turbulent spots. Regarding the issue, researchers from Center of Turbulence Research (CTR) in Stanford University using direct numerical simulation (DNS) as the main research tool together with theoretical analysis, had carried out a series of studies and provided comprehensive explanations on the mechanism of bypass transition [27,28,29,30]. They believe that due to the shear sheltering effect in laminar boundary layer, shear layer acts similar as a low-pass filter, which means low frequency components from disturbances can enter the boundary layer, while the high-frequency component of the disturbance will be suppressed. Those low-frequency disturbances will then be enlarged and stretched due to shear flow along the streamwise direction and eventually form the streamwise velocity streaky structures. And just those streaks form the forward jet and reverse jet in the disturbance velocity field. As shown in Fig. 4.7, when the reverse jet is being lifted to the edge of the boundary layer, shear sheltering no longer exists. Instead, the Kelvin-Helmholtz instability (K-H instability ) in the reverse jet flow velocity profile is triggered by small scale (high frequency band) turbulence. It causes flow instability, and creates random turbulent fragments, which gradually grow into turbulent spots while transporting. The basic form and transport properties of a single turbulent spot have been introduced in the first chapter. Besides, the structural characteristics of the turbulent spots [31] and its evolutionary mechanisms under different environments [32,33,34] and the regulation of interaction [35,36,37] have also been the central topics in the community, while they are out of the scope of this book.

Fig. 4.7

Reverse jet flow in bypass transition [27]

4.2.1.2 Flat Plate Laminar Boundary Layer Separation with Adverse Pressure Gradient

When varying pressure gradient , the temporal evolutions of the boundary layer on turbine blade will be not the same. The adverse pressure gradient after turbine throat will induce boundary layer separation and the resulting load distribution (corresponding to the change in pressure gradient) will play important role in the internal flow field. With the fact that the research on flat plate boundary layer evolution not only can capture the key factor like pressure gradient, but also has the advantages for easy experimental tests and numerical simulations, it becomes important development for the boundary layer in turbomachinery and raised many researchers in this area.

LP turbine, may well encounter laminar boundary layer separation which impacts seriously on aerodynamic performance. First, Laminar boundary layer is gradually decelerating under the influence of adverse pressure gradient , separation may eventually occur. The shear layer after separation will be very sensitive to disturbances due to the inflection point in the velocity profile. It triggers instability and then transition begins. A detailed discussion on this point will be given in the next section. Large scale vortices form in the separated shear layer during transition process. These vortices can entrainment main flow into the boundary layer, which will finally result in the shear layer to reattach and gradually form the turbulent boundary layer. The flow structure formed above is the so-called “laminar separation bubble”. Typical separation bubble structure was first given by the Horton [38] as shown in Fig. 4.8. The streamline which passed the separation point lifts away from wall, then starts to drop to the wall again at the back part of separation bubble and finally reattaches to wall at the reattachment point. This streamline is called the “dividing streamline”. That is, outside this streamline is the main flow, while the inside wedge-shaped region is the separation bubble. The flow velocity is small after the separation point. Just because the flow here almost stagnates and maintains laminar flow condition, this region is named the “dead-air zone”. Displacement of wedge separation bubble can influence main flow by deflecting the flowand, as a result, causing the pressure gradient in this area to change. Figure 4.8 shows that from the separation point till somewhere upstream of the reattachment point, wall pressure remains unchanged, namely the so-called “pressure plateau”. Near the reattachment point, the pressure recoveries to the corresponding re-attached boundary layer level. In this region there is strong reverse flow, whose velocity can reach about 20% of the main flow [2].

Fig. 4.8

Typical structure of laminar separation bubble

As shown in Fig. 4.8, one of the most significant characteristics of separation bubble is the change of the boundary layer pressure distribution, means the existing of plateau. This is a common criterion to determine whether separation happens and its location. Comparing to the physical length of separation bubble, the wall pressure distribution is a more suitable parameter to sort out its type. Accordingly, the separation bubble can be divided into two categories [39]: short bubbles and long bubbles. Short bubbles have weak displacement effect, thus it has limited impact on main stream. It can only influence local pressure distribution and thus less impact on loss. On the contrary, the displacement effect of long separation bubbles is strong, which can affect the distribution of wall pressure in a wide range. Such separation bubble will cause serious profile loss . Besides, it will significantly reduce the flow turning angle and the blade loading . Therefore, long separation bubble must be avoided in the LP turbine blade designs.

Apart from the above classification method which is based on pressure distribution, there is another common classification, which is based on the flow condition at separation point. It divides separation bubbles into three categories [40]: If the Reynolds number based on momentum thickness at the separation point is ${Re}_{\theta s} > 320$, then the transition point prior to the separation, and the separation form as transition separation mode. With decreasing Reynolds number the transition location moves after the separation point. If $240 < {Re}_{\theta s} < 320$ with moderate adverse pressure gradient in the flow, this is laminar boundary layer short bubble separation mode. If the Reynolds number continues decrease, and the flow has a strong adverse pressure gradient, then forms the laminar boundary layer long bubble separation mode. Three kinds of separation modes are shown in Fig. 4.9.

Fig. 4.9

Separation bubble modes in laminar boundary layer [40]

Shear layer transition is one of the most important behaviors of separation bubbles. This process is directly related to the separation bubble type and it impacts the entire flow field. Researchers carried out series studies on separated shear layer transition under steady state, and discussed the influences of Reynolds number, turbulence, pressure distribution and other factors. Though the separated shear layer is, in some ways, similar to the free shear layer, it will inevitably be affected by wall that closes to. Thus its transition process may show combined characteristics of free shear layer transition and attached boundary layer transition. In the experiments on flat plate separated boundary layer shear layer evolution, researchers observed similar instabilities as in the free shear layer, such as spanwise vortex roll, etc. But $\Lambda$ vortex structure is also observed, which proved that both inviscid and viscous instability mechanisms may exist in the separated shear layer transition process [41]. In addition, other researchers also observed in the experimental studies that the large scale spanwise vortex shadding in the transition process has the same dominate frequency as the amplified main frequency in T-S wave [42]. This should due to the interaction between K-H inviscid instability and T-S viscous instability. Based on these observations, to clarify that Which mechanism dominates under which condition, has become the key to further understand separated shear layer transition. Linear stability analysis on reverse velocity profiles shows that the separated shear layer above the bubble has inviscid instability, while inside the bubble there is mainly viscous instability. The strength of the two instabilities depends on the separated shear layer thickness and its distance to wall [43]. Therefore, the bubble thickness can be used to initially determine the dominant transition mechanism. For instance, If the separation bubble has smaller thickness, the wall would have great influence of the shear layer, T-S viscosity instability will dominate. In the contrary, the separation bubble transition will be subject to K-H instability .

Although experiments can capture main transition process of the shear layer, the detailed information of the flow field provided by them is usually limited as the experiments are always limited by factors such as measurement instruments. Recent years, with the rapid development of computer, direct numerical simulation (DNS), large eddy simulation (LES) and other high-precision numerical simulation are widely used and played important role in research of the vortices evolution, which is the key to further investigate separated shear layer transition . DNS results [44] of flat plate boundary layer short bubble showed that flow separation in the transition zone is highly three-dimensional. Transition area is controlled by a series of staggered vortices, which pump the near wall fluid outside to form shear layer on top of the vortex. As shown in Fig. 4.10, $\Lambda$ vortexes can also be observed inside the separated shear layer and then break near the reattachment point, then further develop to turbulence. This also shows that the entire transition process of short separation bubble can be described as the production, development and breakdown of $\Lambda$ vortex structure. Specifically, each characteristic of the short bubble can be presented as following: in the dead-air region and separated shear layer region, $\Lambda$ coherent structures and $\Lambda$ vortices are observed. Near the reattachement point, the dominated flow structure is hairpin vortex. While in the reconstructred turbulent boundary layer, besides hairpin vortices, the quasi-streamwise vortices which show as streamwise streaks are also observed.

Fig. 4.10

$\Lambda$ vortices in the separation bubble and its breakdown process [44]

The vortex structures in separated shear layers vary with inflow conditions. This causes different transition paths. Yaras et al. from Canada Carleton systematically studied laminar flow separation under different conditions using experiments and numerical methods [45,46,47]. They believed that at low FSTI, the transition of separated shear layer is through K-H instabilities. At high FSTI, the receptivity which leads the roll-up of the shear layer is skipped, while instead are the streamwise streaks which are transported via shear layer. Then the strong shear related to the streaks leads to the secondary instability and eventually produces turbulent spots. Different from to the turbulent spots inside the attached flow, the turbulent spots inside shear layer show a series of vortex loops. Figure 4.11 shows the velocity fluctuation contour plot at $\overline{U} \approx 1/2\overline{U}_{e}$ section under a high FSTI flat plate boundary layer. Here, streamwsie streaks can be observed clearly. And turbulent spots which are formed by a series of vortex loops can be seen from contour plots of streamwise and normal velocity fluctuation. In addition, worldwide researchers have continued further investigation using numerical simulations [48,49,50] and provided more complete picture of the transition process. However, there has been no unified conclusion yet, and still requires further under different disturbed environment.

Fig. 4.11

Separation bubble transition under high inlet turbulence, at $\overline{U} \approx 1/2\overline{U}_{e}$ plane [47]

4.2.1.3 Separated Shear Layer Transition Mechanism Under Adverse Pressure Gradient and Unsteady Conditions

Basis on steady flow study, researchers turn attention to the boundary layer under unsteady condition, such as the periodic upstream wakes, which is typical inflow boundary condition that turbine should face in real environment. Using the periodic passing cylinders to simulate unsteady wake is commonly used in experiments. Zhong et al. [51] used liquid crystal to re-produce the bypass transition to turbulent spots which are induced by passing wakes. Figure 4.12 shows experiment set-up and the liquid crystal displayed flow structures. There are three clearly visible turbulent spots induced by wake. The measurements showed that their generation and propagation speed are the same as conventional Emmons turbulent spots. When the turbulent spots transport downstream, their length and width grow, turbulence intensity increases, gradually forms spanwise turbulent strips, between adjacent strips, transition is in natural transition mode.

Fig. 4.12

Liquid crystal visualization of wake induced transition [51]

In order to investigate the boundary layer development under different loads and loading nt loads and loading distributions, Zou Zheng Ping, Liang Yun et al. from Beihang University developed adjustable flexible wall plates rig [52], which achieved to simulate different loading on a flat surface and detailed measured boundary layer under different loading and different inflow conditions. Figure 4.13 shows hot-film signal of typical load distribution on suction face in turbine cascade suction surface under wake effect in Re = 2.1 × 10⁵. Under the wakes, unsteady turbulent region form near plate leading edge and gradually become larger when transporting downstream, in the meantime the random fluctuation increases. From 66.6% chord, hot-film signal shows typical calmed region characteristics. Measurements show that these calmed regions will continue until the plate trailing edge. Furthermore, the natural transition may also occur between wakes under this condition. Comparing different loading distributions, the uniform loaded boundary condition is less sensitivity to Reynolds number changing in steady states. The loss due to low Reynolds number impact is also less, and therefore more suitable for working at low Reynolds number conditions. However, the situation is different with unsteady wakes, the long pressure gradient region in the front part of the aft-loaded profile can prevent boundary layer transition , together with this, if the adverse pressure gradient induced trailing edge separation could be inhibited by wakes , it will leave a smaller loss than the fore-loaded and uniform loaded profile.

Fig. 4.13

Typical hot-film signal, with unsteady wakes, Re = 2.1 × 10⁵ [52]

DNS simulation under the unsteady wake shows that the transition of attached boundary layer occurs first in the isolated turbulent spots. With the increase size of turbulent spots transporting downstream, spots gradually join and eventually form a fully developed turbulent boundary layer [53] as shown in Fig. 4.14. Instantaneous flow field shows that prior to the formation of turbulent spots, the reverse jet appears in fluctuating flow velocity field. These reverse jet, which are near the top of boundary layer, ultimately become turbulent spots under the joint effects of K-H instability and small-scale perturbations in the wake. This process is similar to the mechanism described before of attached boundary layer bypass transition to spots under high FSTI .

Fig. 4.14

Boundary layer transition and turbulent spots growth under unsteady wakes [53]

Compared to velocity stripe and turbulence spots in attached boundary layer, there are more vortex structures in separated shear layer transition process under wakes. Figure 4.15 shows the time series pictures of vortex evolution under interaction between separation bubble and wakes under low Reynolds number using LES [54]. As can be seen, there are a lot of vortices including streamwise vortices, spanwise vortex, hairpin vortex, etc. When the wake just entering the computational domain, there exist positive and negative directions streamwise vortices with ramdom distribution. When transporting downstream, the absolute value of the streamwise vorticity is reducing while the size of vortex is increasing and become more structured. Through shrinking channel, stretched by the main flow, there they become more obvious streamwise vortices. After throat, due to the velocity difference across boundary layer, stretching becomes even stronger. Near the wall there generate a lot of streamwise vortex, as shown in Fig. 4.15(f). Vortices continue moving downstream, and interact with separated boundary layer inducing unstable, rapidly developing spanwise vortices, as shown in Fig. 4.15a. With the development of spanwise vortex, flow finally breakdown to turbulence.

Fig. 4.15

Flow structures and their evolution with interaction between wakes and separation bubbles [54]

4.2.2 LP Turbine Boundary Layer Spatial-Temporal Evolution Under Steady Uniform Inflow

Though flat plate is easy to implement and also to consider most influences of turbine unsteady conditions, the turbine blade has curved surface, which also has an impact on the development of the boundary layer. Therefore, it is necessary to study the LP turbine blade boundary layer evolution based on the understanding of the one for flat plate. This section will discuss suction surface boundary layer evolution in a turbine cascade under steady flow condition.

To discuss the turbine cascade boundary layer development, it is necessary to know the exact condition of boundary layer. Due to the large curvature of cascade, conventional probes may be difficult to implement. Since the hot-film technology can capture high frequency quasi shear stress signals near the wall to characterize the state of the boundary layer, it has become an important method for boundary layer method. Figure 4.16 shows Li Wei, Zhu Junqiang et al. measured surface hot film signals in the transition process of LP turbine cascade suction side boundary layer [55]. The figure shows that: In the laminar flow region, flow and heat exchange is relatively stable, and thus hot-film signal fluctuations are small. After the transition begins, turbulent spots began to appear randomly on internal boundary layer, which leads to the increasing heat exchange and the hot-film signal appears with corresponding peaks. In later transition phase, there is mainly turbulence inside boundary layer with occasional laminar region. So hot-film signal exhibits a higher average value and occasional lower peaks. After transition is complete, signals show high-frequency characteristics in the fully developed turbulent region.

Fig. 4.16

Hot-film signals of a LP turbine suction side boundary layer in the transition process [55]

Reynolds number and incoming turbulence are important factors which impact the development of turbine cascade boundary layer under steady flow condition. A necessary condition to trigger the transition is sufficient disturbance, which in turn depends on the strength of the initial perturbation in the boundary layer and its growth trends. The initial perturbations in the boundary layer generally come from the main stream disturbances, such as wakes, background turbulence, and perturbations from wall surfaces such as roughness. The growth rate of these perturbations is determined largely by the Reynolds number. Therefore, incoming turbulence and the Reynolds number are the key factors that influence the starting position of the transition. With this understanding, researchers developed a mathematical model to predict the transition of separation bubble within a range of Reynolds number and incoming turbulence intensity , based on the measurement of the growth trend of disturbances in the separated shear layer [56]. In addition to the transition initial position, the incoming turbulence may also affect the transition mode of the separated shear layer. It has been pointed out earlier that at high inlet turbulence, the streamwise streaks will appear in the separated shear layer of the plate boundary layer, which also occurs in the turbine cascade. Due to these streaks, turbulent energy of the boundary layer increases rapidly, so the initial position of the transition shifts upstream comparing to inlet low turbulent level. Those streaks may induce vortex roll-up through K-H instability and eventually form a turbulent spot . If the flow turbulence is high enough, both the T-S instability mechanism and the K-H instability mechanism may be skipped during the transition process. Instead, turbulent spots directly appear in the separated shear layer [57]. However, it is also found that the detected dominant frequency in the separated shear layer transition zone may be consistent with the maximum frequency of the T-S wave even with the inflow turbulence is 9% [58]. The above results show that although inflow turbulence is one of the key factors that affect the transition , the transition process is restricted by many factors. To accurately understand the mechanism and the transition path, we should not only consider the impact from turbulence intensity .

The above studies are focusing on the separation bubbles on blade suction side, while with increasing blade loading or reducing Reynolds number, the open separation might happen. Figure 4.17 shows a comparison of flow field for a turbine cascade ($Z_{w} = 1.23$) with a Reynolds number of 1.0 × 10⁵ for different inflow turbulence intensity [59, 60]. The experimental measurements and numerical simulations show that with high turbulence intensity (4.0%), there is small size separation bubble on suction side boundary layer. However, with low turbulence intensity (0.8%), open separation happens and the corresponding profile load reduce about 20%, and the stagnation pressure loss increased significantly.

Fig. 4.17

Flow Structures of High-Loaded Turbine Cascades under Different Operating Conditions [59, 60]

When the separation bubble on the suction surface evolves into an open separation, the aerodynamic performance of the LP turbine will be affected. The transition mechanism on the suction surface , the flow structures and their evolution during the transition process will also be different. Regarding this, many researchers use large eddy simulation to investigate. As the T106D-EIZ low-pressure turbine cascade, which has detailed database available for validation [61, 62], is an ideal research object. Ye Jian, Zhang Weihao, Zou Zhengping et al. from Beihang University studied the evolution of the vortex structure in the open separation of the T106D-EIZ low pressure turbine cascade ($\text{Re = 6.0} \times 10^{4}$) under constant flow conditions [50, 63,64,65,66] and discussed the impact of inflow turbulence on separated shear layer transition. In terms of time-averaged flow, the comparison of the mean stagnation pressure loss coefficients under different incoming turbulence conditions is given in Fig. 4.18. Where the Y-axis is the stagnation pressure loss and the abscissa is the turbulence intensity . It shows that for a cascade with a separating boundary layer on suction side, the stagnation pressure loss coefficient reduces when increasing turbulence comparing to the case of no turbulence. When the FSTI is in range 0–5.0%, it has strong impact on cascade aerodynamic performance. While the freestream turbulence intensity above 5.0%, the influence of turbulence is not obvious.

Fig. 4.18

Cascade aerodynamic performance influenced by free stream turbulence intensity [63]

Regarding the evolution of the shear layer vortex structure, Figure 4.19 shows the coherence structure in the separated shear layer in different turbulence conditions by using iso-surface of $Q = 800$ vorticity identification and dyed with spanwise vorticity $\omega_{z}$. Apparently spanwise vortices can be observed in the separated shear layer without turbulent flow. As the spanwise vortexes show bending during transport process, two-dimensional flow turns to three-dimensional. Then there will be adjacent spanwise vortices interact each other and develop in pairs. In the process, the streamwise vortices developed rapidly then gradually dominant in the downstream of separated shear layer. Due to the inflow turbulence, the shapes of spanwise vortices are no longer clear, while the corresponding iso-surfaces of Q are no longer smooth, and they cannot occupy the whole blade height. It means that the three-dimensional disturbances appear earlier and stronger in the free shear layer. This also results in the streamwise vortex appear earlier. The rolled up spanwise vortex become unstable and they breakdown much faster, so that there are almost no two parallel spanwise vortexes, and there will be no vortex pairing occurs.

