Effect of Geometry and Heat Flux on Turbine Over-Tip Flow Power Extraction Open Access

Over-tip leakage flow is considered one of the most relevant sources of secondary flow losses [1]. Ameri et al. [2] found that up to 2% points efficiency penalty is caused by tip leakage flow with a relative tip gap size of 1.5%. The efficiency penalty was found to correlate linearly with the tip gap height. The over-tip flow is characterized by two main drivers, the pressure differential between the suction side (SS) and pressure side (PS) as well as the relative motion between the blade and the shroud [3,4]. With tip gap sizes of around 1%, it is observed that flow enters at the leading edge (LE) and on the PS following the pressure gradient, separates [5], and passes with high speed through the tip gap [6] in the same direction as the rotor motion. Then the pressure-driven flow enters the flow passage again on the SS to form the tip leakage vortex and mixes with the surrounding fluid in the passage. This process is identified to cause viscous entropy generation [7] in addition to a penalty in flow turning and shaft power extraction. Further, the relative motion of the shroud “scrapes” fluid against the direction of rotor motion. Due to viscous effects and the opposed direction of both driving mechanisms, they work against each other depending on the operating point, tip gap height, and rotational speed of the rotor [8–11].

The complex aerodynamics impact directly the heat transfer in the tip gap [12,13]. Also, an inverse impact of the heat transfer on the flow field was found in transonic tip gap flows [14]. The found separation bubbles and flow reattachment within the tip gap cause zones of increased heat transfer [14–18] on the blade tip. The scraped flow was found to create another source of high heat transfer at the shroud [6]. Hence, the complex tip flow structures also impact the aerothermal performance due to their impact on heat transfer adding more complexity to the individual efficiency contributions.

For very tight clearances close to 0.1% of the passage height, the flow topology changes significantly showing a clear reduction in pressure-driven flow across the tip up to the disappearance of the tip gap vortex as predicted by computational fluid dynamics (CFD) simulations performed by De Maesschalck et al. [6]. Such a drastic change in flow topology implies significant changes in over-tip heat transfer with a reduction in heat transfer over most of the tip gap surface except at the tip corner at the LE and the SS of the blade where heat transfer increases can be noticed [19].

While the effect of surface heat transfer can be evaluated and quantified in the respective surface area, the evaluation of locally generated losses that lead to a reduction in extracted power is challenging. The authors used efficiency comparisons between different tip gap geometries to compute the impact of loss generation and heat transfer by varying tip gap geometries [2,6,20,21]. Adiabatic efficiency formulations as introduced by Ameri et al. [22] or Atkins and Ainsworth [23] were used to assess the performance. However, this method veils the distinct impacts on losses caused by entropy generation and on the tip leakage flow turning. Additionally, the operating point may change slightly when the generated loss is different. Furthermore, a loss coefficient can be computed and analyzed in streamwise planes [11]. This method may evaluate the loss alone, but the effect of tip leakage flow is hard to isolate from other secondary flow phenomena.

The entropy generation method is widely used in the analysis of CFD results in turbomachinery to quantify the generated entropy in volume elements [24–26]. To identify the losses related to the over-tip flow, the respective volume needs to be integrated. Abel et al. [27] used this method to quantify the entropy generation in predefined regions based on the analyzed geometry. Páty et al. [28] analyzed the entropy generation in the tip gap alone. Other methods to trace the tip leakage flow in CFD results are streamlines [9,11,15]. However, to cover the entire domain relevant for tip leakage flow losses, a large number of streamlines would need to be generated and the final selection to consider the over-tip flow might be arbitrary. Furthermore, the entropy generation is not equal to the loss in power extraction. Hence, the contribution of over-tip flow-driven loss terms and their dependency on heat transfer as well as the tip gap height could not be quantified yet.

Recently, the coupling between the entropy generation with the loss of potentially extractable power from the flow was derived. Entropy generation caused by viscous dissipation is linked with a reduction of possible shaft power extraction, while entropy generation due to internal heat transfer impacts the maximum extractable heat transfer [29,30]. The resulting terms can also be calculated locally, and the volume integration directly results in the contribution of the corresponding aerothermal loss [29].