Fig. 4.19

Coherent structure in separated shear layer, iso-surface of $Q = 800$, colored by $\omega_{z}$ [63]

As mentioned above, the separated shear layer transition process may be affected by both the viscous instability mechanism (T-S instability) and the inviscid instability mechanism (K-H instability ). For the conditions discussed here, since the open separation is far from the wall, the effect of viscous instability is comparable weak and the transition process should be dominated by the inviscid instability mechanism. This is confirmed by the roll up of spanwise vortex as shown in instantaneous flow field in Fig. 4.19. In addition, the fastest-growing unstable frequency in the free shear layer under K-H instability can be represented by the dimensionless Strouhal number ($Sr_{\theta }$, based on momentum thickness in shear layer and velocity difference),

$$ Sr_{\theta } = \frac{{f\theta_{s} }}{{U_{es} }} $$

(4.1)

Here, $f$ is the fastest increasing unstable frequency (hereinafter referred to as the most unstable frequency), that is, the vortex rolling up frequency. $\theta_{s}$ and $U_{es}$ are of the boundary layer momentum thickness and boundary layer outer bounder velocity at the separation point, respectively. Study from Ho and Huerre shows that the typical $Sr_{\theta }$ value for the plane free shear layer is 0.016 [67]. Many studies have shown that the separated shear layer Strouhal number $Sr_{\theta s}$ is also roughly in this order. Nevertheless, since the separated shear layer is still affected by the wall somehow, this value may be slightly smaller than the value in the free shear layer. Table 4.2 shows $Sr_{\theta s}$ value from the relevant studies. In addition, the discussion on the influence of inflow turbulence shows there is a significant impact from $Sr_{\theta s}$, which is that the $Sr_{\theta s}$ increases with the increase of turbulence [63].

Table 4.2 $Sr_{\theta s}$ number in studies

4.2.3 Single Stage LP Turbine Boundary Layer Spatial-Temporal Evolution Mechanism

In real turbine stage environment, turbine internal flow has strong unsteady phenomena. In the LP turbine, upstream wakes sweeping is one of the critical unsteady disturbances for downstream cascades, which has significant impact on the transition and separation of the boundary layer.

4.2.3.1 Wake Transport Characteristics in LP Turbine Passages

One of the main characteristics of the incoming wake is the defect of velocity profile, which can be seen as a jet of reverse velocity relative to the main flow, the so-called “negative-jet” [72]. As shown in Fig. 4.20, as the wake enters to next blade row, it is cut by the blade leading edge and becomes independent. Each wake segment can be regarded as a negative-jet. These negative-jets pump fluid from the pressure side to the suction side and affect the profile distributions of pressure and velocity.

Fig. 4.20

Interactions between wake and turbine cascade boundary layer [12]

The phenomenon of wake transport in the cascade can be summarized as the behavior of cutting, bending, stretching and deformation of the wake [73]. Figure 4.21 shows a group of wakes transport in T106 LP turbine cascade measured by LDV [74]. The showing phase-averaged kinetic energy distribution and time-series of the perturbed velocity, clearly describes the transport process: near the cascade inlet plane, the wake is cut by blade leading edge. Due to the high velocity in the middle of the cascade while small velocity near the blade leading edge, the wake, which is transported by the local velocity, forms an arc, which is the so-called “wake bending”. When the bending wake enters bladerows, it transport faster near the suction surface due to there is higher velocity on suction side. It leads to a slow rotation of the wake-axis, and finally become almost parallel to the pressure surface , which is the process of “wake reorientation”. Due to the same reason, the wake is stretched longer along flow direction. In addition, the negative-jet causes the wake near the pressure side to flow towards suction side. And in the vicinity of the suction side, the width of the wake will increase due to the accumulation of fluid. At the same time, the perturbation of the negative jet also accelerates the flow near the suction side wake center and decelerates upstream surface to form a counter-rotating vortices pair. In fact, the wake propagates quickly which means that there is no flow from the pressure side pumped to the suction side during the wake flowing through the bladerow. In addition, as shown in Fig. 4.21, though most of the wake segments has lower the turbulent kinetic energy, there is a region near the suction side appears increased turbulence, which has profound influence on suction side boundary layer development.

Fig. 4.21

The transport of wake in blade passage, phase-averaged mean turbulent kinetic energy, disturbed velocity [74]

In addition to the velocity profile loss , the incoming wake can also be seen as low-energy fluid shedding from upstream blade boundary layer. The transport of wakes in the bladerow also affects the evolution of these vortices. Figure 4.22 shows the evolution of the vortex structure in the wake of LP turbine cascade under the typical operating conditions obtained by LES [63]. It shows the $Q$ method identified vortex structure in the neighboring phase within same wake cycle and dyed by streamwise velocity $u1$. It shows that before entering the bladerow, the wake mainly contains small-scale vortex structures. After entering, under the acceleration by the main stream, the large eddies in streamwise gradually dominate the wake region. Due to the negative-jet, those eddies accumulate on the suction side. Since the flow acceleration on suction side is more obvious, these eddies are almost parallel there. It can be seen that, in addition to the negative-jet, wake-induced vortex pair and some other related mechanisms, those large scale streamwise vortices also have strong impacts on suction side boundary layer.

Fig. 4.22

Wake transport in LPT passages, iso-surface of $Q = 20$, colored by axial velocity $u1$ [63]

4.2.3.2 Wake-Induced Boundary Layer Transition Modes

In general, the flow near normal loaded blade pressure side has pressure gradient that keeps boundary layer in a laminar flow state and does not face severe separation problems. Therefore, more attention is paid to the incoming unsteady wake on the development of the suction side boundary layer. Hodson et al. from Whittle Laboratory in University of Cambridge had most representative of the study [3, 12]. Their results show that wake -induced suction side boundary layer transition will occur through one of the following three pathways:

• (1) When the Reynolds number is high enough, the unsteadiness of the wake causes the laminar boundary layer quickly transit to the turbulent. If the adverse pressure gradient near the trailing edge suction side is not large, the wake prevent separation occurs. Alternatively, if the local adverse pressure gradient is sufficiently weak, the laminar flow boundary layer under will not separate under steady state, but the wake induce boundary layer transition. In this case, the turbulence or other disturbances induced transition will occur at a certain distance downstream of the wake, while there is still no flow separation on the suction side boundary layer as shown in Fig. 4.23a.

  Fig. 4.23

  

  Modes of interaction of wake and separated boundary layer [12, 75,76,77]

• (2) With the decrease of Reynolds number or the increase of the adverse pressure gradient at the suction side trailing edge, the laminar boundary layer under the steady condition will separate. Under unsteady condition, though wake can still induce boundary layer transition, if the wake passing frequency is not high enough, the laminar boundary layer will still separate during the gap between passing wakes. In this case, periodic separation bubbles appear in suction surface boundary layer, and transition may be alternated by wake -induced boundary layer transition and laminar flow separation bubble transition, as shown in Fig. 4.23b.

• (3) If the Reynolds number further reduce or the adverse pressure gradient at the trailing edge is large, the laminar boundary layer separation happens very early and the wake induced transition will occur in the free shear layer of the separation zone, as shown in Fig. 4.23c. In serious condition, wake will not induce re-attachment.

4.2.3.3 Flow Structures in Wake-Induced Boundary Layer Transition

Wake -induced boundary layer transition is common in LP turbines with convention load or those operating under a high Reynolds number. Halstead and Wisler et al. systematically studied the evolution of the attached boundary layer under incoming wakes, and provide the picture of wake-induced transition on suction side [78,79,80,81]. Figure 4.24 shows the spatial-temporal distribution of shear stress fluctuation and the skewness factor in a LP turbine test rig. The region A as shown in the figure is the laminar flow region near the leading edge. Starting from 30% suction surface length, the wake-induced transition zone appears periodically, which is labeled as B. Till around 80% suction surface length, the wake-induced boundary layer transition completes and becomes fully developed turbulence, which is the wake-induced turbulent boundary layer (C region). Between wakes, bypass transition happens near around 50% suction surface length under external disturbances, which is marked as E. The transition process completes near about 65–80% suction surface length, and the downstream F region is the turbulent boundary layer between passing wakes. Upstream of the wake-induced transition zone (B) is a region with relatively weak fluctuation, which is the so-called “calmed region(D)”. (D) which observed weak random fluctuation. It is shown that the presence of this region causes the starting position of bypass transition move significantly downstream. Although it is only measurement of one cascade, the main coherent structures and their evolution can qualitatively characterize the evolution of the attached boundary layer under wakes in typical LP turbine .

Fig. 4.24

Spatial-temporal map of wake -induced boundary layer transition [78]

Similar as flat plate measurements, Figure 4.25 shows the hot-film measured results of the suction surface boundary layer under incoming wakes on a high-loaded LP turbine blade [82]. The increased wall shear stress with strong random fluctuations represent the wake-induced turbulent spots. A calmed region upstream of the turbulence spot is also observed. In this region, the wall shear stress transits from the turbulent state to the laminar state. Comparing to local laminar boundary layer, the calmed region has more full velocity profile, which can effectively suppress the boundary layer separation [83]. Figure 4.25 also shows the variation of the momentum thickness at the trailing edge of suction side with time, and the loss comparison with steady inflow. This profile has a big separation bubble on suction side under steady inflow. It can be seen that profile loss increases with wake passing by, while decreases in the calmed region after wake. Overall, the presence of wakes makes profile loss on suction side become lower.

Fig. 4.25

Wall shear stress and suction side loss distribution under unsteady wakes [82]

It is generally believed that in conventional loaded LP turbine cascade, the transition induced by wakes occurs when the momentum thickness Reynolds number exceeds 90–150 [84]. This is in agreement with the laminar flow boundary layer behavior at high inlet turbulence intensity. However, the situation is different for high and ultra-high loaded cascades. According to Howell’s study, the critical momentum thickness Reynolds number may reach 225, and this difference is considered to be caused by the loading distribution [5]. It should be noted that the acceleration region in the front half of the blade suction side discussed by Howell is longer.

For the situation that separation bubbles show between passing wakes, studies show that the wake can still effectively reduce the profile loss . Figure 4.26 shows the spatial-temporal distribution of the shape factor and dissipation factor of the boundary layer under wakes in turbine blade cascade [85]. There is separation bubble under steady state inflow. As shown in the figure, region between lines B and D represent the wake induced turbulent region, while the immediate following region which between lines D and E is the calmed region. The whole interaction between wake and boundary layer can be described as follows: via bypass transition mode, wakes induce turbulent spots before separation point. The turbulence spots grow and merge into the turbulent patch. Turbulent patch moves into the separation zone, which restraining the separation bubble which is formed after the last wake passing by. Followed by the turbulent zone is the calmed region, full velocity profile can still resist separation. After the calmed region passes, local laminar boundary layer shows inflection point under adverse pressure gradient , and the separation occurs again. Separation indicates an increase in loss. Comparing to Fig. 4.26b, it shows that the region which has higher shape factor region in Fig. 4.26a, has only laminar-like flow energy dissipation, which indicates that this region is just undergoing the initial stage of the formation of the separation bubble. To sum up, the results show that the highly dissipative separation bubbles under adverse pressure gradient in steady flow are replaced by the calmed region and the initial phase of separation flow, which both have low dissipation characteristic. Therefore, under current circumstance, although the wake-induced turbulent zone has a high loss, the time-averaged loss is reduced due to the presence of the calmed zone and the initial stage of separation [86].

Fig. 4.26

Spatial temporal diagram of boundary layer shape factor and dissipation coefficient [85]

If the Reynolds number is low enough or the suction side adverse pressure gradient is very strong, the wake -induced transition cannot occur before the separation point, the wake and laminar separation bubbles will have a very complex interaction, which is facing by suction side boundary layer in most of the LP turbines which operate under low Reynolds number condition. Figure 4.27 shows the interaction between typical wakes with separation bubble [87]. The wall boundary flow downstream of wake center is accelerated by negative-jet, while the upstream flow is decelerated. When the wake reaches the separation zone, the outer region of the boundary layer is accelerated while the inner part, due to viscosity, responds slowly to external disturbances. Therefore, the shear is enhanced in the separation zone. The wake is transported downstream on the outer part of the separation bubble, and the normal component of the negative-jet deforms the shear layer, and makes velocity profile of the separated shear layer unstable. Vortices are rolled-up under the disturbance of wakes. Because of the low transport speed of the roll-up vortices, which is about half of the main flow [88], wakes move downstream of them and continue to disturb the downstream shear layer and create vortices. Finally, the roll-up vortices rapidly break up into turbulence, leading to boundary layer transition . After the roll-up vortices and turbulence boundary layer pass through, the calmed region appears. After the influence of the calmed region disappears, the boundary layer separates again until the next wake comes.

Fig. 4.27

Sketch of laminar boundary layer roll-up under incoming wakes [87]

The wale-induced unsteady evolution of the separation bubbles causes unsteady pressure fluctuation on the wall. Figure 4.28 shows wake-induced unsteady pressure fluctuation in the separation region of T106 LP turbine cascade [85, 88]. It can be seen that the pressure fluctuations first appear upstream of the pressure plateau and then enhance with the development of the flow. When arriving at the upstream part of the time-averaged plateau, the fluctuation reaches its maximum with the amplitude being 30% of the outlet dynamic pressure. Then it maintains this level. The results show that these pressure fluctuations propagate at velocity that is half of the main flow and have the same phase as the influence of the wakes on this region. So it can be deduced that these pressure pulsations are caused by the rolled-up vortices in the shear layer. Other researchers have observed similar pressure fluctuation in their work [89,90,91].

Fig. 4.28

Wake induced pressure fluctuation in the separation boundary layer [88]

For the case of wake-induced shear-layer transition , Figure 4.29 shows the spatial-temporal evolution of the suction side boundary layer shape factor near the trailing edge of the T106 blade [87]. Here, the increasing shape factor indicates the separation bubbles periodically appearing in the boundary layer, and the trajectory of the wake is located between the lines A and E. Line A represents the trajectory of the maximum velocity at boundary outer layer induced by upstream wake. Due to viscosity, the inner region of the boundary layer situated in this trajectory has not achieve the same level acceleration, therefore the shape factor here is larger than the surroundings. Two wedge-shaped regions (C and D) appear at about 70 and 77% suction surface length on B, with propagation velocity which is about half of the main flow. These are the the wall-induced roll-up vortices in the separated shear layer and the resulting turbulent boundary layer after the vortices break down. The results also show that, although there are separated bubbles in the boundary layer, the wake induced attached boundary layer transition structure can be found upstream of the trajectory of wake trailing edge (E). The trajectories F and G in the figure are for the trailing edges of the wake-induced turbulence region and the calmed region, respectively. Stieger and Hodson explained this phenomenon as a result of boundary layer bypass transition induced by turbulent diffusion [87]. The above results show that if the wake does not induce the boundary layer transition before the separation, the unsteady disturbance of the wake may cause the shear layer unstable via inviscid effect, and then roll up vortex and eventually induce transition. In this case, it is still possible to observe a obvious calmed region in the attached boundary layer in downstream part of trailing edge.

Fig. 4.29

Wake-induced separation bubble transition [87]

Similar results have also been obtained from LES of the higher loaded T106D-EIZ turbine cascade with typical wake effects [63, 92]. Though there is open separation on the cascade suction surface under steady condition, this open separation is replaced by periodical small separation bubble under wake condition and thus significantly improve its aerodynamic performance. This, again, proved the potential of using unsteady wake to enhance turbine aerodynamic performance. Figure 4.30 shows the spatial-temporal distribution of the dissipation factor in T106D-EIZ turbine cascade boundary layer. The W, WL and WT represent the trajectories of the wake center, leading and trailing edge, respectively, while the black lines represent the spatial-temporal evolution of the separation region. In contrast to Ref. [87], the separation bubbles in the wake passing region is obviously inhibited, which could be the co-effect of the inviscid negative-jet and wake-induced attached boundary layer transition . In the vicinity of the separation bubble reattachment region (70–80% of the suction surface length), two regions with wake-induced roll-up vortices and their induced transition are observed: T2 and T3. The corresponding calmed regions are labeled C2 and C3. The formation mechanism of these regions may be due to the turbulent diffusion induced boundary layer bypass transition mechanism as described in [87]. As for the location of these flow structures are different, it should due to difference of blade loading and the wake conditions.

Fig. 4.30

Spatial-temporal distribution of T106D-EIZ cascade suction surface dissipation coefficient under the unsteady wake condition [63]

4.2.3.4 Effects of Wake Characteristics on Boundary Layer Evolution

Previous section described the typical mechanism of unsteady wake effect on boundary layer development. There are many parameters of wake can influence the boundary layer, including its width, velocity defect , turbulence intensity , passing frequency and flow coefficient, etc. Till now the studies on influence of wake width, velocity defect and turbulence intensity are not very systematic, while the impacts of the wake passing frequency and flow coefficient have been discussed detailedly by many researchers.

The wake frequency can be measured in terms of the reduced frequency, which has similar physical meaning as Strouhal number , defined as the ratio of the time required for passing through the cascade to the time interval between wakes:

$$ f_{r} = \frac{{f_{b} b}}{{V_{2,is} }} = \frac{{U_{b} b}}{{t_{b} V_{2,is} }} $$

(4.2)

Figure 4.31 shows the T106D-EIZ LP turbine cascade suction side shear stress $\tau_{wall}$ distribution with different unsteady wake conditions [63]. In which the wake flow coefficient is fixed to 0.83, and the reduced frequency is $f_{r} = 0.67,0.335,1.005,1.34,1.675$ respectively. The region where wall shear stress is negative occured the flow separation. The results show that the size of leading edge separation bubbles decreases with the increase of reduced frequency, and after the reduced frequency $f_{r}$ exceeds 1.005, those separation bubbles disappear. Regarding the separation point of the trailing edge separation bubble, the change of wake reduced frequency has a relatively small effect on changing its location, and there are three conditions that the separation point will not change with $f_{r}$ when $f_{r}$ is above 1.005. Wake reducing frequency has more impact on reattachment location comparing to separation point. With the increase of $f_{r}$, reattachment moves upstream. Above influence caused the axial length of separation bubble under $f_{r} = \text{0.335}$ condition increase by 38% when compared to baseline condition. While the other three conditions, the length of separated bubbles were reduced by 15, 24 and 23%. In addition, after the wake reduced frequency exceeds 1.34, the position of the separation point and reattachment point will not move with the change of reduced frequency. This is because the wakes will almost join on affecting the suction side boundary layer when the reduced frequency reaches 1.34. Therefore, there is no significantly changes if further increase the reduced frequency [63]. Figure 4.32 compares the total pressure loss coefficients with the corresponding reduced frequencies. When $f_{r} < 1.005$, the average total pressure loss coefficient decreases with increasing $f_{r}$. However, when $f_{r}$ further increasing to 1.34 and 1.675, the average stagnation pressure loss coefficient increases, indicating that for the specific turbine cascade, there is optimal reduced frequency.