This article uses a novel approach to trace numerically the tip leakage flow with a passive scalar source term and analyses the volumetric terms of decomposed loss contributors to the aerothermal efficiency as well as the volumetric work extraction of the tip leakage vortex. The impact of several thermal conditions on each term is analyzed. Additionally, the impact of geometrical variations on the tip gap height is investigated and the variability with a change in operating conditions is highlighted. Overall, this article aims to expand the knowledge of the ground-laying physical process behind tip leakage losses and penalties in power extraction.

A single-stage transonic turbine with 43 stator vanes and 64 rotor blades [31] is used as geometry for the presented research. One stator vane and one rotor blade are simulated by applying periodic boundary conditions and a mixing plane. Figure 1(a) depicts the flow path of the simulation domain, and Fig. 1(b) shows the airfoil profiles at midspan. The baseline tip gap is flat and measures 0.56 mm, which corresponds to 1.10% of the passage height at the rotor inlet (TG). Three other tip gap geometries with continuously reduced tip gap heights of 75% (TG = 0.83%), 50% (TG = 0.55%), and 25% (TG = 0.28%) of the nominal tip gap heights were assessed in detail during the performed study.

AUTOGRID5 from CADENCE was used to generate structured meshes. To secure mesh independence, a mesh convergence study was conducted. The thickness of the first cell at the wall was maintained to secure $y+$-values below 1 throughout the entire domain for all meshes.

A mesh with 9.28 million cells in the entire domain offers a difference of 0.11% in thermal efficiency and a difference of less than 0.01% in isentropic efficiency in comparison to the finest simulated mesh [29] and was chosen for the following analysis. Of these, 3.23 million cells belong to the stator mesh, and 6.05 million cells belong to the rotor mesh. All tip gap geometries were discretized with 41 cells in the radial direction. The final meshes for the rotor and stator domain are depicted in Fig. 1(c).

The presented analysis was performed using the commercial second-order 3D Reynolds-averaged Navier–Stokes (RANS) solver cfd++ [32] of Metacomp technologies. The k-ω shear stress transport (SST) turbulence model was applied as the turbulence model, as done in numerous other studies concerning high-pressure turbines [15,16,31]. A Courant–Friedrichs–Lewy (CFL) number of 20 was used in all simulations. The total inlet temperature and total inlet pressure were imposed, and a turbulence intensity of 5% was set as an inflow boundary condition. The static pressure at midspan, in combination with radial equilibrium, was imposed at the outlet of the domain. A mixing plane was applied between the stator row and rotor row. The solver was validated with experimental results from the same transonic high-pressure turbine [31] as analyzed in this work and showed excellent agreement with experimental data of isentropic Mach number distribution along the PS and SS [33]. Other studies focusing on over-tip flow and heat transfer validated similar mesh metrics, the used turbulence model, and the use of isothermal walls with experimental data concluding that these over-tip phenomena can be resolved with sufficient precision [18,20,22,34,35]. Additional validation of the shroud over-tip heat transfer against data from a rotating rig [36] using cfd++ is presented in Appendix  A.

To identify the tip leakage flow, the tip gap blocks were treated as a separate zone inside the turbine rotor domain. In this zone, the solver's capability of defining a passive scalar source term was used. An infinite generation rate was implemented so that any flow that enters the tip gap is assigned a passive scalar value of 1. As the over-tip flow mixes with the main flow, the passive scalar value decreases according to the mixing ratio. This way the volume elements containing tip flow relevant for the overall loss generation are identifiable within the rotor domain. The respective over-tip flow concentration is called $ctip$⁠.

For studying the isolated effect of heat transfer, four thermal conditions including adiabatic wall and total temperature inlet-to-wall temperature ratios (TRs) of 1.5, 2.0, and 3.0 were simulated at three inlet temperature levels using the nominal tip gap height. The total-to-static pressure ratio was 3.06, and the reduced rotational speed with $N/Tt,in$ was kept constant at $344.5rpm/K$ for all simulated cases. A summary of the boundary conditions of the performed study is represented in Table 1.

As the concentration $ctip$ allows the identification of the volume elements related to the over-tip flow, volume-based metrics are required to assess work extraction and loss generation.

To validate the method of volumetric power extraction against traditional methods, Eqs. (2) and (3) are integrated over the entire turbine volume and surface, respectively. Table 2 lists the results of the performance parameter comparison. The difference between the area-integrated shaft power $W˙shaft,area,turbine$ and the volume-integrated $W˙shaft,vol,turbine$ is below 0.01% for all observed cases.