Fig. 4.31

Wall shear stress distributions on the suction side at different wake reduced frequencies [63]

Fig. 4.32

Stagnation pressure loss coefficients with different wake reduced frequencies [63]

In case of boundary layer, Figure 4.33 shows the flow structures on suction side in one wake phase under five conditions as mentioned above. The figure shows clearly the instantaneous position of wake location and the shape on cross section $z = 0$, which is dyed using spanwise vorticity $w_{z}$. In the same time, it shows the contour plot of axial velocity $u1 = 0.1$. It is dyed by pitch velocity $u2$, which can identify boundary layer flow structures, and can also give the approximate location of the separated shear layer in the streamwise direction and the normal direction. The figure also shows the pitch velocity $Vt$ distribution on the first-layer mesh and the contours of $Vt = 0$ to accurately identify the separation area within the boundary layer. The contour plot are placed side by side with the iso-surface of the axial velocity $u1 = 0.1$. these contours clearly show the relations between the wake, the coherent structure in the boundary layer and the instantaneous location of the separated bubbles.

Fig. 4.33

Suction side boundary layer flow structures with different wake reduced frequencies [63]

Researches on the effect of reduced frequency $f_{r}$ on the development of boundary layer flow structure can be summarized as follows [63]: The transport characteristics of a single wake in the cascade do not change with $f_{r}$. Within certain range of $f_{r}$, a calmed region can be observed in the boundary layer between wakes, which appears earlier but with a shorter duration when increasing $f_{r}$. Too large or too small of $f_{r}$ will reduce the unsteadiness of the flow inside boundary layer. Correspondingly, the calmed regions in the gap between wakes will disappear. Too large $f_{r}$ will enlarge turbulent boundary layer area, while too small $f_{r}$ will enlarge separation area, which both cause profile loss increases. In addition, the wake itself also increases the dissipation in the main flow. The combined effect of the above effects makes it possible to optimize $f_{r}$ for the aerodynamic performance under certain operating condition. This optimal value is approximately 1 for the T106D-EIZ cascade [63]. Other turbine cascades experiments verify the existence of the optimal reduced frequency [93]. However, the values of the optimal wake frequency may vary due to the blade loading and operating conditions.

Liang Yun and Zou Zhengping et al. did detailed experimental studies [94] on their designed Zw = 1.33 high-lift LP turbine blade with Reynolds number of 1.1 × 10⁵ and inflow turbulence intensity of 1%. Their results as well show that the wake frequency has a great effect on the boundary layer separation control, which is mainly displayed by different action times of turbulent spots or calmed regions and different size of the separation region under different wake frequency. Figure 4.34 shows a spatial-temporal plot of the quasi shear stress on the suction side surface with different wake reduced frequencies. The black dot in the figure represents the measurement location of hot-film probe on the suction surface. In the first half of the blade, the flow is laminar, and the shearing stress is low with less fluctuation. Only when the fluid with high turbulence intensity in the wake sweeps through the blade surface will the shear stress increase. Thus it can identify the leading edge of the wake (WL). its slope in the figure can determine the wake propagation velocity. After the wake passes at about 60% of suction surface, calmed region (C) appears, identified based on shear stress value. The trailing edge of the calmed region has a propagation velocity of 0.3 times the main flow speed. After the calmed region passes, the boundary layer separates again. Before wake arrives, there already appears high shear stress region (P) from the 52% of suction surface. This is because the negative-jet model in the beginning of wake has certain acceleration effect on the downstream flow. This accelerating effect disturbs the shear layer of the separation bubble, prompting its rapid transition to turbulence, resulting in high shear zone. This also indicates that under this condition, the wake-induced boundary layer transition occurs in the separated shear layer. When wake interacts with separation bubbles, the wake negative-jet will form a series of complex vortex structures with the separated shear layer. Stieger [27] verified this point via PIV. On the shear stress spatial-temporal diagram, the low-shear zone (S2) inside the wake-through region also proves the presence of wake induced complex vortices. This phenomenon is more obvious when the wake frequency is low. It can be seen that the separation is not completely inhibited under two wake frequencies, that is in the gap between wakes, there occurs boundary layer separation again. However, when the wake reduced frequency increases from 0.34 to 0.68, the separation region diminishes obviously with the duration decreases as well.

Fig. 4.34

Spatial-temporal diagram of shear stress on suction side with different wake reduced frequencies [94]

Wake can control boundary layer separation. Different wake passing frequencies have different control impact. The most direct expression is the change of the boundary layer loss. By analyzing the trailing edge boundary layer momentum thickness at different frequencies, it can be found that the boundary layer loss is greatest when there is no wake, and it decreases rapidly after adding the wake. With the increase of wake frequency, the boundary layer loss decreases. Further research is needed on how the wake reduces loss, and whether there will always be monotonically correlation on wake frequency and loss.

Liang and Zou [94] show the boundary layer momentum thickness at the suction side trailing edge under different conditions (see Table 4.3). It can be seen that the boundary layer loss decreases monotonically with the increase of wake reduced-frequency, which is consistent with other researchers. Furthermore, Liang and Zou [94] divided the suction surface boundary layer loss into three parts: laminar loss before separation, separation loss and turbulent friction loss. They also provided (see Table 4.4) quantitative analysis on the change of each loss with different wake frequencies. It can be seen that the profile separation loss in the steady state is relatively large in the total loss, so introducing the unsteady effect of wake will inhibit separation and thus can significantly reduce the total loss. In the meantime, the proportion of laminar loss and turbulent friction loss increases, especially the later one. Under this condition, when the reduced wake frequency is 0.68, the ratio of separation loss decreases to 7.04% and the main loss is turbulence friction loss, which indicates that the wake reduced frequency is close to the optimum value.

Table 4.3 Boundary layer momentum thickness near trailing edge

Table 4.4 Profile loss under different reduced frequencies

Similar influence of wake reduced frequency has been shown from even lower Reynolds number experiments [95]. Figure 4.35 shows the spatial-temporal distribution of boundary layer momentum thickness for a turbine cascade with Reynolds number of 2.5 × 10⁴ and inflow turbulence intensity of 0.6%. The solid lines in the figure represent the trajectories of the leading edge of wakes, while the two dotted lines represent the trajectories of the trailing edge of the wakes and the calmed region. When the wake has low passing frequency, the separation bubble will appear near the trailing edge of the profile in the gap between wakes. It should be noted that the definition of the wake reduced frequency ($F = fL_{j - te} /U_{ave}$) is different here, where $f$ is the wake passing frequency , $L_{j - te}$ is the length of the adverse pressure gradient region in the suction side boundary layer, and $U_{ave}$ is the main flow averaged velocity. For the same research object, the specific values of $F$ and $f_{r}$ are different, but the represented physical concepts and their trends are consistent. The results show that the increase of wake reduced frequency can effectively reduce boundary layer momentum thickness at the trailing edge and reduce the profile loss . The discussion on the influence of Reynolds number shows that when Reynolds number is 2.5 × 10⁴, there will not be significantly suppressed separation until $F$ reaches around 0.5. When the Reynolds number is 5.0 × 10⁴, the similar effect can be achieved when $F = 0.3$. This also confirms that the optimum reduced frequency will depend on the cascade specific operating conditions.

Fig. 4.35

Spatial-temporal distribution of suction side boundary layer momentum thickness under different reduced wake frequencies [95], inlet turbulent intensity 0.6%, Re = 25000, $\varphi = 0.7$

In addition to reduced frequency, flow coefficient is also an important parameter to represent incoming wake characteristic, which is defined as the ratio of cascade axial velocity to the wake pitch velocity:

$$ \varphi \text{ = }\frac{{V_{x1} }}{{U_{b} }} $$

(4.3)

With other parameters remain constant, the wake flow coefficient $\varphi$ has obvious influence on wake transport characteristics. The wake entrance angle, width, and relative velocity defect will all change with it [63]. Under same reduced frequency $f_{r}$, the influence of $\varphi$ on the change of time-averaged inlet flow angle is complicated. On one hand, the relative flow angle of the individual wake will increase as $\phi$ increases, the negative incidence will be improved. On the other hand, the relative wake width increases with $\phi$, its impact range is even larger, so that the relative averaged flow angle decreases, negative incidence increases. For a given operating condition, the time-averaged relative flow angle either increased or decreased needs to discuss specifically.

The LES results of T106D-EIZ turbine cascade under different wake flow coefficients show that [63]: In general, the size of the time-averaged separated bubble decreased with the increase of the flow coefficient , but this effect was much weaker compared to reduced frequency. In terms of flow loss, the increase of the flow coefficient makes it impossible to roll up the vortex in the separated shear layer, and thus to reduce the suction side boundary layer flow loss. However, this in turn increases the average disturbance level in the main flow, increasing the main flow loss. The increase of flow coefficient makes the wake act to the boundary layer of the suction surface in a nearly parallel way, thus increasing the influence time of the individual wake to the boundary layer. But the effect of the negative-jet is weakened, which finally increases the ambient turbulence intensity of the boundary layer and reduces the unsteadiness of the wake. This eventually leads the calmed region not to appear under the large flow coefficient condition. On the contrary, the calmed region is more obvious in the small flow coefficient $\varphi$ condition. It can be seen that with constant wake reduced frequency $f_{r}$, flow coefficient $\varphi$ has multiple influences on the turbine cascade performance. For the T106D-EIZ turbine cascade, the total aerodynamic losses of the cascade can be reduced at very large or very low flow coefficient conditions. Figure 4.36 shows the suction side boundary layer flow structures of a phase in wake period under different flow coefficients.

Fig. 4.36

Suction side boundary layer flow with different wake flow coefficients [63]

4.2.4 Unsteady Flow and Boundary Layer Evolution in Multi-stage LP Turbines

LP turbines are generally multiple stages, may reach 5–7 stages, or even more. Though it has been recognized for long that the unsteady nature in multi-stage turbines and their effects may have deeper influences, it is difficult to carry out detailed measurement on the multi-stage turbine test rig, and the cost is very high. The unsteady calculation of the multi-stage turbine also needs a large budget, which has caused some obstacles for research. Thus the research and understanding of unsteady flow phenomena in multi-stage turbines develops slow. With the development of computer hardware and computing technology since 1990s and the improvement of experiment capability, more and more researchers begun to pay attention to the unsteady flow inside multistage turbine , and obtained many important results, which effectively improved the understanding of the relevant issues.

As shown already, potential flow decays fast in space and its influence is generally confined to the adjacent bladerows and the potential field interaction between rows is weak. However, the effective length of wakes is long, which will affect the downstream bladerows. And the maximum turbulence inside the wake may be increased in the blade passages, which may cause the wakes from multiple upstream rows to have a significant effect on the downstream cascade flow, which mainly due to interaction between blades in multi-stages. Figure 4.37 shows the main characteristics of the wake propagation in a multi-stage turbine [96]. It can be seen that the wake from the first-stage rotor (Rotor 1) is cut into segments by the Stator 2. At the same time, wake is distorted in the later channel and the propagation direction is changed. Downstream of Stator 2, the wake from the same first-stage (Stator 1 and Rotor 1) is no longer continuous, but becomes segments which are divided by the Stator 2 wakes, and they transported downstream together. Normally, the blade numbers of two rotor stages are different, which leads the wakes from Rotor 1 enter the Rotor 2 passage from different circumferential positions and eventually cause the non-uniformity in Rotor 2 wakes circumferentially, that is, the so-called “Beating effect” [97, 98]. Figure 4.38 shows the turbulence intensity distribution at third-stage outlet in a multi-stage LP turbine [99]. This figure clearly shows the wake strength in the axial direction due to the different rotor blade numbers from the first two stages. It can be seen that the unsteady flow inside the multistage LP turbine is more complicated with the increase of the turbine stages due to the interaction between wakes from multiple stages.

Fig. 4.37

Transport process and morphology of wakes in multi-stage LP turbines [96]

Fig. 4.38

Turbulence intensity distribution at multi-stage turbine outlet [98]

The wake propagation mechanism described above results in different unsteady flow fields observed at different downstream positions. Figure 4.39 shows the wake trajectories near the first rotor outlet of a three-stage LP turbine and the variation in turbulence intensity over the trajectories obtained from probes at different axial position [78]. It can be seen that the turbulence intensity in the wake of the first stage rotor is about 5%, the turbulence in the wake of the first stator is in the order of 3.5%, while in the gap between wakes, the turbulence intensity drop to 1.5%. The difference in the turbulence intensity may be about 1% for different circumferential positions of the observed results. In addition, the results also show that the turbulence intensity has changed twice in one cycle at both measurement positions, apparently because of the presence of wakes of two rows of blades upstream. A similar phenomenon was observed at the third stage. These phenomena indicate that, although the upstream vanes may be still relatively, their wakes are cut and subsequently deformed by the relative rotation of the cascade passages, and the wake from those comparable still bladerows also has an unsteady effect on the downstream vane.

Fig. 4.39

Circumferential non-uniformity of unsteady fluctuation in multi-stage turbine [78]

The development of suction surface boundary layer in the multi-stage turbine still draws research attention. Hudson and Howell et al. measured the suction side boundary layer in a multistage LP turbine using surface hot-film technology [13, 100]. Figure 4.40 shows the non-dimensional quasi-wall shear stress spatial-temporal distribution on the third-stage suction surface boundary layer in LP turbine of the BR715 engine. Figure 4.40a, b show the results for different positions of second-stage rotor, with vertical axis being the passing period of second-stage rotor blade. It should be noted that the region of high shear stress indicated by H4 is not a turbulent region but only a result of the absence of adhesion between wake and laminar boundary layer. In fact, the transition induced by the upstream second-stage moving blade wake starts at H2 which is about 75% of the camberline. In addition to H2 and H4, another region of high shear stress, H1, which is the transition region induced by second stage wakes, is clearly visible in Fig. 4.40b, confirming that the upstream bladerows’ wakes may also affect the development of the downstream boundary layer. This also clarifies the importance of investigation of wake influences in multi-stage turbine . In addition, transition zone H1 is not evident in Fig. 4.40a, which is caused by circumferential unevenness of wake strength from the upstream row.

Fig. 4.40

Spatial-temporal distribution of the quasi-wall shear stress on third-stage guide vane suction surface in BR 715 LP turbine [13]

It is envisioned that the unsteady effects inside the multi-stage turbine may have a significant effect on the downstream flow. Previous studies pointed out that the relative positional relation of the relative stationary blade row will determine the phase of the upstream wake into the downstream channel, which will affect the downstream flow in the multi-stage turbine and eventually affect the aerodynamic performance of the multi-stage turbine which is so called “clocking effects”. Many researches show that the variation range of the aerodynamic efficiency of the multi-stage turbine blades may reach about 0.5% when the relative pitch position of the turbine blades is changed. Generally the highest efficiency occurs when the wake of the upstream blade row passes over the leading edge of the downstream blade while the lowest efficiency occurs when the wake of the upstream blade passes from the center of the downstream blade channel. Experiments and numerical simulations of 2.5-stage turbine with the same number of blades at different working conditions show that [101, 102] the stator clocking effect varies turbine efficiency by 0.3% in both design and off-design conditions. In design condition, the efficiency may vary in the order of 0.5% (see Fig. 4.41) at different pitch locations, while the highest efficiency change at different radial positions can reach 1%. Accordingly, the researchers believed that if we can efficiently use the time series effect between the bladerows, it is definitely possible to improve aerodynamic performance by 0.8%. Similar results have been reported by other researchers on different multi-stage turbines [103,104,105,106,107].

Fig. 4.41

Circumferential efficiency change in a 2.5-stage turbine [101]

For the multi-row clocking, Figure 4.42 shows the clocking effect of a three-stage (third, fourth, and fifth stage) of a LP turbine component [108]. The efficiency of the turbine components obtained by separately adjusting the circumferential positions of the fourth and fifth stage blades of the turbine are given separately. The adjustment of the circumferential position of the fourth-stage turbine blades results in a variation of the turbine efficiency of 0.5 and 0.6% by adjusting fifth-stage turbine. The figure shows that, no matter the fourth or fifth stage, there is a strong interaction between rotor/stator circumferential positions. This suggests that it is not sufficient to consider only the clocking effect between stationary blades or rotating blades in a multi-stage turbine. In addition, the results in Fig. 4.42 show that the benefits obtained by individually adjusting the circumferential position of a stage turbine blade are of the same order of magnitude. While the efficiency can change 6.8% [108] by adjusting the axial position of the fourth and fifth stage turbine blades simultaneously, indicating that clocking effect in the multistage turbine is not simply superposition. At present, the researches on the interaction of the stator/rotor clocking effect , the clocking between the relative stationary bladerows, and the interaction between the relatively moving bladerows, are neither deep enough nor systematic. The understanding of the above problems has yet to be further improved.

Fig. 4.42

Clocking effects on efficiency in a multi-stage turbine [108]

In addition to the overall performance of turbine components, clocking effect on the evolution of boundary layer structure in multi-stage turbine is also one research topic and many works have been done in recent years [109,110,111]. Figure 4.43 shows the temporal and spatial distribution of the quasi wall shear stress and the total skewness factor of the shear stress at suction side boundary layer of the second stage in a 1.5-stage turbine [112, 113]. In Fig. 4.43a, b, the first-stage stator locates at different circumferential positions, which correspond to the best and worst aerodynamic performances. The circumferential blade positions in these two conditions differs by about 45% of the first stator pitch. In the figure, the solid line FS is the main stream trajectory, S represents the wake trajectory from first stage stator, and R represents the wake trajectory of the first stage rotor. Before about 50% of the camber line, the wake trajectories from upstream two were observed simultaneously. While the trajectories of the first stage vane were more pronounced in Fig. 4.43a, this is one of the main differences of boundary layer structure between these two conditions. In addition, the phase of the upstream rotor wake induced structure appears very different under both conditions, especially near the leading edge. It should be noted that the upstream rotor is in the same phase, and the difference of the wake trajectory is caused by the unsteady evolution mechanism of second stage guide vanes inlet flow angle, which is due to stator clocking. The figures also show that the presence of the first stage stator wake in Fig. 4.43a can lead to the transition happen earlier, thus effectively reducing the length of the separation bubble, which is also the main reason to obtain the optimal aerodynamic performance. Based on the study of the two-dimensional clocking effect , many researchers have studied the three-dimensional clocking effects by numerical simulation or experimental measurement [114,115,116]. Their results show the clocking effect also has significant impact on the intensity and evolution of blade tip leakage flow and end wall secondary flow . This is also an important revenue source for multi-stage turbine timing effect.