From the derivation of the aerothermal efficiency $ηtherm$⁠, a comparison with the isentropic efficiency is only tangible at adiabatic conditions due to inherent limitations of the isentropic efficiency and its foundation in one-dimensionalized equations [29]. This comparison shows that both values are close with deviations below 0.3% points for all investigated temperature levels.

Figure 2(a) shows an isosurface of 25% tip leakage flow concentration $ctip$⁠. Two tip leakage flow vortices can be clearly identified using this method. The first vortex originates close to the LE and immediately moves to lower-span heights. Tallman and Lakshminarayana [9,11] observed a similar behavior by streamline tracking of flow passing the tip gap close to the LE of the blade. Between the first and the second tip leakage vortex, another vortex core can be identified. This vortex is the counterrotating tip horseshoe vortex, which causes the first tip leakage vortex to move downward and suppresses the formation of the second tip vortex in the passage. Newly inflowing over-tip flow stays attached to the SS and feeds the first vortex. As the horseshoe vortex moves to lower-span heights, a second tip leakage vortex forms at around 50% of the chord length.

Additionally, the flow scraped by the shroud friction moving from SS to PS is visible. Over the first half of the chord length, high amounts of scraped flow can be seen. The amount of scraped tip flow reduces rapidly at around half the chord length. The sudden reduction of scrapped flow occurs at the same chord length as the entrance of the second tip leakage vortex.

In Figs. 2(c)–2(e), losses and work extraction are depicted at streamwise passage cross sections. Plane I is located where the tip leakage vortex starts to move noticeably away from the shroud down along the SS of the blade. Overall, both losses show high local loss generation close to the walls. The normal extension of the loss generation at the blade surfaces grows along the chord length due to the boundary layer growth. The tip leakage vortex is one main contributor to the loss system. A local minimum of loss generation can be identified in the cores of both identified vortices in plane IV. Higher losses can be noticed inside the tip gap and where the vortices mix with their surroundings. It is important to note that $lheat$ and $laero$ follow the same qualitative pattern and share the location of local maxima.

Local power extraction is depicted in Fig. 2(e). A positive value at the PS corner at the tip gap indicates where the flow is accelerated into the direction of rotor rotation, implying a penalization of the desired turning of the flow. When the tip leakage flow enters the passage on the SS, the passage flow decelerates the tip cross flow. This causes another redirection of the passage flow and local zones of local positive power close to where the tip flow enters the passage. The positive power that the passage flow experiences by redirecting the tip leakage flow is highlighted in circles. The continuous change of tangential velocity within the tip gap vortex causes the vortex to have zones of high positive but also of high negative power extraction $w˙shaft,vol$⁠.

The highest specific loss generation appears right at the LE, where the first tip leakage flow is about to generate and at around 20–40% chord length due to the scraped flow. However, the tip flow volume is small in these regions. The high specific loss generation between 20% and 40% chord length can be mainly related to the scraped tip flow. Also here, the respective volume of this loss origin is rather low. One other local maximum spans across the blade at 100% tip flow concentration. This is mainly the flow inside the tip gap, which is experiencing the losses due to the blade tip surface and shroud shear layer. Furthermore, two loss streaks can be identified, which are expanding from 100% tip flow concentration up to lower values of $ctip$ with the increasing chord length. The first streak has its origin also at around 20% chord length, while the second originates at 50% chord length coinciding with the penetration of the second tip vortex into the passage. These two loss streaks relate to the shear layer surrounding the first and second tip leakage vortex.

The relative maxima of the volumes are located at higher tip flow concentrations than the streaks of the loss generation marking the tip leakage flow concentrations in the vortex cores. An exponential increase of volume can be noticed at around tip flow concentrations of 10% over the entire chord length. In this work, this is considered as the limit between the over-tip flow responsible for loss generation and surrounding passage flow.

Figure 4(a) depicts the heat transfer on the blade tip surface and the opposing shroud surface for the nominal tip gap size. The surface friction allows us to characterize the flow topology close to the surfaces. At the blade tip close to the LE, a small separation line can be identified. A second separation line can be observed on the PS starting at around 20% chord length and extending up to the trailing edge (TE) caused by a separation bubble to form at the PS [5]. Adjacent to these lines, local heat transfer maxima can be noticed as seen in the literature. With the increasing TR, the flow topology remains unchanged, and the degree of heat transfer increases.