Fig. 4.43

Clocking effects on suction side boundary layer evolution of the second vane in a 1.5-stage turbine [112]

4.2.5 Boundary Layer Losses and Prediction Models

4.2.5.1 Attached Boundary Layer Flow Losses

Since the profile loss is a key contribution for flow loss, the assessment and prediction of the flow loss in the boundary layer is also an important topic of studying the evolution of LP turbine boundary layer. Denton pointed out that the entropy production per unit area of the two-dimensional boundary layer can be expressed as [6]:

$$ \dot{S}_{a} = \frac{d}{dx}\int\limits_{0}^{\delta } \rho V_{x} (S - S_{\delta } )dy = \int\limits_{T}^{\delta } {\frac{1}{T}\tau_{xy} dv_{x} } $$

(4.4)

where, $S$ and $S_{\delta }$, represent the local entropy and the entropy at outer edge of boundary layer in the same reference condition, respectively. $\tau_{xy}$ is the local shear stress. Further, the entropy production yield per unit volume of the two-dimensional boundary layer can be expressed as:

$$ \dot{S}_{v} = \frac{1}{T}\tau_{xy} \frac{{dv_{x} }}{dy} $$

(4.5)

For most of the boundary layer, the velocity changes strongly near the wall, so most of the entropy production is concentrated there. Especially in the turbulent boundary layer, the entropy production mainly exists in viscosity sub-layer layer and the log-law layer. Quantitative studies show that about 90% of the entropy production of boundary layer occurs in the inner layer [117].

To make simple comparison and application, researchers defined the dimensionless dissipation factor to measure the local loss in boundary layer:

$$ C_{d} = \frac{{T\dot{S}_{a} }}{{\rho V_{\delta }^{3} }} $$

(4.6)

where $V_{\delta }$ is the velocity at and outer edge of the boundary layer. Clearly, to calculate boundary layer dissipation factor, it is necessary to know the boundary layer flow details, which is obviously not conducive to the application of dissipation factor. Fortunately, researchers have given the ideal solution to this problem. Schlichting pointed out that for turbulent boundary layer, the value of the loss coefficient is relatively less influenced by the details of the boundary layer flow. He gave the approximate formula for the dissipation factor in turbulent boundary layer with the shape factor $1.2 < H < 2.0$, Reynolds number based on momentum thickness $10^{3} < Re_{\theta } < 10^{5}$ [118]:

$$ C_{d} = 0.0056Re_{\theta }^{ - 1/6} $$

(4.7)

For most LP turbine blades, the Reynolds number based on turbulent momentum thickness is generally in the order of $10^{3}$. Denton and Cumpsty further pointed out that in this case, the turbulent boundary dissipation factor can be approximated by a value of 0.002 [119].

For the laminar boundary layer, Truckenbrodt gave the following formula for calculating the dissipation factor [120]:

$$ C_{d} = \beta {Re}_{\theta }^{ - 1} $$

(4.8)

where coefficient $\beta$ will be affected by the boundary layer shape factor. Truckenbrodt also gave an analytical result for the laminar boundary layer at zero pressure gradient : $\beta = 0.173$. Figure 4.44 shows the variation of the dissipation factor with Reynolds number given by (4.7) and (4.8). It is worth noting that in Reynolds number range where both laminar and the turbulent boundary layer can exsit $300 < \text{Re}_{\theta } < 1000$, the typical dissipation factor in the laminar flow boundary layer is significantly smaller than the one in turbulent boundary layer. This also clarifies the importance to control the turbine blade boundary layer transition .

Fig. 4.44

Boundary layer dissipation coefficient changes with Reynolds number

Base on the local loss, the total loss of the boundary layer is also easy to obtain. The boundary layer total entropy production can be calculated by

$$ \dot{S} = \int\limits_{0}^{{S_{0} }} {\frac{{C_{d} \rho V_{\delta }^{3} }}{T}} ds $$

(4.9)

Similar to above, we define the dimensionless boundary layer entropy loss coefficient to measure the total loss in boundary layer:

$$ \zeta_{s} = \frac{{T\dot{S}}}{{\dot{m}\frac{{V_{ref}^{2} }}{2}}} $$

(4.10)

here, $V_{ref}$ is the reference speed. For turbine , it is the general outlet speed. The total entropy loss coefficient in LP turbine boundary layer can be calculated using the following formula:

$$ \zeta_{s} = 2\sum {\frac{{S_{0} }}{{t\,\cos \alpha_{2} }}\int\limits_{0}^{1} {C_{d} (\frac{{V_{\delta } }}{{V_{2} }})^{3} d(\frac{s}{{S_{0} }})} } $$

(4.11)

semble in the expression represents the summation of boundary layer at the suction side and the pressure side. If the blade surface velocity distribution and dissipation factor $C_{d}$ are known, then Eq. (4.9) can be used to estimate the blade boundary layer loss coefficient. Equation (4.11) also shows that this boundary layer loss is proportional to the cube of the velocity, as the dissipation factor $C_{d}$ does not change much (as previously described, the turbulent boundary layer in the turbine blade basically satisfies this premise). The region with higher velocity will have a greater loss, which also explains that on the same turbine blade, boundary layer loss on the suction surface is generally greater than that on pressure surface .

4.2.5.2 Prediction Models for Interaction Between Unsteady Wake and Boundary Layer

It is also one of the primary goals for researchers to establish physical models that accurately predict the interaction between wake and boundary layers. Thus, many researchers have made detailed investigation on the turbulence spots in the course of wake -induced boundary layer transition [121,122,123]. These results indicate that the propagation velocity of the trailing edge of the turbulent spots , which are induced by unsteady wakes , is about half of the main flow velocity, while the leading edge of them has a propagation velocity almost equal to the velocity of the main flow, which is different from the staedy state. Furthermore, Addison and Hodson assumed that the wake will induce a fully developed turbulence patch even distributing in spanwise on suction side boundary layer. The patch is assumed to originate from somewhere downstream of the leading edge, the propagation velocities of the leading edge and trailing edge are 1 time and 0.5 time of the main flow velocity, respectively. The model is represented in Fig. 4.45a [124]. Based on the above assumptions, local (downstream of the transition position) boundary layer intermittency can be expressed as the ratio of the time that the turbulent patch passing through this place to the passing period of the turbulence patch. The former is determined by the distance from current location to the turbulent spot , while the latter is determined by the wake passing period. Accordingly, Addison and Hodson gave the prediction of the boundary layer intermittency factor:

Fig. 4.45

A simple wake and boundary layer interaction model [124]

$$ \Gamma (s) = \frac{\Delta t}{T} = f\int\limits_{{s_{t} }}^{s} {(\frac{1}{{U_{te} }} - \frac{1}{{U_{le} }})ds,(s_{t} \le s \le \left\{ {\begin{array}{*{20}c} {s_{l} } \\ {s_{m} } \\ \end{array} } \right.)} $$

(4.12)

Here $T$ and $f$ represent the wake passing frequency ($\frac{1}{T} = f$), $\Delta t$ represents the duration of boundary layer being turbulence, $U_{le}$ and $U_{te}$ represent the turbulence patch propagation velocity at leading edge and trailing edge, respectively. $s_{t}$, $s_{l}$ and $s_{m}$ represent the initiation transition location induced by wake , steady boundary layer transition position and transition process completion location induced by wake . Furthermore, if we assume that the velocity of the main flow in the cascade is constant, and the convective velocity at the leading edge and the trailing edge of the turbulence patch are respectively the main velocity and half of it, the above equation can be simplified as

$$ \Gamma (s) = f\frac{{s - s_{t} }}{U},(s_{t} \le s \le \left\{ {\begin{array}{*{20}c} {s_{l} } \\ {s_{m} } \\ \end{array} } \right.) $$

(4.13)

here $U$ is the main flow velocity.

The boundary layer loss can then be predicted base on the intermittency, and Hodson et al. gave the prediction:

$$ \frac{{Y - Y_{l} }}{{Y_{t} - Y_{l} }} = \left\{ {\begin{array}{*{20}l} {1 - \frac{1}{{2\bar{f}}},} \hfill & {\bar{f} \ge 1} \hfill \\ {\frac{1}{{2\bar{f}}},} \hfill & {\bar{f} \ge 1} \hfill \\ \end{array} } \right. $$

(4.14)

where, $Y_{l}$ and $Y_{t}$ represent the boundary layer losses with transition starting at position $s_{l}$ and $s_{t}$, respectively. $\bar{f}$ is calculated using the following equation

$$ \bar{f} = f\int\limits_{{s_{t} }}^{{s_{l} }} {(\frac{1}{{U_{te} }} - \frac{1}{{U_{le} }})ds \approx f\frac{{s_{l} - s_{t} }}{U}} $$

(4.15)

Addison and Hodson used this model to compare with some measurements [125, 126]. The data given in Fig. 4.45b show the good agreement between the model predicted results and the measured data, especially when the wake passing frequency is low.

Although the above model can obtain good prediction under certain conditions, in fact, what the wake induced is not a fully turbulent patch. Instead, they are random appeared independent turbulence events in the wake active region, whose properties are similar as the traditional turbulence spots. Based on this, Hodson et al. [127, 128] modified their former model and proposed that the turbulent spots would be evenly generated along the spanwise direction at a certain rate in the wake active zone, while this zone would propagate downstream along with the wake , as shown in Fig. 4.46a. However, in his model, Hodson used the formula given by Narasimha [129] in 1985 to predict local turbulent spots production rate but did not use the predictive one given by Gostelow and Dey in 1991 [130] which has taken adverse pressure gradient into consideration. This may lead certain errors. Corresponding to the above model, Mayle and Dullenkopf proposed another model based on the turbulence rate [131, 132]. This model no longer considers the wake induced region will move along with the wake , but rather assumes that the wake induced turbulence spot appears at a fixed streamwise location, similar to the approach of Narasimha [133] in studying the transitions of the steady boundary layer. Mayle and Dullenkopf also believed that the wake will induce uniformly distributed turbulent spots in this position, while the transport process of wake on the suction side only impact the production rate of these turbulent spots. Compared to the experimental data, it is shown that these model can all relatively well predict the wake effect on boundary layer without separation bubble under the condition that the transition initial position and the turbulent spots production rate are reasonable estimated. However, there are still no models to predict the wake influence on boundary layer development with the presence of separation bubbles.

Fig. 4.46

Two models on interaction between unsteady wakes and boundary layer based on turbulence spots

4.3 Complex Flow in Shroud and Its Interaction with the Main Flow

In order to reduce leakage in the gap between rotor blade and stationary casing and reduce the drawback of tip leakage on the aerodynamic performance, many turbine implement shroud seal structures, which are more common in LP turbines. From aerodynamic aspect of view, the shroud not only can effectively reduce the leakage near the gap, but also can enhance the load near the gap, thereby improving the turbine work. From mechanical integrate point of view, although the shroud increases the weight of the rotor blade tip which is not preferred by stress, the blade vibration energy can be absorbed by the friction of adjacent blades shrouds. When the adjacent blades are tightened, the twisting and bending deformation of the blade can be reduced, the rigidity of the blade can be enhanced, and the natural frequency of the blade can be improved.

Although the shroud structure has already widely implemented in gas turbine and steam turbine industry, the research mainly focus on the related structural design and strength calculation. But only in the last decade have the mechanism of shroud leakage flow and its interaction with the main flow gradually become one of the hot topics in turbomachinery field. Shroud generally has labyrinth seal, which, however, cannot completely eliminate the gap. Moreover, the predictable and unpredictable factors, such as axial movement and vibration of the rotor, can cause clearance changes in labyrinth seal. To prevent a serious scratch between the rotational and stationary components, the gap between the tooth and casing is designed to have sufficient margin for safety reasons. In addition, as varying operating conditions is generally required for turbomachinary, the deformation of rotational and stationary parts will change with operating conditions. Consequently, the gap between the tooth and casing and the sizes of inlet and outlet cavities as well will change. Therefore, the flow in the shroud region, including the internal flow inside cavity, the leakage flow of the shroud and its interaction with the main flow will be very complex, which is an important source of flow loss. The shroud tip loss may be greater than that of the non-shrouded blade. The study of the flow pattern and the loss mechanism is important for the aerodynamic design of the shrouded turbine.

4.3.1 Leakage Flow in the Shroud Cavities

Turbine labyrinth seal usually constructed by a series of alternating teeth and cavities, inhibiting fluid flow from the high pressure region to the low pressure region. Labyrinth-teeth is an important part of the shroud , which is also a key factor to determine the shroud internal flow. The main aero-thermo characteristics of air flow through the shroud labyrinth seal are similar to those of the general seal structure, but not identical. Shrouded turbine tip flow has its own characteristics and mechanisms, such as inlet and outlet flow from cavity and the interaction with main flow, etc. Denton also briefly described shrouded turbines flow and loss mechanisms when discussing the tubomachinary loss mechanism [6, 119]. Figure 4.47 shows mechanical model and flow diagram of single labyrinth tooth. In this model, the flow area decreases when the leakage flow passes through the tooth tip which forms a jet flow. If there is no significant restriction upstream of the tooth, the flow till the tooth throat can be approximately considered to be isentropic. Based on this assumption, the leakage flow can be considered to be mainly determined by the size of the labyrinth tooth gap, the shrinkage coefficient, the total pressure based on the meridian velocity upstream of tooth, and the static pressure at the exit of the jet. Here, the outlet static pressure is influenced by the leakage flow which entrances the main flow. However, if there is no significant flow restrictor downstream of the tooth, there is no significant difference between this pressure and the corresponding static pressure in the main flow. When the leakage flow entre into the shroud cavity through the tooth tip they will mix, which leads to entropy increase. In most cases, the cavity outlet space is large enough comparing to the leakage jet, so it is possible to approximate that the kinetic energy based on the meridional velocity largely dissipate in the outlet chamber, but this is not always the case in reality. Experimental measurements by Denton and Johnson show that the circumferential velocity of the leak flow is not significantly altered during the mixing process in the outlet chamber [134].

Fig. 4.47

Model of a shrouded blade with single labyrinth tooth [6]

In real shrouded turbine design, multi-tooth labyrinth seal structure is more common. Figure 4.48 shows the flow enthalpy-entropy diagram of three-tooth labyrinth seal [135]. Three labyrinth teeth separate the shroud into four cavities. The airflow isentropically expands in the gap of each tooth (A-B, C-D, E-F). The fluid accelerates and passes from the gap into the adjacent low-pressure cavity. Major kinetic energy will be dissipated in the cavities (B-C, D-E) as the fluid undergoes isobaric mixing in cavities. When flow passes from inlet chamber (A) to the outlet chamber (F), it changes the state from rotor inlet to the rotor outlet flow, followed by the leakage flow into the turbine and mixing with the main flow. It can be seen that the flow near the shroud tip can be seen as consisting of two main parts: one is the shroud and teeth cavity flow, the other one is leakage flow which interact with main flow later. The former determines the shroud leakage flow , which is the seal capability, while the later one determines the flow loss caused by shroud. But of course, these two are inseparable, mutual influencing.

Fig. 4.48

Shroud flow enthalpy-entropy diagram [135]

To study the flow structure and the evolution mechanism of the shroud flow is the basis for estimating the seal ability and predicting their aerodynamic losses, which is also the theoretical basis to design high performance labyrinth seal. For a typical multi-tooth shroud, the internal flow pattern is also very complex due to the geometrical complexity, including various scales eddies which interacting with each other. Figure 4.49 shows a schematic representation of the flow structure in a typical shroud [136]. Due to the presence of the labyrinth teeth, the most leakage flows through the gap in a jet type flow and flows near the stationary wall of the cavity. At the same time, a series multi-scale eddies are induced by the labyrinth cavity and inlet/outlet cavities. In general, the flow in the inlet and outlet cavities of the shroud is the most complex, especially in the exit cavity, which may have main flow locally ingress and strong unsteadiness.

Fig. 4.49

Internal flow structures of the shroud [136]

Because the flow structure in the shroud is affected by its geometry, the corresponding performance parameters such as leakage flow and flow loss will change with the geometry. For the shroud design, it is necessary to understand the geometrical influences and corresponding mechanism. Although there are many researches on general labyrinth parameters design, it is necessary to investigate flow in shroud due to its uniqueness. Rosic et al. [137] carried out detailed experiments and numerical simulations. Their results show that the decrease of the axial length of the inlet and outlet cavities has a very positive effect on the decrease of flow loss inside the shroud. This is mainly due to geometrical changes weaken the main flow ingress and inhibit the generation of complex vortex structures in the two cavities, thereby reducing the mixing loss as shown Fig. 4.50a. In aspect of shroud length, the increase of the inlet plate can weaken the influence of the leading edge potential field and inhibit the high pressure main flow from penetrating into the shroud, which is beneficial for performance. The influence of exit cavity edge length has two sides. On one hand, the increase of the length will increase the leakage velocity, which leads to the increase of mixing loss. On the other hand, it will help to limit the ingression from main flow. The combined effect makes the overall effect of the length of the outlet plate has relatively weak impact on flow loss. The flow structure inside the shroud is not affected by its thickness, but by the size and location of the vortex in the inlet cavity and the position of the leakage jet in the outlet cavity. They will eventually change the shroud thickness, and turbine performance, see Fig. 4.50b. The depth of the shroud cavity is also one of the key factors affecting the shroud internal flow. The decrease of the cavity depth can effectively control the mixing loss inside the inlet cavity and can improve the main flow field to a certain extent. For the outlet cavity, the internal mixing loss also decreases as the cavity depth decreases. However, the velocity of the leakage jet will increase, which may harm to downstream flow field. The effect of different cavity depths on the internal flow structure of the shroud is shown in Fig. 4.50c. Since these results are based on specific blades and shroud, there is no guarantee for a strong generalizability. However, the geometry influences as revealed by those studies provided the guideline on the general shroud design.

Fig. 4.50

Influence of shroud geometries on internal flow structures [137]

In addition to the geometric parameters, the local aerodynamic parameters also influence the flow structure of the shrouds to a great extent. Such as the difference of flow state between the blade suction side and pressure side, which leads to the circumferential inhomogeneity of the aerodynamic parameter, and also determines the non-uniform of the shroud flow in the circumferential direction. This is another important feature of the shroud leakage flow , which is different from the general labyrinth flow. Both experiments and numerical simulations show that this circumferential inhomogeneity is more pronounced in the vicinity of the inlet and outlet cavities [138, 139].

Figure 4.51 shows the cavity exit radial velocity distribution at different circumferential positions of a shroud [140]. It can be seen that the outlet flow is significantly affected by the circumferential pressure gradient distribution. In the vicinity of the blade pressure side, due to the pressure of the main flow area is higher, the main flow ingression is also very obvious. And it mixes with leakage and forms an eddy filling the major space of the outlet cavity. Then the leakage flow can only be emitted from the position close to the bottom surface of the cavity wall. While in the vicinity of the middle passage and blade suction surface , the fluid penetrating into the outlet cavity again enters the cascade passage from the position close to upper wall of the outlet cavity due to a decrease in main flow pressure. And the formed eddy can still occupy most of cavity volume. In addition, it is shown that the secondary flow structures near the blade tip of the main channel, such as the passage vortex strength and position, also affect the flow structure in the shroud outlet cavity [141, 142] as shown in Fig. 4.52. In multi-stage turbine , the unsteady potential field and wake from upstream and downstream bladerow all cause the fluctuating flow in the shroud . In fact, in both the steady and unsteady conditions, the internal flow structure in the shroud is the result of passage flow, including the cascade secondary flow and the interaction from upstream and downstream bladerows. It can be seen that the flow structure inside the shroud and the flow in the turbine channel are tightly coupled, and the understanding of their interaction mechanism is the basis for their accurate prediction.