At the shroud, the surface friction indicates that the flow close to the shroud passes directly from the PS to the SS with a slight turning in the axial direction. Also here, the flow topology does not vary with increased TR. At around 30% chord length, a local minimum can be found for all thermal conditions dividing the tip gap and the shroud into two regions of higher-level heat transfer. Integral heat transfer values are depicted in Fig. 4(b) and show that the heat transfer increases with the growing TR. However, the shroud heat transfer increases less than the tip heat transfer.

A comparison of the chordwise loss generation and work extraction for adiabatic and isothermal cases with TR of 1.5, 2.0, and 3.0 at a total inlet temperature of 1800 K is depicted in Fig. 5 for the volume in the tip block and all remaining over-tip flow.

Laminar viscous losses are generated to a similar degree in the tip gap as well as in the tip vortex and scraped flow as shown in Fig. 3(a). The loss generation inside the tip gap is equal for all TR until 20% rel. chord length. Following, higher TRs cause lower laminar viscous loss generation in the tip. This may be due to the reduction of fluid viscosity related to the wall heat transfer.

All loss terms in the vortex and scraped flow have local maxima at around 25% of the chord length. This increase in losses is caused by the scraped flow toward the PS as shown in Fig. 3(a). The laminar aerodynamic loss $Laero,lam,tip$ shows a second maximum at around 50% chord length. This increase is caused by the penetration of the second tip flow in the main passage. The height of this second maxima depends on the level of heat transfer with a reduction of losses with increased heat transfer. In all other locations, this laminar viscous loss does not vary significantly with heat transfer. From Fig. 5(b), it can be noted that this loss causes up to 45% of the overall losses generated between 20% and 75% of the chord length.

The overall turbulent shear-related loss $Laero,turb,tip$ is of the same order of magnitude as the laminar loss. In the tip gap, however, the loss generation is about a third of the laminar loss with an absolute maximum at 25% rel. chord length. The turbulent shear loss contribution in the vortex is significant and increases with the mixing length in the passage. With increasing heat transfer, this loss increases over most of the passage length. Relative to the overall loss production in the passage, this loss is increasingly responsible for the loss production of $Laero,turb$ up to 58% close to the TE as shown in Fig. 5(d).

Laminar and turbulent losses due to heat transfer $Lheat,lam,tip$ and $Lheat,turbtip$ behave similarly being one order of magnitude lower than the viscous losses at a TR of 1.5 and increasing exponentially over the entire chord length with augmenting TR. These losses increase exponentially because Eqs. (9) and (10) include the square of the temperature gradient and the inverse of the temperature. Both loss terms show a sharp increase due to the scraped flow toward the PS. Both are relatively constant over most of the passage length. With average levels of 20% and 15%, they contribute less to the overall generation of $Laero,turb$ in the passage.

To understand the impact of the over-tip flow on the overall machine performance, the integral values over the entire passage are computed for all thermal conditions listed in Table 1 and are depicted in Fig. 6. At adiabatic conditions, the tip flow is responsible for 26.5–29% of the rotor losses. With increasing TR and increasing heat transfer in the tip gap, the contribution to the overall loss generation decreases so that only 21% of the aerothermal loss in the rotor can be accounted to the tip flow with a total inlet temperature of 1800 K and a TR of 2.0. This trend is caused by other heat transfer-related losses in the rotor passage growing more aggressively than the loss due to heat transfer of the over-tip flow.

Figure 6(b) demonstrates the computed values. The caused efficiency penalty is close to 1.2% points for low TRs and augments with the increasing heat transfer up to 1.6% points at a TR of 3.0. Despite this increasing trend with heat transfer in the tip, the aerothermal efficiency tip flow penalty in the adiabatic simulation shows a high penalty of 1.4% points.

Figure 7(b) shows that the impact on the aerothermal efficiency is rather low between 0.25% and 0.45% points for all thermal conditions. Hence, this considerable penalty in power generation of about 3.2% cannot be identified via typical tip leakage loss analysis using the comparison of efficiencies as a change of over-tip losses also changes the turbine mass flow and hence its operating point.

Figure 4(c) shows the surface heat transfer for a constant TR of 1.5 and decreasing tip gap heights. While the flow topology at the shroud remains unchanged, significant changes can be observed at the blade tip. At a TG of 0.85%, the separation line is right at the LE and the PS. Over the first 20% chord length, wall friction indicates flow in a predominantly axial direction. Following, the flow has pressure-driven characteristics as in the nominal case. The integral value of the tip heat flux shown in Fig. 4(d) reduces only slightly and the shroud heat transfer increases slightly. An additional reduction of a TG of 0.55% changes the flow topology such that the flow enters the tip gap from the SS and enters the tip gap against the blade's orientation of rotation. After a chord length of 30%, the pressure-driven flow dominates again. The separation line close to the PS moves even closer to the PS. There is no separation line identifiable at the LE. This flow topology is consistent with the one identified by De Maesschalck et al. [6]. Both integral heat transfer values of the tip surface reduce by 10% relative to the nominal tip gap size and the opposing shroud surface is on the same level as the nominal tip gap heat flux. With an even further reduced tip gap size of 0.28%, the flow enters the tip gap from the SS up to a chord length of 50%. No separation line is indicated by the wall shear stresses. This causes a more significant drop in tip gap heat flux of 36% in the integral heat transfer as the heat transfer maximum was originated by this feature similar to the findings of Lavagnoli et al. [19].

Figure 8 depicts the isosurfaces of the tip flow concentration $ctip$ at 25% for all simulated tip gap heights ranging from 2.2% down to 0.28% of the rotor inlet passage height. Tip gaps ranging from 2.20% to 0.96% show clearly two developed tip vortices. A global trend of diminishing vortex extension can be noticed with a reduction in TG. A clear flow topology change can be noticed when the tip gap height is reduced to 0.83% of the nominal height. The first tip leakage vortex is reduced and stays attached to the SS until 60% of the chord length before rolling up and the second tip leakage vortex is significantly reduced. Also, the scraped flow extends to chord lengths slightly more downstream than in the nominal case. The further reduction in tip gap height to 0.55% causes the first tip leakage vortex to enter the flow passage further downstream. The second tip vortex does not develop changing the flow topology significantly. Also, the scraped flow extends slightly more toward the TE. With an additional reduction of the tip gap height to 0.28%, the topology changes importantly again. The first tip leakage vortex disappears entirely in the visualization. The second tip leakage vortex moves to the SS and remains at the blade surface expanding toward the center of the passage.

Figure 9 depicts the tip flow loss and power generation along the relative chord length for all simulated tip gap heights at a TR of 1.5. In Fig. 9(a), the laminar shear loss generated in the tip gap changes only slightly despite the considerable geometrical changes and the provoked change in topology. The most significant change in the laminar shear loss can be seen for 0.28% TG from 20% chord length up to 90% chord length inside the gap due to the disappearing separation close to the PS. In tip vortex and scraped flow, only minor variations could be found in the laminar shear loss.

The turbulent shear loss depicted in Fig. 9(c) mostly occurs when the tip vortices enter the passage and are highly dependent on the tip gap height. A rather small reduction can be observed reducing the tip gap height from 1.10% to 0.83%. Higher reductions can be observed with additional reductions in TG due to the change in flow topology. This is consistent with the seen reduction in tip vortex extension in Fig. 8. At mid-cord, the contribution of this loss generated in the tip flow to the overall turbulent shear loss ranges from 45% to around 15%.

The laminar loss caused by heat transfer shown in Fig. 9(e) changes only slightly with tip geometry changes. Also here, the reduction to 0.28% TG gives the clearest reduction over the second half of the chord length. The turbulent loss caused by heat transfer increases significantly at around 25% chord length in the maximum caused by the scraped fluid, while this loss decreases over the remaining parts of the passage due to the general reduction in heat transfer.

The integral contribution of the entire tip flow loss to the complete rotor loss is depicted in Fig. 10(a) for all simulated tip gap geometries and a speedline with expansion ratios spanning from 1.5 up to 3.5. At the design point close to an expansion ratio of 3.05, a reduction to a TG of 0.83% reduces the rotor row loss contribution of the tip gap flow from around 25.5% down to 22%. The reduction can be explained by the reduction of the tip leakage vortex and the related reduction in turbulent shear losses. The additional reduction to 0.55% TG shows a comparable large change of approximately 4% points. This is due to the change of topology shown in Fig. 8, causing the reduced size of the first and second tip gap vortex as well as the significant reduction of turbulent shear-related loss highlighted in Fig. 9(c). The final reduction to 0.28% TG causes a significant reduction in loss contribution by 5% points down to 13% of the rotor loss generation. This important reduction can again be explained by a topology change, which causes the lack of the first tip leakage vortex and the reduction of the tip leakage flow. Overall, the tip leakage flow loss contributes between 1.3% and 0.5% points to the stage efficiency penalty for the design point.