Fig. 4.51

Radial velocity distribution at shroud outlet cavity on different circumferential sections [140]

Fig. 4.52

Sketch of driving mechanisms of flow circumferential nonuniformity in shroud outlet cavity [135]

In addition to the strong three-dimensional nature of the leakage flow , it also has a strong unsteadiness. Pfau used a fast-response probe to measure in detail the inlet cavity of a two-stage turbine rotor shroud . The unsteady evolution of the annular vortex in the inlet cavity and its driving mechanism were described in detail [138]. Figure 4.53 shows the tangential vorticity time series at Z = 0.5 between the second stage rotor and stator. The dotted lines represent the leading edge position and the blade tip position (i.e. R = 1), respectively. The results show that the position, range and strength of the loss core are constantly changing with time. The annular vortex rotates in the inlet cavity with high circumferential speed. The relative rotation between the upstream vane trailing edge and the downstream vane leading edge is the main driving force of the unsteady flow of the inlet cavity. On the basis of the experimental results, Pfau established a simplified flow model. Porreca used experimental and numerical methods to study the unsteady properties of some shroud structures. The mechanism of unsteady loss was discussed in detail. And it is suggested to carry out unsteady analysis when designing shrouded turbine in order to optimize the performance [143].

Fig. 4.53

Contours of tangential vortices and corresponding measuring location [138]

4.3.2 Interaction Between Leakage Flow and Main Flow and Its Effects on the Aerodynamic Performance

Leakage flow injected from the shroud outlet cavity will mix with turbine channel flow with complex process. An accurate understanding of this physical process will provide an important theoretical basis for rational organizing of the flow near the tip and reduce the corresponding flow loss. At shroud inlet, the leakage flow is designed to have basically the same velocity triangle as the main stream. However, since there is no blade to turn the flow in the shroud cavity, the leakage flow circumferential velocity has similar flow condition as mainflow at blade tip. At the same time, the leakage axial velocity is complicatedly changed due to the labyrinth teeth action and the obstruction from cavity wall, and thus there is complicated exchange between axial velocity and radial velocity. These factors eventually lead to the velocity triangles of the leakage flow and the main flow no longer match when the leakage re-enters into the turbine channel. It means that the leakage flow relative to the main flow occurs over or under turning phenomena, and then mixing, resulting in flow losses. Mixing causes the leakage flow velocity triangle constantly changing in the flow direction near the end wall. In contrast, the mainflow velocity triangle does not change significantly [139], as shown in Fig. 4.54. The difference between the endwall velocity triangle and the designed velocity triangle also causes the downstream bladerow to have actual larger or smaller incidence than the design one and thereby affect the aerodynamic performance of the downstream bladerow. It should be mentioned that there is no general law governing the relative velocity relation between the leakage flow and the main flow, which includes both the axial velocity and its magnitude. They all depend on turbine aerodynamic environment and the geometrical structure of the shroud.

Fig. 4.54

Circumferential mass averaged velocity triangle near shroud outlet cavity [139]

Due to the uneven circumferential aerodynamic condition in the turbine channel, the leakage flow entering the main flow is not a uniform air curtain, but has a certain distribution along the circumference. The actual distribution pattern is related to the main flow flow field and the shroud structure. In extreme condition, the leakage flow may not be able to flow from the vicinity of the pressure side but only from the vicinity of the suction side that forming a circumferentially distributed jet structure. In addition, under the effects from the secondary flow near the tip and the unsteady condition of upstream and downstream, the shroud leakage flow near blade tip can also show a strong unsteady condition [144, 145]. Leaving from the shroud , leakage flow may also induce a small size but strong streamwise vortex near the blade suction side. The shape of this vortex is similar to the leakage vortex near the tip of the non-shrouded turbine. It will not only accelerate the mixing between the leakage flow and the main flow, its corresponding steady and unsteady effects are likely to show up in the downstream bladerow flow field.

In terms of loss caused by shroud leakage flow, Wallis and Denton et al. first analyzed it in detail [146], which including the following contributions: Leakage fluid does not participate in turbine work and results in loss of turbine work; Mixing loss in the inlet cavity, and drag loss along the shroud; Mixing loss in the outlet cavity; Loss due to mixing between the leakage flow and the main flow, and the corresponding loss caused by the change of the downstream bladerow velocity triangle . Base on those, Rosic and Denton further analyzed the flow loss caused by the shroud using low-speed turbine test rig and numerical simulation method [8]. With different design variants, they gave a quantitative comparison as shown in Fig. 4.55. It can be seen that the total efficiency reduction due to the shroud flow is 2.6%; within it, the efficiency reduction due to mixing in the shroud inlet and outlet cavities was 0.2 and 0.25%, respectively. The decrease of aerodynamic efficiency caused by drag loss of the shroud is the least, about 0.05%. In this study, the leakage flow rate was 1.7%, and it was considered that the resulting turbine work loss and the corresponding decrease in the aerodynamic efficiency were also the same value. The remaining 0.4% decrease in the aerodynamic efficiency is caused by the mixing of the leakage flow with the main stream and the change in the row angle of the downstream blade. The above results indicate that the reduction of turbine performance due to the shroud flow needs to be controlled. In addition to minimizing leakage flow rate, emphasis should be on reducing interactions when leakage flow re-enter into main flow.

Fig. 4.55

Loss decomposition of shroud leakage flow [8]

Regarding if shrouds are required for turbine rotor or not, it is generally believed that the shrouded turbine has certain aerodynamic performance advantage over the unshrouded turbine with same amount of clearance (labyrinth teeth clearance for the shrouded one). And with the reduction of tip clearance, the aerodynamic performance of the shrouded turbine will decrease gradually and become the same once the clearance reaches to 0. However, it has been found in studies that shrouded turbines and unshrouded turbines may have the same aerodynamic efficiency when the tip clearance decreases to a certain non-zero critical value, which is referred as “Break-even clearance”. Further reducing the clearance, the aerodynamic efficiency of the shrouded turbine will be lower than that of the non-shrouded turbine, and the difference of the efficiency when the gap is 0 is called “offset loss” as shown in Fig. 4.56. The loss is mainly caused by the mixing in upstream and downstream of the shroud, the drag loss in the cavity and the shroud induced secondary flow loss in the channel. Yoon et al. [147] verified the existence of “break-even clearance” in their experiments. The results show that the values of “break-even clearance” and “compensation loss” of different turbines may be around 0.5%. As the turbine reaction degree decreases, both quantities are reduced. The significance of the existence of the “break-even clearance” is to provide a basis for the choice of whether to use shrouded turbine or not. If turbine tip clearance can be controlled to a smaller level, for example close to or even smaller than the “break-even clearance”, and if factors such as flutter or blade deformation are not taken into account, then turbine without shroud is preferred. Because at this time, aerodynamic performance of un-shrouded turbine will not show significant disadvantages or even better, but it benefit in structural stress and weight.

Fig. 4.56

Efficiency trends of turbine with and without shroud [147]

As shown in the previous section that the geometrical parameters of the shroud directly affect the internal flow structure. This will definitely affect the associated flow loss, and thus affect turbine aerodynamic performance and corresponding parameters. In this area there has already been lots of publications [148,149,150]. Rosic and Denton et al. further discussed the influence of the main geometrical parameters of the shroud on the leakage flow and the corresponding flow loss based on the discussion of the flow structure [137]. It can be seen from Fig. 4.57 that leakage flow shows a monotonically increasing tendency with the increase of the axial length of inlet and outlet cavity in a wide range. Only when the axial length of inlet cavity is reduced to a certain value or the axial length of outlet cavity is increased to a certain value, the variation range of the leakage flow is gradually reduced, that is, the variation trend is nonlinear. The specific variation amplitude is related to the specific shroud structure selected in the study. Due to the decrease of the leakage flow and the decrease of the mixing loss of the leakage flow in the inlet and outlet cavities, the turbine aerodynamic efficiency decreases monotonically with the increase of cavity axial length. And when the axial length of the cavity is in a certain range with smaller value, the aerodynamic efficiency change is more severe. In terms of the influence of the shroud inlet plate, the leakage flow rate generally decreases with the increase of the length of the edge plate, which is mainly because that the elongation of the inlet edge can keep the inlet cavity away from high pressure region near the blade leading edge. At the same time, it can be seen that the change in leakage flow rate is not very pronounced when the edge plate is very long or very short. While the aerodynamic efficiency of the turbine shows a different trend, which increases linearly with the elongation of the inlet edge plate. The influence of the outlet plate length on the leakage flow rate is small. With the elongation of the outlet edge, the leakage flow rate first increases slightly and then decreases slightly. Correspondingly, the efficiency of the turbine is also decrease and then increase. The change amplitude is smaller than the change due to inlet edge. Increasing the thickness of the shroud edge causes a greater axial velocity of the leakage flow in the vicinity of the first labyrinth tooth, resulting in increased leakage flow rate, which is against the turbine efficiency. On the other hand, an increase in the thickness of the edge plate can also reduce the cavity volume, which is possible to reduce the dissipation of its internal flow. Under the combined influences of these two factors, the aerodynamic efficiency of the turbine increases and then decreases with the increases of the plate thickness. The influence of the shroud cavity height also showed two sides. Increasing the cavity height enhanced the mixing in the cavity, which increases the kinetic loss. But on the other hand, it also reduces the leakage flow rate. Therefore, the turbine efficiency shows a trend of first decrease and then increase.

Fig. 4.57

Effects of geometrical parameters of the shroud on turbine efficiency [137]

4.3.3 Shroud Leakage Flow Low-Dimensional Models and Multi-dimensional Coupling Simulation

The function of the shroud is sealing , which determines that it has generally more complex structure and narrow geometric space. It indeed makes the detailed measurement of the flow field of the shroud area difficult to achieve. Numerical simulation has become one of the important investigation methods. Although the numerical simulation can provide detailed flow field, this requires meshing for the complex shroud geometry, which increases the difficulty of pre-processing and time consumption. Moreover, the complicated calculation also increases the difficulty of convergence and accuracy, and puts forward the strict requirements for the selection of the boundary conditions, initial conditions, as well as the turbulence model and the transition model. Even if the above issues can be proper resolved, the increased amount of cells will also greatly increase the computational cost. Therefore, the full three-dimensional numerical simulation which considers the actual geometry of the shroud is still mainly used in the study of the flow mechanism. Regarding the performance evaluation and flow diagnosis for industrial purpose, a more practical method which can take into account both the calculation accuracy and efficiency is needed. The multi-dimensional correlation is a very effective method, which is to establish a low-order mathematical-physical model for the flow in the shroud and its interaction with the main flow, and then couple it with the three-dimensional calculation of the main flow and finally achieve effective results. This is an important research area with shrouded turbines.

The prerequisite for the evaluation of shroud flow field is the precise prediction of the leakage flow , which mainly depends on labyrinth characteristic. So far, there are many studies on the mathematical model of the labyrinth flow, especially the modeling of the labyrinth leakage. Ideally, the flow through the labyrinth seal structure can be calculated according to the isentropic flow theory [151]:

$$ \dot{m}_{ideal} = \left\{ {\begin{array}{*{20}l} {\frac{{p_{0} A}}{{\sqrt {kRT_{0} } }}\sqrt {\frac{{2k^{2} }}{k - 1}\left( {\frac{{p_{n} }}{{p_{0} }}} \right)^{2/k} \left[ {1 - (\frac{{p_{n} }}{{p_{0} }})^{{\frac{k - 1}{k}}} } \right]} } \hfill & {\frac{{p_{n} }}{{p_{0} }} > \left( {\frac{2}{k + 1}} \right)^{{\frac{k}{k - 1}}} } \hfill \\ {\sqrt {\frac{k}{{RT_{0} }}} p_{0} A\left( {\frac{k + 1}{2}} \right)^{{\frac{k + 1}{2(k - 1)}}} } \hfill & {\frac{{p_{n} }}{{p_{0} }} \le \left( {\frac{2}{k + 1}} \right)^{{\frac{k}{k - 1}}} } \hfill \\ \end{array} } \right. $$

(4.16)

where A is the clearance cross-sectional area at tooth tip, the subscripts “0” and “n” represent the corresponding inlet and outlet cross-sections, respectively. In general, the flow through the throttling element results in an actual flow rate smaller than the theoretical flow rate due to inlet loss, friction loss, and outlet sudden expansion losses. Normally flow rate coefficient $C_{D}$ is introduced for labyrinth seal, which is defined as the ratio of the actual mass flow rate $\dot{m}_{real}$ and the ideal mass flow rate $\dot{m}_{ideal}$:

$$ C_{D} = \frac{{\dot{m}_{real} }}{{\dot{m}_{ideal} }} $$

(4.17)

This is a key parameter to evaluate the efficiency of labyrinth seal. This parameter can reflect the contraction effect of gas flow through the gap and the friction influence on the velocity which leads the outlet mass flow rate less than the theoretical value. Its value depends on many labyrinth seal factors such as tooth geometry, gap, number of teeth, inlet and outlet pressure, and other parameters.

In engineering applications, Martin gave the formula to calculate seal leakage in 1908 [152], as shown in Eq. (4.18). Based on adiabatic flow assumption, this method assumes each labyrinth is a series of nozzles and assumes the flow is isothermal. Each element which is formed by a tooth and the followed cavity has the same flow resistance, and the kinetic energy in each chamber is completely converted to heat. Stodola used Bernoulli equation and continuity equation for each section of the tooth, and integrates it along the seal length under the assumption that the pressure difference is not large and then obtained the approximation formula for calculating the leakage of the labyrinth seal [153], as shown in Eq. (4.19). The formula also believes that it is ideal throttling process, no ventilation effect. The formula is only validated for sub-critical condition and the number of teeth is greater than 4.

$$ \dot{m} = C_{D} A\frac{{p_{0} }}{{\left( {RT_{0} } \right)^{1/2} }}[\frac{{1 - (p_{n} /p_{0} )^{2} }}{{N - \text{In(}p_{n} \text{/}p_{0} \text{)}}}]^{1/2} $$

(4.18)

$$ \dot{m} = C_{D} A(\frac{{p_{0}^{2} - p_{n}^{2} }}{{NRT_{0} }})^{1/2} $$

(4.19)

The above two methods are relatively simple, but with large errors. Thus they are difficult to meet engineering requirement. Later researchers revised the formula using analytical method and experimental data and gave a relatively high accuracy calculation method. For example, Vermes modified the non-adiabatic condition based on the Martin formula and deduced the correction coefficient of permeability effect. The ratio of the tooth tip width and tooth cavity width to the tip clearance height was taken into account [154]. The resulting formula is shown below:

$$ \dot{m} = C_{D} A\frac{{p_{0} }}{{\left( {RT_{0} } \right)^{1/2} }}\left[ {\frac{{1 - (p_{n} /p_{0} )^{2} }}{{N - \text{In(}p_{n} \text{/}p_{0} \text{)}}}} \right]^{1/2} (1 - \alpha^{{\prime }} )^{ - 1/2} $$

(4.20)

where term $(1 - \alpha^{{\prime }} )^{ - 1/2}$ is the ventilation effect correction term, $\alpha^{{\prime }}$ is kinetic energy residual factor, which is related with the shroud cavity geometry. Although the ventilation effect is revised here, the wall effect is still not considered. Thus the accuracy is still needed to be improved [155, 156].

Denton theoretically analyzed the shroud leakage in single labyrinth and assumed the flow does not deflect in the shroud. The leakage flow rate is therefore calculated as below:

$$ \frac{{\dot{m}}}{{\dot{m}_{mainflow} }} = C_{D} (\frac{g}{h})\sqrt {\sec^{2} \beta_{2} - \tan^{2} \beta_{1} } $$

(4.21)

Here $h$ is the height of the blade. Regarding the shroud loss, early prediction model generally believe that the change in efficiency is proportional to the leakage flow . Based on this understanding, Denton and Johnson presented a very simple model to estimate the turbine shroud loss [134], suggesting that the change in efficiency is proportional to the relative height $\gamma$ of the gap:

$$ \Delta \eta = k\gamma $$

(4.22)

where $\gamma$ is defined as the ratio of the shroud clearance to the blade height, and the proportionality factor $k$ is determined by the actual geometrical parameters of blade and shroud, and the aerodynamic condition of the turbine . Obviously, this simple proportional correlation is too rough, and the proportion coefficient $k$ is also mainly dependent on experience. So this type of prediction models has difficulty to provide satisfactory results for turbine aerodynamic design and optimization process. Later, based on the prediction of the shroud leakage mass flow rate, Denton proposed a mathematical model for single tooth labyrinth shroud to calculate the mixing loss coefficient $\zeta$ of the shroud-leakage flow and the main-flow [119]:

$$ \zeta = \frac{{T\Delta s}}{{0.5V_{2}^{2} }} = 2\frac{{\dot{m}}}{{\dot{m}_{mainflow} }}\left( {1 - \frac{{V_{t} }}{{V_{t2,mainflow} }}\sin^{2} \beta_{2} } \right) $$

(4.23)

where $V_{t}$ and $V_{t2,mainflow}$ are the circumferential velocity of the leak flow and the turbine main flow, respectively. This equation shows that the difference in circumferential velocity is the main source of loss due to leakage and main flow mixing at shroud outlet. In order to reduce this loss, the designer needs to make the circumferential velocity of the leak flow as consistent as possible with the main flow.

Base on above model, Yoon et al. further gave an aerodynamic efficiency correlation to evaluate turbine stage aerodynamic efficiency influenced by shroud [147]:

$$ \Delta \eta = \frac{{T\Delta s}}{{\Delta h_{stage}^{*} }} = \frac{\zeta }{{2H_{T} }}(\frac{{W_{2} }}{U})^{2} $$

(4.24)

This formula shows that the quantitative relationship between the turbine-stage aerodynamic efficiency and the loss caused by shroud flow is determined by turbine load factor $H_{T}$, outlet relative velocity $W_{2}$ and turbine circumferential velocity $U$. Further, the change of turbine work can also be primary predicted:

$$ \frac{{\Delta P_{T} }}{{P_{T,noleak} }}/\frac{{\Delta \eta }}{{\eta_{noleak} }} \approx \frac{{\varphi (\tan \alpha_{1} - 2\tan \beta_{2} ) - 1}}{{\varphi (\tan \alpha_{1} - \tan \beta_{2} ) - 1}} $$

(4.25)

The results predicted by above formula show good agreement with experiments. For a conventional turbine layout, the relative change in the turbine output power, which is affected by the shroud obtained from (4.25), is generally 1.5–2 times of the relative change in efficiency. Based on above analysis, Yoon et al. pointed out that the leakage-caused decrease of aerodynamic performance is more reasonable attributed to dissipation of the relative kinetic energy rather than an increase in leakage flow or associated loss factor. At the same time, Yoon et al.’s results show that the stage reaction rate is also one of the key factors affecting the shroud . For high reaction rate turbine, its aerodynamic efficiency and power are more affected by shroud loss, which is due to the high relative velocity at stage outlet under high reaction rate, i.e., higher relative kinetic energy [147].

Along with deeper studies, researchers expect to obtain more accurate prediction of the influence of the shroud and the flow field in the main channel in order for further accurate investigation and evaluation. It cannot be achieved by only one-dimensional mathematical model. As a compromise between a simple one-dimensional mathematical model and a full three-dimensional numerical simulation, it is an ideal choice to couple the low-order mathematical model with the main flow 3D simulation, which is to add an additional source term in three-dimensional numerical simulation. When the parameters such as the shroud geometry, position, and the flow conditions at inlet and outlet cavities are determined, using one-dimensional model to evaluate the parameters such as the leakage flow rate, the temperature rise and the momentum change, and then apply the corresponding flow parameters at cavity inlet and outlet. Because this method can consider the influence of the leakage flow in the three-dimensional passage flow without increasing computational cost, thus has attracted more attention. At present, this kind of multi-dimensional coupling numerical method has become a new development direction, which plays an important role in the field of shrouded turbine, air system and so on.

Rosic and Denton et al. [157] applied this method to the numerical simulation of one three-stage shrouded turbine test rig. The implementation of the shroud model in the simulation is given in Fig. 4.58. The comparison between experimental results and numerical simulations with/without shroud structure shows that the coupling method is better than the numerical simulation which does not consider the shroud in terms of capturing turbine inner flow structures. But there is still gap comparing to the simulation which fully considering the shroud. Gier et al. have developed a similar coupling method for shrouded turbine . They studied one shrouded turbine, focusing on the influence of the leakage flow on the main flow [158]. Figure 4.59 shows the numerical results comparison between the multi-dimensional coupling method and the full shroud geometry simulation. It shows that the coupling method can effectively analyze the influence of the shroud leakage flow on the main flow except variation of circumferential parameters. The main disadvantage of using the one-dimensional shroud model is that the circumferential parameters variation of the leakage flow cannot be considered. For this, Hunter analyzed the results of three-dimensional full simulation, extracted the relevant distribution characteristics, focusing on the non-uniform distribution of the flow parameters at the interface between the shroud flow and the main flow on the multi-dimensional coupling calculation [159]. Comparing to experimental results, due to the consideration of 2D characteristics on the interface, this research provided more accurate prediction on shrouded leakage flow and channel endwall secondary flow than traditional one dimensional shroud model. These multidimensional studies have already achieved some useful results. However, the factors considered are not comprehensive and detailed, such as the one-dimensional model did not take specific shroud inlet cavity into consideration, even this local flow field has important impact on shrouded flow. Also, the two dimensional character of the leakage is still need to be modeled and applied in engineering.

Fig. 4.58

Rosic’s shroud model coupled with the main flow [157]

Fig. 4.59

Comparison of shrouded turbine passage flow between different methods [158]

Based on the research of Rosic et al., Wang Peng and Zou Zhengping from Beihang University carried out further research on shrouded turbine multidimensional coupling method CFD [160, 161]. Since the prediction accuracy of the low-dimensional shroud leakage flow model is critical to the success of the multidimensional coupling calculation, Wang et al. established a new shrouded leakage flow model. The model can take into account the shroud inlet cavity flow and the key features such as mass/momentum/energy of the leakage flow and predict the two-dimensional distribution of the leakage flow [160]. Details are as follows: In the aspect of the leakage flow rate, according to the jet characteristic of the leakage flow, reasonable modeling is carried out based on classical jet theory to represent the residual kinetic energy in each part of the shroud. Respectively based on the wall jet and free jet, residual-energy factors of the flow between the labyrinth tooth are evaluated and correlated based on virtual jet theory and then coupled together. The inlet cavities are also modeled and its flow is equivalent to the flow pass several labyrinths. The virtual labyrinth tooth number is used to consider the residual kinetic energy in the inlet cavity when predicting the leakage flow rate. On the basis of the leakage flow rate model, the momentum and energy characteristics of the leakage flow, the effective width of the leakage flow at the interface and the temperature correction of the outlet cavity are modeled respectively based on the angular momentum theorem and the non-isothermal jet theory. The turbulence characteristics of the leakage flow are taken into account. The following is the expression for the number of virtual tooth on the shroud in this leakage model:

$$ N^{*} = N_{e} + (1 - \alpha_{1} ) + (N_{seal} - 1)(1 - \alpha_{2} ) $$

(4.26)

Here the inlet cavity equivalent tooth number and the residual-energy factor of the flow between tooth are:

$$ N_{e} = \frac{{p_{0}^{2} - p_{1}^{2} }}{{p_{1}^{2} - p_{N}^{2} }}\left[ {N_{seal} - \text{In(}\frac{{p_{N} }}{{p_{1} }}\text{)}} \right] + \text{In(}\frac{{p_{1} }}{{p_{0} }}\text{)} $$

(4.27)

$$ \alpha_{1} = \frac{{10.390b_{0} \sqrt {N_{e} } }}{(3.382L01 + L02)}\quad \quad \alpha_{2} \frac{{2.009b_{0} }}{0.296L1 + L2} $$

(4.28)

The size parameters like L1 and L2 characterize the key geometrical features of the shroud cavity. Regarding the two-dimensional characteristics of the leakage flow , an approximate relation is given on account of the circumference inhomogeneity at the interface. That is, the shifting correlation between the circumferential distribution of the main flow pressure in the vicinity of the outlet cavity and the circumferential distribution of the leakage jet strength. After shifting downstream by an axial scale along wake trace, the circumferential distribution of the strength characteristic of pressure shows negative correlation with that of the leakage flow strength. Considering the rationality and veracity of multi-dimensional scaling in multi-dimensional coupling computation, a mathematical description of the two-dimensional circumferential characteristics of the outlet cavity is established based on this correspondence relationship. Taking the flow flux as an example, assuming that the circumferential coordinate at the interface is $x$: Let $F_{{\dot{m}}} (x)$ be the flow flux at $x$, and $\bar{F}_{{\dot{m}}}$ be the circumferential averaged flow flux of the leakage jet at the interface, we can get:

$$ F_{{\dot{m}}} \left( x \right) = C_{{\dot{m}}} \bar{F}_{{\dot{m}}} \frac{{\bar{P}}}{{P(x - \frac{H}{\tan \,\theta })}} - \left( {C_{{\dot{m}}} - 1} \right)\bar{F}_{{\dot{m}}} $$

(4.29)

where $P(x)$ is the upstream pressure at circumferential coordinates $x$ of the interface, $\bar{P}$ is the corresponding average pressure, $\theta$ is the trailing edge angle of the rotor blade, $H$ is the axial size of the interface, $C_{{\dot{m}}}$ is the correction factor. At the same time, in terms of the axial direction, there is also no uniform value at the interface. Therefore, the detailed axial characteristics within the leakage flow are also considered when expanding the models to 2D distributions using the multi-dimensional scaling method. The specific modeling method for the axial distribution is still based on the jet theory.

Based on this new shroud leakage flow, numerical simulations of flow in shrouded turbine based on multi-dimensional coupling method achieve higher accuracy. Figure 4.60a compares the results of different methods for predicting flow angle downstream of the leakage flow. Figure 4.60b compares the radial distribution of total pressure downstream of the leakage flow under variable operating conditions (different backpressures). In the figure, CNM represents multi-dimensional coupling calculation, and CTM represents the result of full 3D shroud calculation. The figures shows that the results predicted by multi-dimensional coupling method agree well with the full three-dimensional shroud results, no matter for the design or off-design conditions. It shows the accuracy of the shroud leakage model and the implement potential of the multi-dimensional coupling method. Figure 4.60c shows that this method also achieves a more accurate prediction in the specific leakage flow region and its downstream flow fields.

Fig. 4.60

Multi-dimension coupling simulation results in a shrouded turbine [160]

4.3.4 Shroud Leakage Flow Control Technology

Since turbine flow loss due to leakage flow cannot be ignored, how to effectively suppress the loss and improve the aerodynamic performance of the shrouded turbine becomes an important research topic. A lot of works have been carried out and some effective flow control methods have been obtained.

As Denton pointed out that the significant difference between the circumferential velocity of the leak flow and the main flow is one of the main sources of mixing loss [119]. This will not only bring a strong mixing near the cavity outlet, but also influences the incidence of the downstream blade row, which in turn affects its flow loss. Studies have shown that various losses caused by the different circumferential velocity would account for 60% of the total loss due to leakage flow [162]. Therefore, effective control the outflow angle of shroud leakage flow may reduce the flow loss. Wallis and Denton et al. [146] first attempted this approach by mounting a series of small vanes on the shroud where concentrated by leakage flow and reversing the leakage. Due to the mechanical constraints of equipment and other reasons, this method did not achieve the desired results, even on the contrary, the turbine efficiency decreased. However, using plates instead of small vanes to turn the leakage flow into axial direction achieved better results. It effectively reduced the rotor outlet circumferential velocity near casing and reduced the downstream stator secondary flow region, thereby increased the efficiency of entire turbine. They pointed out that using diversion devices to control shroud leakage losses is feasible. But it needs deep understanding of the unsteadiness and three-dimensional flow structures near the shroud in order to develop good control method. Based on these observations, Rosic and Denton further investigated this idea [8] by mounting with different turning vanes on the casing at rotor outlet as shown in Fig. 4.61. The experimental results show that current control methodology can effectively reduce the loss of flow and improve the aerodynamic performance of the shrouded turbine . This effect depends on the geometrical parameters of the turning vanes and their circumferential distance. Setting eight vanes for each blade channel obviously reduced the loss and improved situations of incidence for the downstream stator blades, resulting 0.4% increase in the turbine brake efficiency. In addition, the experimental study also shows that if change the rectangular vanes to the smaller L-shaped vanes; it is possible to obtain better aerodynamic efficiency and can also effectively avoid axial scratching. Unlike Rosic and Denton, Gao et al. carried out numerical simulations using a row of turning vanes inside the cavity, say between the two teeth (Fig. 4.62), to turn the leakage flow and consequently achieved certain control effects [163].

Fig. 4.61

Implementation of turning vane at shroud outlet cavity [8]

Fig. 4.62

Implementation of turning vane inside shroud cavity [8, 163]

In addition of using turning vanes, the exit geometry of the shroud cavity is one of the important parameter to control leakage influence. Rosic and Denton et al. [164] made systematic studies on three manufacture methods for exit shape by experiments and numerical simulations. The first method is using proper curvature surface downstream of cavity, such as chamfering as shown in Fig. 4.63a. This method can suppress backflow from main stream, and can reduce corner separation and loss at cavity outlet by controlling radial velocity of the leakage flow , and further reduce the secondary flow intensity. Other researchers also investigated this method and received benefits [165, 166]. The second method is to mount annular axial turning plates at the interface between the outlet cavity and the main channel as shown in Fig. 4.63b. This ring-shaped baffle structure can move the location where leakage flow into mainflow towards upstream, thereby suppressing the penetration of the main flow. This can effectively reduce the associated mixing loss and improve downstream flow conditions. Experiment results also show that the length of plate is the main factor that determines the effectiveness of this method, while the appropriate slope of the plate may lead to better results. Both of the above-mentioned methods increase the rotor axial movement which may cause a risk of scratching. But the plate of the latter method may be of a modular design, which means it can be replaced after damage. Thus it has better engineering applicability. Based on these two methods, Rosic and Denton et al. also developed a method of arranging ring-shaped radial deflectors in the outlet cavity [164] as shown in Fig. 4.63c. The method has the advantages of the former two methods, can effectively reduce the leakage flow mass flow rate, its radial velocity, and shift the location where leakage flow penetrate into main flow to move forward. Thus it can achieve better control results, and also avoid the rotor axial movement which may lead the scratching problem. Experimental results also show that this method is less affected by the location of the deflector/plate, which lead to a strong engineering practicality. And thus it is recommended by researchers. In addition, the effectiveness of the above control methods is enhanced with the increase of the leakage mass flow rate, which is very favorable for turbine component performance under off-design condition and the performance maintenance of wearing labyrinth teeth after long working time. Pfau et al. [167] also proposed a new idea in the shape of the shroud-outlet cavity, which used the idea of non-axisymmetric endwall. They designed circumferential uneven shroud structure and corresponding cavity outlet based on the circumferential inhomogeneity flow induced by the interaction between the leak flow and the main flow, as shown in Fig. 4.63d. The application of this method on a two-stage turbine test rig showed that this method can effectively adjust the circumferential distribution of leakage flow and improve its interaction with the main flow, and finally achieve the purpose of reducing flow loss and improving aerodynamic performance.

Fig. 4.63

Flow control based on outlet cavity modelling [8, 164, 167]

The design of the shroud inlet cavity would also affect the flow field and the loss with a certain impact. Many researchers studied the size of the inlet cavity, the shape of the leading plate, the arrangement of the guiding plates in the inlet cavity, etc. [134, 167, 168]. These methods can benefit in terms of reducing leakage mass flow rate and improving flow field in the main passage. Based on the above work, Rosic and Denton et al. optimized the shrouds of a shrouded turbine. They optimized the inlet and outlet cavity geometries with the leakage flow rates of 1.48% and 1.78%, respectively. The turbine aerodynamic efficiency gained 0.60% and 0.75% [164].

The above described shroud flow control methods are carried out under full shroud conditions. Correspondingly, there is also partial shroud, as shown in Fig. 4.64. The main purpose of using partial shroud design is to reduce the shroud’s weight and thus reduce stress of blades and discs, and further to ensure the reliability of turbine components. Although the introduction of partial shroud can make the flow field near the shroud more complicated, which often leads to an increase in flow loss. However, if the structure/geometry of partial shroud can be optimized to achieve reasonable flow field and also reasonably use of some particular characteristics of shroud flow, it may reduce the loss of component aerodynamic efficiency and maintain it in an acceptable range. Its multi-disciplinary performance may be better than the fully shroud design. It may increase aerodynamic efficiency while reducing weight. Although a number of studies have been carried out on partial shrouds [143, 167, 169] and gained certain understanding of the flow characteristics for some typical labyrinth shrouds and their effects on aerodynamic performance, until now, the research on the flow control of partial shroud has still been limited, and lack of universal applicable conclusions and methods.

Fig. 4.64

Partial labyrinth shroud geometries

Apart from using labyrinth shroud geometry modification, aerodynamic method also plays an important role in shroud flow control . Comparing with the conventional labyrinth seal structure, the pneumatic sealing method has certain advantages in structure and aerodynamics. It will not be limited by the mechanical restriction of scratching between rotational and stationary components, and it is not sensitive to the displacement between those two components either. Also, by adjusting the labyrinth seal inject speed and angle can balance the pressure difference, which drives leakage flow , between pressure side and suction side and thus no need to reduce reaction rate of the blade tip region. The thought of using aerodynamic way to seal this direction has existed for long, both for the un-shroud [170] and the shrouded cases [171]. In addition to the direct using injection to weaken the leakage, it is also generate “air curtain” to reduce cross section area thus to reduce leakage [172]. Curtis and Denton et al. [173] verified the aerodynamic seal method on a turbine test rig. The set-up is shown in Fig. 4.65. In this set-up, the air curtain injects towards leakage flow direction and forms 45° angle with the axial direction. Numerical simulation and experimental study on the mass flow rate of different air curtains show that increasing jet flow rate can generally achieve the goal of eliminate leakage. However, at this time the turbine aerodynamic efficiency decreases significantly comparing to non-injection condition. The optimal efficiency occurs when the jet flow rate is 1/3 of the leakage flow rate without labyrinth seal. Curits and Denton et al. proposed two predictive models based on the momentum conservation and the streamline curvature of the injection. Predicted results show good agreement with the experimental results [173]. The use of air curtain in the labyrinth seal has its limitations. On one hand, it is difficult to control the circumferential distribution so do the results. On the other hand, whether it is advantages or disadvantages to the overall turbine performance depends on the source of the air curtain, which means the “extraction” may harm to performance. Besides, using air curtain flow may not only reduce certain amount of leakage flow , but also reduce the originally cooling air amount in this region. Therefore, final results depend on the balance of many factors.

Fig. 4.65

An Air curtain in a turbine blade shroud [173]

4.4 Secondary Flow in High-Loaded LP Turbine Endwall Region

Because the aspect ratio of LP turbine blades is relatively large, the secondary flow loss in the endwall region is relatively small, so it did not attracted much attention. But with the increasing load and requirements of LP turbine, the influence of secondary flow in endwall region has attracted more and more attention in recent years. The evolution mechanism of the secondary flow and the control of its loss in the endwall region of the high load/ultrahigh load LP turbine are important for the development of the high efficiency and high load LP turbine aerodynamic design method.

4.4.1 Secondary Flow Structure and Loss Characteristics in LP Turbine Endwall Region

The secondary flow structure and its evolution of the LP turbine endwall are similar to that of the high-pressure turbine. However, comparing to the HP turbine, in addition to the geometric characteristics like high aspect ratio , the LP turbine blade passage generally has more aerodynamic characteristics like high flow turning angle, low Reynolds number , large traverse pressure gradient , adverse pressure gradient near trailing edge, etc. These characteristics determine that the secondary flow structure and the loss configuration of the LP turbine are different from those in HP turbine.

Hodson and Dominy studied the secondary flow structure in a LP turbine cascade [174]. The main characteristics that distinguish the endwall flow in LP turbines from the one in HP turbines are the evolution mechanisms of passage vortices and horseshoe vortices, the formation of corner vortices, the influence of trailing edge shedding vortices and the effects of separation bubble on the blade suction side. Figure 4.66 shows typical secondary flow structures in turbine endwall region. Figure 4.66a, b are the oil visualization results of secondary flow structures on the endwall region and suction surface , respectively. Figure 4.66c shows topology schematic of secondary flow structures. In the figure, “S” and “R” represent the separation and reattachment lines of various separation flow structures. “0” and “5” respectively correspond to the leading edge separation bubble and the trailing edge separation bubble on the suction surface , “1” and “2” represent the incoming boundary layer separation corresponded to the rolled-up horseshoe vortexes (“s” and “p” denote suction side branch and pressure side branch, respectively). “3” represents corner vortex induced separation. “4” represents the separation caused by the passage vortex. The results show that the suction side leg of the horseshoe vortex is not eaten by the passage vortex in the LP turbine , and its trajectory has not been significantly affected by the passage vortex. Basically it locates close to the suction surface of the blade and develops till trailing edge, which may be due to the strong transverse pressure gradient in LP turbine . In addition, the fluid in the suction side of the horseshoe vortex is not drawn into the separation bubble at the trailing edge, yet its strength is weakened by the separation bubble. The appearance of the corner vortex in the LP turbine channel may also be different from that in HP turbine. Observation shows that the corner vortex of this cascade is formed in a position where the suction side leg of the horseshoe vortex hit the suction surface of the blade, which may be closer to the upstream comparing to HP turbine. Since the transverse pressure gradient near the leading edge of the blade determines the position where the suction side leg of the horseshoe vortex hits the suction surface , the blade loading and its distribution, as well as the incidence will directly affect the formation position of the corner vortex. In terms of the evolution of the passage vortex, the LP turbine passage vortex may still be very close to the suction side near the exit of the cascade channel, rather than in a HP turbine that would propagate to a location near the middle of the passage, which would also be due to the greater transverse pressure gradient . Just because the passage vortex of the LP turbine is closer to the blade, its interaction with the trailing edge shedding vortex is more significant. Those trailing edge shedding vortexes will evolve into the counter rotating streamwise vortexes, and eventually may result in flow after leaving the passage shows more serious underturning, which is not seen in HP turbine .