It is worth highlighting that the tip geometries are differently sensitive to the operating point. The nominal tip gap shows a reduced loss generation with a reduced expansion ratio. The two intermediate tip heights show a slightly lower decrease of loss generation up to an expansion ratio of 2.5, where loss generation increases again to drop again when the expansion ratio is reduced even further. The tip loss generation with the geometry of 0.28% TG has a predominantly increasing trend toward lower pressure ratios and hence behaves against the trends shown by the bigger tip gap heights. Additionally, the variation between the different geometries reduces going to lower pressure ratios. At an ER of 1.5, the aerothermal efficiency penalty is even the same for all geometries.

The integrated penalty of the tip leakage flow on the power extraction and its dependence on the tip gap geometry is shown in Fig. 11. Similar to the loss generation, the different flow topologies can be identified due to the three different levels. The most relevant reduction in power generation penalty is obtained from the nominal tip gap height down toward 0.83% TG with a reduction by one-third. The closest tip gap height of 0.28% has only a penalty of 0.3% of the turbine power generation at design conditions. This low value may be explained by the small amount of flow following the pressure gradient on the SS of the blade. Additionally, the flow that enters the flow passage on the SS aligns with the blade surface so that the tip leakage flow is turned similarly to the passage flow.

Over the entire speedline, it can be noticed that the relative power penalty and its contribution to the aerothermal efficiency reduce once moving away from the design point. Also, the power penalties of the reduced tip gap geometries shrink up to an ER of 2 and increase again at an expansion ratio of 1.5.

Laminar viscous loss is directly related to flow separation. Hence, eliminating the pressure side separation bubble implies a significant loss reduction and reduces tip heat transfer. Rounded edges on the blade tip pressure side are recommended to mitigate flow separation.

The prime loss contributor is the “turbulent viscous loss,” generated when the over-tip flow enters the flow tip gap passage and creates the second tip leakage vortex at around mid-chord, where the over-tip flow separates at relatively high speed. To reduce this loss, design solutions that maintain the second tip vortex flow attached to the suction side are implemented. Previously, Ameri and Bunker [18], Ameri [37], and Ade et al. [38] used rounded edges or chamfer at the front of the suction side to maintain this vortex flow attached. An alternative strategy is to open the tip gap from PS toward SS, reducing the tip flow velocity and counteracting flow detachment. A combination of the above-listed design features was observed by De Maesschalck et al. [20] as a result of multiobjective optimization.

The presented understanding of flow features' impact on decoupled losses implies that the design parameters should be allowed more freedom than traditional designs. Vortex generators, cavities, pedestals, or flow injection may be used to align the over-tip flow along the blade suction side and reduce the normal jet exit velocity component, which is conducive to avoiding flow separation. It is also shown that the over-tip flow significantly impacts the shaft power generation depending on its topology. Such variations are barely identifiable using the traditional efficiency analysis, leading to essential loss penalties being unnoticed. Hence, future tip geometry optimizations should consider our detailed tip loss breakdown.

An innovative method to trace flow passing through the over-tip region is combined with a novel loss analysis for volumetric loss generation in CFD results. The losses generated in the tip gap and the tip vortices are decoupled into four relevant loss terms relevant for aerothermal efficiency. The terms are quantified for varying total temperature inlet-to-wall ratios from 1.5 to 3.0 and four tip gap heights ranging from 0.28% to 1.10% of the rotor passage height. Losses in the rotor passage generated by over-tip flow vary between 12% and 28% depending on the thermal condition and the tip gap height, resulting in efficiency penalties between 0.5% points and 1.6% points of thermal efficiency.

Additionally, a model to estimate the volume-specific work extraction penalty due to the underturning of the over-tip flow was derived. Power generation is overwhelmingly reduced by up to 3.4% although the corresponding efficiency penalty remains small with values between 0.05 and 0.28% points. Hence, this important penalty cannot be evaluated with common comparisons of stage efficiencies.

Losses caused by laminar shear stresses occur in the same magnitude in the tip gap as well as in the tip vortex, while losses due to turbulent dissipation mainly occur in the tip vortex. Losses caused by internal heat transfer occur in the same order of magnitude in the tip gap as well as in the tip vortices. The scraped flow from SS to PS forms the feature dominantly responsible for the laminar and turbulent loss due to heat transfer.