Fig. 4.66

Typical LP turbine endwall region secondary flow structures [174], Endwall flow visualization, blade suction surface flow visualization, flow structures

The flow losses in the LP turbine endwall region can generally be divided into endwall boundary layer losses and secondary flow mixing losses, with the later losses dominating, especially for high-loaded LP turbines. There not only exists a variety of large scale vortex structures in the endwall region, but also be lots of turbulences due to the breakdown of vortexes. Therefore, the study of the secondary flow loss in endwall more focus on the evolution of the secondary kinetic energy (SKE) dissipation and turbulent dissipation. Based on the analysis of measurements on three cross sections downstream of one turbine casescade, Moore and Adhere believed that flow losses downstream of the cascade mainly come from the reduction of SKE [175]. However, MacIsaac and other researchers concluded different from another cascade. Their results show that the contribution of SKE reduction to the total loss is only about half [176]. Note that the two researches have different aspect ratios, flow conditions, load and so on, which may be the reason that cause different conclusions. It also indicates that the geometrical aerodynamic characteristics of cascades need to be taken into account when measuring the SKE loss. In terms of the contribution of turbulence dissipation, results from Gregory-Smith et al. show that although turbulent kinetic energy (TKE) in the cascade may increase tenfold, it can only account for about 25% of the total pressure loss [177]. A similar result was given by MacIsaac et al. [176], in which the total pressure loss of the turbine cascade is present, but the TKE measured at three cross sections downstream of turbine cascade does not increase at all. These results show that there is no clear correlation between the increase of TKE and the increase of flow loss, and therefore the TKE cannot be simply used to measure the flow loss. When measuring the Reynolds stress in the mixing process downstream of cascades, Moore and others found that the production term of turbulence shows good agreement with the total pressure loss. And they believed that the production of turbulence plays a key role in the secondary flow mixing in turbine [178]. This conclusion is also supported by MacIsaac et al. [176] and Lyall et al. [179]. In the meantime Lyall et al. pointed out that after being divided into the secondary flow dissipation and the turbulence dissipation, the Secondary flow loss inside the turbine can be further divided into the reversible term and irreversible term. They also gave expressions for each term through the theoretical derivation.

4.4.2 Pressure Surface Separation and Its Impacts on Flow in LP Turbine Endwall Region

It is difficult to avoid the existence of strong adverse pressure gradient in the trailing edge of the suction face due to the increasing load of the blade. This will cause the suction surface boundary layer to thicken rapidly and increase the risk of boundary layer separation. The influence mechanism of the boundary layer thickening and even separation on the endwall secondary flow region is basically the same as that of the HP turbine . For high-loaded low-weight LP turbines, the effect of increasing blade loading is not limited to the suction side, but may also change the pressure surface flow field, resulting in significant change of endwall losses.

There are two options to reduce LP turbine weight: hollow thick blades or solid thin blades. Since the latter have obvious advantages in the processing, manufacturing and maintenance, they are widely used. But with the increasing load, there may be adverse pressure gradient near the leading edge of pressure surface on the solid thin LP turbine blade. As the boundary layer is still in laminar condition at this time, it is easy to form separation bubbles in the design state, that is, pressure side separation. Therefore, it is important to control the separation of the pressure side of the high load/ultrahigh load LP turbine and its effect on the endwall secondary flow .

Although this pressure side separation phenomenon only occur at large negative incidence in conventional turbines or in high load turbines with thick blades, preliminary discussions have been made [180,181,182]. These studies mainly focus on the impact of the incidence. As Yamamoto found in the study, only the incidence is larger than a certain threshold, the conclusion that the reduction of incidence will inhibit the strength of secondary flow was established. When the incidence is less than this threshold, the decrease of the incidence causes the secondary flow to intensify. The reason is that the pressure surface is separated [183]. However, as the load distribution of the blade changes significantly at negative incidence, these results could not provide satisfactory description to the possibility of a pressure-surface separation bubble and corresponding impact to endwall flow in the high-loaded LP turbine with a thin blade design.

Brear and Hodson tested a cascade of high load LP turbines with a solid thin blade design. Their results show that the pressure surface separation bubble is still evident at +10° incidence, while under the negative incidence the separation bubble can occupy about 65% of the chord length [184], as shown in Fig. 4.67. To discuss the effect of the bubble on the endwall region, Brear and Hodson compared and analyzed the endwall flow of blade (blade A) and the other three blades which have same load and same suction side shape but different pressure surfaces (blade B, C and D) [184]. The geometry of those four blades is shown in Fig. 4.68a. Blade D, like blade A, is uniform in spanwise, while the change of profile is the pressure surface which is a significant thickening of the blade, i.e., a thick blade design. Blade B and C are non-uniform in spanwise, but has the same profile as blade A at midspan location. Blade C is the same as blade D at the root while blade B is slightly thinner comparing to blade D at the root. The limiting streamlines on the pressure side of the four blades are given in Fig. 4.68b. It can be seen that the blade D removes the separation bubble from the pressure surface side of entire height, while the blade C can eliminate separation in the endwall, but the separation is still present in the middle of the blade. Blade B cannot eliminate the separation bubble from root, but only to a certain extent, weaken the separation. In terms of the secondary flow, the flow visualization results on the suction surface of the blade A and the blade D are given in Fig. 4.68c. The results show that the formation of the passage vortex in blade D endwall region moves to about 10% of the arc length downstream when compared to blade A, and its spanwise spread near the trailing edge is also significantly reduced. These differences are mostly due to the decreasing of the separation bubble on the pressure surface of blade D when compared with blade A. For blade D, the passage vortex is formed mainly by the fluid coming from the endwall boundary layer and the boundary layer attached to the pressure surface , which is the same as HP turbine . The formation of passage vortex will be affected by the presence of separation bubble on blade A pressure surface. Figure 4.68d shows the three-dimensional streamlines near the endwall on blade A. Due to the pressure surface flow separation, the low-energy fluid in the separation bubble cannot resist the transverse pressure gradient in the passage, causing the fluid transport toward the suction side. The results in Fig. 4.68d show that the fluid within the separation bubble flows upstream and mix into the boundary layer near the endwall after leaving the pressure surface from near the reattachment point, which is due to the interaction between streamwise adverse pressure gradient in main flow near pressure surface and the pressure side leg of the horseshoe vortex. Those fluids which are from the vicinity of the pressure surface will eventually reach near the suction face and get rolled into passage vortex. Since the momentum of the fluid in the separation bubble is much lower than that in the boundary layer but contains much more fluid than the boundary layer, the vortex in the endwall region will be significantly enhanced once the pressure surface is separating. It can be seen that the key to determine the structure and strength of the secondary flow near the endwall region is the low energy containt fluid and the pressure gradient , and the most direct negative effect of pressure surface separation is the provision of a large number of low energy fluids.

Fig. 4.67

Pressure surface separation in high-loaded LP turbine [184]

Fig. 4.68

LP turbine blades and flow structures at different pressure surfaces [184]

Obviously, Brear and Hodson’s work is about the cascade. The actual pressure gradient inside turbine is more complicated due to bending and other three-dimensional shape of the blade, so the impact of pressure surface separation bubble will be more complex. In addition, the pressure gradients in the radial direction may also be the key factor that influences pressure surface separation bubbles in the rotor blade row. Yamamoto found that the low-energy fluid in the pressure-surface separation bubble would transport towards the blade tip due to radial pressure gradient , and finally leak into the suction side through the gap, which affects the leakage flow near the tip, the passage vortex and other secondary flow structures [183]. This is obviously another way for the pressure-surface separation flow to interact with the endwall secondary flow structure. However, the research on the evolution of pressure surface separation and its interaction with secondary flow in complex pressure gradient environment is still insufficient, and the understanding of this phenomenon needs to be further improved.

In terms of the endwall flow loss, Figure 4.69 shows the distribution of the dissipation coefficient at endwall to 10% span of blade A and D in Fig. 4.68a [184]. Blade A has a large loss in the shear zone of the pressure-separated bubble with the maximum loss appears near the 25% axial chord position, which is the transition position of the separating shear layer. In the blade D, since the pressure-surface separation bubble completely disappears, the corresponding loss does not exist. When focusing on the vicinity of the suction surface , it can be seen that the high loss regions of blade D are obviously reduced and the position is closer to the downstream, which is consistent with the evolution characteristics of passage vortex. The quantitative analysis of the four blades indicated that the flow loss near blade endwall can be reduced by 15% by eliminating the separation bubble on the pressure surface , and the reduction in the flow loss near the pressure and the suction sides can contribute about 80%. This is due to the disappearance of the separation bubble on the pressure side and the weakening of passage vortex in the vicinity of suction surface .

Fig. 4.69

Comparison of flow loss near endwall regions in a high loaded LP turbine [184]

4.4.3 Endwall Boundary Layer Evolution Mechanism and Its Impacts

The development of LP turbine endwall boundary layer is also a key factor to determine the flow structure in endwall region. It has been shown that near the turbine endwall mainly exist the boundary layer of upstream flow and the newly formed boundary layer in the passage. These boundary layers are generally dominated by laminar flow or the boundary layers in the transition process [185, 186]. For LP turbines, Vera et al. [187] measured the flow structure and evolution of the endwall boundary layer in a LP turbine cascade and the second and fourth stage stator cascade in one four-stage LP turbine using hot-film. Figure 4.70 shows the arrangement of the hot-film measuring points on the endwall near the leading edge of the fourth stage guide vane and the corresponding results. The experimental results show that the Reynolds number is 8.6 × 10⁴ based on measurement condition and axial chord. The original signal and the ensemble-averaged results of the wall shear stress are shown in the figure. Here, measurement points S00 and S01 are located in the sealed cavity. The amplitude of the averaged signal is small and not periodic, which indicates that the development of the boundary layer is dominated by the sealing flow, which has less impact from upstream flow. The measuring points S1, S2 and S3 are located on the endwall of the turbine channel, and the ensemble-averaged results of these three points show obvious periodicity due to the periodic passing upstream rotor row. The pulse of the instantaneous signal at the measuring point S1 is not symmetrical about the ensemble average signal, but exhibits an occasional spike. These spikes correspond to turbulent burst in the boundary layer, indicating that the boundary layer is at the beginning of the transition process. Comparing the signals S1, S2 and S3, it can be found that the large-scale pulsations gradually evolve into high-frequency and low-intensity pulsations, which is typical of the boundary layer transition process. It should be noted out that, S3 still measures significant pulsation, which indicates the transition process of the boundary layer has not been completed, where the boundary layer is still not completely turbulent state.

Fig. 4.70

Development of LPT inlet endwall boundary layer [187]

Figure 4.71 shows the hot-film measurement results of pressure side endwall boundary layer [187]. It should be noted that the measurement results shows no presence of separation bubbles on the pressure side. The pressure surface branch of the horseshoe vortex rolls up the boundary layer of the endwall during its transportation towards to suction surface , causes it to lift away the wall surface, then re-establishes new boundary layer downstream of the endwall. The newly formed boundary layer will re-evolve under the streamwise and transverse pressure gradient . Measurement points S4-S7 located downstream of horseshoe vortex pressure surface branch, which is the interior of the new boundary layer. Results show that the newly developed boundary layer exhibits obvious laminar flow characteristics near the pressure surface , and its evolution to the suction surface side is accompanied by flow transition . In addition, in the region close to the pressure surface . The passing frequency of the upstream rotor was found obvious in the ensemble average results, which indicates that the flow will be affected stongly by the upstream blade row. With the development of the flow, the passing frequency of the upstream blade is weakened in the boundary layer, and disappear in the middle of the passage. This phenomenon is as well the results of strengthen of the horseshoe vortex pressure surface branch which causes the lifting of the fluid near endwall, thus makes the impact of the inflow away from the endwall . For the suction side, the inflow boundary layer near the leading edge is affected by the suction side leg of the horseshoe vortex. If the inflow boundary layer is laminar, it may induce the boundary layer to transit. The boundary layer near the trailing edge, however, is mainly from the new formed boundary layer which was caused by the pressure side leg of the horseshoe vortex and has developed from upstream and the pressure surface. This part of boundary layer usually shows the similar characteristics to fully developed turbulence.

Fig. 4.71

Development of the boundary layer on the pressure side endwall [187]

Because the state and thickness of the endwall boundary layer near the leading edge of the blade are largely dependent on the incoming boundary layer, while the endwall boundary layer of the leading edge endwall determines the rolling position and strength of the horseshoe vortex, so the endwall inflow boundary layer will have a significant impact on the flow of the turbine endwall. De la Rosa Blanco and Hodson studied the flow structures and losses of the LPT cascade under different endwall incoming boundary layer conditions [188]. The research subjects are two blades, one of which is thick while the other is thin. Three kinds of inflow boundary layers are given, including a “laminar boundary layer” with shape factor of 2.18, a “turbulent boundary layer” with shape factor of 1.64 and a “thick turbulent boundary layer” with shape factor of 1.44. Similar to the results given in Ref. [184], there is no separation bubble on the pressure surface of the thick blade, whereas there is a wide range of pressure surface separation in the thin blade. In addition, when the endwall inflow boundary layer is turbulent, the pressure surface separation bubble transports apparently towards the endwall , and the root cause of which is the interaction between the pressure side branch of the horseshoe vortex and the separation bubble. When the inflow boundary layer is laminar, the radial transportation phenomenon does not occur because the rolling position and the evolution trajectory of the pressure surface branch of the horseshoe vortex are more deviated from the pressure surface . Figure 4.72 shows cascade endwall flow visualization results under different conditions. In the figure, the main flow structures such as the saddle point P₂ caused by the rolling up of the horseshoe vortex, the pressure surface branch of the horseshoe vortex S_e1, the position P₁ where the horseshoe vortex pressure surface striking the suction surface , and the corner vortex CV near the trailing edge of the suction surface are shown. It can be seen that the saddle point P₂ is closer to the suction surface under the condition that the endwall incoming boundary layer is laminar, regardless of whether it is a thick blade or a thin blade. Correspondingly, the position of the horseshoe vortex pressure surface branch S_e1 is closer to the suction surface and away from the pressure surface , which results that there is no direct interaction between the pressure surface branching S_e1 of the horseshoe vortices and the pressure surface separation. In addition, flow visulization results also show that in the case the endwall incoming boundary layer is turbulent, another separation line S_e2 can be observed near the pressure side. In the thin blade cascade, this separation line is more obvious due to the interaction between the pressure surface branch of the horseshoe vortex and the pressure side separation bubble. In the case of corner vortex, its evolution trajectory also shows a significant correlation with the endwall incoming boundary layer flow. No matter the thick blade or the thin blade, the corner vortex under the turbulent boundary layer is more obvious.

Fig. 4.72

Flow visulization of LPT endwalls with different inlet endwall boundary conditions [188]

In terms of flow losses, Fig. 4.73 compares total pressure loss coefficient distribution for the 50% axial chord length section downstream of the cascade. The results show that the influence of the state of endwall incoming boundary layer on the total pressure loss distribution is more obvious, while the influence due to blade thickness is relatively small. When the inflow boundary layer is turbulent, there is a loss kernel/core in the turbine endwall region, and the maximum total pressure loss locates in the center of the core. When the incoming boundary layer is laminar, there are two loss cores appear in the endwall region, but the maximum value of the total pressure loss is less and its position is radially close to endwall. The difference on loss distribution characteristics is caused by the difference on secondary flow evolution mechanisms, which, at the root, is due to the different states of endwall incoming boundary layers. For laminar boundary layer, the passage vortex near endwall region is relatively weak. Then the mixing with the trailing edge shedding vortex after leaving the cascade passage is also weak. Therefore, two vortices can be observed at the 50% axial chord length downstream of the cascade, corresponding to two loss cores. For the incoming turbulent boundary layer, since the passage vortex is very strong, it has strong mixing process with trailing edge shedding vortex. The trailing edge shedding vortex is not observed after mixing, so there is only one vortex downstream of cascade, but with more obvious loss. Through the analysis of the total flow loss of the cascade, the flow loss increases with the endwall incoming turbulent boundary layer when compared to laminar one, no matter thick or thin blades. And with increased turbulent boundary layer thickness, this loss increases more obvious. Comparing the effects due to blade thickness, under the incoming turbulent boundary layer, the aerodynamic performance of the thin blade significantly decreases. When compared to the thick blade this is mainly due to the interaction between the pressure side branch of the horseshoe vortex and the pressure side separation, so that more low-energy fluid is involved in the passage vortex, so that the passage vortex becomes strong and the loss increases. However, under the incoming laminar boundary layer, the aerodynamic characteristics of the thin blade have no obvious change compared to the thick blade. This is because there is no strong interaction between the branch of the horseshoe vortex and the pressure side separation of the thin blades.

Fig. 4.73

Total pressure loss distributions at the cascade outlet [188]

For the development of LPT blade boundary layer, Reynolds number has a very important effect. Results show that Reynolds number is also important for the secondary flow near endwall region. With Reynolds number increases, the blockage effect due to the secondary flow in the endwall region decreases [189], which is the reason why turbine aerodynamic performance increases with Reynolds number. If there is boundary layer separation near the blade endwall, the change of the Reynolds number may change the characteristics of these separated flows, leading to obvious change of the secondary flow near blade endwall [190]. If there is no boundary layer separation, the transition process is affected with changing of the Reynolds number, but the boundary layer flow structure is not affected by that. At high Reynolds number, the transition process accelerates no matter for the incoming boundary layer near the leading edge or the new formed boundary layer in the passage. Therefore, the boundary layer near the leading edge and near the suction surface is closer to the turbulent state. Then, since the state and thickness of the inflow boundary layer will directly influence the position and strength of the horseshoe vortex while the new formed boundary layer affects the amount of fluid entrained in the passage vortex, and intensity of the whole secondary flow in the endwall region will be affected. This is the main influence mechanism of Reynolds number on flow near the endwall region in this circumstance.

4.4.4 LP Turbine Endwall Flow Under Unsteady Conditions

The LP turbine stage is working in the unsteady flow conditions. The upstream periodic wake and other unsteady effects will not be only limited to the blade boundary layer, but will also affect the evolution of the endwall flow structure. Although the current understanding of unsteady flow conditions on the HP turbine endwall region is relatively comprehensive, the research on the unsteady evolution mechanism of flow structures in LP turbine endwall region is less and the understanding is not comprehensive enough. It is gratifying that in recent years this problem has aroused the concerns by more researchers with corresponding works.

Schneider from ILA cooperated with MTU made experimental measurements and numerical simulation to investigate the endwall secondary flow in a two-stage LP turbine . This two-stage turbine test rig was used to simulate the third and fourth stages of typical LPT, and the Mach number and Reynolds number were consistent with the actual components [191]. Figure 4.74 shows the flow structure near the endwall in the first-stage rotor under two different phases in a cycle. Here uses contour plot of streamwise vorticity to identify the vortex. The results show that the passage vortex and the wake from upstream guide vanes will have strong unsteady interaction after entering into the downstream blade rows. The main process can be summarized as two stages. Firstly, the shear layer from the upstream wake become unstable in the downstream channel and is cut into independent segments in the spanwise direction. These segments are then rolled up in the channel and gradually formed into streamwsie vortex structure. In the diagram, two streamwise vortices, which rolled up by upstream wake shear layer, are respectively identified as TSSL_{up, h} for the one further to the endwall and TSSL_{low, h} for the closer one. Between these two streamwise vortexes are the passage vortex PV_h from upstream blade rows. This PV_h plays an important role in the instability and the subsequent evolution of the upstream wake shear layer. Just because the blockage of the PV_h, the part of the upstream wake shear layer which is near the endwall will not to transport towards blade suction side but is stretched and eventually cut there. On the other hand, the induction of PV_h is also an important reason for the upstream shear layer to roll up and evolve the streamwise vortices. At 25/16T, the passage vortex rPV_h in the rotor blade passage is observed near the trailing edge in the endwall region. The direction of rotation is the same as that of TSSL_{low, h}, which is rolled up by the upstream wake shear layer near the endwall , and the two vortexes appear very close to each other, so they are possible to merge in the evolution process and enhance the strength. Note that the passage vortex rPV_h cannot be clearly identified at 12/16T, while at the same time the upstream wake is acting near the trailing edge of the suction surface. That is to say, the passage vortex rPV_h is likely to merge with the streamwise vortex TSSL_{low, h}, which indicates that the generation and evolution of passage vortex under the unsteady condition will be significantly affected by the unsteady upstream flow condition. Figure 4.75 shows the measured circumferential absolute flow angle changing over time with a dual-filament hot-film. The data are processed using the ensemble average method. It can be seen that the vortices causes the rotor outlet flow angle showing significant over and under turning near the endwall region (corresponding to the white and black areas, respectively). The main vortex pair also results in a large fluctuation of the rotor outlet flow angle with fluctuation amplitude can be as high as ±10°. These unsteady fluctuations affect not only the local blade row but also cause the incidence loss downstream blade rows undergoing obvious fluctuation.

Fig. 4.74

Endwall flow structure of a LPT with unsteady wakes [191]

Fig. 4.75

Ensemble average of circumferential absolute flow angle time series [191]

It can be seen that the LPT endwall flow under the turbine stage environment is significantly different from that under steady flow condition. The vortices generated from the upstream flow field are still strong in the blade passage, while the passage vortex of the blade itself is weakened under the influence of the inflow vortex structures. Similar results have been obtained from unsteady experiments and numerical simulations of the T106 turbine cascade [192]. The discussion of the wake passing frequency on the T106 turbine cascade shows that the intensity of the secondary vortices in the endwall region decreases with the wake frequency increasing. Correspondingly, the radial distribution of the outlet flow angle of the cascade is also more uniform. It should be noted, however, that the flow coefficient and reduced frequency of the wake in this study are different from those of typical LP turbines, which may affect the propagation characteristics of the wake in the cascade and may affect its interaction with the flow structures near endwall .

Due to the significant changes in the flow structures in LPT stage environment, the flow loss in the endwall region is also changed accordingly. The flow loss caused by the inflow vortex structures will play an important role. Also the interaction between the upstream wakes and the upstream passage vortices near the endwall region, the influence of the incoming vortex structures on endwall boundary layer, and the secondary flow structures, and other unsteady interactions will all have profound impacts on the flow loss in the LPT endwall region. Although the flow evolution and loss mechanism of the LPT endwall region under unsteady conditions have attracted more and more attention, until now, the research on this problem is not comprehensive and deep. In particular, there is a lack of targeted research that can fully consider the geometrical and aerodynamic characteristics as well as the working environment of high/ultrahigh load LPT. In addition, due to the complex multi-scale vortex structures and turbulent mixing in the endwall flow, experimental methods or RANS numerical simulation are difficult to give sufficiently fine flow structures and their evolution law, which seriously restricts the understanding of turbine endwall flow. With the development of computer technology and hardware, the high-precision numerical methods such as DES, LES and even DNS have been applied in the study of three-dimensional flow in turbomachinery [193,194,195]. It also provides a powerful tool for further study the flow structures of the LPT endwall region in the unsteady environment [196].

4.5 Low Reynolds Number Effects in LP Turbines

Reynolds number, which represents the ratio of typical inertial forces to typical viscous forces, is the key dimensionless parameter that determines the flow condition. Researches have been shown that when the Reynolds number is lower than a certain value, the amplitude of the perturbation will gradually decrease and eventually disappear, and the flow field will return to the state before the disturbance. It means flow is stable that it will always maintain laminar flow state without transition . If the Reynolds number is high, the disturbance may gradually increase or always exist, so that the flow does not return to the state before disturbed, the flow is unstable, and will eventually induce laminar flow to transit to turbulent flow. For typical tubomachinary, since the Reynolds number is sufficiently high (mostly above the self-similarity Reynolds number), the boundary layer is considered to be turbulent, regardless laminar condition and transition processes. It brings grate convenience to the design and numerical simulation. However, this may not always the case in reality. Such as aircraft engines in the high-altitude state lower inflow density may lead LPT characteristic Reynolds number to be much lower than the self-similarity Reynolds number. This situation is more obvious in the final stage of LPT in small-scale engines that used in high-altitude unmanned aerial vehicles. Because of low Reynolds number , the blade boundary layer will remain laminar in a large range. The transition , separation and reattachment characteristics of the laminar boundary layer are the key factors that determine the flow characteristics of the LPT. The deeply understanding of these effects is the key for LPT aerodynamic design under low Reynolds number conditions.

4.5.1 Effects of Reynolds Number on the LP Turbine Aerodynamic Performance

Ever since 1989, Hourmouziadis analyzed the flow pattern and the flow losses of the LP turbine blade suction surface boundary layer under different Reynolds numbers [197], as shown in Fig. 4.76. In the figure, red color represents the separation loss and the blue color represents the friction loss. It as can be seen that at high Reynolds number, the boundary layer is turbulent. Although the friction loss is higher than laminar flow, the resistance to separation is stronger. Only when the trailing edge adverse pressure gradient increases to a certain extent, the turbulent separation occurs and the corresponding loss increases. When the blades are operated at very low Reynolds number , the boundary layer maintains the laminar flow state. Although the friction loss is small, its anti-separation ability is weak and the large scale laminar flow separation is prone to occur under the adverse pressure gradient , leading to a sharp increase in separation loss. Researches show that the laminar boundary layer thickness increases with the decrease of Reynolds number [198, 199]. The thicker the boundary layer before separation, the larger the separation scale and the greater the separation loss is. As the boundary layer loss dominates in the LP turbine , how to control the LP turbine blade suction surface boundary layer development in low Reynolds number conditions has become a research focus.

Fig. 4.76

Effects of Reynolds number on suction side boundary layer and losses in LP turbines [197]

When the Reynolds number is lower than self-similarity range, the aerodynamic performance like the flow capacity and the aerodynamic efficiency , are affected significantly by the Reynolds number. For typical high-thrust civil large-bypass ratio turbofan engines, studies have shown that LP turbine efficiency may drop by 2% from takeoff to cruise altitude [200]. For smaller aircraft engines, the low Reynolds number effect will be more obvious. Figure 4.77 shows the plot of the LPT efficiency versus the Reynolds number in PW545 engine. Both the experimental and the numerical simulations show that when the Reynolds number is lower than the self-similarity range, the efficiency decreases with the decrease of the Reynolds number, and this trend is nonlinear, that is, in the low Reynolds number range the drop is more obvious. The lower left corner of the map corresponds to 18,288 m above sea level working Reynolds number. We can see that efficiency decreased approximately by 6% [201]. A similar situation also exists in the “Global Hawk” high altitude unmanned surveillance aircraft engine AE3007H. Data show that during high altitude 19,800 m cruise operation, the engine’s fuel consumption significant increases due to degradation of LP turbine performance [202].

Fig. 4.77

Reynolds number effects on PW545 LP turbine efficiency [201]

MTU and the University of Stuttgart ILA laboratory investigated the low Reynolds number impact in detail on a two-stage LPT test rig and got similar impact of Reynolds number [203]. Based on this work, researchers further optimized the turbine blades and endwall of the test rig. The aerodynamic performance of the LP turbine in the design and off-design state was measured with the change of Reynolds number. In the off-design condition, the expansion ratio and rotational speed are reduced by 21% and 60%, respectively, compared with the design state, in order to realize the high attack angle in off-design case [204]. Figure 4.78 shows the mass flow rate and component isentropic efficiency changing with Reynolds number. All the data are non-dimensionalized under the corresponding conditions $\text{Re}_{v1} = 1.8 \times 10^{5}$. It should be noted that, in order to compare between different conditions, the Reynolds number is defined with the characteristic parameters of the first stage guide vane. In terms of flow capacity, test results show that the flow of turbine components operating in both the design and off-design states decreased by approximately 2.9 and 3.7%, respectively, over the tested Reynolds number range. It needs to be noted that the decline in the flow capacity shows obviously with Reynolds number lower than $\text{Re}_{v1} = 0.75 \times 10^{5}$. While in the range of the Reynolds number above this value, the variation of the flow capacity of turbine parts is basically the same in both design state and off-design state. The measurement results show that since the flow angle of the first stage turbine guide vanes does not change, its flow capacity varying with Reynolds number is basically consistent in different working conditions. But the second stage guide vane has a positive angle of attack of about 18° in off-design condition, which results in a greater separation at low Reynolds number conditions, and thus a greater reduction in the flow capacity of entire turbine component. In terms of aerodynamic efficiency , experimental results show that the isentropic efficiency of the turbine components drop approximately 3.9% while working in the design state at low Reynolds number . But the component efficiency has consistent trend with the changing of Reynolds number under off-design condition, though the decrease is even larger, reaching 4.6%. The authors believe that the main reason for the efficiency change is that the second-stage guide vanes are working at a large angle of attack, and the evolution of the boundary layer and the secondary flow structures in endwall region are more sensitive to the decrease of the Reynolds number.

Fig. 4.78

The flow capacity and aerodynamic efficiency changing with Reynolds number in a two-stage turbine [204]

The above research shows that it has great importance to study the internal flow structures and aerodynamic performance changing under low Reynolds number condition within LP turbine. It is the key to predict the aerodynamic performance of low-pressure turbine components under different working conditions. On the basis of this, the development of effective flow control methodologies will be an important way to improve the performance of LPT and even the whole engine under low Reynolds number condition. In order to consider low Reynolds number impact on the performance in design phase, researchers provided aerodynamic performance correction for turbine components under low-Reynolds number conditions [205]. This correction method relies on large number of experimental data for empirical correlation, and has good applicability in a certain range. But it is worth noting that with the development of design method considering low-Reynolds number, the sensitivity of LP turbine aerodynamic performance to Reynolds number is no longer as serious as before. Thus the old correlation is not fully applicable. Thus we need to be careful [190].

4.5.2 LP Turbine Internal Flow Under Low Reynolds Number Conditions

Reynolds number and loading distribution, which are the key parameters to determine the state of the boundary layer, have important influence on the flow diagram in the turbine . Figure 4.79 compares the suction surface pressure distribution and the surface hot-film data of the $Z_{w} = 1.08$ Pack-B low-pressure turbine cascade with different background turbulence conditions in $\text{Re} = 0.5 \times 10^{5}$ and $\text{Re} = 1.0 \times 10^{5}$ operation conditions [206]. At higher Reynolds number ($\text{Re} = 1.0 \times 10^{5}$), the separation bubble appears near the trailing edge of the turbine blade. The influence of the turbulence on the flow structure of the suction surface is limited and the size of the separation bubble does not change. When the Reynolds number is reduced to $\text{Re} = 0.5 \times 10^{5}$ working condition, the size of the separation bubble near the trailing edge of the cascade increases obviously, and so do the transition position, which resulting in the re-attachment location moves downstream. In particular, when the FSTI is 0.4%, the separation of the trailing edge of the cascade will no longer re-attach but open, and the separation position moves obviously advanced. The flow loss of the Pack-B turbine cascade under different background turbulence conditions is also affected by the changes in the flow structure as shown in Fig. 4.80. In general, the effect of Reynolds number shows mainly in two aspects. On one hand, as the Reynolds number decreases, the separation bubble increases, thus the separation loss increases. On the other hand, the decrease of the Reynolds number can reduce the wetting area of the turbulent boundary layer and reduce the friction loss. For the low-pressure turbine with low Reynolds number , generally the separation loss dominates, so the decrease of the Reynolds number leads to monotonous increased profile loss . As shown in Fig. 4.80, a decrease in the Reynolds number results in an open separation on the trailing edge of the Pack-B leaves under low FSTI, which results in a sharp increase in blade loss. It means that at low Reynolds number , profile loss is more sensitive to background turbulence.

Fig. 4.79

Suction surface loading and hot-film data under different Reynolds numbers, Pack-B cascade [206]

Fig. 4.80

Effects of Reynolds Number on Pack-B cascade loss [206]

As the blade loading increases, the risk of large-scale separation of the turbine cascade under low Reynolds number conditions will increase. Figure 4.81 shows a flow diagram of the turbine cascade under different Reynolds numbers [59, 60]. Experimental results and numerical simulations show that, at low turbulence (0.8%), after the Reynolds number is reduced to 1.0 × 10⁵, the small-size separation bubble in the boundary layer of the cascade suction surface is replaced by open separation. The profile aerodynamic loading decreased and the total pressure loss increased significantly. The boundary layer separation of the suction surface is suppressed at higher turbulence.

Fig. 4.81

Flow structures of high-loaded turbine cascade under different operating conditions [59, 60]

In the unsteady condition, the periodical sweeping wakes from upstream can effectively restrain the suction surface boundary layer separation, and the influence from wake will be affected by the Reynolds number. Figure 4.82 shows the temporal-special evolution of the boundary layer of the Pack-B cascade suction surface with $\text{Re} = 0.5 \times 10^{5}$ and $\text{Re} = 1.0 \times 10^{5}$ two Reynolds numbers [207]. The time-averaged separation and reattachment lines are shown, as well as transient separation zone, wake induced turbulence zone, and calmed region. It can be seen that the transition in wake -induced separation occurred later at the lower Reynolds number (corresponding to region C). So the separation bubble size is overall larger, which is consistent with the trend under steady-state conditions. Downstream of the separation bubble, both for the wake -affected period and the gap between wakes, the areas with high shear stress values expanded obviously as the Reynolds number increased. Correspondingly, the skewness factor values of the shear stress appear in a greater range with negative and near zero values downstream of the separation bubble. This indicates that the boundary layer either in last stage of transition or in full turbulence is dominant near the cascade trailing edge under high Reynolds number condition, and the turbulent boundary layer is mainly in the wake period while the one in end phase of transition mainly exists between wakes. In contrast, at low Reynolds number , the skewness factor of the shear stress near the trailing edge of the cascade is positive in a large area, indicating that the transition process has not yet complete.

Fig. 4.82

Suction side boundary layer spatial-temporal evolution of the Pack-B cascade under different Reynolds numbers [207]

Researches show that the decrease of the Reynolds number for low pressure turbine components at high altitude not only makes the laminar flow in a larger area on the blade surface, but also increases the intensity and the area of the adverse pressure gradient in the blade loading distribution. The influence of these two factors drastically reduces the efficiency of low-pressure turbines at high altitudes [208, 209]. The effect of the low Reynolds number on turbine blade components is not only limited to profile loss , but also the development of the secondary flow in endwall region [210], which is also an important manifestation of the low Reynolds number effect. Figure 4.83 shows flow field of one LPT rotor channel with corresponding working heights of 11 and 20 km [190]. It can be seen clearly from the comparison of the limited streamlines and the pressure contour on blade surface that the boundary layer of most of the blade height, changing from trailing edge separation bubble to un-reattached separation condition, when the working height of the engine from 11 to 20 km. At the same time, the small separation bubbles near the leading edge of the pressure surface are also significantly enlarged under the harsh conditions of high altitude, which has been able to affect more than half of the pressure surface at 20 km altitude. Correspondingly, the secondary flow in the endwall region of this cascade is obviously enhanced also the influence range is enlarged, which causes the related flow loss increased rapidly.

Fig. 4.83

Flow field of a LPT cruising at different altitudes [190]

MTU and ILA laboratory in Stuttgart University measured the flow in the second stage stator passage under different Reynolds number on its two-stage low-pressure turbine test rig [203]. The distribution of the cascade efficiency is shown in Fig. 4.84 [211], with both experimental and numerical results. From the distribution of the cascade efficiency, it can be distinguish the losses induced by wake of the cascade and the endwall secondary flow . Comparing the two Reynolds number cases it shows that the wake widened obviously at low Reynolds number and the efficiency of the cascade in the wake region also decreased obviously, which is consistent with the increase of the boundary layer loss of the suction surface in low Reynolds number as mentioned before. To further quantify the width and strength of the vane wakes, which are affected by Reynolds number, Fig. 4.85, which shows the change in wake thickness ($\delta_{1}$) with the Reynolds number [211]. Here, the displacement thickness is defined as:

Fig. 4.84

Contour plot of blade row efficiency at exit of vane 2 under different Reynolds numbers [211]

Fig. 4.85

Relative displacement thickness of wake and loss cores at different Reynolds numbers at about 41% relative annulus height [211]

$$ \delta_{1,wake} = \int\limits_{0}^{t} {(1 - \frac{{V_{2} }}{{V_{2,is} }})dt} $$

where $V_{2}$ and $V_{2,is}$ represent the outlet speed and the ideal speed at cascade outlet, respectively. $t$ is the pitch. It can be seen from the definition that the parameter represents the decrease of the flow capacity due to the loss of the wake velocity and has similar physical meaning to the velocity loss coefficient of the blade. The data given in Fig. 4.85 are data near the midspan (about 41% blade height) and are non-dimensional using the results of the Reynolds number $\text{Re}_{v2} = 0.88 \times 10^{5}$. The results show that when the Reynolds number decreases from $\text{Re}_{v2} = 0.88 \times 10^{5}$ to $\text{Re}_{v2} = 0.67 \times 10^{5}$ and $\text{Re}_{v2} = 0.35 \times 10^{5}$, the flow loss near the vane increases by about 8 and 40%, respectively. The results also show that the loss of the blade loss is larger at lower Reynolds number. In terms of the secondary flow near endwall, the position of the loss core with respect to the Reynolds number change is not obvious in the experimental results and the numerical simulation results given in Fig. 4.84 (although the numerical simulation results differ slightly from the experimental results). However, those loss core regions are significantly increased, which is due to the enhanced secondary flow in endwall under low Reynolds number conditions. In addition, a high-Reynolds number of conditions at about 57% of blade height can be observed a decreased cascade efficiency region. This is mainly due to the secondary flow in the first-stage vane passage, but it vanishes due to strong mixing in the channel at lower Reynolds number condition.