Increasing the temperature ratio results in a reduction of laminar viscous losses in the tip gap, while the same loss changes significantly in the vortex and scraped flow. Respective turbulent losses increase slightly with increasing heat transfer. All losses related to internal heat flux augment exponentially with heat transfer.

The analysis of tip flow concentration in the passage allowed us to identify three different flow topologies for different tip gaps. For tip gap heights down to 1.10% of the passage height, the tip horseshoe vortex causes the formation of two separate tip leakage vortices. With reduced tip gap height, the first tip gap vortex originating at the LE reduces its size significantly until it disappears entirely for a tip gap size of around 0.288% of the passage height. At such a close tip gap, the remaining over-tip flow does not form a vortex and orientates along the SS of the blade surface.

The loss generation is governed by the vortex flow topology inside the passage. Laminar shear losses in the tip gap barely vary between 1% and 0.5% tip gap height. Only, a reduction to 0.28% causes a considerable drop in laminar shear losses within the gap. Turbulent shear losses reduce continuously with tip gap height; however, they are of a lower order of magnitude inside the gap. The main reduction in loss generation is caused by a reduced turbulent shear loss in the over-tip vortices. The internal heat transfer-related losses inside the tip gap and the tip vortices reduce with the reducing tip gap height. However, the heat transfer-related loss generation increases in the scraped-over-tip flow when the tip gap is tighter. Design recommendations are formulated based on the identified topologies and loss mechanisms.

The authors thank the US Department of Energy for the appointment of Professor Paniagua to the Faculty Research Participation Program at the NETL.

• The European Union's Horizon 2020 Research and Innovation Program.

• The Marie Sklodowska-Curie (Grant No. 893251).

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

$k$ =

thermal conductivity

$l$ =

specific loss

$p$ =

pressure

$u$ =

rotational velocity

$v$ =

absolute velocity

$w$ =

relative velocity

$x$ =

Cartesian coordinate

$γ$ =

heat capacity ratio

$q˙$ =

local heat flux

$w˙$ =

local power extraction

$A$ =

area

$L$ =

integral loss

$N$ =

rotational speed

$R$ =

radial coordinate

$T$ =

temperature

$V$ =

volume

$Z$ =

axial coordinate

$Q˙$ =

heat flux integral

$W˙$ =

power integral

$ctip$ =

tip flow concentration

$cp$ =

isobaric heat capacity

$η$ =

efficiency

$Θ$ =

circumferential coordinate

$μ$ =

kinematic viscosity

$ρ$ =

density

$τ$ =

shear stress tensor

$∂$ =

partial derivative

$adi$ =

adiabatic

$aero$ =

aerodynamic

$heat$ =

heat-related loss

i =

Cartesian coordinate index

$in$ =

at inlet plane

j =

Cartesian coordinate index

$lam$ =

laminar

$mean$ =

arithmetic mean

$ort$ =

orthogonal velocity component

$out$ =

at outlet plane

p =

pressure based

$rev$ =

reversible

s =

isentropic

t =

total condition

$TWall$ =

isothermal

$turb$ =

turbulent

$shaft$ =

shaft

$vol$ =

calculation based on volume integral

$wall$ =

value at the wall

$Δ$ =

difference

LE =

leading edge

PS =

pressure side

SS =

suction side

TE =

trailing edge

TG =

tip gap height

TR =

temperature ratio

The solvers capability of predicting the heat transfer in the tip region of the blade was further validated using experimental data provided by Castillo Sauca et al. [36]. The heat transfer is measured at the shroud during rotation using atomic layer thermopile sensors. The two-stage turbine with squealer tips was simulated with cfd++ using identical solver settings, and the mesh resolution in the tip gap is comparable to the one used in the present study. Numerical results were filtered with a moving average to take the circular shape of the sensor into account and to achieve comparability. The comparison of the data is depicted in Fig. 12. The values of local maxima of averaged CFD results show exceptional agreement with experimental values in the tip gap region. Also, the shroud heat transfer prediction aligns with high quality. The underprediction within the tip gap region is of the order of magnitude as it can be seen in other studies concerning over-tip heat transfer of squealer tip [22]. Hence, the solver is deemed to be suitable for the prediction of over-tip heat transfer.